- Email: [email protected]
Group size and cooperation among strangers
Accepted Manuscript Title: Group Size and Cooperation among Strangers Author: John Duffy Huan Xie PII: DOI: Reference:
S0167-2681(16)00037-8 http://dx.doi.org/doi:10.1016/j.jebo.2016.02.007 JEBO 3753
To appear in:
Journal
Received date: Revised date: Accepted date:
7-4-2015 16-2-2016 22-2-2016
of
Economic
Behavior
&
Organization
Please cite this article as: John Duffy, Huan Xie, Group Size and Cooperation among Strangers, (2016), http://dx.doi.org/10.1016/j.jebo.2016.02.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
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cr us an M d te
We model an n-person PD game with anonymous and randomly matched agents. We show that it can be easier to sustain society-wide cooperation with a larger group size. Strong experimental evidence is provided for this theoretical prediction when there is no strategic uncertainty. The introduction of strategic uncertainty may diminish this group size effect.
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ip t
Group Size and Cooperation among Strangers Huan Xie
Department of Economics University of California, Irvine 3151 Social Science Plaza Irvine CA 92697-5100, USA Email: [email protected]
Corresponding author Department of Economics Concordia University, CIREQ and CIRANO 1455 de Maisonneuve Blvd. O., H1155 Montreal, Quebec, H3G 1M8, Canada Email: [email protected]
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te
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an
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John Duffy
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Group Size and Cooperation among Strangers∗
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John Duy† and Huan Xie‡
cr
February 16, 2016
Abstract
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an
us
We study how group size aects cooperation in an in nitely repeated n-player Prisoner's Dilemma (PD) game. In each repetition of the game, groups of size n ≤ M are randomly and anonymously matched from a xed population of size M to play the n-player PD stage game. We provide conditions for which the contagious strategy (Kandori, 1992) sustains a social norm of cooperation among all M players. Our main nding is that if agents are su ciently patient, a social norm of society-wide cooperation becomes easier to sustain under the contagious strategy as n increases toward M . In an experiment where the population size M is xed and conditions identi ed by our theoretical analysis hold, we nd strong evidence that cooperation rates are higher with larger group sizes than with smaller group sizes in treatments where each subject interacts with M − 1 robot players who follow the contagious strategy. When the number of human subjects increases in the population, the cooperation rates decrease signi cantly, indicating that it is the strategic uncertainty among the human subjects that hinders cooperation. JEL Classi cation Codes: C72, C73, C78, Z13. Keywords: Cooperation, Social Norms, Group Size, Repeated Games, Random Matching, Prisoner's Dilemma, Experiment.
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∗
We have bene ted from comments by two anonymous referees, Bram Cadsby, Gabriele Camera, Prosper Dovonon,
Erik Kimbrough, Ming Li, Quang Nguyen and seminar participants at Nanyang Technological University, Shanghai University of Finance and Economics, the 2012 China International Conference on Game Theory and Applications and the 4th Annual Xiamen University International Workshop on Experimental Economics.
Duy gratefully ac-
knowledges research support from the UCI School of Social Sciences. Xie gratefully acknowledges the hospitality of the Economic Growth Center at the Division of Economics in Nanyang Technological University and funding support from FQRSC (2010-NP-133118). We thank Tyler Boston for his great help in programming the experiment. † University of California, Irvine, du[email protected] ‡ Concordia University, CIREQ and CIRANO, [email protected]
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1
Introduction
What choice of group size maximizes (or minimizes) the possibility of achieving a social norm of cooperation in a nite population of self-interested strangers? This question would seem to be of some importance for the design of ad hoc committees, juries and teams.
It is also of interest to
experimentalists interested in understanding how the extent of pro-social behavior might depend on
Speci cally, we consider a population of players of xed size
ip t
the matching group size of subject participants. In this paper we oer an answer to this question.
M.
In every period,
players in this population are randomly and anonymously matched to form groups
n-person Prisoner's Dilemma game with all the members of their groups, M/n, is assumed to be an integer (i.e., M is a multiple of n).
The
n=2
group. The total
person version of this environment has been previously studied by Kandori (1992),
us
number of
t = 1,2,...∞, of size n and
cr
then play an
who shows that a social norm of cooperation among anonymous, randomly matched players is sustainable under certain conditions on the game.
n=2
and the possibility vanishes in the limit as for what value(s) of
n>2
Kandori further shows that a social norm of
M gets large x M and ask:
case becomes more di cult to sustain as
an
cooperation among strangers in the
M → ∞.
By contrast, in this paper we
is a social norm of cooperation among strangers easiest to achieve? In
M
other words, is there an optimal group size for maximizing the likelihood of cooperative outcomes? Our answer is that under certain conditionsspeci cally if agents are su ciently patient a social norm of cooperation among strangers, which is sustained by universal play of a contagious trigger gets larger, and becomes easiest to achieve when
That is, we nd that cooperation can be easiest to sustain when the group size is as large
as possible.
This seemingly counterintuitive nding readily follows from the rational-choice logic
te
n = M.
n
d
strategy, becomes steadily easier to achieve as
of the contagious trigger strategy that is used to support cooperation among randomly matched, non-communicative and anonymous strangers. Intuitively, if agents are su ciently patient, then
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the costs of igniting a contagion toward mutual defection are greatest when the matching group size,
n,
equals the population size,
M.
On the other hand, once a defection has started in the
community, the bene ts to slowing down the contagious process are also minimized in this same case where
n = M.
Therefore, the players' incentives to follow the contagious strategy are easiest
to satisfy when the group size is as large as possible. Our main nding is consistent with Kandori's (1992) result. a social norm of cooperation is easier to achieve as be viewed as showing that for the case where a smaller
M.
n
n
While we x
increases towards
is xed at
2,
M,
M
Kandori's result can
cooperation is easier to sustain for
Still, our ndings serve to generalize Kandori's (1992) extension of the folk theorem
for repeated games with random, anonymous matchings to the multiple-player (n Dilemma game. The variety of
and show that
n-player
> 2)
Prisoner's
version of the Prisoner's Dilemma game is widely used to model a
social dilemmas including, e.g., the tragedy of the commons (Hardin (1968)).
we show that our monotonicity result holds in an
n-player
In addition,
binary public good game.
We also provide an empirical test of our main theoretical results by designing and implementing an experiment. In this experiment, we x the population size,
M,
and the discount factor,
δ,
and
2
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study play of an inde nitely repeated game in which players from the population are randomly and anonymously matched in each repetition to play an game. Within the population of size
M,
n-player version of the Prisoner's Dilemma stage
some fraction of players are robot players programmed to
play according to the contagious strategy while the remaining fraction of players are human subjects and this ratio is public knowledge. In this setting we nd strong evidence that, consistent with our
as compared with smaller matching groups, e.g., of size
n=2
n=6
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theoretical predictions, cooperation rates are higher with larger matching groups, e.g. of size
as subjects learn, with experience,
the more immediate consequences of triggering an infectious wave of defection when the group size
cr
is larger. We show further how dierences in cooperation rates between dierent group sizes vary in ways that re ect the predictions of our theory as we vary the payo incentives of the game as well. Finally we show how our theoretical results nd the strongest support when we eliminate strategic
us
uncertainty by having subjects interact only with robot players.
Our paper contributes to the theoretical and experimental literature on sustaining cooperation among anonymous, randomly matched players.
While this is an admittedly stark environment,
an
it is an important benchmark case in both the theoretical and experimental literature and one
1 In addition to the original
that naturally characterizes many types of socio-economic interactions.
seminal paper by Kandori (1992), Ellison (1994) and Dal Bó (2007) provide further generalizations of
M
how a social norm of cooperation may be sustained among anonymous, randomly matched players in 2-player Prisoner's Dilemma games. Xie and Lee (2012) extend Kandori's result to 2-player trust games under anonymous random matchings. Camera and Giore ( 2014) oers a tractable analysis
d
of the contagious equilibria by characterizing a key statistic of contagious punishment processes and
te
deriving closed-form expressions for continuation payos o the equilibrium path. Experimentally, Duy and Ochs (2009) report on an experiment that examines play in an inde nitely repeated, two-player Prisoner's Dilemma game and nd that a cooperative norm does not emerge in the treatments with anonymous random matching but does emerge under xed pairings as players
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gain more experience. Camera and Casari (2009) examine cooperation under random matching by focusing on the role of private or public monitoring of the anonymous (or non-anonymous) players' choices.
They nd that such monitoring can lead to a signi cant increase in the frequency of
cooperation relative to the case of no monitoring. Duy et al. (2013) test the contagious equilibrium in the lab using trust games and nd that information on past play signi cantly increases the level of trust and reciprocity under random matchings. Camera et al. (2013) report wide heterogeneity in strategies employed at the individual level in an experiment in which anonymous randomly matched subjects play the Prisoner's Dilemma game in sequences of inde nite duration. Compared with this previous literature, our paper is the rst to theoretically and experimentally extend the analysis of the contagious equilibrium from a 2-player stage game to an
n-player
stage game.
Our main
theoretical nding, that a cooperative social norm is easier to sustain with a larger rather than
1
There is also an experimental literature that studies cooperation in repeated Prisoner's Dilemma games of in-
de nite duration among xed pairs of players (partners) e.g., Dal Bó (2007), Aoyagi and Fréchette (2009), Dal Bó and Fréchette (2011), Fudenberg et al. (2012). Engle-Warnick and Slonim (2006) examines a trust game of inde nite duration with xed pairs.
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a smaller group size, is new to the literature but nds support both in our own experiment and qualitatively in the previous literature as well, if one considers a large group size to be a partial substitute for public monitoring or xed matching.
2
There are several experimental papers that study the consequences of group size for contributions to a public good using the liner voluntary contribution mechanism (VCM). Isaac and Walker (1988)
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and Isaac et al. (1994) examine how groups of size 4, 10, 40 and 100 play a repeated public good game. One of their main ndings is that, holding the marginal per capita return (MPCR) to the public good constant, an increase in the number of players,
n,
leads to no change or an increase
cr
(depending on the MPCR) in the mean percentage of each player's xed and common endowment that is contributed toward the public good, and this eect is strongest with group sizes of 40 and 100 in comparison with group sizes of 4 and 10. Carpenter (2007) studies contributions under the
us
VCM when players are randomly matched into groups of size 5 or 10, and he allows participants to monitor and punish each other following each repetition.
In this setting, he also nds that
contributions are higher with a larger group size. Xu et al. (2013) examine the eectiveness of an
an
individual-punishment mechanism in larger groups of 40 participants compared with smaller groups of size four. They nd that the individual punishment mechanism is eective when the MPCR is constant but not when the marginal group return (MGR) is held constant (in which case the MPCR
M
is decreasing). Similarly, Nosenzo et al. (2015) nd that it is more di cult to sustain cooperation in larger groups with high MPCR, however, with a low MPCR, considerations of the social bene ts may dominate the negative eect of group size.
Weimann et al.
(2014) report that rst-round
d
contributions to a public good increases with the MPCR distance, which is de ned as the dierence
te
between the actual MPCR and the minimal MPCR necessary to create a social dilemma for a given group size. They further demonstrate that small groups behave similar to large groups when the MPCR distance is controlled.
There are several important dierences between our experimental results and linear VCM games
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(public good games). First, in our model and experimental design, we impose a payo normalization under which players' payos are always equal to 1 when full cooperation is achieved, regardless of the group size.
Equivalently, this corresponds to a public goods experiment where the MPCR
decreases as the group size increases. Therefore, our monotonicity result and the evidence from our experiment show that cooperation increases with group size, even as the MPCR is decreasing, a result that we believe is new to this literature. Second, in our treatments where human subjects only interact with robot players, cooperation does not rely on any other-regarding preferences, which is an important explanatory variable in the experimental public goods literature.
Third,
the game we have implemented has several dierences from the public good game: 1) the strategy space is continuous under the VCM and not binary as in the
n-player
Prisoner's Dilemma game
that we study; 2) subjects in many of these public goods experiments (Carpenter (2007) being an exception) are in
2
xed
matches of size
We note that when the group size,
n,
n
for all repetitions of the public good game whereas in
M , then our than M , one can
is set equal to the largest possible value, the population size
model converges to one of perfect public monitoring and xed matching. Thus for group sizes less
view larger group sizes as being closer approximations to perfect public monitoring and xed matchings.
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our setup players are randomly and anonymously matched into groups of size
n
in each repetition
of the game; and 3) perhaps most importantly, in all of these public good game experiments, the game is
nitely-repeated
so that in theory, positive public good provision (cooperation) or
e ciency is not possible according to standard backward induction arguments if players have the usual self-regarding preferences. By contrast, we study in nitely repeated, binary choice
n-player
ip t
prisoner's dilemma games where cooperation and e ciency are theoretically possible even if players are randomly and anonymously matched and have purely self-regarding preferences. aware of any prior experimental study of inde nitely repeated
n-player
We are not
Prisoner's Dilemma games
cr
under anonymous random matching and our use of programmed robot players in this setting is also new to the literature. The rest of the paper is organized as follows.
Section 2 presents our model and section 3
us
presents our main theoretical results on the consequences of group size for the sustainability of social norms of cooperation among anonymous and randomly matched strangers. Section 4 shows how our framework maps into the classic public good game of Isaac and Walker (1988). Section 5
an
describes the experimental design and section 6 reports on the ndings of an experiment testing our main theoretical results. Finally, section 7 concludes with a brief summary and some suggestions
The Model
Consider a nite population of
M
players. Time is discrete, the horizon is in nite and all players
d
2
M
for future research.
δ ∈ [0, 1]. In each period, the M players are randomly of size n ≤ M , with all matchings being equally likely.
and
te
have a common period discount factor,
m groups We assume that M is a multiple, m, of n. The randomly matched group members then simultaneously and without communication play an n-player Prisoner's Dilemma game where each player chooses a strategy from the set {C, D}, with C representing cooperation and D representing defection. Let i denote the number of members of the group choosing to cooperate (i.e., the number of cooperators) other than the representative player himself so that 0 ≤ i ≤ n − 1. Let Ci and Di denote the payos to cooperation and defection, respectively, when there are i cooperators. An n-player
anonymously matched into
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Prisoner's Dilemma game is de ned by the following three assumptions regarding these payos: A1: A2: A3:
Di > Ci for 0 ≤ i ≤ n − 1. Ci+1 > Ci and Di+1 > Di for 0 ≤ i < n − 1. Cn−1 > D0 . Assumption A1 says that defection is always a dominant strategy.
Assumption A2 says that
payos are increasing with the number of cooperators. Finally, assumption A3 says that if all participants adopt the dominant strategy, the outcome is sub-optimal relative to the mutual cooperation outcome. These conditions are standard in the literature on
n-person
Prisoner's Dilemma games
(See, e.g., Okada (1993, Assumption 2.1)). We further suppose that the payo matrix is symmetric for each player in the group and is as given in Table 1.
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number of cooperators in the group
0
1
2
...
n−1
C
C0 C1 C2
...
Cn−1
D
D0 D1 D2
...
Dn−1
Table 1: The Payo Matrix of the
n-Player
Prisoner's Dilemma Game
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We next de ne the contagious strategy following Kandori (1992) and show that a social norm of cooperation can be sustained as a sequential equilibrium if all players adopt this strategy. De ne a player as a c-type if in all previous repetitions of the game this player and all of the other
cr
group members with whom he has interacted in all prior periods have never chosen outcome of the stage game played in every prior period has been cooperation,
Otherwise, the player is a d-type player.
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member the player has encountered.
i.e., the
by every group (Note that the
d-type players in the same period among the population (or community) of players of size M ≥ n). The contagious strategy can now be de ned as follows: A player chooses C if he is c-type and chooses D if he is d-type. players in any period does not preclude the presence of
an
presence of
c-type
C,
D,
n−1
We next provide a set of su cient conditions that sustains the contagious strategy as a sequential
n.
We rst introduce some notation. Let
M
equilibrium when the group size is
Xt
denote the number
(anij ) to be an
d-type players at time t. De ne An = M × M transition probability matrix n where aij = Pr(Xt+1 = j|Xt = i and all players follow the contagious strategy) given group size n. De ne Bn = (bnij ) as an M × M transition probability matrix where bnij = Pr(Xt+1 = j|Xt = i and one d-type player deviates to playing C while all other players follow the contagious strategy) given group size n. Let Hn = Bn − An , which indicates how the diusion of defection is delayed n n n by the unilateral deviation of one of the d-type players. De ne Zn = (ρ0 ρ1 . . . ρn−1 ), where ρn0 , ρn1 , . . . , ρnn−1 are M × 1 vectors such that the ith element of ρnj is the conditional probability that a d-type player meets j c-type players in the group when there are i d-type players in the n n community given that the group size is n (i.e., Zn = (zij ) is an M × n matrix where zij = Pr(a d-type player meets j − 1 c-type players in his group in period t|Xt = i) given a group size of n). De ne ei as a 1 × M vector whose ith element is 1 and with zeros everywhere else. Finally, de ne T T column vectors vn = (D0 , D1 , . . . , Dn−1 ) and un = (C0 , C1 , . . . , Cn−1 ) , whose ith element is the payo for a player from choosing D and C respectively, given that there are i − 1 other players in the group who choose C .
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of
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Next we show that a one-shot deviation from the contagious strategy is unpro table after any history. On the equilibrium path, a one-shot deviation is unpro table if
∞ Cn−1 X ≥ δ t e1 Atn Zn vn . 1−δ t=0
(1)
The left hand side of (1) is the payo from cooperating forever and the right-hand side of (1) is the payo that the player earns if the player initiates a defection and defects forever afterward. O the equilibrium path, following Kandori (1992), we identify a su cient condition for a one-shot
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deviation to be unpro table under any consistent beliefs. Suppose there are
∞ X
δ t ek Atn Zn vn ≥ ek Zn un + δ
t=0
k d-type
δ t ek Bn Atn Zn vn .
(2)
t=0
The left hand side of (2) is the payo that a are
∞ X
d-type
player earns from playing
D
forever when there
players including the player himself, while the right hand side of (2) is what a
player receives when he deviates from the contagious strategy, playing back to playing
D
forever after.
C
d-type
ip t
k=
k d-type players, where
n, n + 1, . . . , M .3 Then a one-shot deviation o the equilibrium path is unpro table if
today and then reverting
Inequalities (1) and (2) can be manipulated into equilibrium
cr
conditions 1 and 2 in the following lemma.
Lemma 1 The contagious strategy constitutes a sequential equilibrium if the following two condiEquilibrium Condition 1:
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tions are satis ed:
Cn−1 ≥ (1 − δ)e1 (I − δAn )−1 Zn vn ,
an
Equilibrium Condition 2: ek Zn (vn − un ) ≥ δek Hn (I − δAn )−1 Zn vn . The intuition behind equilibrium conditions 1 and 2 is similar to that for the
n = 2 case studied
When a player is on the equilibrium path, he has no incentive to deviate
from cooperation when
δ
M
by Kandori (1992).
is su ciently large. When a player is o the equilibrium path, he has no
incentive to deviate from continued play of the contagious strategy if the extra payo from defection
vn − un , is large enough.
Using Lemma 1 we can prove the following theorem.
d
in the current period,
Theorem 1 Under uniformly random matching, the contagious strategy described above constitutes
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a sequential equilibrium strategy for any nite population size, M , if δ , Cn−1 − Do , and all components of vn − un are su ciently large.
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Proof: See Appendix A.
3
A Monotonicity Result
In this section we ask the following question: Fixing the population size at
n ≤ M
M,
which group size
4
maximizes the possibility of achieving a social norm of cooperation among strangers?
Although we can characterize the equilibrium conditions for the contagious strategy, we cannot derive closed-form solutions since the formulas for the elements of the transition matrices become too complicated to derive for group sizes
n>
A
and
B
2. 5 Therefore, in this section we switch to
6 the use of numerical methods. 3 4
n
Since the player under consideration is a
d-type,
there must be at least
n d-type
players in the community.
In Appendix B, we also ask how the answer to this question changes if instead of xing
but in such a way that the number of groups,
m = M/n,
M,
we vary both
M
and
is held constant. We show there that we obtain similar
results with respect to satisfaction of the equilibrium conditions needed for cooperation to be sustained as a social norm, if we vary
5
6
n
but hold the number of groups constant.
Kandori (1989) provides transition matrix formulas for the
n=2
case only.
The Mathematica program used for our numerical results is available upon request.
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Page 9 of 50
Furthermore, for greater tractability, we focus on a simple symmetric speci cation for the payo parameters that satisfy assumptions A1-A3.
Speci cally, we impose the following additional as-
7 sumptions:
A5: A6:
ip t
C0 = 0. Di − Ci = α for 0 ≤ i ≤ n − 1. Ci+1 − Ci = Di+1 − Di = βn for 0 ≤ i < n − 1.
A4:
Under these assumptions, the payo matrix (Table 1) now takes on the speci c form shown in
with a larger group size since the payo from full cooperation,
n.
(n − 1)βn ,
may grow with the group
To properly correct for this dependency, we also normalize the payo matrix in such a way
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size,
cr
Table 2. Finally, we note that under our parameterization it may be easier to achieve full cooperation
that the payo from full cooperation is xed and constant by the following assumption:
(n − 1)βn = 1
A7:
for any
n.
n ≥ 2. 0
C
0
D
βn
...
n−1
2βn
...
(n − 1)βn
...
α + (n − 1)βn
α α + βn α + 2βn
d
Table 2: The Simpler Payo Matrix for the
α < Cn−1 = 1
2
1
M
number of cooperators in the group
an
Note that under this normalization, to satisfy assumption A3, we must have
n-Player
te
equilibrium conditions as the group size takes on the increases toward
M
for all
Prisoner's Dilemma Game
M = 12 and examine values n = 2, 3, 4, 6, 12.
In order to examine the question raised above, we x
changes in the two
n
We nd that as
both the on-equilibrium and o-equilibrium conditions become easier to satisfy,
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which we refer as the
Monotonicity Result.
3.1 Equilibrium Condition 1
We rst examine the eect of increases in the group size,
n,
on equilibrium condition 1. Although
we are mainly interested in the case where payos are normalized to eliminate the dependency on
n,
for the moment we keep payos for equilibrium condition 1 in their original unnormalized form
(i.e.,
Cn−1 = (n − 1)βn ),
so that we can derive some intuition on the discounted summation of the
probability of earning each payo outcome. Given a group of size
n,
we can write equilibrium condition 1 as:
(n − 1)βn ≥ pn0 α + pn1 (α + βn ) + . . . + pnn−1 (α + (n − 1)βn ) = (
n−1 X
pnj )α +
j=0 7
A slightly dierent normalization, for instance,
n−1 X
jpnj βn ,
j=1
D0 = 0, Di = iβ , Ci = Di − α, Cn−1 = 1,
gives similar results.
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Page 10 of 50
pnj ≡ (1 − δ)e1 (I − δAn )−1 ρnj denotes the discounted summation of the probability of meeting j cooperators (c-types) in a group of size n once a player has initiated a defection. Then given Pn−1 n that j=0 pj = 1 as shown in Lemma 2 in Appendix A and the normalization assumption A7, (n − 1)βn = 1, equilibrium condition 1 becomes where
α ≤ pn ,
(3)
ip t
where
Pn−1
jpnj . n−1 j=1
p ≡1−
Equation (3) says that the net payo from defection (which is
α)
must be less than or equal to the
n net loss from initiating a defection (which is p ).
cr
n
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Proposition 1 If pn is increasing in n, then Condition (3) (Equilibrium condition 1) is monotonically less restrictive as the group size n increases.
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3.2 Equilibrium Condition 2
We next examine the eects of increases in the group size,
n,
on equilibrium condition 2. Given our
Di −Ci = α for i = 0, 1, . . . , n−1, the left hand side of equilibrium condition 2, representing the extra payo from defection, is equal to α. The right hand side of equilibrium condition 2, the payo to a d-type player from slowing down the contagious process, achieves its highest value when the number of d-type players are at a minimum, i.e., when k = n. Thus it is su cient to compare equilibrium condition 2 at k = n for dierent group sizes, n = 2, 3, 4, 6, 12.
d
M
payo speci cation that
te
Similar to equilibrium condition 1, we rst present equilibrium condition 2 with the original payo parameters and then we impose our normalization later. Given a group of size
n,
we can write equilibrium condition 2 as:
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
n α ≥ q0n α + q1n (α + βn ) + . . . + qn−1 (α + (n − 1)βn )
= (
n−1 X j=0
qjn )α +
n−1 X
jqjn βn ,
j=1
qjn ≡ δen Hn (I −δAn )−1 ρnj denotes the change in the discounted summation of the probability of meeting j c-type players in the group when the d-type player reverts back to playing cooperation instead of defection given that the group size is n and there are k = n d-type players in the Pn−1 n population. Given that j=0 qj = 0 as shown in Lemma 2 and the normalization assumption A7, (n − 1)βn = 1, equilibrium condition 2 becomes where
α ≥ qn,
(4)
where
Pn−1 n
q ≡
jqjn . n−1 j=1
α) must be greater than n (which is q ).
Equation (4) says that the net payo from continuing a defection (which is or equal to the net bene t from slowing down a contagious defection
9
Page 11 of 50
Proposition 2 If qn is decreasing in n, then Condition (4) (Equilibrium condition 2) is monoton-
ically less restrictive as the group size n increases.
3.3 Numerical Findings for Dierent Values of δ Propositions 1-2 require that
pn
qn
is increasing and
n so that the su cient equilib-
is decreasing in
n,
becomes monotonically less restrictive as the group size,
n conditions hold by computing p and
pn δ
0.01
0.1
n
for given 0.3
and
δ
0.5
n,
and for dierent discount
cr
M=
12.8
0.7
0.9
0.99
us
all under a xed
increases. We next ask whether these
q n for dierent group sizes,
0.000925 0.011028 0.051716 0.141773 0.331384 0.704922 0.966359
n=3
0.001859 0.022562 0.100818 0.239648 0.460745 0.788192 0.977226
n=4
0.002785 0.033155 0.137519 0.298782 0.524404 0.821914 0.981270
n=6
0.004600 0.050899 0.185380 0.363489 0.585281 0.850813 0.984588
n = 12
0.010000 0.100000 0.300000 0.500000 0.700000 0.900000 0.990000
qn δ
0.01
0.1
an
n=2
M
δ,
factors,
ip t
rium conditions for the contagious strategy to sustain a social norm of cooperation among strangers
for given
n
0.3
0.5
and
δ 0.7
0.9
0.99
0.000839 0.009584 0.039064 0.088720 0.168391 0.290770 0.363919
n=3
0.001195 0.012416 0.040398 0.072725 0.109571 0.151112 0.171385
n=4
0.000926 0.009293 0.028096 0.047192 0.066581 0.086265 0.095219
n=6
0.000059 0.000590 0.001771 0.002952 0.004132 0.005313 0.005844
n = 12
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
te
d
n=2
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 3: Numerical Results on
Our numerical exercises on
n Table 3 reveals that p and
pn
and
qn
pn
qn
and
for Dierent
n
and
δ (M = 12) n,
illustrate some interesting results. Given any group size
q n increase with
δ.
Therefore, if a player cares more about the future,
then the extra loss from initiating a contagious wave of defection and the extra bene t from slowing down a contagion both become larger. Next we ask: given a xed
δ,
how do
pn
and
qn
change with increases in the group size
n?
First, in all cases, the contagious equilibrium always exists, i.e., the numerical value in each cell for
pn
is always larger than the value in the corresponding cell for
between
qn
and
pn
both equilibrium conditions 1 and 2 hold.
monotonically increasing in
n
qn.
Thus by choosing any
α
pn
is
Furthermore, we nd that
n given any δ , and q is monotonically decreasing in
n if δ
is su ciently
large enough (greater than 0.5). Given the numerical results in Table 3, we conjecture that there exists a threshold value for the discount factor, increasing in
8
Notice that
n
δ¯
such that, for any
n and q is monotonically decreasing in
pn
and
qn
are both functions of
δ, n
and
n.
δ > δ¯, pn
is monotonically
This observation is veri ed in Figure 1.
M.
10
Page 12 of 50
ip t cr us an and
M
pn
qn
as functions of
δ (M = 12)
te
d
Figure 1:
Intuitively, with a larger group size, an initial defection spreads to more innocent (c-type) players.
Furthermore, via the random re-matching each period, defection spreads to the entire
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
population of
M
players much faster since there are fewer groups given the xed population size,
M , and a larger group size, n. These two eects together imply that the contagious process is faster with a larger group size n and thus the payo from starting a defection is reduced (i.e., the net loss pn from starting a defection is increased), making the condition on the equilibrium path easier to satisfy. O the equilibrium path, condition 2 becoming less restrictive with a larger group size is also due to the faster contagious process associated with a larger group size. When the speed of contagion is faster, the eectiveness of a single becomes smaller. Thus, the
d-type
d-type
player slowing down the contagious process
player has less of an incentive to deviate from the contagious
strategy o the equilibrium path by reverting back to playing cooperation again (i.e., the net bene t from slowing down the contagion
qn
monotonically decreases).
Summarizing, our main nding is that, for a xed population,
M,
and a su ciently high
δ,
the
conditions under which the contagious strategy sustains play of the cooperative strategy in a
n-
player Prisoner's Dilemma game by all anonymously and randomly matched players in each period are monotonically more easily satis ed as the group size,
n
increases toward
M.
11
Page 13 of 50
4
An Application to a Public Goods Game
In this section we show that with a slightly dierent normalization of the payo matrix, the
n-
person Prisoner's Dilemma game can be re-interpreted as a public goods game so that our previous monotonicity result continues to hold in this public goods game version of the stage game.
M players, who are anonymously and randomly assigned to groups of size n in each period to play an n-player public goods game. Here we study a binary choice
ip t
As before, we assume a population of
version of the classic public good game (Issac and Walker (1988)) where each player is endowed with a single token and must decide whether or not to invest that token in his own privately held account
µn for of γ , but
Each token invested in the public account yields a payo of
cr
or in a public account.
group member. A token invested in the private account yields an additional payo
each only
Table 4 represents the payo matrix for the
us
to the player associated with that private account.
player from choosing to invest in the public account (C ) or in the private account (D ) given the number of other contributors to the public account in the group of size
The standard public
an
and
M
µn > 0
n.
γ > 0, so that non-contribution to the public good is always a dominant strategy in the one-shot, n-player game, and further that γ + µn < nµn , which implies that the social optimum is achieved when all n players contribute to the public good. Notice that these restrictions also satisfy assumptions A1-A3, as de ned in Section 2 for an n-player Prisoner's good game setup has
Dilemma game.
0
1
2
...
n−1
µn
2µn
3µn
...
nµn
...
γ + nµn
number of contributors in the group (invest in the public account)
D
(invest in the private account)
te
d
C
γ + µn γ + 2µn γ + 3µn
Table 4: The Payo Matrix for the
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
n-Player
Public Goods Game
When this public goods game serves as the stage game played by a population of are randomly divided up into groups of size
n
M
players, who
in every period, the su cient conditions to sustain
the contagious equilibrium are very similar to those shown before. On the equilibrium path we must have:
nµ ≥ pn0 (γ + µn ) + pn1 (γ + 2µn ) + . . . + pnn−1 (γ + nµn ),
while o the equilibrium path we require that:
n γ ≥ q0n (γ + µn ) + q1n (γ + 2µn ) + . . . + qn−1 (γ + nµn ). De ne
p˜n ≡
n−1 n n−1 p = − n n
and
n−1 n q˜n ≡ q = n
Pn−1 j=1
jpnj
n
Pn−1 j=1
n
jqjn
.
12
Page 14 of 50
Then with the normalization that
µn = 1/n,
equilibrium condition 1 becomes
γ ≤ p˜n and equilibrium condition 2 becomes
γ ≥ q˜n .
δ
still holds for the public goods game when
ip t
Based on the previous numerical results (Table 3), it is easy to show that the monotonicity pattern is su ciently large, with the threshold value for
Experimental Design
us
5
cr
slightly increased.
δ¯
In the next two sections we report on an experiment that tests the monotonicity results of Propositions 1-2. For this purpose we return to the simpli ed parameterization of the model used in Section
M = 12 and a discount factor δ = 0.75. Our main two treatment variables are the group size, n and the value of α. With the 1 normalized payo in Table 2, βn = n−1 for dierent groups of size n. For instance, the stage game
an
3. In our experimental design, we always consider communities of size
M
payo for the 2-person PD and the 6-person PD are shown in Table 5.
D
C
C
0
1
D
α α+1
d
Other Player's Choice
0
1
2
3
4
5
C
0
0.2
0.4
0.6
0.8
1
D
te
Number of Others Playing C
α α + 0.2 α + 0.4 α + 0.6 α + 0.8 α + 1
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 5: The Payo Matrices for the 2-Player and 6-Player Prisoner's Dilemma Game
A third treatment variable in our experimental design is the number of human subjects in each 12-player community. We start with the treatments in which each 12-player community consists of just one human subject who interacts with 11 other robot players as opposed to allowing 12 human subjects to interact with one another. We employ this design in order to avoid the coordination problem of strategy selection among 12 human players and thereby remove strategic uncertainty. As with other folk-theorem type results, the contagious equilibrium is not the unique equilibrium of the in nitely repeated
n-player
PD game that we implement in our experiment.
There exist
many other non-cooperative equilibria including the one where all players choose to defect in every round of the supergame. Empirically, when players face both the selection of their own strategy and the uncertainty of strategy selection by other players, the outcome of play can be far from that predicted by the contagious equilibrium.
9 Furthermore, we would expect that this problem of
strategic uncertainty is naturally more severe as the group size
9
n
gets larger. For these reasons, we
See for example, Duy et al. (2013) and Duy and Ochs (2009).
13
Page 15 of 50
chose to rst eliminate the strategic uncertainty dimension from our experimental design by having our players interact with robot players programmed to play according to the contagious strategy so as to provide a cleaner test of our monotonicity results. human subject and 11 robots per community of size
10 While our baseline design has just 1
M = 12,
in other treatments, we increase the
number of human subjects in each community to 2 and 6 while decreasing the number of robots to
M
xed at 12.
ip t
10 and 6, respectively, i.e. holding
Using the three treatment variables discussed above, i.e., the group size
n,
the value of
α
and
the number of human subjects in each community, we summarize all of our experimental treatments
cr
in Table 6. The treatments are denoted using labels of the form aHbR_X, where a is the number of human subjects and b is the number of robots in each community of 12 players. X∈{A, B, C, D, E} represents dierent pairs of values for the group size,
n,
and payo parameter,
α,
as indicated
us
in Table 6. This same table also provides the on-equilibrium and o-equilibrium conditions on
α,
given the group size chosen for each treatment. Using these conditions, and the treatment value for
α,
Table 6 also indicates (under the heading Cooperation?)
whether or not cooperation by
an
all players can be sustained as a sequential equilibrium of the inde nitely repeated game where players are randomly and anonymously matched in groups of size
n
in each round. We conducted
one session each for the treatments involving 1H11R or 2H10R and two sessions for the treatment
provides further details.
α
Players
d
Treatment Session Group Size
M
involving 6H6R. Note that each session gives multiple independent observations Table 7 below
On-equm O-equm
Cooperation?
1
n=6
α = 0.5
1 HS 11 robots
α ≤ 0.65 α ≥ 0.004
Yes
1H11R_B
2
n=2
α = 0.5
1 HS 11 robots
α ≤ 0.40 α ≥ 0.19
No
1H11R_C
3
n=2
α = 0.3
1 HS 11 robots
α ≤ 0.40 α ≥ 0.19
Yes
1H11R_D
4
n=3
α = 0.6
1 HS 11 robots
α ≤ 0.53 α ≥ 0.12
No
te
1H11R_A
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
5
n=6
α = 0.6
1 HS 11 robots
α ≤ 0.65 α ≥ 0.004
Yes
6, 7
n=6
α = 0.5
6 HS 6 robots
α ≤ 0.65 α ≥ 0.004
Yes
8
n=6
α = 0.5
2 HS 10 robots
α ≤ 0.65 α ≥ 0.004
Yes
1H11R_E 6H6R_A
2H10R_A
Table 6: Summary of Treatments
The value of
α
in each treatment is varied based on several considerations. First, we wanted
a parameterization that could sustain the contagious strategy as an equilibrium in a community of a xed population size
under a larger group size but not under a smaller group size
so as to
test our main monotonicity result. Second, we chose to focus on the on-equilibrium-path condition rather than the o-equilibrium-path condition; if
α
was instead chosen in such a way that the
on-equilibrium-path condition (but not the o-equilibrium-path condition) was always satis ed for both group size treatments, e.g., a choice of
α = 0.1,
then we might observe that subjects seldom
chose to defect (with the consequence that they were seldom actually o the equilibrium path)
10
We note that this type of experimental design involving robot players has not previously been implemented to
test the contagious equilibrium prediction.
14
Page 16 of 50
under either group size, making it di cult to detect any treatment eect. Therefore, we chose to
n = 2 (Sessions 1 and 2). By the same principle, we chose α = 0.6 to test the group size eect between n = 3 and n = 6 (Sessions 4 and 5). In addition, in Session 3 we set α = 0.3 for a group size n = 2 so that cooperation via the contagious equilibrium is supported in this case, as opposed to α = 0.5 for group size n = 2 as in set
α = 0.5
to test the group size eect between
n=6
and
ip t
Session 2. In sessions 1-5, there is just one human subject per community of size 12. By contrast, in sessions 6, 7, and 8, the number of human subjects in the 12-player community is varied but we
α = 0.5
and
Given our choice of
n=6 α
as in Session 1 so that the contagious equilibrium can be sustained.
in each treatment and a xed
δ = 0.75
and
M = 12,
we are able to test
cr
always set
the following hypotheses using our experimental data.
Hypothesis 1:
us
Hypotheses 1-3 concern aggregate treatment eects on cooperation.
The overall cooperation rate is higher with a larger group size than with a
smaller group size given the same value of
α
in the 1H11R treatments (Session 1 vs. 2, Session 5
Hypothesis 2: with a smaller
an
vs. 4).
The overall cooperation rate is higher when the equilibrium condition is satis ed
α than when the equilibrium condition is not satis ed with a larger α given the same
Hypothesis 3:
M
group size in the 1H11R treatments (Session 3 vs. 2).
The overall cooperation rate is lower with an increase in the ratio of human
subjects to robots (playing the contagious strategy) in a community given the same group size,
α
n
(Sessions 1 vs. 6 and 7 vs. 8).
d
and value of
Hypotheses 4-7 concern the individual behavior of the human subjects: The frequency with which subjects are on the equilibrium path is larger when
te
Hypothesis 4:
the on-equilibrium-path condition is satis ed than when the condition is not satis ed.
Hypothesis 5:
The cooperation rate when subjects are
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
c-types
(on-the-equilibrium-path) is
higher with a larger group size than with a smaller group size given the same value of
Hypothesis 6:
The cooperation rate when subjects are
d-types
α.
(o-the-equilibrium-path) is
not dierent between a larger group size and a smaller group size.
Hypothesis 7:
The cooperation rate when subjects are
c-types
is higher when the number of
human subjects in the community decreases, given the same group size
n
and value of
α.
In our 1H11R treatments, we explicitly told our subjects that in each round of a supergame (or sequence as it was referred to in the experiment) they would be randomly matched with
n−1
other robot players (out of a total population of 11 robot players) and not with any other human subjects. Since
n < M,
subjects were told that there would also be robot-robot group interactions
that they would not be a part of given the uniform random matching that we used. Subjects were further instructed that the robots in each community played according to the rules of the contagious strategy. Speci cally subjects were told: The robots are programmed to make their choices according to the following rules:
•
choose X in the rst round of each new sequence;
15
Page 17 of 50
•
if, during the current sequence, any of a robot's group members, including you or any other robot players have chosen Y in any prior round of that sequence, then the robot will switch to choosing Y in all remaining rounds of the sequence;
•
otherwise, the robot will continue to choose X.
C,
while Y refers to the defect action
D.11
Thus subjects
ip t
Here X refers to the cooperative action
had complete knowledge of the strategies to be played by their opponents.
We did not provide
subjects with any further information, such as the number of periods it might take for them to meet
cr
a defecting player once they (the human subject) had initiated a defection, as this calculation was one that we wanted subjects to make on their own.
In our 2H10R and 6H6R treatments, everything remained the same except that subjects knew
us
that there would be 2(6) human subjects and 10(6) robot players in their 12-player community for the duration of the session. Subjects remained in the same community with the same 1 other or the same 5 other human subjects in all supergames (sequences) of a session. Further, subjects were
size
M = 12.
an
told that there would be no spillovers from human subjects interacting in dierent communities of Therefore, as in the 1H11R treatments, each community of size 12 in the 2H10R and
6H6R treatments constitutes an independent observation.
M
One may be concerned that explicitly telling subjects the contagious strategy used by robots will induce a kind of experimenter demand eect by which the human subjects will also follow this same contagious strategy. However, if that were the case, then our hypotheses that the group size
n
d
matters would not nd any support since subjects in all of our treatments were told the same
information about the contagious strategy played by the robot players. On the other hand, if we
or
n = 3,
te
observe a higher cooperation rate under a group size of
n=6
than under a group size of
n=2
then it implies that subjects rationally choose to follow the contagious strategy more
frequently when the equilibrium conditions were satis ed.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
To implement an in nite-horizon
n-player
PD game in the laboratory, we use the standard
random termination methodology (Roth and Murnighan 1978) in which subjects participate in supergames that consist of an inde nite number of rounds, where the probability of continuation from one round to the next is a known constant equal to the discount factor, choice of
δ = 0.75,
δ ∈ (0, 1).
With our
the expected duration of a supergame is 4 rounds.
To enable subjects to gain some experience with the play of an inde nitely repeated game, we had them participate in multiple supergames in a session. As noted above, subjects were informed that at the start of each and every new supergame (sequence) all of the robot players in their community of size 12 would start out each new supergame as c-types playing the cooperative strategy (X) and that robot players would only change to playing the defect strategy (Y) if they became d-types during that supergame. This feature allows subjects to treat each new supergame as a fresh start rather than viewing the entire session as one long inde nitely repeated game, with the possibility of switching to a dierent strategy at the start of each new supergame. We did not x the number
11 the
We used the neutral labels X and Y rather than (Cooperate and Defect) in our experimental implementation of
n-player
game.
16
Page 18 of 50
of supergames played in advance.
Instead, during the experiment, we allowed subjects to play
for at least one hour, and the supergame in progress beyond one hour was determined to be the last supergame (we did not inform subjects of our stopping rule); when that nal supergame was completed, the session was declared to be over. Following the completion of the experiment, three sequences were randomly selected from all played and subjects were paid their total earnings from
12 All sessions were completed within the two
ip t
those three sequences in addition to a $7 show-up fee.
hour time-horizon for which we recruited subjects; a typical session required about 30 minutes for the reading of instructions followed by 1 hour of play of multiple supergames.
cr
The experiment was conducted at the Experimental Social Science Laboratory (ESSL) of the University of California, Irvine using undergraduate students with no prior experience with our experimental design. Instructions were read aloud and then subjects completed a brief comprehension
n=6
treatment are provided in Appendix C; instructions for
us
quiz. The instructions used in the other treatments are similar.
Experimental Findings
an
6
Table 7 reports some details about all of our experimental sessions.
As this table reveals, on
M
average, each session involved 18 supergames and 76 rounds. In total, 93 subjects participated in our experiment, with average earnings of USD $16.35. In the 1H11R treatments, since each subject interacted with an independent group of 11 other robots all playing according to the contagious
d
strategy, each subject's behavior amounts to a single, independent observation. Thus, the number of
te
subjects we have for each of sessions 1-5 corresponds to the number of independent observations. For treatment 6H6R_A, there are 6 human subjects in each community and 12 subjects in each session, so each session produces two independent observations, and as we have two sessions of 6H6R_A, we have a total of 4 independent observations of this treatment. For treatment 2H10R_A, there
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are 2 human subjects in each community; since we recruited 18 human subject for this treatment, we thus have 9 independent observations of this treatment. In the remainder of this section, we will begin with an aggregate data analysis, followed by analysis of data at the individual level. For the aggregate results, we will rst provide a summary of ndings on the group size eect, the eect of
α,
and the eect of the number of human subjects
within a community. After that we will present results from Mann-Whitney tests that compare the cooperation rates in the dierent treatments all together in Table 9. We rst examine results from the treatments with one human subject in each community (1H11R). Figure 2 shows the average cooperation rate per round over time for treatments 1H11R_A (B, D, E). The blue line (with diamonds) in each panel is the average cooperation rate of the human subjects only, while the red line (with squares) in each panel is the average cooperation rate by the entire 12-member community as a whole, consisting of 1 human subject and 11 robots. The start of each new supergame is indicated by a vertical line.
12
We chose to pay for three randomly sequences, as opposed to just one, so as to avoid the possibility that a short
(e.g., 1-round) supergame was chosen for payment under the pay-one-sequence-only protocol.
17
Page 19 of 50
Treatment Session
No. of
No. of
Subjects
Obs.
No. of
No. of
Supergames Rounds
Average Earnings
1
10
10
18
75
USD 22.59
1H11R_B
2
12
12
20
76
USD 15.33
1H11R_C
3
12
12
19
75
USD 13.11
1H11R_D
4
10
10
22
77
USD 20.21
1H11R_E
5
7
7
18
77
USD 15.31
6H6R_A
6
12
2
19
81
USD 11.49
6H6R_A
7
12
2
15
83
USD 14.63
2H10R_A
8
18
9
15
65
USD 18.09
N/A
12
N/A
18
76
USD 16.35
cr
us
Average
ip t
1H11R_A
Table 7: Description of Experimental Sessions
n=2
and
respectively with
an
The two top panels of Figure 2 show the average cooperation rates for the group size of
α = 0.5.
For Session 2 (the
n=2
n=6
group size), we observe a decline in
the cooperation rate over time by the human subjects in almost all supergames lasting more than
M
2 rounds, which indicates that more human subjects began to switch from cooperation to defection from round 2 if they did not choose to defect from the beginning of the supergame. Consistently, the cooperation rate at the community level shows a similar pattern but remains above the cooperation
d
rate by the human subjects alone as it takes some time for the contagious strategy, as played by the
with
n = 2,
te
robot players, to spread throughout the population of size 12. Across all supergames of the session there is no obvious learning eect or convergence.
By contrast, the cooperation rates in Session 1 under a group size of
n = 6
exhibit a very
dierent pattern over time. Indeed, consistent with our theory, the overall cooperation rate is higher
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in Session 1 (n
= 6)
than in Session 2 (n
= 2).
Although there is also a decline in cooperation
over the course of each sugergames at the beginning of Session 1, the cooperation rate eventually becomes high, at around 90%, following the fourth supergame of this session (approximately after the rst one-third of the session has been completed) and remains high for the remaining supergames of that session. This nding indicates that, given the payo parameters we have chosen and the strategy followed by the robots, most subjects learn over time that it is in their best interest to follow the contagious strategy when
n is large (n = 6),
relative to the case where
A comparison of the cooperation rates between groups of size
n=2
and
n=6
n is small (n = 2). indicates that the
human subjects responded to the payo incentives of the game. They choose to start defecting more frequently when the contagious eect of a single defection is much slower in Session 2 (n
= 2)
and
this tendency to defect was not diminished by experience. By contrast, when the contagious eect of a defection is more immediate, as in Session 1 (n
= 6),
subjects learned to avoid triggering a
wave of defection. The two bottom panels of Figure 2 show the average cooperation rates for the group of size and
n = 3 respectively with the choice of α = 0.6.
n=6
This treatment comparison provides a robustness
18
Page 20 of 50
Cooperation Rate (Session 1 - 1H11R_A, n=6, α=0.5)
Human Subjects Group Total
1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 1 2 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 910111 2 1 2 3 1 2 1 2 3 4 5 6 7 1 2 1 2 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1 2 1
ip t
Round
Cooperation Rate (Session 2 - 1H11R_B, n=2, α=0.5)
Human Subjects Group Total
1
cr
0.8 0.6 0.4 0.2
us
0
1231234512341234567123123456123456781121231234567891112111234123456781212345
an
Round
Cooperation Rate (Session 5 - 1H11R_E, n=6, α=0.6) 1 0.8
Human Subjects Group Total
M
0.6 0.4 0.2 0
1 2 3 4 5 1 2 3 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 1 2 3 4 1 2 3 4 5 6 1 1 2 3 4 5 1 2 3 4 5 6 7 8 91011121 2 3 1 2 3 4 5 6 1 2 3 1 2 3 4 1 2 3 1 1 2 3 1 2
d
Round
Cooperation Rate (Session 4 - 1H11R_D, n=3, α=0.6)
Human Subjects Group Total
te
1 0.8 0.6 0.4
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
0.2 0
1 2 3 4 5 6 7 8 1 1 2 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 9 1011121314151617 1 2 3 1 1 2 1 2 1 2 3 1 1 2 3 1 2 1 2 1 1 1 1 2 3 4 1 1 2 3 4 5 6 7 8 1 2 3 4 5
Figure 2:
Round
Cooperation Rate by Human Subjects and Communities Over Time (Treatment
1H11R_A,B,D,E)
check as to whether the group size eect continues to hold with a dierent parameterization for and
α.
Indeed, comparing the dierent group sizes
n = 6
and
n = 3,
we continue to nd that,
consistent with our theory, cooperation rates are higher with the larger group of size under the smaller group of size
n = 3.
n
n=6
than
We summarize the group size eect in Finding 1, which
supports Hypothesis 1. Detailed statistical support can be found in Table 9.
Finding 1 Cooperation rates are higher under larger groups than under smaller groups given the same α.
19
Page 21 of 50
Not only did our subjects respond to the incentives induced by having dierent group sizes, but
α when holding the group size (α = 0.3, n = 2) and Session 2 (α = 0.5,
also they responded to the incentives associated with changes in constant. Figure 3 shows a comparison between Session 3
n = 2).
With a larger
α, players have a larger temptation to initiate a defection, thus making it more
di cult to sustain cooperation within the community. Figure 3 and Finding 2 con rm Hypothesis
α = 0.3,
the cooperation rate is sustained at a level above
50%. By contrast, as seen in the bottom panel of the same gure where
ip t
2. In the top panel of Figure 3 where
α = 0.5,
the cooperation
tests regarding Finding 2 are presented in Table 9.
us
Cooperation Rate (Session 3 - 1H11R_C, n=2, α=0.3)
cr
rate drops below 20% if the supergame lasts for 4 or more rounds. Results from Mann-Whitney
1 0.8 0.6
an
0.4
Human Subjects Group Total
0.2 0
123451123451234561234512345678123111234567812341234512345612345611231234121
M
Round
Cooperation Rate (Session 2 - 1H11R_B, n=2, α=0.5) 1 0.8
Human Subjects Group Total
0.6
d
0.4 0.2 0
te
1231234512341234567123123456123456781121231234567891112111234123456781212345
Round
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 3:
Cooperation Rates by Human Subjects and Communities Over Time (Treatments
1H11R_B,C)
Finding 2 Cooperation rates are higher under a smaller α than under a larger α given the same group size, n.
We next compare the 1H11R treatment with the 2H10R and 6H6R treatments, restricting attention to the case where
n=6
and
α = 0.5,
so that the contagious equilibrium is supported by
the same set of parameters in the three treatments and we can examine whether or not the ratio of humans to robots in each community of size 12 will aect the possibility that the contagious equilibrium is selected.
13 The comparison is made using Figure 4 and statistical support is included
in Table 9. The rst panel of this gure shows cooperation rates for the 1H11R_A treatment, the
13
We did not study the case of a group size of
humans to robots, since when
n=2
and
α = 0.5,
n = 2
across these three treatments involving dierent ratios of
our theory predicts that a social norm of mutual cooperation is
not sustainable using the contagious strategy.
20
Page 22 of 50
second panel for the 2H10R_A treatment and the last two panels show cooperation rates for the 6H6R_A treatment (2 sessions). As the ratio of human subjects to robots in a community increases, we nd that the average cooperation rate monotonically decreases. Recall that for the 1H11R_A treatment, the cooperation rate converges to about 90% after subjects gained some experience in the rst four supergames. By contrast, in the 2H10R_A treatment, the cooperation rate is closer
14 Finally, in the 6H6R_A treatment, the cooperation rate almost
ip t
to 50% throughout the session.
always drops to zero after a supergame lasts for 2 or 3 rounds. Based on the results of the 6H6R treatment, we did not think it necessary to run treatments where the community consisted entirely
cr
of 12 human subjects and 0 robots, as we expect that cooperation rates would likely have been close to 0 in these treatments as well.
us
Finding 3 Cooperation rates monotonically decrease as the number of human subjects in a com-
an
munity increases, given the same group size n and value of α.
Tables 8 and 9 provide further detailed evidence that support Findings 1-3.
Table 8 reports
cooperation rates calculated based on human subjects' choices as well as on community-wide action choices (humans plus robot players) over all rounds of all supergames. We further calculated the
M
cooperation rates over the rst and second halves of each session. Table 9 reports
p-values
from
two-tailed Mann-Whitney tests on the cooperation rates between treatments. The rst two rows
n = 6 vs. n = 2 given α = 0.5 (treatment 1H11R_A vs. 1H11R_B), and n = 6 vs. n = 3 given α = 0.6 (treatment 1H11R_D vs. 1H11R_E). The third and fourth rows test Hypothesis 2 (the eect of α) for α = 0.5 vs. α = 0.3 given n = 2 (treatment 1H11R_B vs. 1H11R_C) and α = 0.5 vs. α = 0.6 given n = 6 (treatment 1H11R_A vs. 1H11R_E), where we only expect a signi cant dierence in the case of n = 2. The last three rows test Hypothesis 3 (the number of human subjects in a community) given n = 6 and α = 0.5.
te
d
test Hypothesis 1 (the group size eect) for
The
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
p-values
shown in Table 9 are all consistent with a careful examination of the cooperation
rates shown in Figures 2-4. and
n=6
We nd the most signi cant group size eect between the
n = 2
treatments. On the other hand, strategic uncertainty (about the play of other human
subjects) also plays an important role in de-stabilizing cooperation in the environment we study here where players are randomly and anonymously matched in each repetition of the stage game. Although nearly full cooperation can be sustained in the second half of our sessions where
n=6
and each human subject interacts only with 11 robot players programmed to play according to the contagious strategy, the cooperation rate drops to close zero when
n
remains equal to 6, but the
community consists of 6 human and 6 robot players, even though the equilibrium conditions for sustenance of a social norm of cooperation continue to be satis ed. In order to further ensure that these treatment eects are not driven by subjects' reactions to the instructions used in dierent
14
Note that for the 2H10R treatment, the aggregate cooperation rate masks some heterogeneity at the community
level. Focusing on behavior in the last three sequences of the 2H10R_A treatment, we found that 3 out of 9 human groups (each consisting of 2 human subjects) always chose to cooperate, another 3 out of 9 human groups always chose to defect, and the remaining 3 groups cooperated at average levels of 10%, 20%, and 70%, respectively.
21
Page 23 of 50
Cooperation Rate (Session 1 - 1H11R_A, n=6, α=0.5)
Human Subjects Group Total
1 0.8 0.6 0.4 0.2 0
Round
Human Subjects Group Total
cr
Cooperation Rate (Session 8 - 2H10R_A, n=6, α=0.6)
ip t
1 2 3 4 5 6 7 8 1 2 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 910111 2 1 2 3 1 2 1 2 3 4 5 6 7 1 2 1 2 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1 2 1
1 0.8
0.4 0.2 0
us
0.6
1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1011 1 1 2 3 4 5 6 7 8 9 10 1 2 1 2 1 2 3 1 2 3 4 5 6 7 8 9 1011 1 1 1 1 2 1 2
an
Round
Cooperation Rate (Session 6 - 6H6R_A, n=6, α=0.6) 1 0.8
Human Subjects Group Total
M
0.6 0.4 0.2 0
1 2 3 4 5 6 1 1 2 1 2 3 1 2 3 4 5 6 7 8 910111213141 2 1 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 1 2 1 2 3 4 5 6 7 8 9 1 1 2 1 2 3 1 2 3 4 5 6 7 1 2 3 4 1 2 1 2 3 4 5 6 7
d
Round
te
Cooperation Rate (Session 7 - 6H6R_A, n=6, α=0.6)
1 0.8 0.6
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Human Subjects Group Total
0.4 0.2 0
1 1 2 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 1 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 1 2 1 2 3 4 5 6 7 8 9
Figure 4:
Round
Cooperation Rate by Human Subjects and Communities Over Time (Treatments
1H11R_A, 2H10R_A, 6H6R_A)
treatments, we compared the subjects' choice in the very rst period of the sessions and did not found any signi cant dierence across treatments. Importantly, our results indicate that subjects understand how to evaluate payos in the inde nitely repeated games we induce despite the relatively complicated environment, and they correctly respond to the incentives provided. Our ndings from varying the ratio of humans to robots suggest that it is
strategic uncertainty
about the play of others and not bounded rationality with regard
to payo calculations that may explain the failure of human subjects to achieve the e cient equi-
22
Page 24 of 50
Treatment Session
Human Subjects
Communities
Whole Session 1st half 2nd half Whole Session 1st half 2nd half 1
0.800
0.732
0.870
0.846
0.776
0.918
1H11R_B
2
0.364
0.373
0.355
0.743
0.754
0.733
1H11R_C
3
0.692
0.732
0.651
0.883
0.898
0.869
1H11R_D
4
0.444
0.379
0.508
0.667
0.534
0.797
1H11R_E
5
0.800
0.812
0.788
0.865
6H6R_A
6
0.062
0.079
0.047
0.148
6H6R_A
7
0.104
0.152
0.058
0.147
2H10R_A
8
0.418
0.368
0.466
0.519
ip t
1H11R_A
0.855
0.159
0.142
0.204
0.090
cr
0.875
0.441
0.594
No. of Obs.
1H11R_A vs.
Human Subjects
Whole Session 1st half 2nd half Whole Session 1st half 2nd half
10 vs. 12
0.006
0.008
0.003
0.015
0.391
0.005
10 vs. 7
0.039
0.009
0.128
0.044
0.005
0.403
12 vs. 12
0.067
0.097
0.045
0.017
0.067
10 vs. 7
0.767
0.488
0.290
1.000
0.348
0.291
0.000
0.000
0.005
0.005
0.003
0.034
0.034
0.029
0.049
0.059
0.034
0.001
0.001
0.003
0.005
0.020
0.005
1H11R_E 1H11R_B vs.
1H11R_E 1H11R_A vs.
10 vs. 24
6H6R_A
(10 vs. 4)
te
1H11R_A vs.
1H11R_A vs.
10 vs. 18
2H10R_A
(10 vs. 9)
2H10R_A vs.
18 vs. 24
6H6R_A
(9 vs. 4)
Table 9:
0.018
d
1H11R_C
M
1H11R_B 1H11R_D vs.
Communities
an
Treatment
us
Table 8: Cooperation Rates for Each Session
0.000
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
p-values
from Mann-Whitney Tests on Cooperation Rates
librium in inde nitely repeated games where Kandori's contagious equilibrium construction can be used to support a social norm of cooperative play by all players. We next ask whether there is a signi cant learning eect when the contagious equilibrium condition is or is not satis ed. In our experiment, we nd such a learning eect for some but not for all of our treatments/sessions. Table 10 reports
p-values
for the two-tailed Wilcoxon matched-
pair signed ranks tests between the rst and the second halves of each session. Learning is most signi cant in treatment 1H11R_A (n
= 6,
session 1), in which subjects learn to cooperate with the
11 robot players who are playing according to the contagious strategy, and in treatment 6H6R_A (n
= 6,
sessions 6 and 7), where cooperation rates converge to zero as half of the community
23
Page 25 of 50
members are human subjects. In sessions 3 and 5, the cooperation rates decrease signi cantly from the 1st half of sessions to the 2nd half of sessions, but the magnitude of this decrease is small. Treatment Session Cooperation? Human Subjects Communities Pattern over Time 1
Yes
0.081
0.018
increasing
1H11R_B
2
No
0.692
0.346
at
1H11R_C
3
Yes
0.010
0.054
decreasing
1H11R_D
4
No
0.185
0.006
increasing
1H11R_E
5
Yes
0.050
0.050
decreasing
6, 7
Yes
0.001
0.144
decreasing
8
Yes
0.614
0.056
increasing
2H10R_A
p-value of Wilcoxon Matched-Pair Signed Ranks Tests on Cooperation Rates (1st vs.
us
Table 10:
cr
6H6R_A
ip t
1H11R_A
half of the Session)
2nd
an
We next analyze individual strategic behavior. Speci cally, we ask to what extent the subjects in each treatment behaved according to the contagious strategy that is needed to support a social norm of cooperation among strangers. Table 11 shows the average of individual frequencies with
M
which subjects in each treatment are on the equilibrium path, the average of individual frequencies with which subjects chose to cooperate when on the equilibrium path and the average of individual
15 Finally, for each human subject,
frequencies of choosing to defect when o the equilibrium path.
d
we also calculated the frequency with which that subject played the contagious strategy, which is
te
given by:
freq. of following the contagious strategy
=
freq. on the equilibrium path
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
+
× cooperation
freq. o the equilibrium path
In Table 12 we report the
p-values
× defection
rate on the equilibrium path
rate o the equilibrium path.
from two-tailed Mann Whitney tests comparing all of these
frequencies across various treatments. The tests are conducted by using the average of the individual frequencies of human subjects within each community, which constitutes an independent observation since human subjects in dierent communities did not have any interactions with one another by our experimental design.
Most of the results shown in Tables 11- 12 are consistent with Hypotheses 4-7 regarding individual strategic behavior.
For the treatments with 1H11R, the average frequency with which
subjects are found to be on the equilibrium path is always signi cantly larger in sessions where the on-equilibrium-path condition is satis ed than in those treatments where this same condition is not satis ed. On the other hand, an increase in the number of human subjects in a community, as in treatments 2H10R_A and 6H6R_A, signi cantly lowers the frequency of on-equilibrium-path play
15
Recall that a player is de ned to be on the equilibrium path in the rst round of a supergame or when the player
has never experienced a defection by his group members or himself in the past rounds of a supergame. Otherwise, a player is o the equilibrium path.
24
Page 26 of 50
Treatment Session
Frequency
Cooperate Rate
On Equm Path
On Equm Path
Defect Rate
Frequency using
O Equm Path Contagious Strategy
1
82.27%
86.49%
74.48%
91.60%
1H11R_B
2
50.99%
56.50%
94.46%
79.50%
1H11R_C
3
76.11%
74.35%
93.96%
89.00%
1H11R_D
4
57.40%
60.35%
97.48%
84.42%
1H11R_E
5
84.42%
90.43%
85.76%
6H6R_A
6
23.66%
11.31%
95.41%
6H6R_A
7
19.08%
37.58%
96.40%
2H10R_A
8
50.68%
61.67%
96.53%
ip t
1H11R_A
91.47% 75.51%
cr
85.54% 87.69%
Cooperate Rate
On Equm Path
On Equm Path
0.008
0.008
(10 vs. 12)
1H11R_B
0.118
0.015
(10 vs. 12)
(8 vs. 11)
(10 vs. 12)
0.043
0.127
0.017
0.301
(10 vs. 7)
(10 vs. 7)
(9 vs. 4)
(10 vs. 7)
0.090
0.086
0.479
0.061
(12 vs. 12)
(12 vs. 12)
(11 vs. 6)
(12 vs. 12)
0.693
1.000
1.000
(10 vs. 7)
(8 vs. 4)
(10 vs. 7)
0.005
0.024
0.494
0.047
(10 vs. 4)
(10 vs. 4)
(8 vs. 4)
(10 vs. 4)
0.085
0.163
0.381
0.251
(10 vs. 9)
(10 vs. 9)
(8 vs. 7)
(10 vs. 9)
0.009
0.031
0.706
0.216
(9 vs. 4)
(9 vs. 4)
(7 vs. 4)
(9 vs. 4)
1H11R_D vs. 1H11R_E 1H11R_B vs. 1H11R_C 1H11R_A vs.
0.843 (10 vs. 7)
te
1H11R_E 1H11R_A vs. 6H6R_A 1H11R_A vs. 2H10R_A
Frequency using
O Equm Path Contagious Strategy
M
1H11R_A vs.
Defect Rate
an
Frequency
d
Treatment
us
Table 11: Strategy Analysis of Human Subjects
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2H10R_A vs. 6H6R_A
Table 12:
p-value
of Mann-Whitney Tests on Individual Strategy
of the contagious strategy. The average frequency of on-equilibrium-path play moves from about 80% to 50% and further declines to 20% when comparing across the 1H11R_A, 2H10R_A and 6H6R_A treatments, respectively (p
< 0.1
or
p < 0.01).
Conditional on being on the equilibrium path (column 3 in Table 12), the dierence in coopera-
n = 6 and n = 2 treatments is signi cant (session 1 vs. 2, p < 0.01) but there dierence between the n = 6 and n = 3 treatments (session 4 vs. 5, p = 0.127).
tion rates between the is no signi cant
The cooperation rates on the equilibrium path are not signi cantly dierent between 1H11R_A and 2H10R_A treatments (p
= 0.163),
but are signi cantly dierent between 1H11R_A and 6H6R_A
treatments and between 2H10R_A and 6H6R_A treatments (p
< 0.05).
Conditional on being o
25
Page 27 of 50
the equilibrium path (column 4 in Table 12), as Hypothesis 6 states, the cooperation rates are not signi cantly dierent between treatments except for treatments 1H11R_D and 1H11R_E, which may be due to the small number of subjects o the equilibrium path in treatment 1H11R_E. Finally, the frequency with which subjects followed the contagious strategy (column 5 in Table 12) is signi cantly larger when the on-equilibrium-path condition is satis ed and for a 1H11R commu-
ip t
nity. For the 2H10R_A treatment, the frequency that subjects followed the contagious strategy is in between those in found for the 1H11R_A and 6H6R_A treatments, although it is not signi cantly dierent. However, this frequency is signi cantly dierent between 1H11R_A and 6H6R_A
< 0.05).
cr
treatments (p
Finding 4 With a larger group size n = 6 and with a 1H11R community, subjects are more often
an
us
on the equilibrium path, choose cooperation more frequently when on the equilibrium path, and more often follow the contagious strategy, as compared with the corresponding 1H11R treatment with a smaller group size n = 2 and the treatment with n = 6 and 6H6R. Summarizing our experimental results, we nd that the behavior of the human subjects is consistent with our theoretical predictions on the impact of group size for cooperative play. Given the same payo parameter
α,
cooperation rates increase when the group size increases.
Under
M
a small group size, the cooperation rate declines over time and this pattern repeats itself across supergames even as subjects gain repeated experience with the environment. By contrast, under a larger group size, subjects learn to stick with cooperation after experiencing the much quicker
d
consequences of triggering a contagious wave of defection in their community.
te
When there is more than one human subject in each community, cooperation rates decrease as the ratio of human subjects to robots increases. Comparing the 1H11R treatment with the 6H6R treatment with the same group size
n=6
and
α
that always supports the contagious equilibrium,
the cooperation rate by the human subjects drops from 80% in 1H11R to less than 10% in 6H6R,
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
suggesting that it is greater strategic uncertainty that works to destroy cooperation in communities with randomly and anonymously matched players.
7
Conclusions
We have examined the eect of group size,
n,
on the equilibrium conditions needed to sustain
cooperation via the contagious strategy as a sequential equilibrium in repeated play of an Prisoner's Dilemma game, given a nite population of players of size
M ≥ n
n-player
and random and
anonymous matching of players in each repetition of the game. We nd that, if agents are su ciently patient, the equilibrium conditions, both on the equilibrium path and o the equilibrium path, become less restrictive, and thus more easily satis ed as the group size
n
increases toward
M.
This
result arises from the faster speed with which a contagious wave of defections can occur as the group size becomes larger.
Our ndings expand upon Kandori's (1992) idea that a social norm
of cooperative behavior among randomly matched strangers can be policed by community-wide enforcement. Speci cally, we show that communitywide enforcement becomes easier to sustain as
26
Page 28 of 50
the speed with which information travels becomes faster, which is here proxied by increases in the matching group size,
n.
Other interpretations of
n
are possible. Consider, for example, two towns with similar popula-
tions. The tra c rules are commonly known to all but there is no o cial police force that regulates drivers to follow these tra c rules in either town.
It's always more e cient if everyone follows
ip t
the tra c rules than if no one does. Suppose the only dierence between the two towns is in the structure of the roads. One town has several main streets with a few crosses. The other town has many small streets with a lot of crosses. The rst town's road structure corresponds to our large
cr
group structure whereas the second town's road structure corresponds to our small group structure. Disobedience of tra c laws has a more immediate impact in the rst town than in the second town, and so, by the logic of our theory, one might expect greater adherence to tra c laws in the rst As another example, centralized communication or monitoring mecha-
us
town than in the second.
nisms (credit bureaus) might also perform the same role played by larger group sizes in easing the conditions under which a social norm of cooperation is sustained in a large population of players.
n
increases to
M,
we eectively achieve increasingly perfect monitoring of the actions
an
After all, as
chosen by all players.
We have also empirically evaluated our theory by designing and reporting on an experiment
we found that subjects
were
M
exploring some of the comparative statics implications of the theory. Consistent with the theory, able to achieve higher rates of cooperation when randomly and anony-
mously matched into larger rather than smaller groups of size
n
to play a inde nitely repeated
Prisoner's Dilemma game. In many of our experimental treatments, we removed strate-
d
n-player
gic uncertainty by having human participants play only against robot players who were known to
te
always play according to the contagious strategy. By contrast, Camera et al. (2012) found large heterogeneity in strategies adopted by participants in a 2-player Prisoner's Dilemma game with random and anonymous matching. When there is heterogeneity in strategies or a belief that strategies
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are heterogeneous, the incentives for agents to play according to the contagious strategy may be greatly weakened or become non-existent. We have found evidence for this phenomenon as well, as we observed declines in cooperation rates as we increased the ratio of human subjects to robots in the population of players from which groups were randomly formed. A next step in this literature might be to examine whether the fraction of robot players playing according to the contagious strategy could be very gradually reduced, e.g., from
n−1
robots on
down to 0, and gradually replaced with human subjects, so as to keep the total population size,
M,
xed.
Alternatively, one could rst have 12 human subjects play for a time in our 1H11R
treatment condition and subsequently have those same 12 subjects interact with one another (no robots -12H0R) playing the same inde nitely repeated n-player PD game. It would be of interest to know whether the human subjects, who would be free to choose any strategy, could learn to coordinate on the cooperative outcome in such settings. We leave these projects to future research.
27
Page 29 of 50
References
Camera,
G. and A. Giore (2014), A Tractable Analysis of Contagious Equilibria,
Camera,
G, Casari M and Bigoni M. (2012), Cooperative Strategies in Anonymous Economies:
Journal of
Mathematical Economics, 50, 290300.
Carpenter,
Games and Economic Behavior, 75, 570-586.
J.P., (2007), Punishing Free-riders: How Group Size Aects Mutual Monitoring and
Games and Economic Behavior, 60, 31-51.
cr
the Provision of Public Goods,
Dal
ip t
An Experiment,
Bó, P. (2005), Cooperation under the Shadow of the Future: Experimental Evidence from
The American Economic Review, 95(5):
1591-1604.
us
In nitely Repeated Games,
Dal
Bó, P. (2007), Social Norms, Cooperation and Inequality,
Dal
Bó, P. and Fréchette G R., (2011), The Evolution of Cooperation in In nitely Repeated
Economic Theory, 30, 89-105.
an
Games: Experimental Evidence,
The American Economic Review, 101(1):
411-429
Duy,
J. and J. Ochs (2009), Cooperative Behavior and the Frequency of Social Interaction,
Duy,
J., H. Xie and Y-J. Lee (2013), Social Norms, Information, and Trust Among Strangers:
Ellison,
Economic Theory, 52(2), 669-708.
d
Theory and Evidence,
M
Games and Economic Behavior, 66, 785-812.
G. (1994), Cooperation in the Prisoner's Dilemma with Anonymous Random Matching,
Engle-Warnick,
te
Review of Economic Studies, 61, 567-588.
J. and Slonim, R.L., (2006), Inferring Repeated-Game Strategies from Actions:
Ac ce p
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Evidence from Trust Game Experiments,
Fudenberg,
D, Dreber A, Rand DG. (2012), Slow to Anger and Fast to Forgive: Cooperation in
an Uncertain World,
Hardin, Isaac,
American Economic Review, 102:720-749.
G. (1968), The Tragedy of the Commons,
Science, 162, 1243-1248.
M., and J. Walker, (1988), Group Size Eects in Public Goods Provision: The Voluntary
Contribution Mechanism,
Isaac,
Economic Theory, 28, 603632.
Quarterly Journal of Economics, 103, 179-199.
M., J. Walker, and A. Williams (1994), Group Size and the Voluntary Provision of Public
Goods,
Kandori,
Journal of Public Economics, 54, 1-36.
M. (1992), Social Norms and Community Enforcement,
Review of Economic Studies,
59, 63-80.
Nosenzo,
D., S. Quercia, and M. Sefton (2015), Cooperation in Small Groups:
Group Size
Experimental Economics
The Eect of
18, 4-14.
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Okada,
A. (1993), The Possibility of Cooperation in an n-Person Prisoners' Dilemma with Insti-
tutional Arrangements,
A.E. and J.K. Murnighan (1978), Equilibrium Behavior and Repeated Play of the Prisoners'
Dilemma,
Weimann,
Journal of Mathematical Psychology, 17, 189-198.
J., J. Brosig-Koch, T. Heinrich, H. Hennig-Schmidt, C. Keser, C. Stahr (2014),
An
ip t
Roth,
Public Choice, 77, 629-656.
Explanation of (First Round) Contributions in Public-Good Experiments, CESifo working paper no. 5039, October.
cr
Games and Economic
us
Behavior, 76(2), 548-555.
B, B. Cadsby, L. Fan and F. Song, (2013), Group Size, Coordination, and the Eectiveness of Punishment in the Voluntary Contributions Mechanism: An Experimental Investigation, 89-105.
M
an
Games, 4(1):
d
Xu,
H. and Y-J. Lee (2012), Social Norms and Trust Among Strangers,
te
Xie,
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29
Page 31 of 50
Appendix A We rst show that the equilibrium conditions in Lemma 1 are equivalent to the equilibrium con-
n = 2 . Translating our notation to T T that used by Kandori (1992), Cn−1 = 1, un = (−l, 1) , vn = (0, 1 + g) , Zn = (iM − ρ ρ) 1 T (ρ = M −1 (M − 1, M − 2, . . . , 1, 0) , in which the ith element of ρ is the conditional probability that a d-type player meets a c-type when there are i d-types, and iM is a 1 × M vector with all elements equal to 1), An = A, Bn = B , Hn = H . Thus condition 1, equation 1 can be written as:
us
= (1 − δ)e1 (I − δA)−1 ρ(1 + g),
!
cr
1 ≥ (1 − δ)e1 (I − δA)−1 (iM − ρ ρ)
0 1+g
ip t
ditions provided by Kandori (1992) when the group size
which is the same as equilibrium condition 1 in Kandori (1992). Condition 2, equation 2 can be
ek (iM − ρ ρ)
(
!
0 1+g
≥ δek H(I − δA)−1 (iM − ρ ρ)
!
,
k−1 M −k )g + ( )l ≥ δek H(I − δA)−1 ρ(1 + g), M −1 M −1
M
i.e.,
l g
an
written as
which is the same as equilibrium condition 2 in Kandori (1992).
Pn−1 n j=0 pj = 1
Proof.
and
By de nition,
Pn−1 n j=0 qj = 0.
pnj
te
then
d
Lemma 2 De ne pnj ≡ (1 − δ)e1 (I − δAn )−1 ρnj and qjn ≡ δen Hn (I − δAn )−1 ρnj (j = 0, . . . , n − 1), denotes the discounted summation of the probability of meeting
j c-type
n players in the group once a defection has started when the group size is n, and qj denotes the change
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in the discounted summation of the probability of meeting
d-type
j c-type
players in the group when the
player reverts back to playing cooperation instead of defection given that the group size is
n
k = n d-type players. Notice that by de nition the summation of the elements in each row of matrix Zn , An , and Bn is always equal to 1. Denote ik as a 1 × k vector with all elements t t equal to 1. Thus Zn in = iM , An iM = iM and Bn iM = iM for any group size n and t = 0, 1, . . . , ∞. and there are
Therefore we have
Pn−1 n P∞ t −1 −1 t t=0 δ e1 An iM = 1, j=0 pj = (1 − δ)e1 (I − δAn ) Zn in = (1 − δ)e1 (I − δAn ) iM = (1 − δ) Pn−1 n P ∞ −1 −1 t t t=0 δ en (Bn − An )An iM = 0. j=0 qj = en Hn (I − δAn ) Zn in = en Hn (I − δAn ) iM =
Proof.
[Theorem 1]
limδ→1 (I − δAn )−1 ρnj < ∞ for j = 1, . . . , n − 1. (Therefore, limδ→1 pnj = 0 for j = 1, . . . , n − 1 and limδ→1 pn0 = 1.) The proof is similar as in Kandori's (1992) proof for Theorem n 1. Since Xt = M is the absorbing state and the M th element of ρj is zero for j = 1, . . . , n − 1, We rst show that
(I − δAn )−1 ρnj =
∞ X t=0
δ t Atn ρnj =
∞ X
δ t A˜tn ρnj = (I − δ A˜n )−1 ρnj ,
for
j = 1, . . . , n − 1
t=0 30
Page 32 of 50
where
A˜n
is a matrix obtained by replacing the last column of
only to show the existence of triangular and so is
(I − A˜n ).
(I − A˜n )−1 .
Since the number of
An
by zeros. Given this, we have
d-types
never declines,
A˜n
is upper-
The determinant of an upper-triangular matrix is the products of its
(I − A˜n ). Therefore, limδ→1 pnj = limδ→1 (1 − δ)e1 (I − δAn )−1 ρnj → 0 and qjn = δen Hn (I − δAn )−1 ρnj is nite for j = 1, . . . , n − 1. −1 Z v = pn D + Pn−1 pn D ≤ Now the r.h.s. of equilibrium condition 1, (1 − δ)e1 (I − δAn ) n n o o j=1 j j Pn−1 n Pn−1 n n Do + j=1 pj Dj , where the inequality comes from j=0 pj = 1 and po ≤ 1. Therefore, equilibrium Pn−1 n condition 1 is satis ed if Cn−1 − Do ≥ j=1 pj Dj , which is satis ed when Cn−1 − Do and δ are
ip t
diagonal elements, which are all strictly positive for
su ciently large.
is
Therefore, equilibrium condition 2 is satis ed
is su ciently large.
d
M
an
vn − un
P n = − n−1 j=1 qj .
te
when
n and so qo
Pn−1 n j=1 qj Dj ,
us
nite because
Pn−1 n j=0 qj = 0
δen Hn (I − δAn )−1 Zn vn = qon Do +
cr
Similarly, the r.h.s. of equilibrium condition 2,
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
31
Page 33 of 50
Appendix B: Fixing the Number of Groups, m = M/n M
In this appendix we examine the case where
and
n
are varied in such a way that the number of
m = M/n is held constant. In particular, we compare the equilibrium conditions in three cases where m = 3: 1) M = 12 and n = 4; 2) M = 9 and n = 3; and 3) M = 6 and n = 2. Our aim here is to understand whether variations in the group size n continue to matter for satisfaction of the
groups
ip t
equilibrium conditions needed for cooperation to be sustained as a social norm, when the number of groups is held constant. The following numerical results are obtained holding xed
δ = 0.9.
cr
Equilibrium Condition 1
M/n = 3:
us
First we consider equilibrium condition 1 for each of the three cases where
M =6
and
n = 2: βn ≥ 0.773647α + 0.226353(α + βn );
M =9
and
n = 3: 2βn ≥ 0.774597α + 0.0696314(α + βn ) + 0.155771(α + 2βn );
M = 12 and n = 4: 3βn ≥ 0.783465α+0.030344(α+βn )+0.0546601(α+2βn )+0.131531(α+3βn ).
an
βn = (n − 1)−1 ,
Imposing the normalization condition follows: and
n = 2: α ≤ 0.773647
for
βn = 1;
M =9
and
n = 3: α ≤ 0.809415
for
βn = 1/2;
M = 12
and
n = 4: α ≤ 0.821913
for
these conditions can be simpli ed as
M
M =6
βn = 1/3.
We observe that when the number of groups
d
to those in the case where the population size
m = M/n is xed (at 3) the results are very similar M is xed: equilibrium condition 1 is observed to
n
becomes larger. Intuitively, again, this is driven by the
te
become less restrictive as the group size
faster contagious process with a larger group size. The extent of the tendency for cooperation to become more easily sustainable as
n
increases is smaller than in the case where
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
M
is xed, since in
the latter case the contagious process becomes faster not only due to a larger group size but also due to there being a smaller number of groups as
n
increases.
Equilibrium Condition 2
Finally, we consider equilibrium condition 2 for each of the three cases where
M/n = 3:
M =6
and
k = n = 2: α ≥ −0.270439α + 0.270439(α + βn );
M =9
and
k = n = 3: α ≥ −0.219582α + 0.15486(α + βn ) + 0.064722(α + 2βn );
M = 12 and k = n = 4: α ≥ −0.187568α + 0.126324(α + βn ) + 0.0512613(α + 2βn ) + 0.00998238(α + 3βn ). If we further impose our payo normalization, then we have:
M =6
and
k = n = 2: α ≥ 0.270439
for
βn = 1;
M =9
and
k = n = 3: α ≥ 0.142152
for
βn = 1/2;
M = 12
and
k = n = 4: α ≥ 0.086265
for
βn = 1/3.
As with equilibrium condition 1, the results for equilibrium condition 2 under a xed ratio for
M/n
are similar to those found under a xed
M.
Equilibrium condition 2 becomes less restrictive
32
Page 34 of 50
n.
te
d
M
an
us
cr
ip t
with increases in the group size,
Ac ce p
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33
Page 35 of 50
Appendix C: Instructions (Treatment n = 6) [Instructions for the
n=2
treatment are similar]
Overview This is an experiment in economic decision-making. The Department of Economics has provided You are guaranteed $7 for showing up and completing this experiment.
ip t
funds for this research.
During the course of this experiment, you will be called upon to make a series of decisions. The decisions you make determine your additional earnings for the experiment, beyond the $7 show-up
cr
payment. Your total earnings, including the show-up payment, will be paid to you in cash and in private at the end of the session. We ask that you not talk with one another and that you silence any mobile devices for the duration of this experiment.
us
Speci cs
In today's experiment, you will play with 11 computerized robots denoted by R1, R2, . . . ,
an
R11. You and the 11 robots consist of a single community. In this room, we have ____ human participants. So in total we have ___ communities. The play in each community will not in uence other communities in any way.
Each sequence consists of an inde nite
M
The experiment consists of a number of sequences.
number of rounds. At the start of each round of a sequence, you and every robot in your community will be randomly and anonymously assigned to one of two groups of size 6. All possible divisions
d
of the 12 community members into two groups of size 6 are equally likely at the start of each new round. Thus, the composition of your 6 group members is very likely to change from one round to
te
the next. For example, you may have robots R1, R3, R4, R6, R10 in your group in one round, and have robots R2, R3, R6, R8, R9 in your group in another round. However, you will not know which robot is in your group in a given round since the matching is anonymous. Note also that in addition
Ac ce p
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to the group you are in, there will be a second, 6-member group in your community consisting of all robot players.
In each round, all members of both 6-member groups for that round must simultaneously choose between two options: X or Y. Your earnings for the round will be decided by your own choice and by how many of the other 5 robot members in your group choose X and Y in the round. The same is true for the robot players in your group and the robot players in the other group. Speci cally, the determination of a player's earning (you or the robots) is explained in the payo table that appears on your decision screen which is shown in Table 1 on the next page. In Table 1, the second and third rows show your earnings (in dollars) from choosing option X or option Y respectively, given the number of the 5 robots in your group who choose option X or option Y, as indicated by the rst row.
Example: Suppose that 3 of the 5 robot group members choose X while 2 choose Y: 3X,2Y. If you choose X, your earnings for the round is $0.60 (row two, column four). If you choose Y, your earnings for the round is $1.10 (row three, column four).
34
Page 36 of 50
ip t cr us an M
Figure 5: Payo Table
d
Note that you and your robot group members all face the same payo table as shown in Table 1. This payo table not only shows Your earnings but it also shows how Others' earnings are
te
aected by your choice and the choices of others. To see this, suppose again that 3 of the others choose X and 2 of the others choose Y: 3X,2Y. In this case, if you choose X, then the earnings for the round for the 3 other group members who (like you) choose X is $0.60, and the earnings
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
for the round for the 2 other group members who choose Y is $1.30. If, in this same scenario (3 others choose X and 2 others choose Y: 3X,2Y), you instead choose Y, then the earnings for the round for the 3 other group members who choose X is $0.40, and the earnings for the round for the 2 other group members who (like you) choose Y is $1.10.
Robot Rules
The robots are programmed to make their choices according to the following rules: - choose X in the rst round of each new sequence; - if, during the current sequence, any of a robot's group members, including you or any other robot players have chosen Y in any prior round of that sequence, then the robot will switch to choosing Y in all remaining rounds of the sequence. - otherwise, the robot will continue to choose X. Notice that the choice of each robot may be dierent in each round if their experience in previous rounds of the sequence is dierent. As a human participant, you are always free to choose X or Y. Notice further that no robot will start choosing Y unless the human subject in their community has previously chosen Y in a round of the current sequence.
35
Page 37 of 50
When you make your choice in each round, you will be reminded of the results in all previous rounds of the current sequence. You will be shown your own choice, the number of your robot group members who chose X, the number of your robot group members who chose Y, your earnings for the round and your cumulative earnings for the current sequence of rounds. To complete your choice in each round, simply click on the radio button next to option X or Y (as shown at the bottom of Table
ip t
1) and then click the red Submit button. You can change your mind anytime prior to clicking the Submit button. After you have clicked the red Submit button, the computer program will record your choice and the choices made by your robot group members and determine your earnings for the
cr
round. Then the results of the round will appear on your computer screens. You will be reminded of your own choice and will be informed about how many of your robot group members have chosen X or Y, as well as the payos that you have earned for the round.
Please record the results of
us
the round on your RECORD SHEET under the appropriate headings. Immediately after you have received this information on choices and payos for the round, a random number from 1 to 100 will be drawn by the computer to determine whether the sequence continues or not. If a number from
an
1 to 75 is chosen, the sequence will continue with another round. If a number from 76 to 100 is chosen, the sequence ends. Therefore, after each round there is 75% chance that the sequence will continue with another round and a 25% chance that the sequence will end. Suppose that a number
M
less than or equal to 75 has appeared. Then you will play the same game as in the previous round, but you and your robot community members will be randomly and anonymously assigned to two new groups of size 6. The robots always play according to the robot rules stated above, even in
d
the all-robot group. You only see the outcome of your own group's decisions each round. After you have made your decision for the new round and learned the outcome, record the results and your
te
earnings for the round on your record sheet under the appropriate headings. Then another random number will be drawn to decide whether the sequence continues for another round. If a number greater than 75 appears, then the sequence ends. Depending on the time available, a new sequence
Ac ce p
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may be played. The new sequence will again consist of an inde nite number of rounds of play of the same game as described above. Recall that, according to the robot rules, all robots play X in the rst round of each new sequence, and will continue doing so until they experience play of Y by one or more players in their group, at which point they will switch over to playing Y for the duration of that sequence.
Earnings
Following completion of the last sequence, the experimenter will randomly draw three sequences to determine your earnings from today's experiment. Your earnings from each of the three chose sequences will be the accumulated earnings from all the rounds played in those three sequences. Since you don't know which three sequences will be chosen for payment you will want to do your best in every round of every sequence.
At the end of the session you will be shown your total
earnings on your computer screen. This total amount will include your $7 show-up payment. Please write down the total amount you are owed on your receipt.Then you will be paid your earnings in cash and in private.
36
Page 38 of 50
Final Comments First, do not discuss your decisions or your results with anyone at any time during the experiment. Second, each of you will play with 11 robots in today's session. Your play will not aect the earnings of any other human subject participant.
ip t
Third, your identity and the identity of your robot group members are never revealed. Fourth, at the end of every round of a sequence, there is 75% chance that the sequence will continue with another round and a 25% chance that the sequence will end.
cr
Fifth, remember that at the start of each round of a sequence you and your 11 robot community members are randomly and anonymously assigned to one of two groups of size 6, so the composition of your robot group members is very likely to change from one round to the next.
us
Sixth, the robots always play according to the rules stated in the instructions. Speci cally, they always start out a sequence playing X and only switch to playing Y if a member of their group has chosen to play Y in any prior round of the sequence; otherwise they continue to play X.
an
Finally, we will randomly draw three sequences at the end of today's session to determine your earnings. Your total earnings will be the sum of your accumulated earnings from the three chosen sequences.
M
Questions?
te
d
Now is the time for questions. Does anyone have any questions?
Ac ce p
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37
Page 39 of 50
Table
H* program to produce the transition matrices with fixed population size n=12, compare three cases with group size = 2,3,4,6*L H*Choose parameters*L
ip t
H*Number of players*L n = 12;
"D "D
an
Print@" Print@" Print@ "BELOW Print@"
us
cr
Print@"POPULATION SIZE M=12"D Print@"MATRIX biga IS THE MATRIX A DEFINED IN THE PAPER"D Print@"MATRIX bigb IS THE MATRIX B DEFINED IN THE PAPER"D Print@"MATRIX bigh IS THE MATRIX H=B-A DEFINED IN THE PAPER"D
IS THE TRANSITION MATRIX FOR 12-PERSON PRISONER'S DILEMMA WITH 1 GROUP"D "D
d te
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
M
H*Matrix I=in*L in = IdentityMatrix@nD;
H*Transition matrix A, B, Z*L biga12 = Array@a, 8n, n
Ac ce p
In[642]:=
bigb12 = Array@b, 8n, n
H*bigz6=Array@z,8n,6
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga12@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
biga12 = ReplacePart@biga12, 881, 12< -> 1, 82, 12< -> 1, 83, 12< -> 1, 84, 12< -> 1, 85, 12< -> 1, 86, 12< -> 1, 87, 12< ® 1, 88, 12< ® 1, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
Page 40 of 50
TransitionMatrices.nb
H*Initialize B matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb12@@i, jDD = 0;D;D;
bigb12 = ReplacePart@biga12, 881, 1< -> 1, 81, 12< ® 0
us
H*Matrix H*L Print@"bigh12 = ", MatrixForm@bigb12 - biga12DD; Print@" "D
cr
Print@"bigb12 = ", MatrixForm@bigb12DD; Print@" "D;
ip t
H*Now insert the correct nonzero values of b_ij*L
H*Initialize Z matrix so that all elements are zero*L
an
For@i = 1, i £ n, i ++, For@j = 1, j £ 12, j ++, bigz12@@i, jDD = 0;D;D;
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
te
H*Matrix I=in*L in = IdentityMatrix@nD;
d
M
Print@ "BELOW IS THE TRANSITION MATRIX FOR 6-PERSON PRISONER'S DILEMMA WITH 2 GROUPS"D Print@" "D
Ac ce p
2
H*Transition matrix A, B, Z*L biga6 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga6@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
biga6 = ReplacePart@biga6, 881, 6< -> 1, 82, 6< ® 5 11, 82, 12< ® 6 11, 83, 6< ® 2 11, 83, 12< ® 9 11, 84, 6< ® 2 33, 84, 12< ® 31 33, 85, 6< ® 1 66, 85, 12< ® 65 66, 86, 6< ® 1 462, 86, 12< ® 461 462, 87, 12< ® 1, 88, 12< ® 1, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
ip t
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb6@@i, jDD = 0;D;D;
Print@"bigb6 = ", MatrixForm@bigb6DD; Print@" "D;
M
an
H*Matrix H*L Print@"bigh6 = ", MatrixForm@bigb6 - biga6DD; Print@" "D
us
cr
bigb6 = ReplacePart@bigb6, 881, 1< -> 1, 82, 6< ® 5 11, 82, 7< ® 6 11, 83, 6< ® 2 11, 83, 7< ® 3 11, 83, 12< ® 6 11, 84, 6< ® 2 33, 84, 7< ® 4 33, 84, 12< ® 27 33, 85, 6< ® 1 66, 85, 7< ® 1 22, 85, 12< ® 31 33, 86, 6< ® 1 462, 86, 7< ® 1 77, 86, 12< ® 65 66, 87, 7< ® 1 462, 87, 12< ® 461 462, 88, 12< ® 1, 89, 12< ® 1, 810, 12< ® 1, 811, 12< ® 1, 812, 12< ® 1
H*Below is the calculation for 4-person prisoner's dilemma with 3 groups*L
Ac ce p
H*Matrix I=in*L in = IdentityMatrix@nD;
te
d
Print@ "BELOW IS THE TRANSITION MATRIX FOR 4-PERSON PRISONER'S DILEMMA WITH 3 GROUPS"D Print@" "D
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
H*Transition matrix A, B, Z*L biga4 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga4@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
Page 42 of 50
TransitionMatrices.nb
H*Initialize B matrix so that all elements are zero*L
cr
Print@"biga4 = ", MatrixForm@biga4DD; Print@" "D;
ip t
biga4 = ReplacePart@biga4, 881, 4< -> 1, 82, 4< ® 3 11, 82, 8< ® 8 11, 83, 4< ® 3 55, 83, 8< ® 36 55, 83, 12< ® 16 55, 84, 4< ® 1 165, 84, 8< ® 68 165, 84, 12< ® 32 55, 85, 8< ® 7 33, 85, 12< ® 26 33, 86, 8< ® 1 11, 86, 12< ® 10 11, 87, 8< ® 1 33, 87, 12< ® 32 33, 88, 8< ® 1 165, 88, 12< ® 164 165, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb4@@i, jDD = 0;D;D;
us
H*Now insert the correct nonzero values of b_ij*L
M
an
bigb4 = ReplacePart@bigb4, 881, 1< -> 1, 82, 4< ® 3 11, 82, 5< ® 8 11, 83, 4< ® 3 55, 83, 5< ® 12 55, 83, 8< ® 24 55, 83, 9< ® 16 55, 84, 4< ® 1 165, 84, 5< ® 8 165, 84, 8< ® 4 11, 84, 9< ® 16 55, 84, 12< ® 16 55, 85, 5< -> 1 165, 85, 8< ® 34 165, 85, 9< ® 34 165, 85, 12< ® 96 165, 86, 8< -> 1 11, 86, 9< ® 4 33, 86, 12< ® 26 33, 87, 8< ® 1 33, 87, 9< ® 2 33, 87, 12< ® 10 11, 88, 8< ® 1 165, 88, 9< ® 4 165, 88, 12< ® 32 33, 89, 9< -> 1 165, 89, 12< ® 164 165, 810, 12< ® 1, 811, 12< -> 1, 812, 12< -> 1
d
Print@"bigb4 = ", MatrixForm@bigb4DD; Print@" "D;
te
H*Matrix H*L Print@"bigh4 = ", MatrixForm@bigb4 - biga4DD; Print@" "D
Ac ce p
4
H*Below is the calculation for 3-person prisoner's dilemma with 4 groups*L
Print@ "BELOW IS THE TRANSITION MATRIX FOR 3-PERSON PRISONER'S DILEMMA WITH 4 GROUPS"D Print@" "D H*Transition matrix A, B, Z*L biga3 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga3@@i, jDD = 0;D;D;
Page 43 of 50
TransitionMatrices.nb
5
H*Now insert the correct nonzero values of a_ij*L
H*Initialize B matrix so that all elements are zero*L
cr
Print@"biga3 = ", MatrixForm@biga3DD; Print@" "D;
ip t
biga3 = ReplacePart@biga3, 881, 3< -> 1, 82, 3< ® 2 11, 82, 6< ® 9 11, 83, 3< -> 1 55, 83, 6< ® 27 55, 83, 9< ® 27 55, 84, 6< ® 2 11, 84, 9< ® 36 55, 84, 12< ® 9 55, 85, 6< -> 1 22, 85, 9< ® 6 11, 85, 12< ® 9 22, 86, 6< -> 1 154, 86, 9< ® 27 77, 86, 12< ® 9 14, 87, 9< ® 2 11, 87, 12< ® 9 11, 88, 9< ® 4 55, 88, 12< ® 51 55, 89, 9< -> 1 55, 89, 12< ® 54 55, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
us
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb3@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of b_ij*L
d
M
an
bigb3 = ReplacePart@bigb3, 881, 1< -> 1, 82, 3< ® 2 11, 82, 4< ® 9 11, 83, 3< -> 1 55, 83, 4< ® 9 55, 83, 6< ® 18 55, 83, 7< ® 27 55, 84, 4< ® 1 55, 84, 6< ® 9 55, 84, 7< ® 18 55, 84, 9< ® 18 55, 84, 10< ® 9 55, 85, 6< -> 1 22, 85, 7< ® 3 22, 85, 9< ® 9 22, 85, 10< ® 27 110, 85, 12< ® 9 55, 86, 6< -> 1 154, 86, 7< ® 3 77, 86, 9< ® 24 77, 86, 10< ® 18 77, 86, 12< ® 9 22, 87, 7< ® 1 154, 87, 9< ® 27 154, 87, 10< ® 27 154, 87, 12< ® 9 14, 88, 9< ® 4 55, 88, 10< ® 6 55, 88, 12< ® 9 11, 89, 9< -> 1 55, 89, 10< ® 3 55, 89, 12< ® 51 55, 810, 10< -> 1 55, 810, 12< ® 54 55, 811, 12< -> 1, 812, 12< -> 1
te
Print@"bigb3 = ", MatrixForm@bigb3DD; Print@" "D;
Ac ce p
H*Matrix H*L Print@"bigh3 = ", MatrixForm@bigb3 - biga3DD; Print@" "D; H*Below is the calculation for 2-person prisoner's dilemma with 6 groups*L
Print@ "BELOW IS THE TRANSITION MATRIX FOR 2-PERSON PRISONER'S DILEMMA WITH 6 GROUPS"D Print@" "D H*define s function*L
s@num_D := Product@2 * i - 1, 8i, 1, num 2
Print@"s@nD = ", s@nDD; Print@" "D
Page 44 of 50
TransitionMatrices.nb
biga2 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
H*Now insert the correct nonzero values of a_ij*L
cr
Print@"Matrix A initialized."D;
ip t
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga2@@i, jDD = 0;D;D;
Print@"biga2 = ", MatrixForm@biga2DD; Print@" "D
an
us
For@i = 1, i £ n, i ++, If@Mod@i, 2D 0, H*if i is even*LFor@j = i, j £ Min@2 i, nD, j += 2, biga2@@i, jDD = HBinomial@i, j - iD * Binomial@n - i, j - iD * HHj - iL !L * s@2 i - jD * s@n - jDL denom;D;, H*if i is odd*L For@j = i + 1, j £ Min@2 i, nD, j += 2, biga2@@i, jDD = HBinomial@i, j - iD * Binomial@n - i, j - iD * HHj - iL !L * s@2 i - jD * s@n - jDL denom;D;D;D;
M
H*Initialize H matrix so that all elements are zero*L
d
bigh2 = Array@h, 8n, n
te
Print@"Matrix H initialized."D;
H*Now insert the correct nonzero values of h_ij*L
For@i = 1, i £ n, i ++, If@Mod@i, 2D 0, H*if i is even*L For@j = i + 2, j £ Min@2 i, nD, j += 2, bigh2@@i, jDD = - HHj - iL iL * biga2@@i, jDD; bigh2@@i, j - 1DD = Abs@bigh2@@i, jDDD;D;, H*if i is odd*L For@j = i + 1, j £ Min@2 i, nD, j += 2, bigh2@@i, jDD = - HHj - iL iL * biga2@@i, jDD; bigh2@@i, j - 1DD = Abs@bigh2@@i, jDDD;D;D;D;
Ac ce p
6
Print@"bigh2 = ", MatrixForm@bigh2DD; Print@" "D H*end of program*L
POPULATION SIZE M=12 MATRIX biga IS THE MATRIX A DEFINED IN THE PAPER MATRIX bigb IS THE MATRIX B DEFINED IN THE PAPER MATRIX bigh IS THE MATRIX H=B-A DEFINED IN THE PAPER
Page 45 of 50
TransitionMatrices.nb
7
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1
bigb12 =
1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
te 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
Ac ce p
bigh12 =
cr
0 0 0 0 0 0 0 0 0 0 0 0
us
0 0 0 0 0 0 0 0 0 0 0 0
an
0 0 0 0 0 0 0 0 0 0 0 0
M
0 0 0 0 0 0 0 0 0 0 0 0
d
biga12 =
0 0 0 0 0 0 0 0 0 0 0 0
ip t
BELOW IS THE TRANSITION MATRIX FOR 12-PERSON PRISONER'S DILEMMA WITH 1 GROUP
0 0 0 0 0 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BELOW IS THE TRANSITION MATRIX FOR 6-PERSON PRISONER'S DILEMMA WITH 2 GROUPS
Page 46 of 50
TransitionMatrices.nb
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1 0 0 0 0
0
0
5 11 2 11 2 33 1 66 1 462
6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bigb6 =
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0 -1 0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
bigh6 =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0
0
0 0 0 0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 0
6 11 9 11 31 33 65 66 461 462
0 0 0 0 0
te
0 0 0 0 0 0
6 11 9 11 31 33 65 66 461 462
ip t
0 0 0 0 0
0
0 0 0 0 0
cr
0 0 0 0 0 biga6 =
0 0 0 0 0
us
0 0 0 0 0
5 11 2 11 2 33 1 66 1 462
an
0 0 0 0 0
1
M
0 0 0 0 0
1 1 1 1 1
d
0 0 0 0 0
0 0 0 0
0 6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 -
Ac ce p
8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
BELOW IS THE TRANSITION MATRIX FOR 4-PERSON PRISONER'S DILEMMA WITH 3 GROUPS
Page 47 of 50
TransitionMatrices.nb
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0
0
0
3 11 3 55 1 165
8 11 12 55 8 165 1 165
0 0 0 0
0 0 0 0 0 0
0 0 0 0
0
0 0 0
8 11 36 55 68 165 7 33 1 11 1 33 1 165
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
0 0 0 0
0
0 0
0
0
0
0 0
0
24 55 4 11 34 165 1 11 1 33 1 165
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0 0 0 0 0 0
0 0 0
0
0
0 0
0 0 0
0
0
0 0
0 0 0
0
0
0 0
0
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
Ac ce p
bigh4 =
0 0 0 0 0 0 0 0 0 0 0 0 0 0
d
0
0
1 1 1 1
0
0
0 0 0
0 0 0 0
0 0
0 0 0
8 11 12 55 8 165 1 165
0 0 0 0
0 0
0
0
0 16 55 32 55 26 33 10 11 32 33 164 165
0 0 0
0 0 0
1 0 0 -1
0
0 0 0
ip t
0 0 0
0 0 0
0 0 0
bigb4 =
0 0 0
0 0 0
0 0 0 0 0 0
te
biga4 =
0 0 0
cr
0 0 0
3 11 3 55 1 165
0 0 0 0 0 0
0
8 11 12 55 8 165 1 165
0 0 0
0
0 0 0
0
0 0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0
us
0 0 0
1
an
0 0 0
0
16 55 32 55 26 33 10 11 32 33 164 165
M
0 0 0
1 1 1
0
0 0
0
0
0 0
0
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0
9
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
BELOW IS THE TRANSITION MATRIX FOR 3-PERSON PRISONER'S DILEMMA WITH 4 GROUPS
Page 48 of 50
TransitionMatrices.nb
9 11 27 55 2 11 1 22 1 154
0 0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0 0
0
0 0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
1 0
0
0
2 11 1 55
9 11 9 55 1 55
0 0 0 0
0 0 0 0 0 0 0 0 0 0
9 55 9 22 9 14 9 11 51 55 54 55
1 1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
18 55 9 55 1 22 1 154
27 55 18 55 3 22 3 77 1 154
0
0
0
18 55 9 22 24 77 27 154 4 55 1 55
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
0
0 0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
0
0 0 0 0
0 0
0
0
9 11 9 55 1 55
0 0 0 -
0
9 11 9 55 1 55
0 0
0 0
9 55 9 22 9 14 9 11 51 55 54 55
M
0
0
0 0
1 0 -1
bigh3 =
0 0
d
0 0
0 0
27 55 36 55 6 11 27 77 2 11 4 55 1 55
ip t
0
0 0
0 0
0 0
bigb3 =
0 0
te
biga3 =
2 11 1 55
cr
0 0
1
us
0 0
an
0 0
Ac ce p
10
0 0 0 0 0
0 0
0 0
1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
27 55 18 55 3 22 3 77 1 154
0 0
27 55 18 55 3 22 3 77 1 154
0 0
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0 0
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0 0 0 0 0 0 0 0 0
0 0
BELOW IS THE TRANSITION MATRIX FOR 2-PERSON PRISONER'S DILEMMA WITH 6 GROUPS
Page 49 of 50
TransitionMatrices.nb
11
s@nD = 10 395
0
1
0
1 11
0 0 0
0 0
0 0 0 0
0 10 11 3 11 1 33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
8 11 16 33 5 33 5 231
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8 33 80 231 20 77 4 33 1 33
8 33 80 231 20 77 4 33 1 33
0
0
0 0
0
0 16 231 8 33 16 33 8 11 10 11
0 0 0 0
0 0
0 0
1 1
0 0 0
0 0 0 0
0 10 11 1 11
0 0
0 -
10 11 1 11
0 0
8 11 8 33 1 33
-
8 11 8 33 1 33
0
0
16 33 4 11 10 77 5 231
16 33 4 11 10 77 5 231
0
0
0
d
0
te
1 -1
M
Matrix H initialized.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Ac ce p
bigh2 =
0
cr
0
8 33 40 77 20 33 16 33 3 11 1 11
us
0
16 33 20 33 30 77 5 33 1 33
an
biga2 =
0
0
ip t
Matrix A initialized.
16 231 40 231 8 33 8 33 2 11 1 11
0
16 231 40 231 8 33 8 33 2 11 1 11
-
0
Page 50 of 50
S0167-2681(16)00037-8 http://dx.doi.org/doi:10.1016/j.jebo.2016.02.007 JEBO 3753
To appear in:
Journal
Received date: Revised date: Accepted date:
7-4-2015 16-2-2016 22-2-2016
of
Economic
Behavior
&
Organization
Please cite this article as: John Duffy, Huan Xie, Group Size and Cooperation among Strangers, (2016), http://dx.doi.org/10.1016/j.jebo.2016.02.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
ip t
cr us an M d te
We model an n-person PD game with anonymous and randomly matched agents. We show that it can be easier to sustain society-wide cooperation with a larger group size. Strong experimental evidence is provided for this theoretical prediction when there is no strategic uncertainty. The introduction of strategic uncertainty may diminish this group size effect.
Ac ce p
Page 1 of 50
ip t
Group Size and Cooperation among Strangers Huan Xie
Department of Economics University of California, Irvine 3151 Social Science Plaza Irvine CA 92697-5100, USA Email: [email protected]
Corresponding author Department of Economics Concordia University, CIREQ and CIRANO 1455 de Maisonneuve Blvd. O., H1155 Montreal, Quebec, H3G 1M8, Canada Email: [email protected]
Ac ce p
te
d
M
an
us
cr
John Duffy
Page 2 of 50
Group Size and Cooperation among Strangers∗
ip t
John Duy† and Huan Xie‡
cr
February 16, 2016
Abstract
te
d
M
an
us
We study how group size aects cooperation in an in nitely repeated n-player Prisoner's Dilemma (PD) game. In each repetition of the game, groups of size n ≤ M are randomly and anonymously matched from a xed population of size M to play the n-player PD stage game. We provide conditions for which the contagious strategy (Kandori, 1992) sustains a social norm of cooperation among all M players. Our main nding is that if agents are su ciently patient, a social norm of society-wide cooperation becomes easier to sustain under the contagious strategy as n increases toward M . In an experiment where the population size M is xed and conditions identi ed by our theoretical analysis hold, we nd strong evidence that cooperation rates are higher with larger group sizes than with smaller group sizes in treatments where each subject interacts with M − 1 robot players who follow the contagious strategy. When the number of human subjects increases in the population, the cooperation rates decrease signi cantly, indicating that it is the strategic uncertainty among the human subjects that hinders cooperation. JEL Classi cation Codes: C72, C73, C78, Z13. Keywords: Cooperation, Social Norms, Group Size, Repeated Games, Random Matching, Prisoner's Dilemma, Experiment.
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∗
We have bene ted from comments by two anonymous referees, Bram Cadsby, Gabriele Camera, Prosper Dovonon,
Erik Kimbrough, Ming Li, Quang Nguyen and seminar participants at Nanyang Technological University, Shanghai University of Finance and Economics, the 2012 China International Conference on Game Theory and Applications and the 4th Annual Xiamen University International Workshop on Experimental Economics.
Duy gratefully ac-
knowledges research support from the UCI School of Social Sciences. Xie gratefully acknowledges the hospitality of the Economic Growth Center at the Division of Economics in Nanyang Technological University and funding support from FQRSC (2010-NP-133118). We thank Tyler Boston for his great help in programming the experiment. † University of California, Irvine, du[email protected] ‡ Concordia University, CIREQ and CIRANO, [email protected]
Page 3 of 50
1
Introduction
What choice of group size maximizes (or minimizes) the possibility of achieving a social norm of cooperation in a nite population of self-interested strangers? This question would seem to be of some importance for the design of ad hoc committees, juries and teams.
It is also of interest to
experimentalists interested in understanding how the extent of pro-social behavior might depend on
Speci cally, we consider a population of players of xed size
ip t
the matching group size of subject participants. In this paper we oer an answer to this question.
M.
In every period,
players in this population are randomly and anonymously matched to form groups
n-person Prisoner's Dilemma game with all the members of their groups, M/n, is assumed to be an integer (i.e., M is a multiple of n).
The
n=2
group. The total
person version of this environment has been previously studied by Kandori (1992),
us
number of
t = 1,2,...∞, of size n and
cr
then play an
who shows that a social norm of cooperation among anonymous, randomly matched players is sustainable under certain conditions on the game.
n=2
and the possibility vanishes in the limit as for what value(s) of
n>2
Kandori further shows that a social norm of
M gets large x M and ask:
case becomes more di cult to sustain as
an
cooperation among strangers in the
M → ∞.
By contrast, in this paper we
is a social norm of cooperation among strangers easiest to achieve? In
M
other words, is there an optimal group size for maximizing the likelihood of cooperative outcomes? Our answer is that under certain conditionsspeci cally if agents are su ciently patient a social norm of cooperation among strangers, which is sustained by universal play of a contagious trigger gets larger, and becomes easiest to achieve when
That is, we nd that cooperation can be easiest to sustain when the group size is as large
as possible.
This seemingly counterintuitive nding readily follows from the rational-choice logic
te
n = M.
n
d
strategy, becomes steadily easier to achieve as
of the contagious trigger strategy that is used to support cooperation among randomly matched, non-communicative and anonymous strangers. Intuitively, if agents are su ciently patient, then
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the costs of igniting a contagion toward mutual defection are greatest when the matching group size,
n,
equals the population size,
M.
On the other hand, once a defection has started in the
community, the bene ts to slowing down the contagious process are also minimized in this same case where
n = M.
Therefore, the players' incentives to follow the contagious strategy are easiest
to satisfy when the group size is as large as possible. Our main nding is consistent with Kandori's (1992) result. a social norm of cooperation is easier to achieve as be viewed as showing that for the case where a smaller
M.
n
n
While we x
increases towards
is xed at
2,
M,
M
Kandori's result can
cooperation is easier to sustain for
Still, our ndings serve to generalize Kandori's (1992) extension of the folk theorem
for repeated games with random, anonymous matchings to the multiple-player (n Dilemma game. The variety of
and show that
n-player
> 2)
Prisoner's
version of the Prisoner's Dilemma game is widely used to model a
social dilemmas including, e.g., the tragedy of the commons (Hardin (1968)).
we show that our monotonicity result holds in an
n-player
In addition,
binary public good game.
We also provide an empirical test of our main theoretical results by designing and implementing an experiment. In this experiment, we x the population size,
M,
and the discount factor,
δ,
and
2
Page 4 of 50
study play of an inde nitely repeated game in which players from the population are randomly and anonymously matched in each repetition to play an game. Within the population of size
M,
n-player version of the Prisoner's Dilemma stage
some fraction of players are robot players programmed to
play according to the contagious strategy while the remaining fraction of players are human subjects and this ratio is public knowledge. In this setting we nd strong evidence that, consistent with our
as compared with smaller matching groups, e.g., of size
n=2
n=6
ip t
theoretical predictions, cooperation rates are higher with larger matching groups, e.g. of size
as subjects learn, with experience,
the more immediate consequences of triggering an infectious wave of defection when the group size
cr
is larger. We show further how dierences in cooperation rates between dierent group sizes vary in ways that re ect the predictions of our theory as we vary the payo incentives of the game as well. Finally we show how our theoretical results nd the strongest support when we eliminate strategic
us
uncertainty by having subjects interact only with robot players.
Our paper contributes to the theoretical and experimental literature on sustaining cooperation among anonymous, randomly matched players.
While this is an admittedly stark environment,
an
it is an important benchmark case in both the theoretical and experimental literature and one
1 In addition to the original
that naturally characterizes many types of socio-economic interactions.
seminal paper by Kandori (1992), Ellison (1994) and Dal Bó (2007) provide further generalizations of
M
how a social norm of cooperation may be sustained among anonymous, randomly matched players in 2-player Prisoner's Dilemma games. Xie and Lee (2012) extend Kandori's result to 2-player trust games under anonymous random matchings. Camera and Giore ( 2014) oers a tractable analysis
d
of the contagious equilibria by characterizing a key statistic of contagious punishment processes and
te
deriving closed-form expressions for continuation payos o the equilibrium path. Experimentally, Duy and Ochs (2009) report on an experiment that examines play in an inde nitely repeated, two-player Prisoner's Dilemma game and nd that a cooperative norm does not emerge in the treatments with anonymous random matching but does emerge under xed pairings as players
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gain more experience. Camera and Casari (2009) examine cooperation under random matching by focusing on the role of private or public monitoring of the anonymous (or non-anonymous) players' choices.
They nd that such monitoring can lead to a signi cant increase in the frequency of
cooperation relative to the case of no monitoring. Duy et al. (2013) test the contagious equilibrium in the lab using trust games and nd that information on past play signi cantly increases the level of trust and reciprocity under random matchings. Camera et al. (2013) report wide heterogeneity in strategies employed at the individual level in an experiment in which anonymous randomly matched subjects play the Prisoner's Dilemma game in sequences of inde nite duration. Compared with this previous literature, our paper is the rst to theoretically and experimentally extend the analysis of the contagious equilibrium from a 2-player stage game to an
n-player
stage game.
Our main
theoretical nding, that a cooperative social norm is easier to sustain with a larger rather than
1
There is also an experimental literature that studies cooperation in repeated Prisoner's Dilemma games of in-
de nite duration among xed pairs of players (partners) e.g., Dal Bó (2007), Aoyagi and Fréchette (2009), Dal Bó and Fréchette (2011), Fudenberg et al. (2012). Engle-Warnick and Slonim (2006) examines a trust game of inde nite duration with xed pairs.
3
Page 5 of 50
a smaller group size, is new to the literature but nds support both in our own experiment and qualitatively in the previous literature as well, if one considers a large group size to be a partial substitute for public monitoring or xed matching.
2
There are several experimental papers that study the consequences of group size for contributions to a public good using the liner voluntary contribution mechanism (VCM). Isaac and Walker (1988)
ip t
and Isaac et al. (1994) examine how groups of size 4, 10, 40 and 100 play a repeated public good game. One of their main ndings is that, holding the marginal per capita return (MPCR) to the public good constant, an increase in the number of players,
n,
leads to no change or an increase
cr
(depending on the MPCR) in the mean percentage of each player's xed and common endowment that is contributed toward the public good, and this eect is strongest with group sizes of 40 and 100 in comparison with group sizes of 4 and 10. Carpenter (2007) studies contributions under the
us
VCM when players are randomly matched into groups of size 5 or 10, and he allows participants to monitor and punish each other following each repetition.
In this setting, he also nds that
contributions are higher with a larger group size. Xu et al. (2013) examine the eectiveness of an
an
individual-punishment mechanism in larger groups of 40 participants compared with smaller groups of size four. They nd that the individual punishment mechanism is eective when the MPCR is constant but not when the marginal group return (MGR) is held constant (in which case the MPCR
M
is decreasing). Similarly, Nosenzo et al. (2015) nd that it is more di cult to sustain cooperation in larger groups with high MPCR, however, with a low MPCR, considerations of the social bene ts may dominate the negative eect of group size.
Weimann et al.
(2014) report that rst-round
d
contributions to a public good increases with the MPCR distance, which is de ned as the dierence
te
between the actual MPCR and the minimal MPCR necessary to create a social dilemma for a given group size. They further demonstrate that small groups behave similar to large groups when the MPCR distance is controlled.
There are several important dierences between our experimental results and linear VCM games
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(public good games). First, in our model and experimental design, we impose a payo normalization under which players' payos are always equal to 1 when full cooperation is achieved, regardless of the group size.
Equivalently, this corresponds to a public goods experiment where the MPCR
decreases as the group size increases. Therefore, our monotonicity result and the evidence from our experiment show that cooperation increases with group size, even as the MPCR is decreasing, a result that we believe is new to this literature. Second, in our treatments where human subjects only interact with robot players, cooperation does not rely on any other-regarding preferences, which is an important explanatory variable in the experimental public goods literature.
Third,
the game we have implemented has several dierences from the public good game: 1) the strategy space is continuous under the VCM and not binary as in the
n-player
Prisoner's Dilemma game
that we study; 2) subjects in many of these public goods experiments (Carpenter (2007) being an exception) are in
2
xed
matches of size
We note that when the group size,
n,
n
for all repetitions of the public good game whereas in
M , then our than M , one can
is set equal to the largest possible value, the population size
model converges to one of perfect public monitoring and xed matching. Thus for group sizes less
view larger group sizes as being closer approximations to perfect public monitoring and xed matchings.
4
Page 6 of 50
our setup players are randomly and anonymously matched into groups of size
n
in each repetition
of the game; and 3) perhaps most importantly, in all of these public good game experiments, the game is
nitely-repeated
so that in theory, positive public good provision (cooperation) or
e ciency is not possible according to standard backward induction arguments if players have the usual self-regarding preferences. By contrast, we study in nitely repeated, binary choice
n-player
ip t
prisoner's dilemma games where cooperation and e ciency are theoretically possible even if players are randomly and anonymously matched and have purely self-regarding preferences. aware of any prior experimental study of inde nitely repeated
n-player
We are not
Prisoner's Dilemma games
cr
under anonymous random matching and our use of programmed robot players in this setting is also new to the literature. The rest of the paper is organized as follows.
Section 2 presents our model and section 3
us
presents our main theoretical results on the consequences of group size for the sustainability of social norms of cooperation among anonymous and randomly matched strangers. Section 4 shows how our framework maps into the classic public good game of Isaac and Walker (1988). Section 5
an
describes the experimental design and section 6 reports on the ndings of an experiment testing our main theoretical results. Finally, section 7 concludes with a brief summary and some suggestions
The Model
Consider a nite population of
M
players. Time is discrete, the horizon is in nite and all players
d
2
M
for future research.
δ ∈ [0, 1]. In each period, the M players are randomly of size n ≤ M , with all matchings being equally likely.
and
te
have a common period discount factor,
m groups We assume that M is a multiple, m, of n. The randomly matched group members then simultaneously and without communication play an n-player Prisoner's Dilemma game where each player chooses a strategy from the set {C, D}, with C representing cooperation and D representing defection. Let i denote the number of members of the group choosing to cooperate (i.e., the number of cooperators) other than the representative player himself so that 0 ≤ i ≤ n − 1. Let Ci and Di denote the payos to cooperation and defection, respectively, when there are i cooperators. An n-player
anonymously matched into
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Prisoner's Dilemma game is de ned by the following three assumptions regarding these payos: A1: A2: A3:
Di > Ci for 0 ≤ i ≤ n − 1. Ci+1 > Ci and Di+1 > Di for 0 ≤ i < n − 1. Cn−1 > D0 . Assumption A1 says that defection is always a dominant strategy.
Assumption A2 says that
payos are increasing with the number of cooperators. Finally, assumption A3 says that if all participants adopt the dominant strategy, the outcome is sub-optimal relative to the mutual cooperation outcome. These conditions are standard in the literature on
n-person
Prisoner's Dilemma games
(See, e.g., Okada (1993, Assumption 2.1)). We further suppose that the payo matrix is symmetric for each player in the group and is as given in Table 1.
5
Page 7 of 50
number of cooperators in the group
0
1
2
...
n−1
C
C0 C1 C2
...
Cn−1
D
D0 D1 D2
...
Dn−1
Table 1: The Payo Matrix of the
n-Player
Prisoner's Dilemma Game
ip t
We next de ne the contagious strategy following Kandori (1992) and show that a social norm of cooperation can be sustained as a sequential equilibrium if all players adopt this strategy. De ne a player as a c-type if in all previous repetitions of the game this player and all of the other
cr
group members with whom he has interacted in all prior periods have never chosen outcome of the stage game played in every prior period has been cooperation,
Otherwise, the player is a d-type player.
us
member the player has encountered.
i.e., the
by every group (Note that the
d-type players in the same period among the population (or community) of players of size M ≥ n). The contagious strategy can now be de ned as follows: A player chooses C if he is c-type and chooses D if he is d-type. players in any period does not preclude the presence of
an
presence of
c-type
C,
D,
n−1
We next provide a set of su cient conditions that sustains the contagious strategy as a sequential
n.
We rst introduce some notation. Let
M
equilibrium when the group size is
Xt
denote the number
(anij ) to be an
d-type players at time t. De ne An = M × M transition probability matrix n where aij = Pr(Xt+1 = j|Xt = i and all players follow the contagious strategy) given group size n. De ne Bn = (bnij ) as an M × M transition probability matrix where bnij = Pr(Xt+1 = j|Xt = i and one d-type player deviates to playing C while all other players follow the contagious strategy) given group size n. Let Hn = Bn − An , which indicates how the diusion of defection is delayed n n n by the unilateral deviation of one of the d-type players. De ne Zn = (ρ0 ρ1 . . . ρn−1 ), where ρn0 , ρn1 , . . . , ρnn−1 are M × 1 vectors such that the ith element of ρnj is the conditional probability that a d-type player meets j c-type players in the group when there are i d-type players in the n n community given that the group size is n (i.e., Zn = (zij ) is an M × n matrix where zij = Pr(a d-type player meets j − 1 c-type players in his group in period t|Xt = i) given a group size of n). De ne ei as a 1 × M vector whose ith element is 1 and with zeros everywhere else. Finally, de ne T T column vectors vn = (D0 , D1 , . . . , Dn−1 ) and un = (C0 , C1 , . . . , Cn−1 ) , whose ith element is the payo for a player from choosing D and C respectively, given that there are i − 1 other players in the group who choose C .
te
d
of
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Next we show that a one-shot deviation from the contagious strategy is unpro table after any history. On the equilibrium path, a one-shot deviation is unpro table if
∞ Cn−1 X ≥ δ t e1 Atn Zn vn . 1−δ t=0
(1)
The left hand side of (1) is the payo from cooperating forever and the right-hand side of (1) is the payo that the player earns if the player initiates a defection and defects forever afterward. O the equilibrium path, following Kandori (1992), we identify a su cient condition for a one-shot
6
Page 8 of 50
deviation to be unpro table under any consistent beliefs. Suppose there are
∞ X
δ t ek Atn Zn vn ≥ ek Zn un + δ
t=0
k d-type
δ t ek Bn Atn Zn vn .
(2)
t=0
The left hand side of (2) is the payo that a are
∞ X
d-type
player earns from playing
D
forever when there
players including the player himself, while the right hand side of (2) is what a
player receives when he deviates from the contagious strategy, playing back to playing
D
forever after.
C
d-type
ip t
k=
k d-type players, where
n, n + 1, . . . , M .3 Then a one-shot deviation o the equilibrium path is unpro table if
today and then reverting
Inequalities (1) and (2) can be manipulated into equilibrium
cr
conditions 1 and 2 in the following lemma.
Lemma 1 The contagious strategy constitutes a sequential equilibrium if the following two condiEquilibrium Condition 1:
us
tions are satis ed:
Cn−1 ≥ (1 − δ)e1 (I − δAn )−1 Zn vn ,
an
Equilibrium Condition 2: ek Zn (vn − un ) ≥ δek Hn (I − δAn )−1 Zn vn . The intuition behind equilibrium conditions 1 and 2 is similar to that for the
n = 2 case studied
When a player is on the equilibrium path, he has no incentive to deviate
from cooperation when
δ
M
by Kandori (1992).
is su ciently large. When a player is o the equilibrium path, he has no
incentive to deviate from continued play of the contagious strategy if the extra payo from defection
vn − un , is large enough.
Using Lemma 1 we can prove the following theorem.
d
in the current period,
Theorem 1 Under uniformly random matching, the contagious strategy described above constitutes
te
a sequential equilibrium strategy for any nite population size, M , if δ , Cn−1 − Do , and all components of vn − un are su ciently large.
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Proof: See Appendix A.
3
A Monotonicity Result
In this section we ask the following question: Fixing the population size at
n ≤ M
M,
which group size
4
maximizes the possibility of achieving a social norm of cooperation among strangers?
Although we can characterize the equilibrium conditions for the contagious strategy, we cannot derive closed-form solutions since the formulas for the elements of the transition matrices become too complicated to derive for group sizes
n>
A
and
B
2. 5 Therefore, in this section we switch to
6 the use of numerical methods. 3 4
n
Since the player under consideration is a
d-type,
there must be at least
n d-type
players in the community.
In Appendix B, we also ask how the answer to this question changes if instead of xing
but in such a way that the number of groups,
m = M/n,
M,
we vary both
M
and
is held constant. We show there that we obtain similar
results with respect to satisfaction of the equilibrium conditions needed for cooperation to be sustained as a social norm, if we vary
5
6
n
but hold the number of groups constant.
Kandori (1989) provides transition matrix formulas for the
n=2
case only.
The Mathematica program used for our numerical results is available upon request.
7
Page 9 of 50
Furthermore, for greater tractability, we focus on a simple symmetric speci cation for the payo parameters that satisfy assumptions A1-A3.
Speci cally, we impose the following additional as-
7 sumptions:
A5: A6:
ip t
C0 = 0. Di − Ci = α for 0 ≤ i ≤ n − 1. Ci+1 − Ci = Di+1 − Di = βn for 0 ≤ i < n − 1.
A4:
Under these assumptions, the payo matrix (Table 1) now takes on the speci c form shown in
with a larger group size since the payo from full cooperation,
n.
(n − 1)βn ,
may grow with the group
To properly correct for this dependency, we also normalize the payo matrix in such a way
us
size,
cr
Table 2. Finally, we note that under our parameterization it may be easier to achieve full cooperation
that the payo from full cooperation is xed and constant by the following assumption:
(n − 1)βn = 1
A7:
for any
n.
n ≥ 2. 0
C
0
D
βn
...
n−1
2βn
...
(n − 1)βn
...
α + (n − 1)βn
α α + βn α + 2βn
d
Table 2: The Simpler Payo Matrix for the
α < Cn−1 = 1
2
1
M
number of cooperators in the group
an
Note that under this normalization, to satisfy assumption A3, we must have
n-Player
te
equilibrium conditions as the group size takes on the increases toward
M
for all
Prisoner's Dilemma Game
M = 12 and examine values n = 2, 3, 4, 6, 12.
In order to examine the question raised above, we x
changes in the two
n
We nd that as
both the on-equilibrium and o-equilibrium conditions become easier to satisfy,
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which we refer as the
Monotonicity Result.
3.1 Equilibrium Condition 1
We rst examine the eect of increases in the group size,
n,
on equilibrium condition 1. Although
we are mainly interested in the case where payos are normalized to eliminate the dependency on
n,
for the moment we keep payos for equilibrium condition 1 in their original unnormalized form
(i.e.,
Cn−1 = (n − 1)βn ),
so that we can derive some intuition on the discounted summation of the
probability of earning each payo outcome. Given a group of size
n,
we can write equilibrium condition 1 as:
(n − 1)βn ≥ pn0 α + pn1 (α + βn ) + . . . + pnn−1 (α + (n − 1)βn ) = (
n−1 X
pnj )α +
j=0 7
A slightly dierent normalization, for instance,
n−1 X
jpnj βn ,
j=1
D0 = 0, Di = iβ , Ci = Di − α, Cn−1 = 1,
gives similar results.
8
Page 10 of 50
pnj ≡ (1 − δ)e1 (I − δAn )−1 ρnj denotes the discounted summation of the probability of meeting j cooperators (c-types) in a group of size n once a player has initiated a defection. Then given Pn−1 n that j=0 pj = 1 as shown in Lemma 2 in Appendix A and the normalization assumption A7, (n − 1)βn = 1, equilibrium condition 1 becomes where
α ≤ pn ,
(3)
ip t
where
Pn−1
jpnj . n−1 j=1
p ≡1−
Equation (3) says that the net payo from defection (which is
α)
must be less than or equal to the
n net loss from initiating a defection (which is p ).
cr
n
us
Proposition 1 If pn is increasing in n, then Condition (3) (Equilibrium condition 1) is monotonically less restrictive as the group size n increases.
an
3.2 Equilibrium Condition 2
We next examine the eects of increases in the group size,
n,
on equilibrium condition 2. Given our
Di −Ci = α for i = 0, 1, . . . , n−1, the left hand side of equilibrium condition 2, representing the extra payo from defection, is equal to α. The right hand side of equilibrium condition 2, the payo to a d-type player from slowing down the contagious process, achieves its highest value when the number of d-type players are at a minimum, i.e., when k = n. Thus it is su cient to compare equilibrium condition 2 at k = n for dierent group sizes, n = 2, 3, 4, 6, 12.
d
M
payo speci cation that
te
Similar to equilibrium condition 1, we rst present equilibrium condition 2 with the original payo parameters and then we impose our normalization later. Given a group of size
n,
we can write equilibrium condition 2 as:
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n α ≥ q0n α + q1n (α + βn ) + . . . + qn−1 (α + (n − 1)βn )
= (
n−1 X j=0
qjn )α +
n−1 X
jqjn βn ,
j=1
qjn ≡ δen Hn (I −δAn )−1 ρnj denotes the change in the discounted summation of the probability of meeting j c-type players in the group when the d-type player reverts back to playing cooperation instead of defection given that the group size is n and there are k = n d-type players in the Pn−1 n population. Given that j=0 qj = 0 as shown in Lemma 2 and the normalization assumption A7, (n − 1)βn = 1, equilibrium condition 2 becomes where
α ≥ qn,
(4)
where
Pn−1 n
q ≡
jqjn . n−1 j=1
α) must be greater than n (which is q ).
Equation (4) says that the net payo from continuing a defection (which is or equal to the net bene t from slowing down a contagious defection
9
Page 11 of 50
Proposition 2 If qn is decreasing in n, then Condition (4) (Equilibrium condition 2) is monoton-
ically less restrictive as the group size n increases.
3.3 Numerical Findings for Dierent Values of δ Propositions 1-2 require that
pn
qn
is increasing and
n so that the su cient equilib-
is decreasing in
n,
becomes monotonically less restrictive as the group size,
n conditions hold by computing p and
pn δ
0.01
0.1
n
for given 0.3
and
δ
0.5
n,
and for dierent discount
cr
M=
12.8
0.7
0.9
0.99
us
all under a xed
increases. We next ask whether these
q n for dierent group sizes,
0.000925 0.011028 0.051716 0.141773 0.331384 0.704922 0.966359
n=3
0.001859 0.022562 0.100818 0.239648 0.460745 0.788192 0.977226
n=4
0.002785 0.033155 0.137519 0.298782 0.524404 0.821914 0.981270
n=6
0.004600 0.050899 0.185380 0.363489 0.585281 0.850813 0.984588
n = 12
0.010000 0.100000 0.300000 0.500000 0.700000 0.900000 0.990000
qn δ
0.01
0.1
an
n=2
M
δ,
factors,
ip t
rium conditions for the contagious strategy to sustain a social norm of cooperation among strangers
for given
n
0.3
0.5
and
δ 0.7
0.9
0.99
0.000839 0.009584 0.039064 0.088720 0.168391 0.290770 0.363919
n=3
0.001195 0.012416 0.040398 0.072725 0.109571 0.151112 0.171385
n=4
0.000926 0.009293 0.028096 0.047192 0.066581 0.086265 0.095219
n=6
0.000059 0.000590 0.001771 0.002952 0.004132 0.005313 0.005844
n = 12
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
te
d
n=2
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 3: Numerical Results on
Our numerical exercises on
n Table 3 reveals that p and
pn
and
qn
pn
qn
and
for Dierent
n
and
δ (M = 12) n,
illustrate some interesting results. Given any group size
q n increase with
δ.
Therefore, if a player cares more about the future,
then the extra loss from initiating a contagious wave of defection and the extra bene t from slowing down a contagion both become larger. Next we ask: given a xed
δ,
how do
pn
and
qn
change with increases in the group size
n?
First, in all cases, the contagious equilibrium always exists, i.e., the numerical value in each cell for
pn
is always larger than the value in the corresponding cell for
between
qn
and
pn
both equilibrium conditions 1 and 2 hold.
monotonically increasing in
n
qn.
Thus by choosing any
α
pn
is
Furthermore, we nd that
n given any δ , and q is monotonically decreasing in
n if δ
is su ciently
large enough (greater than 0.5). Given the numerical results in Table 3, we conjecture that there exists a threshold value for the discount factor, increasing in
8
Notice that
n
δ¯
such that, for any
n and q is monotonically decreasing in
pn
and
qn
are both functions of
δ, n
and
n.
δ > δ¯, pn
is monotonically
This observation is veri ed in Figure 1.
M.
10
Page 12 of 50
ip t cr us an and
M
pn
qn
as functions of
δ (M = 12)
te
d
Figure 1:
Intuitively, with a larger group size, an initial defection spreads to more innocent (c-type) players.
Furthermore, via the random re-matching each period, defection spreads to the entire
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
population of
M
players much faster since there are fewer groups given the xed population size,
M , and a larger group size, n. These two eects together imply that the contagious process is faster with a larger group size n and thus the payo from starting a defection is reduced (i.e., the net loss pn from starting a defection is increased), making the condition on the equilibrium path easier to satisfy. O the equilibrium path, condition 2 becoming less restrictive with a larger group size is also due to the faster contagious process associated with a larger group size. When the speed of contagion is faster, the eectiveness of a single becomes smaller. Thus, the
d-type
d-type
player slowing down the contagious process
player has less of an incentive to deviate from the contagious
strategy o the equilibrium path by reverting back to playing cooperation again (i.e., the net bene t from slowing down the contagion
qn
monotonically decreases).
Summarizing, our main nding is that, for a xed population,
M,
and a su ciently high
δ,
the
conditions under which the contagious strategy sustains play of the cooperative strategy in a
n-
player Prisoner's Dilemma game by all anonymously and randomly matched players in each period are monotonically more easily satis ed as the group size,
n
increases toward
M.
11
Page 13 of 50
4
An Application to a Public Goods Game
In this section we show that with a slightly dierent normalization of the payo matrix, the
n-
person Prisoner's Dilemma game can be re-interpreted as a public goods game so that our previous monotonicity result continues to hold in this public goods game version of the stage game.
M players, who are anonymously and randomly assigned to groups of size n in each period to play an n-player public goods game. Here we study a binary choice
ip t
As before, we assume a population of
version of the classic public good game (Issac and Walker (1988)) where each player is endowed with a single token and must decide whether or not to invest that token in his own privately held account
µn for of γ , but
Each token invested in the public account yields a payo of
cr
or in a public account.
group member. A token invested in the private account yields an additional payo
each only
Table 4 represents the payo matrix for the
us
to the player associated with that private account.
player from choosing to invest in the public account (C ) or in the private account (D ) given the number of other contributors to the public account in the group of size
The standard public
an
and
M
µn > 0
n.
γ > 0, so that non-contribution to the public good is always a dominant strategy in the one-shot, n-player game, and further that γ + µn < nµn , which implies that the social optimum is achieved when all n players contribute to the public good. Notice that these restrictions also satisfy assumptions A1-A3, as de ned in Section 2 for an n-player Prisoner's good game setup has
Dilemma game.
0
1
2
...
n−1
µn
2µn
3µn
...
nµn
...
γ + nµn
number of contributors in the group (invest in the public account)
D
(invest in the private account)
te
d
C
γ + µn γ + 2µn γ + 3µn
Table 4: The Payo Matrix for the
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
n-Player
Public Goods Game
When this public goods game serves as the stage game played by a population of are randomly divided up into groups of size
n
M
players, who
in every period, the su cient conditions to sustain
the contagious equilibrium are very similar to those shown before. On the equilibrium path we must have:
nµ ≥ pn0 (γ + µn ) + pn1 (γ + 2µn ) + . . . + pnn−1 (γ + nµn ),
while o the equilibrium path we require that:
n γ ≥ q0n (γ + µn ) + q1n (γ + 2µn ) + . . . + qn−1 (γ + nµn ). De ne
p˜n ≡
n−1 n n−1 p = − n n
and
n−1 n q˜n ≡ q = n
Pn−1 j=1
jpnj
n
Pn−1 j=1
n
jqjn
.
12
Page 14 of 50
Then with the normalization that
µn = 1/n,
equilibrium condition 1 becomes
γ ≤ p˜n and equilibrium condition 2 becomes
γ ≥ q˜n .
δ
still holds for the public goods game when
ip t
Based on the previous numerical results (Table 3), it is easy to show that the monotonicity pattern is su ciently large, with the threshold value for
Experimental Design
us
5
cr
slightly increased.
δ¯
In the next two sections we report on an experiment that tests the monotonicity results of Propositions 1-2. For this purpose we return to the simpli ed parameterization of the model used in Section
M = 12 and a discount factor δ = 0.75. Our main two treatment variables are the group size, n and the value of α. With the 1 normalized payo in Table 2, βn = n−1 for dierent groups of size n. For instance, the stage game
an
3. In our experimental design, we always consider communities of size
M
payo for the 2-person PD and the 6-person PD are shown in Table 5.
D
C
C
0
1
D
α α+1
d
Other Player's Choice
0
1
2
3
4
5
C
0
0.2
0.4
0.6
0.8
1
D
te
Number of Others Playing C
α α + 0.2 α + 0.4 α + 0.6 α + 0.8 α + 1
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 5: The Payo Matrices for the 2-Player and 6-Player Prisoner's Dilemma Game
A third treatment variable in our experimental design is the number of human subjects in each 12-player community. We start with the treatments in which each 12-player community consists of just one human subject who interacts with 11 other robot players as opposed to allowing 12 human subjects to interact with one another. We employ this design in order to avoid the coordination problem of strategy selection among 12 human players and thereby remove strategic uncertainty. As with other folk-theorem type results, the contagious equilibrium is not the unique equilibrium of the in nitely repeated
n-player
PD game that we implement in our experiment.
There exist
many other non-cooperative equilibria including the one where all players choose to defect in every round of the supergame. Empirically, when players face both the selection of their own strategy and the uncertainty of strategy selection by other players, the outcome of play can be far from that predicted by the contagious equilibrium.
9 Furthermore, we would expect that this problem of
strategic uncertainty is naturally more severe as the group size
9
n
gets larger. For these reasons, we
See for example, Duy et al. (2013) and Duy and Ochs (2009).
13
Page 15 of 50
chose to rst eliminate the strategic uncertainty dimension from our experimental design by having our players interact with robot players programmed to play according to the contagious strategy so as to provide a cleaner test of our monotonicity results. human subject and 11 robots per community of size
10 While our baseline design has just 1
M = 12,
in other treatments, we increase the
number of human subjects in each community to 2 and 6 while decreasing the number of robots to
M
xed at 12.
ip t
10 and 6, respectively, i.e. holding
Using the three treatment variables discussed above, i.e., the group size
n,
the value of
α
and
the number of human subjects in each community, we summarize all of our experimental treatments
cr
in Table 6. The treatments are denoted using labels of the form aHbR_X, where a is the number of human subjects and b is the number of robots in each community of 12 players. X∈{A, B, C, D, E} represents dierent pairs of values for the group size,
n,
and payo parameter,
α,
as indicated
us
in Table 6. This same table also provides the on-equilibrium and o-equilibrium conditions on
α,
given the group size chosen for each treatment. Using these conditions, and the treatment value for
α,
Table 6 also indicates (under the heading Cooperation?)
whether or not cooperation by
an
all players can be sustained as a sequential equilibrium of the inde nitely repeated game where players are randomly and anonymously matched in groups of size
n
in each round. We conducted
one session each for the treatments involving 1H11R or 2H10R and two sessions for the treatment
provides further details.
α
Players
d
Treatment Session Group Size
M
involving 6H6R. Note that each session gives multiple independent observations Table 7 below
On-equm O-equm
Cooperation?
1
n=6
α = 0.5
1 HS 11 robots
α ≤ 0.65 α ≥ 0.004
Yes
1H11R_B
2
n=2
α = 0.5
1 HS 11 robots
α ≤ 0.40 α ≥ 0.19
No
1H11R_C
3
n=2
α = 0.3
1 HS 11 robots
α ≤ 0.40 α ≥ 0.19
Yes
1H11R_D
4
n=3
α = 0.6
1 HS 11 robots
α ≤ 0.53 α ≥ 0.12
No
te
1H11R_A
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
5
n=6
α = 0.6
1 HS 11 robots
α ≤ 0.65 α ≥ 0.004
Yes
6, 7
n=6
α = 0.5
6 HS 6 robots
α ≤ 0.65 α ≥ 0.004
Yes
8
n=6
α = 0.5
2 HS 10 robots
α ≤ 0.65 α ≥ 0.004
Yes
1H11R_E 6H6R_A
2H10R_A
Table 6: Summary of Treatments
The value of
α
in each treatment is varied based on several considerations. First, we wanted
a parameterization that could sustain the contagious strategy as an equilibrium in a community of a xed population size
under a larger group size but not under a smaller group size
so as to
test our main monotonicity result. Second, we chose to focus on the on-equilibrium-path condition rather than the o-equilibrium-path condition; if
α
was instead chosen in such a way that the
on-equilibrium-path condition (but not the o-equilibrium-path condition) was always satis ed for both group size treatments, e.g., a choice of
α = 0.1,
then we might observe that subjects seldom
chose to defect (with the consequence that they were seldom actually o the equilibrium path)
10
We note that this type of experimental design involving robot players has not previously been implemented to
test the contagious equilibrium prediction.
14
Page 16 of 50
under either group size, making it di cult to detect any treatment eect. Therefore, we chose to
n = 2 (Sessions 1 and 2). By the same principle, we chose α = 0.6 to test the group size eect between n = 3 and n = 6 (Sessions 4 and 5). In addition, in Session 3 we set α = 0.3 for a group size n = 2 so that cooperation via the contagious equilibrium is supported in this case, as opposed to α = 0.5 for group size n = 2 as in set
α = 0.5
to test the group size eect between
n=6
and
ip t
Session 2. In sessions 1-5, there is just one human subject per community of size 12. By contrast, in sessions 6, 7, and 8, the number of human subjects in the 12-player community is varied but we
α = 0.5
and
Given our choice of
n=6 α
as in Session 1 so that the contagious equilibrium can be sustained.
in each treatment and a xed
δ = 0.75
and
M = 12,
we are able to test
cr
always set
the following hypotheses using our experimental data.
Hypothesis 1:
us
Hypotheses 1-3 concern aggregate treatment eects on cooperation.
The overall cooperation rate is higher with a larger group size than with a
smaller group size given the same value of
α
in the 1H11R treatments (Session 1 vs. 2, Session 5
Hypothesis 2: with a smaller
an
vs. 4).
The overall cooperation rate is higher when the equilibrium condition is satis ed
α than when the equilibrium condition is not satis ed with a larger α given the same
Hypothesis 3:
M
group size in the 1H11R treatments (Session 3 vs. 2).
The overall cooperation rate is lower with an increase in the ratio of human
subjects to robots (playing the contagious strategy) in a community given the same group size,
α
n
(Sessions 1 vs. 6 and 7 vs. 8).
d
and value of
Hypotheses 4-7 concern the individual behavior of the human subjects: The frequency with which subjects are on the equilibrium path is larger when
te
Hypothesis 4:
the on-equilibrium-path condition is satis ed than when the condition is not satis ed.
Hypothesis 5:
The cooperation rate when subjects are
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
c-types
(on-the-equilibrium-path) is
higher with a larger group size than with a smaller group size given the same value of
Hypothesis 6:
The cooperation rate when subjects are
d-types
α.
(o-the-equilibrium-path) is
not dierent between a larger group size and a smaller group size.
Hypothesis 7:
The cooperation rate when subjects are
c-types
is higher when the number of
human subjects in the community decreases, given the same group size
n
and value of
α.
In our 1H11R treatments, we explicitly told our subjects that in each round of a supergame (or sequence as it was referred to in the experiment) they would be randomly matched with
n−1
other robot players (out of a total population of 11 robot players) and not with any other human subjects. Since
n < M,
subjects were told that there would also be robot-robot group interactions
that they would not be a part of given the uniform random matching that we used. Subjects were further instructed that the robots in each community played according to the rules of the contagious strategy. Speci cally subjects were told: The robots are programmed to make their choices according to the following rules:
•
choose X in the rst round of each new sequence;
15
Page 17 of 50
•
if, during the current sequence, any of a robot's group members, including you or any other robot players have chosen Y in any prior round of that sequence, then the robot will switch to choosing Y in all remaining rounds of the sequence;
•
otherwise, the robot will continue to choose X.
C,
while Y refers to the defect action
D.11
Thus subjects
ip t
Here X refers to the cooperative action
had complete knowledge of the strategies to be played by their opponents.
We did not provide
subjects with any further information, such as the number of periods it might take for them to meet
cr
a defecting player once they (the human subject) had initiated a defection, as this calculation was one that we wanted subjects to make on their own.
In our 2H10R and 6H6R treatments, everything remained the same except that subjects knew
us
that there would be 2(6) human subjects and 10(6) robot players in their 12-player community for the duration of the session. Subjects remained in the same community with the same 1 other or the same 5 other human subjects in all supergames (sequences) of a session. Further, subjects were
size
M = 12.
an
told that there would be no spillovers from human subjects interacting in dierent communities of Therefore, as in the 1H11R treatments, each community of size 12 in the 2H10R and
6H6R treatments constitutes an independent observation.
M
One may be concerned that explicitly telling subjects the contagious strategy used by robots will induce a kind of experimenter demand eect by which the human subjects will also follow this same contagious strategy. However, if that were the case, then our hypotheses that the group size
n
d
matters would not nd any support since subjects in all of our treatments were told the same
information about the contagious strategy played by the robot players. On the other hand, if we
or
n = 3,
te
observe a higher cooperation rate under a group size of
n=6
than under a group size of
n=2
then it implies that subjects rationally choose to follow the contagious strategy more
frequently when the equilibrium conditions were satis ed.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
To implement an in nite-horizon
n-player
PD game in the laboratory, we use the standard
random termination methodology (Roth and Murnighan 1978) in which subjects participate in supergames that consist of an inde nite number of rounds, where the probability of continuation from one round to the next is a known constant equal to the discount factor, choice of
δ = 0.75,
δ ∈ (0, 1).
With our
the expected duration of a supergame is 4 rounds.
To enable subjects to gain some experience with the play of an inde nitely repeated game, we had them participate in multiple supergames in a session. As noted above, subjects were informed that at the start of each and every new supergame (sequence) all of the robot players in their community of size 12 would start out each new supergame as c-types playing the cooperative strategy (X) and that robot players would only change to playing the defect strategy (Y) if they became d-types during that supergame. This feature allows subjects to treat each new supergame as a fresh start rather than viewing the entire session as one long inde nitely repeated game, with the possibility of switching to a dierent strategy at the start of each new supergame. We did not x the number
11 the
We used the neutral labels X and Y rather than (Cooperate and Defect) in our experimental implementation of
n-player
game.
16
Page 18 of 50
of supergames played in advance.
Instead, during the experiment, we allowed subjects to play
for at least one hour, and the supergame in progress beyond one hour was determined to be the last supergame (we did not inform subjects of our stopping rule); when that nal supergame was completed, the session was declared to be over. Following the completion of the experiment, three sequences were randomly selected from all played and subjects were paid their total earnings from
12 All sessions were completed within the two
ip t
those three sequences in addition to a $7 show-up fee.
hour time-horizon for which we recruited subjects; a typical session required about 30 minutes for the reading of instructions followed by 1 hour of play of multiple supergames.
cr
The experiment was conducted at the Experimental Social Science Laboratory (ESSL) of the University of California, Irvine using undergraduate students with no prior experience with our experimental design. Instructions were read aloud and then subjects completed a brief comprehension
n=6
treatment are provided in Appendix C; instructions for
us
quiz. The instructions used in the other treatments are similar.
Experimental Findings
an
6
Table 7 reports some details about all of our experimental sessions.
As this table reveals, on
M
average, each session involved 18 supergames and 76 rounds. In total, 93 subjects participated in our experiment, with average earnings of USD $16.35. In the 1H11R treatments, since each subject interacted with an independent group of 11 other robots all playing according to the contagious
d
strategy, each subject's behavior amounts to a single, independent observation. Thus, the number of
te
subjects we have for each of sessions 1-5 corresponds to the number of independent observations. For treatment 6H6R_A, there are 6 human subjects in each community and 12 subjects in each session, so each session produces two independent observations, and as we have two sessions of 6H6R_A, we have a total of 4 independent observations of this treatment. For treatment 2H10R_A, there
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are 2 human subjects in each community; since we recruited 18 human subject for this treatment, we thus have 9 independent observations of this treatment. In the remainder of this section, we will begin with an aggregate data analysis, followed by analysis of data at the individual level. For the aggregate results, we will rst provide a summary of ndings on the group size eect, the eect of
α,
and the eect of the number of human subjects
within a community. After that we will present results from Mann-Whitney tests that compare the cooperation rates in the dierent treatments all together in Table 9. We rst examine results from the treatments with one human subject in each community (1H11R). Figure 2 shows the average cooperation rate per round over time for treatments 1H11R_A (B, D, E). The blue line (with diamonds) in each panel is the average cooperation rate of the human subjects only, while the red line (with squares) in each panel is the average cooperation rate by the entire 12-member community as a whole, consisting of 1 human subject and 11 robots. The start of each new supergame is indicated by a vertical line.
12
We chose to pay for three randomly sequences, as opposed to just one, so as to avoid the possibility that a short
(e.g., 1-round) supergame was chosen for payment under the pay-one-sequence-only protocol.
17
Page 19 of 50
Treatment Session
No. of
No. of
Subjects
Obs.
No. of
No. of
Supergames Rounds
Average Earnings
1
10
10
18
75
USD 22.59
1H11R_B
2
12
12
20
76
USD 15.33
1H11R_C
3
12
12
19
75
USD 13.11
1H11R_D
4
10
10
22
77
USD 20.21
1H11R_E
5
7
7
18
77
USD 15.31
6H6R_A
6
12
2
19
81
USD 11.49
6H6R_A
7
12
2
15
83
USD 14.63
2H10R_A
8
18
9
15
65
USD 18.09
N/A
12
N/A
18
76
USD 16.35
cr
us
Average
ip t
1H11R_A
Table 7: Description of Experimental Sessions
n=2
and
respectively with
an
The two top panels of Figure 2 show the average cooperation rates for the group size of
α = 0.5.
For Session 2 (the
n=2
n=6
group size), we observe a decline in
the cooperation rate over time by the human subjects in almost all supergames lasting more than
M
2 rounds, which indicates that more human subjects began to switch from cooperation to defection from round 2 if they did not choose to defect from the beginning of the supergame. Consistently, the cooperation rate at the community level shows a similar pattern but remains above the cooperation
d
rate by the human subjects alone as it takes some time for the contagious strategy, as played by the
with
n = 2,
te
robot players, to spread throughout the population of size 12. Across all supergames of the session there is no obvious learning eect or convergence.
By contrast, the cooperation rates in Session 1 under a group size of
n = 6
exhibit a very
dierent pattern over time. Indeed, consistent with our theory, the overall cooperation rate is higher
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in Session 1 (n
= 6)
than in Session 2 (n
= 2).
Although there is also a decline in cooperation
over the course of each sugergames at the beginning of Session 1, the cooperation rate eventually becomes high, at around 90%, following the fourth supergame of this session (approximately after the rst one-third of the session has been completed) and remains high for the remaining supergames of that session. This nding indicates that, given the payo parameters we have chosen and the strategy followed by the robots, most subjects learn over time that it is in their best interest to follow the contagious strategy when
n is large (n = 6),
relative to the case where
A comparison of the cooperation rates between groups of size
n=2
and
n=6
n is small (n = 2). indicates that the
human subjects responded to the payo incentives of the game. They choose to start defecting more frequently when the contagious eect of a single defection is much slower in Session 2 (n
= 2)
and
this tendency to defect was not diminished by experience. By contrast, when the contagious eect of a defection is more immediate, as in Session 1 (n
= 6),
subjects learned to avoid triggering a
wave of defection. The two bottom panels of Figure 2 show the average cooperation rates for the group of size and
n = 3 respectively with the choice of α = 0.6.
n=6
This treatment comparison provides a robustness
18
Page 20 of 50
Cooperation Rate (Session 1 - 1H11R_A, n=6, α=0.5)
Human Subjects Group Total
1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 1 2 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 910111 2 1 2 3 1 2 1 2 3 4 5 6 7 1 2 1 2 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1 2 1
ip t
Round
Cooperation Rate (Session 2 - 1H11R_B, n=2, α=0.5)
Human Subjects Group Total
1
cr
0.8 0.6 0.4 0.2
us
0
1231234512341234567123123456123456781121231234567891112111234123456781212345
an
Round
Cooperation Rate (Session 5 - 1H11R_E, n=6, α=0.6) 1 0.8
Human Subjects Group Total
M
0.6 0.4 0.2 0
1 2 3 4 5 1 2 3 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 1 2 3 4 1 2 3 4 5 6 1 1 2 3 4 5 1 2 3 4 5 6 7 8 91011121 2 3 1 2 3 4 5 6 1 2 3 1 2 3 4 1 2 3 1 1 2 3 1 2
d
Round
Cooperation Rate (Session 4 - 1H11R_D, n=3, α=0.6)
Human Subjects Group Total
te
1 0.8 0.6 0.4
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
0.2 0
1 2 3 4 5 6 7 8 1 1 2 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 9 1011121314151617 1 2 3 1 1 2 1 2 1 2 3 1 1 2 3 1 2 1 2 1 1 1 1 2 3 4 1 1 2 3 4 5 6 7 8 1 2 3 4 5
Figure 2:
Round
Cooperation Rate by Human Subjects and Communities Over Time (Treatment
1H11R_A,B,D,E)
check as to whether the group size eect continues to hold with a dierent parameterization for and
α.
Indeed, comparing the dierent group sizes
n = 6
and
n = 3,
we continue to nd that,
consistent with our theory, cooperation rates are higher with the larger group of size under the smaller group of size
n = 3.
n
n=6
than
We summarize the group size eect in Finding 1, which
supports Hypothesis 1. Detailed statistical support can be found in Table 9.
Finding 1 Cooperation rates are higher under larger groups than under smaller groups given the same α.
19
Page 21 of 50
Not only did our subjects respond to the incentives induced by having dierent group sizes, but
α when holding the group size (α = 0.3, n = 2) and Session 2 (α = 0.5,
also they responded to the incentives associated with changes in constant. Figure 3 shows a comparison between Session 3
n = 2).
With a larger
α, players have a larger temptation to initiate a defection, thus making it more
di cult to sustain cooperation within the community. Figure 3 and Finding 2 con rm Hypothesis
α = 0.3,
the cooperation rate is sustained at a level above
50%. By contrast, as seen in the bottom panel of the same gure where
ip t
2. In the top panel of Figure 3 where
α = 0.5,
the cooperation
tests regarding Finding 2 are presented in Table 9.
us
Cooperation Rate (Session 3 - 1H11R_C, n=2, α=0.3)
cr
rate drops below 20% if the supergame lasts for 4 or more rounds. Results from Mann-Whitney
1 0.8 0.6
an
0.4
Human Subjects Group Total
0.2 0
123451123451234561234512345678123111234567812341234512345612345611231234121
M
Round
Cooperation Rate (Session 2 - 1H11R_B, n=2, α=0.5) 1 0.8
Human Subjects Group Total
0.6
d
0.4 0.2 0
te
1231234512341234567123123456123456781121231234567891112111234123456781212345
Round
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 3:
Cooperation Rates by Human Subjects and Communities Over Time (Treatments
1H11R_B,C)
Finding 2 Cooperation rates are higher under a smaller α than under a larger α given the same group size, n.
We next compare the 1H11R treatment with the 2H10R and 6H6R treatments, restricting attention to the case where
n=6
and
α = 0.5,
so that the contagious equilibrium is supported by
the same set of parameters in the three treatments and we can examine whether or not the ratio of humans to robots in each community of size 12 will aect the possibility that the contagious equilibrium is selected.
13 The comparison is made using Figure 4 and statistical support is included
in Table 9. The rst panel of this gure shows cooperation rates for the 1H11R_A treatment, the
13
We did not study the case of a group size of
humans to robots, since when
n=2
and
α = 0.5,
n = 2
across these three treatments involving dierent ratios of
our theory predicts that a social norm of mutual cooperation is
not sustainable using the contagious strategy.
20
Page 22 of 50
second panel for the 2H10R_A treatment and the last two panels show cooperation rates for the 6H6R_A treatment (2 sessions). As the ratio of human subjects to robots in a community increases, we nd that the average cooperation rate monotonically decreases. Recall that for the 1H11R_A treatment, the cooperation rate converges to about 90% after subjects gained some experience in the rst four supergames. By contrast, in the 2H10R_A treatment, the cooperation rate is closer
14 Finally, in the 6H6R_A treatment, the cooperation rate almost
ip t
to 50% throughout the session.
always drops to zero after a supergame lasts for 2 or 3 rounds. Based on the results of the 6H6R treatment, we did not think it necessary to run treatments where the community consisted entirely
cr
of 12 human subjects and 0 robots, as we expect that cooperation rates would likely have been close to 0 in these treatments as well.
us
Finding 3 Cooperation rates monotonically decrease as the number of human subjects in a com-
an
munity increases, given the same group size n and value of α.
Tables 8 and 9 provide further detailed evidence that support Findings 1-3.
Table 8 reports
cooperation rates calculated based on human subjects' choices as well as on community-wide action choices (humans plus robot players) over all rounds of all supergames. We further calculated the
M
cooperation rates over the rst and second halves of each session. Table 9 reports
p-values
from
two-tailed Mann-Whitney tests on the cooperation rates between treatments. The rst two rows
n = 6 vs. n = 2 given α = 0.5 (treatment 1H11R_A vs. 1H11R_B), and n = 6 vs. n = 3 given α = 0.6 (treatment 1H11R_D vs. 1H11R_E). The third and fourth rows test Hypothesis 2 (the eect of α) for α = 0.5 vs. α = 0.3 given n = 2 (treatment 1H11R_B vs. 1H11R_C) and α = 0.5 vs. α = 0.6 given n = 6 (treatment 1H11R_A vs. 1H11R_E), where we only expect a signi cant dierence in the case of n = 2. The last three rows test Hypothesis 3 (the number of human subjects in a community) given n = 6 and α = 0.5.
te
d
test Hypothesis 1 (the group size eect) for
The
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
p-values
shown in Table 9 are all consistent with a careful examination of the cooperation
rates shown in Figures 2-4. and
n=6
We nd the most signi cant group size eect between the
n = 2
treatments. On the other hand, strategic uncertainty (about the play of other human
subjects) also plays an important role in de-stabilizing cooperation in the environment we study here where players are randomly and anonymously matched in each repetition of the stage game. Although nearly full cooperation can be sustained in the second half of our sessions where
n=6
and each human subject interacts only with 11 robot players programmed to play according to the contagious strategy, the cooperation rate drops to close zero when
n
remains equal to 6, but the
community consists of 6 human and 6 robot players, even though the equilibrium conditions for sustenance of a social norm of cooperation continue to be satis ed. In order to further ensure that these treatment eects are not driven by subjects' reactions to the instructions used in dierent
14
Note that for the 2H10R treatment, the aggregate cooperation rate masks some heterogeneity at the community
level. Focusing on behavior in the last three sequences of the 2H10R_A treatment, we found that 3 out of 9 human groups (each consisting of 2 human subjects) always chose to cooperate, another 3 out of 9 human groups always chose to defect, and the remaining 3 groups cooperated at average levels of 10%, 20%, and 70%, respectively.
21
Page 23 of 50
Cooperation Rate (Session 1 - 1H11R_A, n=6, α=0.5)
Human Subjects Group Total
1 0.8 0.6 0.4 0.2 0
Round
Human Subjects Group Total
cr
Cooperation Rate (Session 8 - 2H10R_A, n=6, α=0.6)
ip t
1 2 3 4 5 6 7 8 1 2 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 910111 2 1 2 3 1 2 1 2 3 4 5 6 7 1 2 1 2 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1 2 1
1 0.8
0.4 0.2 0
us
0.6
1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1011 1 1 2 3 4 5 6 7 8 9 10 1 2 1 2 1 2 3 1 2 3 4 5 6 7 8 9 1011 1 1 1 1 2 1 2
an
Round
Cooperation Rate (Session 6 - 6H6R_A, n=6, α=0.6) 1 0.8
Human Subjects Group Total
M
0.6 0.4 0.2 0
1 2 3 4 5 6 1 1 2 1 2 3 1 2 3 4 5 6 7 8 910111213141 2 1 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 1 2 1 2 3 4 5 6 7 8 9 1 1 2 1 2 3 1 2 3 4 5 6 7 1 2 3 4 1 2 1 2 3 4 5 6 7
d
Round
te
Cooperation Rate (Session 7 - 6H6R_A, n=6, α=0.6)
1 0.8 0.6
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Human Subjects Group Total
0.4 0.2 0
1 1 2 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 1 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 1 2 1 2 3 4 5 6 7 8 9
Figure 4:
Round
Cooperation Rate by Human Subjects and Communities Over Time (Treatments
1H11R_A, 2H10R_A, 6H6R_A)
treatments, we compared the subjects' choice in the very rst period of the sessions and did not found any signi cant dierence across treatments. Importantly, our results indicate that subjects understand how to evaluate payos in the inde nitely repeated games we induce despite the relatively complicated environment, and they correctly respond to the incentives provided. Our ndings from varying the ratio of humans to robots suggest that it is
strategic uncertainty
about the play of others and not bounded rationality with regard
to payo calculations that may explain the failure of human subjects to achieve the e cient equi-
22
Page 24 of 50
Treatment Session
Human Subjects
Communities
Whole Session 1st half 2nd half Whole Session 1st half 2nd half 1
0.800
0.732
0.870
0.846
0.776
0.918
1H11R_B
2
0.364
0.373
0.355
0.743
0.754
0.733
1H11R_C
3
0.692
0.732
0.651
0.883
0.898
0.869
1H11R_D
4
0.444
0.379
0.508
0.667
0.534
0.797
1H11R_E
5
0.800
0.812
0.788
0.865
6H6R_A
6
0.062
0.079
0.047
0.148
6H6R_A
7
0.104
0.152
0.058
0.147
2H10R_A
8
0.418
0.368
0.466
0.519
ip t
1H11R_A
0.855
0.159
0.142
0.204
0.090
cr
0.875
0.441
0.594
No. of Obs.
1H11R_A vs.
Human Subjects
Whole Session 1st half 2nd half Whole Session 1st half 2nd half
10 vs. 12
0.006
0.008
0.003
0.015
0.391
0.005
10 vs. 7
0.039
0.009
0.128
0.044
0.005
0.403
12 vs. 12
0.067
0.097
0.045
0.017
0.067
10 vs. 7
0.767
0.488
0.290
1.000
0.348
0.291
0.000
0.000
0.005
0.005
0.003
0.034
0.034
0.029
0.049
0.059
0.034
0.001
0.001
0.003
0.005
0.020
0.005
1H11R_E 1H11R_B vs.
1H11R_E 1H11R_A vs.
10 vs. 24
6H6R_A
(10 vs. 4)
te
1H11R_A vs.
1H11R_A vs.
10 vs. 18
2H10R_A
(10 vs. 9)
2H10R_A vs.
18 vs. 24
6H6R_A
(9 vs. 4)
Table 9:
0.018
d
1H11R_C
M
1H11R_B 1H11R_D vs.
Communities
an
Treatment
us
Table 8: Cooperation Rates for Each Session
0.000
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
p-values
from Mann-Whitney Tests on Cooperation Rates
librium in inde nitely repeated games where Kandori's contagious equilibrium construction can be used to support a social norm of cooperative play by all players. We next ask whether there is a signi cant learning eect when the contagious equilibrium condition is or is not satis ed. In our experiment, we nd such a learning eect for some but not for all of our treatments/sessions. Table 10 reports
p-values
for the two-tailed Wilcoxon matched-
pair signed ranks tests between the rst and the second halves of each session. Learning is most signi cant in treatment 1H11R_A (n
= 6,
session 1), in which subjects learn to cooperate with the
11 robot players who are playing according to the contagious strategy, and in treatment 6H6R_A (n
= 6,
sessions 6 and 7), where cooperation rates converge to zero as half of the community
23
Page 25 of 50
members are human subjects. In sessions 3 and 5, the cooperation rates decrease signi cantly from the 1st half of sessions to the 2nd half of sessions, but the magnitude of this decrease is small. Treatment Session Cooperation? Human Subjects Communities Pattern over Time 1
Yes
0.081
0.018
increasing
1H11R_B
2
No
0.692
0.346
at
1H11R_C
3
Yes
0.010
0.054
decreasing
1H11R_D
4
No
0.185
0.006
increasing
1H11R_E
5
Yes
0.050
0.050
decreasing
6, 7
Yes
0.001
0.144
decreasing
8
Yes
0.614
0.056
increasing
2H10R_A
p-value of Wilcoxon Matched-Pair Signed Ranks Tests on Cooperation Rates (1st vs.
us
Table 10:
cr
6H6R_A
ip t
1H11R_A
half of the Session)
2nd
an
We next analyze individual strategic behavior. Speci cally, we ask to what extent the subjects in each treatment behaved according to the contagious strategy that is needed to support a social norm of cooperation among strangers. Table 11 shows the average of individual frequencies with
M
which subjects in each treatment are on the equilibrium path, the average of individual frequencies with which subjects chose to cooperate when on the equilibrium path and the average of individual
15 Finally, for each human subject,
frequencies of choosing to defect when o the equilibrium path.
d
we also calculated the frequency with which that subject played the contagious strategy, which is
te
given by:
freq. of following the contagious strategy
=
freq. on the equilibrium path
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
+
× cooperation
freq. o the equilibrium path
In Table 12 we report the
p-values
× defection
rate on the equilibrium path
rate o the equilibrium path.
from two-tailed Mann Whitney tests comparing all of these
frequencies across various treatments. The tests are conducted by using the average of the individual frequencies of human subjects within each community, which constitutes an independent observation since human subjects in dierent communities did not have any interactions with one another by our experimental design.
Most of the results shown in Tables 11- 12 are consistent with Hypotheses 4-7 regarding individual strategic behavior.
For the treatments with 1H11R, the average frequency with which
subjects are found to be on the equilibrium path is always signi cantly larger in sessions where the on-equilibrium-path condition is satis ed than in those treatments where this same condition is not satis ed. On the other hand, an increase in the number of human subjects in a community, as in treatments 2H10R_A and 6H6R_A, signi cantly lowers the frequency of on-equilibrium-path play
15
Recall that a player is de ned to be on the equilibrium path in the rst round of a supergame or when the player
has never experienced a defection by his group members or himself in the past rounds of a supergame. Otherwise, a player is o the equilibrium path.
24
Page 26 of 50
Treatment Session
Frequency
Cooperate Rate
On Equm Path
On Equm Path
Defect Rate
Frequency using
O Equm Path Contagious Strategy
1
82.27%
86.49%
74.48%
91.60%
1H11R_B
2
50.99%
56.50%
94.46%
79.50%
1H11R_C
3
76.11%
74.35%
93.96%
89.00%
1H11R_D
4
57.40%
60.35%
97.48%
84.42%
1H11R_E
5
84.42%
90.43%
85.76%
6H6R_A
6
23.66%
11.31%
95.41%
6H6R_A
7
19.08%
37.58%
96.40%
2H10R_A
8
50.68%
61.67%
96.53%
ip t
1H11R_A
91.47% 75.51%
cr
85.54% 87.69%
Cooperate Rate
On Equm Path
On Equm Path
0.008
0.008
(10 vs. 12)
1H11R_B
0.118
0.015
(10 vs. 12)
(8 vs. 11)
(10 vs. 12)
0.043
0.127
0.017
0.301
(10 vs. 7)
(10 vs. 7)
(9 vs. 4)
(10 vs. 7)
0.090
0.086
0.479
0.061
(12 vs. 12)
(12 vs. 12)
(11 vs. 6)
(12 vs. 12)
0.693
1.000
1.000
(10 vs. 7)
(8 vs. 4)
(10 vs. 7)
0.005
0.024
0.494
0.047
(10 vs. 4)
(10 vs. 4)
(8 vs. 4)
(10 vs. 4)
0.085
0.163
0.381
0.251
(10 vs. 9)
(10 vs. 9)
(8 vs. 7)
(10 vs. 9)
0.009
0.031
0.706
0.216
(9 vs. 4)
(9 vs. 4)
(7 vs. 4)
(9 vs. 4)
1H11R_D vs. 1H11R_E 1H11R_B vs. 1H11R_C 1H11R_A vs.
0.843 (10 vs. 7)
te
1H11R_E 1H11R_A vs. 6H6R_A 1H11R_A vs. 2H10R_A
Frequency using
O Equm Path Contagious Strategy
M
1H11R_A vs.
Defect Rate
an
Frequency
d
Treatment
us
Table 11: Strategy Analysis of Human Subjects
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2H10R_A vs. 6H6R_A
Table 12:
p-value
of Mann-Whitney Tests on Individual Strategy
of the contagious strategy. The average frequency of on-equilibrium-path play moves from about 80% to 50% and further declines to 20% when comparing across the 1H11R_A, 2H10R_A and 6H6R_A treatments, respectively (p
< 0.1
or
p < 0.01).
Conditional on being on the equilibrium path (column 3 in Table 12), the dierence in coopera-
n = 6 and n = 2 treatments is signi cant (session 1 vs. 2, p < 0.01) but there dierence between the n = 6 and n = 3 treatments (session 4 vs. 5, p = 0.127).
tion rates between the is no signi cant
The cooperation rates on the equilibrium path are not signi cantly dierent between 1H11R_A and 2H10R_A treatments (p
= 0.163),
but are signi cantly dierent between 1H11R_A and 6H6R_A
treatments and between 2H10R_A and 6H6R_A treatments (p
< 0.05).
Conditional on being o
25
Page 27 of 50
the equilibrium path (column 4 in Table 12), as Hypothesis 6 states, the cooperation rates are not signi cantly dierent between treatments except for treatments 1H11R_D and 1H11R_E, which may be due to the small number of subjects o the equilibrium path in treatment 1H11R_E. Finally, the frequency with which subjects followed the contagious strategy (column 5 in Table 12) is signi cantly larger when the on-equilibrium-path condition is satis ed and for a 1H11R commu-
ip t
nity. For the 2H10R_A treatment, the frequency that subjects followed the contagious strategy is in between those in found for the 1H11R_A and 6H6R_A treatments, although it is not signi cantly dierent. However, this frequency is signi cantly dierent between 1H11R_A and 6H6R_A
< 0.05).
cr
treatments (p
Finding 4 With a larger group size n = 6 and with a 1H11R community, subjects are more often
an
us
on the equilibrium path, choose cooperation more frequently when on the equilibrium path, and more often follow the contagious strategy, as compared with the corresponding 1H11R treatment with a smaller group size n = 2 and the treatment with n = 6 and 6H6R. Summarizing our experimental results, we nd that the behavior of the human subjects is consistent with our theoretical predictions on the impact of group size for cooperative play. Given the same payo parameter
α,
cooperation rates increase when the group size increases.
Under
M
a small group size, the cooperation rate declines over time and this pattern repeats itself across supergames even as subjects gain repeated experience with the environment. By contrast, under a larger group size, subjects learn to stick with cooperation after experiencing the much quicker
d
consequences of triggering a contagious wave of defection in their community.
te
When there is more than one human subject in each community, cooperation rates decrease as the ratio of human subjects to robots increases. Comparing the 1H11R treatment with the 6H6R treatment with the same group size
n=6
and
α
that always supports the contagious equilibrium,
the cooperation rate by the human subjects drops from 80% in 1H11R to less than 10% in 6H6R,
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
suggesting that it is greater strategic uncertainty that works to destroy cooperation in communities with randomly and anonymously matched players.
7
Conclusions
We have examined the eect of group size,
n,
on the equilibrium conditions needed to sustain
cooperation via the contagious strategy as a sequential equilibrium in repeated play of an Prisoner's Dilemma game, given a nite population of players of size
M ≥ n
n-player
and random and
anonymous matching of players in each repetition of the game. We nd that, if agents are su ciently patient, the equilibrium conditions, both on the equilibrium path and o the equilibrium path, become less restrictive, and thus more easily satis ed as the group size
n
increases toward
M.
This
result arises from the faster speed with which a contagious wave of defections can occur as the group size becomes larger.
Our ndings expand upon Kandori's (1992) idea that a social norm
of cooperative behavior among randomly matched strangers can be policed by community-wide enforcement. Speci cally, we show that communitywide enforcement becomes easier to sustain as
26
Page 28 of 50
the speed with which information travels becomes faster, which is here proxied by increases in the matching group size,
n.
Other interpretations of
n
are possible. Consider, for example, two towns with similar popula-
tions. The tra c rules are commonly known to all but there is no o cial police force that regulates drivers to follow these tra c rules in either town.
It's always more e cient if everyone follows
ip t
the tra c rules than if no one does. Suppose the only dierence between the two towns is in the structure of the roads. One town has several main streets with a few crosses. The other town has many small streets with a lot of crosses. The rst town's road structure corresponds to our large
cr
group structure whereas the second town's road structure corresponds to our small group structure. Disobedience of tra c laws has a more immediate impact in the rst town than in the second town, and so, by the logic of our theory, one might expect greater adherence to tra c laws in the rst As another example, centralized communication or monitoring mecha-
us
town than in the second.
nisms (credit bureaus) might also perform the same role played by larger group sizes in easing the conditions under which a social norm of cooperation is sustained in a large population of players.
n
increases to
M,
we eectively achieve increasingly perfect monitoring of the actions
an
After all, as
chosen by all players.
We have also empirically evaluated our theory by designing and reporting on an experiment
we found that subjects
were
M
exploring some of the comparative statics implications of the theory. Consistent with the theory, able to achieve higher rates of cooperation when randomly and anony-
mously matched into larger rather than smaller groups of size
n
to play a inde nitely repeated
Prisoner's Dilemma game. In many of our experimental treatments, we removed strate-
d
n-player
gic uncertainty by having human participants play only against robot players who were known to
te
always play according to the contagious strategy. By contrast, Camera et al. (2012) found large heterogeneity in strategies adopted by participants in a 2-player Prisoner's Dilemma game with random and anonymous matching. When there is heterogeneity in strategies or a belief that strategies
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
are heterogeneous, the incentives for agents to play according to the contagious strategy may be greatly weakened or become non-existent. We have found evidence for this phenomenon as well, as we observed declines in cooperation rates as we increased the ratio of human subjects to robots in the population of players from which groups were randomly formed. A next step in this literature might be to examine whether the fraction of robot players playing according to the contagious strategy could be very gradually reduced, e.g., from
n−1
robots on
down to 0, and gradually replaced with human subjects, so as to keep the total population size,
M,
xed.
Alternatively, one could rst have 12 human subjects play for a time in our 1H11R
treatment condition and subsequently have those same 12 subjects interact with one another (no robots -12H0R) playing the same inde nitely repeated n-player PD game. It would be of interest to know whether the human subjects, who would be free to choose any strategy, could learn to coordinate on the cooperative outcome in such settings. We leave these projects to future research.
27
Page 29 of 50
References
Camera,
G. and A. Giore (2014), A Tractable Analysis of Contagious Equilibria,
Camera,
G, Casari M and Bigoni M. (2012), Cooperative Strategies in Anonymous Economies:
Journal of
Mathematical Economics, 50, 290300.
Carpenter,
Games and Economic Behavior, 75, 570-586.
J.P., (2007), Punishing Free-riders: How Group Size Aects Mutual Monitoring and
Games and Economic Behavior, 60, 31-51.
cr
the Provision of Public Goods,
Dal
ip t
An Experiment,
Bó, P. (2005), Cooperation under the Shadow of the Future: Experimental Evidence from
The American Economic Review, 95(5):
1591-1604.
us
In nitely Repeated Games,
Dal
Bó, P. (2007), Social Norms, Cooperation and Inequality,
Dal
Bó, P. and Fréchette G R., (2011), The Evolution of Cooperation in In nitely Repeated
Economic Theory, 30, 89-105.
an
Games: Experimental Evidence,
The American Economic Review, 101(1):
411-429
Duy,
J. and J. Ochs (2009), Cooperative Behavior and the Frequency of Social Interaction,
Duy,
J., H. Xie and Y-J. Lee (2013), Social Norms, Information, and Trust Among Strangers:
Ellison,
Economic Theory, 52(2), 669-708.
d
Theory and Evidence,
M
Games and Economic Behavior, 66, 785-812.
G. (1994), Cooperation in the Prisoner's Dilemma with Anonymous Random Matching,
Engle-Warnick,
te
Review of Economic Studies, 61, 567-588.
J. and Slonim, R.L., (2006), Inferring Repeated-Game Strategies from Actions:
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Evidence from Trust Game Experiments,
Fudenberg,
D, Dreber A, Rand DG. (2012), Slow to Anger and Fast to Forgive: Cooperation in
an Uncertain World,
Hardin, Isaac,
American Economic Review, 102:720-749.
G. (1968), The Tragedy of the Commons,
Science, 162, 1243-1248.
M., and J. Walker, (1988), Group Size Eects in Public Goods Provision: The Voluntary
Contribution Mechanism,
Isaac,
Economic Theory, 28, 603632.
Quarterly Journal of Economics, 103, 179-199.
M., J. Walker, and A. Williams (1994), Group Size and the Voluntary Provision of Public
Goods,
Kandori,
Journal of Public Economics, 54, 1-36.
M. (1992), Social Norms and Community Enforcement,
Review of Economic Studies,
59, 63-80.
Nosenzo,
D., S. Quercia, and M. Sefton (2015), Cooperation in Small Groups:
Group Size
Experimental Economics
The Eect of
18, 4-14.
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Okada,
A. (1993), The Possibility of Cooperation in an n-Person Prisoners' Dilemma with Insti-
tutional Arrangements,
A.E. and J.K. Murnighan (1978), Equilibrium Behavior and Repeated Play of the Prisoners'
Dilemma,
Weimann,
Journal of Mathematical Psychology, 17, 189-198.
J., J. Brosig-Koch, T. Heinrich, H. Hennig-Schmidt, C. Keser, C. Stahr (2014),
An
ip t
Roth,
Public Choice, 77, 629-656.
Explanation of (First Round) Contributions in Public-Good Experiments, CESifo working paper no. 5039, October.
cr
Games and Economic
us
Behavior, 76(2), 548-555.
B, B. Cadsby, L. Fan and F. Song, (2013), Group Size, Coordination, and the Eectiveness of Punishment in the Voluntary Contributions Mechanism: An Experimental Investigation, 89-105.
M
an
Games, 4(1):
d
Xu,
H. and Y-J. Lee (2012), Social Norms and Trust Among Strangers,
te
Xie,
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
29
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Appendix A We rst show that the equilibrium conditions in Lemma 1 are equivalent to the equilibrium con-
n = 2 . Translating our notation to T T that used by Kandori (1992), Cn−1 = 1, un = (−l, 1) , vn = (0, 1 + g) , Zn = (iM − ρ ρ) 1 T (ρ = M −1 (M − 1, M − 2, . . . , 1, 0) , in which the ith element of ρ is the conditional probability that a d-type player meets a c-type when there are i d-types, and iM is a 1 × M vector with all elements equal to 1), An = A, Bn = B , Hn = H . Thus condition 1, equation 1 can be written as:
us
= (1 − δ)e1 (I − δA)−1 ρ(1 + g),
!
cr
1 ≥ (1 − δ)e1 (I − δA)−1 (iM − ρ ρ)
0 1+g
ip t
ditions provided by Kandori (1992) when the group size
which is the same as equilibrium condition 1 in Kandori (1992). Condition 2, equation 2 can be
ek (iM − ρ ρ)
(
!
0 1+g
≥ δek H(I − δA)−1 (iM − ρ ρ)
!
,
k−1 M −k )g + ( )l ≥ δek H(I − δA)−1 ρ(1 + g), M −1 M −1
M
i.e.,
l g
an
written as
which is the same as equilibrium condition 2 in Kandori (1992).
Pn−1 n j=0 pj = 1
Proof.
and
By de nition,
Pn−1 n j=0 qj = 0.
pnj
te
then
d
Lemma 2 De ne pnj ≡ (1 − δ)e1 (I − δAn )−1 ρnj and qjn ≡ δen Hn (I − δAn )−1 ρnj (j = 0, . . . , n − 1), denotes the discounted summation of the probability of meeting
j c-type
n players in the group once a defection has started when the group size is n, and qj denotes the change
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in the discounted summation of the probability of meeting
d-type
j c-type
players in the group when the
player reverts back to playing cooperation instead of defection given that the group size is
n
k = n d-type players. Notice that by de nition the summation of the elements in each row of matrix Zn , An , and Bn is always equal to 1. Denote ik as a 1 × k vector with all elements t t equal to 1. Thus Zn in = iM , An iM = iM and Bn iM = iM for any group size n and t = 0, 1, . . . , ∞. and there are
Therefore we have
Pn−1 n P∞ t −1 −1 t t=0 δ e1 An iM = 1, j=0 pj = (1 − δ)e1 (I − δAn ) Zn in = (1 − δ)e1 (I − δAn ) iM = (1 − δ) Pn−1 n P ∞ −1 −1 t t t=0 δ en (Bn − An )An iM = 0. j=0 qj = en Hn (I − δAn ) Zn in = en Hn (I − δAn ) iM =
Proof.
[Theorem 1]
limδ→1 (I − δAn )−1 ρnj < ∞ for j = 1, . . . , n − 1. (Therefore, limδ→1 pnj = 0 for j = 1, . . . , n − 1 and limδ→1 pn0 = 1.) The proof is similar as in Kandori's (1992) proof for Theorem n 1. Since Xt = M is the absorbing state and the M th element of ρj is zero for j = 1, . . . , n − 1, We rst show that
(I − δAn )−1 ρnj =
∞ X t=0
δ t Atn ρnj =
∞ X
δ t A˜tn ρnj = (I − δ A˜n )−1 ρnj ,
for
j = 1, . . . , n − 1
t=0 30
Page 32 of 50
where
A˜n
is a matrix obtained by replacing the last column of
only to show the existence of triangular and so is
(I − A˜n ).
(I − A˜n )−1 .
Since the number of
An
by zeros. Given this, we have
d-types
never declines,
A˜n
is upper-
The determinant of an upper-triangular matrix is the products of its
(I − A˜n ). Therefore, limδ→1 pnj = limδ→1 (1 − δ)e1 (I − δAn )−1 ρnj → 0 and qjn = δen Hn (I − δAn )−1 ρnj is nite for j = 1, . . . , n − 1. −1 Z v = pn D + Pn−1 pn D ≤ Now the r.h.s. of equilibrium condition 1, (1 − δ)e1 (I − δAn ) n n o o j=1 j j Pn−1 n Pn−1 n n Do + j=1 pj Dj , where the inequality comes from j=0 pj = 1 and po ≤ 1. Therefore, equilibrium Pn−1 n condition 1 is satis ed if Cn−1 − Do ≥ j=1 pj Dj , which is satis ed when Cn−1 − Do and δ are
ip t
diagonal elements, which are all strictly positive for
su ciently large.
is
Therefore, equilibrium condition 2 is satis ed
is su ciently large.
d
M
an
vn − un
P n = − n−1 j=1 qj .
te
when
n and so qo
Pn−1 n j=1 qj Dj ,
us
nite because
Pn−1 n j=0 qj = 0
δen Hn (I − δAn )−1 Zn vn = qon Do +
cr
Similarly, the r.h.s. of equilibrium condition 2,
Ac ce p
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31
Page 33 of 50
Appendix B: Fixing the Number of Groups, m = M/n M
In this appendix we examine the case where
and
n
are varied in such a way that the number of
m = M/n is held constant. In particular, we compare the equilibrium conditions in three cases where m = 3: 1) M = 12 and n = 4; 2) M = 9 and n = 3; and 3) M = 6 and n = 2. Our aim here is to understand whether variations in the group size n continue to matter for satisfaction of the
groups
ip t
equilibrium conditions needed for cooperation to be sustained as a social norm, when the number of groups is held constant. The following numerical results are obtained holding xed
δ = 0.9.
cr
Equilibrium Condition 1
M/n = 3:
us
First we consider equilibrium condition 1 for each of the three cases where
M =6
and
n = 2: βn ≥ 0.773647α + 0.226353(α + βn );
M =9
and
n = 3: 2βn ≥ 0.774597α + 0.0696314(α + βn ) + 0.155771(α + 2βn );
M = 12 and n = 4: 3βn ≥ 0.783465α+0.030344(α+βn )+0.0546601(α+2βn )+0.131531(α+3βn ).
an
βn = (n − 1)−1 ,
Imposing the normalization condition follows: and
n = 2: α ≤ 0.773647
for
βn = 1;
M =9
and
n = 3: α ≤ 0.809415
for
βn = 1/2;
M = 12
and
n = 4: α ≤ 0.821913
for
these conditions can be simpli ed as
M
M =6
βn = 1/3.
We observe that when the number of groups
d
to those in the case where the population size
m = M/n is xed (at 3) the results are very similar M is xed: equilibrium condition 1 is observed to
n
becomes larger. Intuitively, again, this is driven by the
te
become less restrictive as the group size
faster contagious process with a larger group size. The extent of the tendency for cooperation to become more easily sustainable as
n
increases is smaller than in the case where
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
M
is xed, since in
the latter case the contagious process becomes faster not only due to a larger group size but also due to there being a smaller number of groups as
n
increases.
Equilibrium Condition 2
Finally, we consider equilibrium condition 2 for each of the three cases where
M/n = 3:
M =6
and
k = n = 2: α ≥ −0.270439α + 0.270439(α + βn );
M =9
and
k = n = 3: α ≥ −0.219582α + 0.15486(α + βn ) + 0.064722(α + 2βn );
M = 12 and k = n = 4: α ≥ −0.187568α + 0.126324(α + βn ) + 0.0512613(α + 2βn ) + 0.00998238(α + 3βn ). If we further impose our payo normalization, then we have:
M =6
and
k = n = 2: α ≥ 0.270439
for
βn = 1;
M =9
and
k = n = 3: α ≥ 0.142152
for
βn = 1/2;
M = 12
and
k = n = 4: α ≥ 0.086265
for
βn = 1/3.
As with equilibrium condition 1, the results for equilibrium condition 2 under a xed ratio for
M/n
are similar to those found under a xed
M.
Equilibrium condition 2 becomes less restrictive
32
Page 34 of 50
n.
te
d
M
an
us
cr
ip t
with increases in the group size,
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
33
Page 35 of 50
Appendix C: Instructions (Treatment n = 6) [Instructions for the
n=2
treatment are similar]
Overview This is an experiment in economic decision-making. The Department of Economics has provided You are guaranteed $7 for showing up and completing this experiment.
ip t
funds for this research.
During the course of this experiment, you will be called upon to make a series of decisions. The decisions you make determine your additional earnings for the experiment, beyond the $7 show-up
cr
payment. Your total earnings, including the show-up payment, will be paid to you in cash and in private at the end of the session. We ask that you not talk with one another and that you silence any mobile devices for the duration of this experiment.
us
Speci cs
In today's experiment, you will play with 11 computerized robots denoted by R1, R2, . . . ,
an
R11. You and the 11 robots consist of a single community. In this room, we have ____ human participants. So in total we have ___ communities. The play in each community will not in uence other communities in any way.
Each sequence consists of an inde nite
M
The experiment consists of a number of sequences.
number of rounds. At the start of each round of a sequence, you and every robot in your community will be randomly and anonymously assigned to one of two groups of size 6. All possible divisions
d
of the 12 community members into two groups of size 6 are equally likely at the start of each new round. Thus, the composition of your 6 group members is very likely to change from one round to
te
the next. For example, you may have robots R1, R3, R4, R6, R10 in your group in one round, and have robots R2, R3, R6, R8, R9 in your group in another round. However, you will not know which robot is in your group in a given round since the matching is anonymous. Note also that in addition
Ac ce p
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to the group you are in, there will be a second, 6-member group in your community consisting of all robot players.
In each round, all members of both 6-member groups for that round must simultaneously choose between two options: X or Y. Your earnings for the round will be decided by your own choice and by how many of the other 5 robot members in your group choose X and Y in the round. The same is true for the robot players in your group and the robot players in the other group. Speci cally, the determination of a player's earning (you or the robots) is explained in the payo table that appears on your decision screen which is shown in Table 1 on the next page. In Table 1, the second and third rows show your earnings (in dollars) from choosing option X or option Y respectively, given the number of the 5 robots in your group who choose option X or option Y, as indicated by the rst row.
Example: Suppose that 3 of the 5 robot group members choose X while 2 choose Y: 3X,2Y. If you choose X, your earnings for the round is $0.60 (row two, column four). If you choose Y, your earnings for the round is $1.10 (row three, column four).
34
Page 36 of 50
ip t cr us an M
Figure 5: Payo Table
d
Note that you and your robot group members all face the same payo table as shown in Table 1. This payo table not only shows Your earnings but it also shows how Others' earnings are
te
aected by your choice and the choices of others. To see this, suppose again that 3 of the others choose X and 2 of the others choose Y: 3X,2Y. In this case, if you choose X, then the earnings for the round for the 3 other group members who (like you) choose X is $0.60, and the earnings
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
for the round for the 2 other group members who choose Y is $1.30. If, in this same scenario (3 others choose X and 2 others choose Y: 3X,2Y), you instead choose Y, then the earnings for the round for the 3 other group members who choose X is $0.40, and the earnings for the round for the 2 other group members who (like you) choose Y is $1.10.
Robot Rules
The robots are programmed to make their choices according to the following rules: - choose X in the rst round of each new sequence; - if, during the current sequence, any of a robot's group members, including you or any other robot players have chosen Y in any prior round of that sequence, then the robot will switch to choosing Y in all remaining rounds of the sequence. - otherwise, the robot will continue to choose X. Notice that the choice of each robot may be dierent in each round if their experience in previous rounds of the sequence is dierent. As a human participant, you are always free to choose X or Y. Notice further that no robot will start choosing Y unless the human subject in their community has previously chosen Y in a round of the current sequence.
35
Page 37 of 50
When you make your choice in each round, you will be reminded of the results in all previous rounds of the current sequence. You will be shown your own choice, the number of your robot group members who chose X, the number of your robot group members who chose Y, your earnings for the round and your cumulative earnings for the current sequence of rounds. To complete your choice in each round, simply click on the radio button next to option X or Y (as shown at the bottom of Table
ip t
1) and then click the red Submit button. You can change your mind anytime prior to clicking the Submit button. After you have clicked the red Submit button, the computer program will record your choice and the choices made by your robot group members and determine your earnings for the
cr
round. Then the results of the round will appear on your computer screens. You will be reminded of your own choice and will be informed about how many of your robot group members have chosen X or Y, as well as the payos that you have earned for the round.
Please record the results of
us
the round on your RECORD SHEET under the appropriate headings. Immediately after you have received this information on choices and payos for the round, a random number from 1 to 100 will be drawn by the computer to determine whether the sequence continues or not. If a number from
an
1 to 75 is chosen, the sequence will continue with another round. If a number from 76 to 100 is chosen, the sequence ends. Therefore, after each round there is 75% chance that the sequence will continue with another round and a 25% chance that the sequence will end. Suppose that a number
M
less than or equal to 75 has appeared. Then you will play the same game as in the previous round, but you and your robot community members will be randomly and anonymously assigned to two new groups of size 6. The robots always play according to the robot rules stated above, even in
d
the all-robot group. You only see the outcome of your own group's decisions each round. After you have made your decision for the new round and learned the outcome, record the results and your
te
earnings for the round on your record sheet under the appropriate headings. Then another random number will be drawn to decide whether the sequence continues for another round. If a number greater than 75 appears, then the sequence ends. Depending on the time available, a new sequence
Ac ce p
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may be played. The new sequence will again consist of an inde nite number of rounds of play of the same game as described above. Recall that, according to the robot rules, all robots play X in the rst round of each new sequence, and will continue doing so until they experience play of Y by one or more players in their group, at which point they will switch over to playing Y for the duration of that sequence.
Earnings
Following completion of the last sequence, the experimenter will randomly draw three sequences to determine your earnings from today's experiment. Your earnings from each of the three chose sequences will be the accumulated earnings from all the rounds played in those three sequences. Since you don't know which three sequences will be chosen for payment you will want to do your best in every round of every sequence.
At the end of the session you will be shown your total
earnings on your computer screen. This total amount will include your $7 show-up payment. Please write down the total amount you are owed on your receipt.Then you will be paid your earnings in cash and in private.
36
Page 38 of 50
Final Comments First, do not discuss your decisions or your results with anyone at any time during the experiment. Second, each of you will play with 11 robots in today's session. Your play will not aect the earnings of any other human subject participant.
ip t
Third, your identity and the identity of your robot group members are never revealed. Fourth, at the end of every round of a sequence, there is 75% chance that the sequence will continue with another round and a 25% chance that the sequence will end.
cr
Fifth, remember that at the start of each round of a sequence you and your 11 robot community members are randomly and anonymously assigned to one of two groups of size 6, so the composition of your robot group members is very likely to change from one round to the next.
us
Sixth, the robots always play according to the rules stated in the instructions. Speci cally, they always start out a sequence playing X and only switch to playing Y if a member of their group has chosen to play Y in any prior round of the sequence; otherwise they continue to play X.
an
Finally, we will randomly draw three sequences at the end of today's session to determine your earnings. Your total earnings will be the sum of your accumulated earnings from the three chosen sequences.
M
Questions?
te
d
Now is the time for questions. Does anyone have any questions?
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37
Page 39 of 50
Table
H* program to produce the transition matrices with fixed population size n=12, compare three cases with group size = 2,3,4,6*L H*Choose parameters*L
ip t
H*Number of players*L n = 12;
"D "D
an
Print@" Print@" Print@ "BELOW Print@"
us
cr
Print@"POPULATION SIZE M=12"D Print@"MATRIX biga IS THE MATRIX A DEFINED IN THE PAPER"D Print@"MATRIX bigb IS THE MATRIX B DEFINED IN THE PAPER"D Print@"MATRIX bigh IS THE MATRIX H=B-A DEFINED IN THE PAPER"D
IS THE TRANSITION MATRIX FOR 12-PERSON PRISONER'S DILEMMA WITH 1 GROUP"D "D
d te
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
M
H*Matrix I=in*L in = IdentityMatrix@nD;
H*Transition matrix A, B, Z*L biga12 = Array@a, 8n, n
Ac ce p
In[642]:=
bigb12 = Array@b, 8n, n
H*bigz6=Array@z,8n,6
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga12@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
biga12 = ReplacePart@biga12, 881, 12< -> 1, 82, 12< -> 1, 83, 12< -> 1, 84, 12< -> 1, 85, 12< -> 1, 86, 12< -> 1, 87, 12< ® 1, 88, 12< ® 1, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
Page 40 of 50
TransitionMatrices.nb
H*Initialize B matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb12@@i, jDD = 0;D;D;
bigb12 = ReplacePart@biga12, 881, 1< -> 1, 81, 12< ® 0
us
H*Matrix H*L Print@"bigh12 = ", MatrixForm@bigb12 - biga12DD; Print@" "D
cr
Print@"bigb12 = ", MatrixForm@bigb12DD; Print@" "D;
ip t
H*Now insert the correct nonzero values of b_ij*L
H*Initialize Z matrix so that all elements are zero*L
an
For@i = 1, i £ n, i ++, For@j = 1, j £ 12, j ++, bigz12@@i, jDD = 0;D;D;
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
te
H*Matrix I=in*L in = IdentityMatrix@nD;
d
M
Print@ "BELOW IS THE TRANSITION MATRIX FOR 6-PERSON PRISONER'S DILEMMA WITH 2 GROUPS"D Print@" "D
Ac ce p
2
H*Transition matrix A, B, Z*L biga6 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga6@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
biga6 = ReplacePart@biga6, 881, 6< -> 1, 82, 6< ® 5 11, 82, 12< ® 6 11, 83, 6< ® 2 11, 83, 12< ® 9 11, 84, 6< ® 2 33, 84, 12< ® 31 33, 85, 6< ® 1 66, 85, 12< ® 65 66, 86, 6< ® 1 462, 86, 12< ® 461 462, 87, 12< ® 1, 88, 12< ® 1, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
ip t
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb6@@i, jDD = 0;D;D;
Print@"bigb6 = ", MatrixForm@bigb6DD; Print@" "D;
M
an
H*Matrix H*L Print@"bigh6 = ", MatrixForm@bigb6 - biga6DD; Print@" "D
us
cr
bigb6 = ReplacePart@bigb6, 881, 1< -> 1, 82, 6< ® 5 11, 82, 7< ® 6 11, 83, 6< ® 2 11, 83, 7< ® 3 11, 83, 12< ® 6 11, 84, 6< ® 2 33, 84, 7< ® 4 33, 84, 12< ® 27 33, 85, 6< ® 1 66, 85, 7< ® 1 22, 85, 12< ® 31 33, 86, 6< ® 1 462, 86, 7< ® 1 77, 86, 12< ® 65 66, 87, 7< ® 1 462, 87, 12< ® 461 462, 88, 12< ® 1, 89, 12< ® 1, 810, 12< ® 1, 811, 12< ® 1, 812, 12< ® 1
H*Below is the calculation for 4-person prisoner's dilemma with 3 groups*L
Ac ce p
H*Matrix I=in*L in = IdentityMatrix@nD;
te
d
Print@ "BELOW IS THE TRANSITION MATRIX FOR 4-PERSON PRISONER'S DILEMMA WITH 3 GROUPS"D Print@" "D
H*e_i=en@@iDD*L en = IdentityMatrix@nD;
H*Transition matrix A, B, Z*L biga4 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga4@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of a_ij*L
Page 42 of 50
TransitionMatrices.nb
H*Initialize B matrix so that all elements are zero*L
cr
Print@"biga4 = ", MatrixForm@biga4DD; Print@" "D;
ip t
biga4 = ReplacePart@biga4, 881, 4< -> 1, 82, 4< ® 3 11, 82, 8< ® 8 11, 83, 4< ® 3 55, 83, 8< ® 36 55, 83, 12< ® 16 55, 84, 4< ® 1 165, 84, 8< ® 68 165, 84, 12< ® 32 55, 85, 8< ® 7 33, 85, 12< ® 26 33, 86, 8< ® 1 11, 86, 12< ® 10 11, 87, 8< ® 1 33, 87, 12< ® 32 33, 88, 8< ® 1 165, 88, 12< ® 164 165, 89, 12< ® 1, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb4@@i, jDD = 0;D;D;
us
H*Now insert the correct nonzero values of b_ij*L
M
an
bigb4 = ReplacePart@bigb4, 881, 1< -> 1, 82, 4< ® 3 11, 82, 5< ® 8 11, 83, 4< ® 3 55, 83, 5< ® 12 55, 83, 8< ® 24 55, 83, 9< ® 16 55, 84, 4< ® 1 165, 84, 5< ® 8 165, 84, 8< ® 4 11, 84, 9< ® 16 55, 84, 12< ® 16 55, 85, 5< -> 1 165, 85, 8< ® 34 165, 85, 9< ® 34 165, 85, 12< ® 96 165, 86, 8< -> 1 11, 86, 9< ® 4 33, 86, 12< ® 26 33, 87, 8< ® 1 33, 87, 9< ® 2 33, 87, 12< ® 10 11, 88, 8< ® 1 165, 88, 9< ® 4 165, 88, 12< ® 32 33, 89, 9< -> 1 165, 89, 12< ® 164 165, 810, 12< ® 1, 811, 12< -> 1, 812, 12< -> 1
d
Print@"bigb4 = ", MatrixForm@bigb4DD; Print@" "D;
te
H*Matrix H*L Print@"bigh4 = ", MatrixForm@bigb4 - biga4DD; Print@" "D
Ac ce p
4
H*Below is the calculation for 3-person prisoner's dilemma with 4 groups*L
Print@ "BELOW IS THE TRANSITION MATRIX FOR 3-PERSON PRISONER'S DILEMMA WITH 4 GROUPS"D Print@" "D H*Transition matrix A, B, Z*L biga3 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga3@@i, jDD = 0;D;D;
Page 43 of 50
TransitionMatrices.nb
5
H*Now insert the correct nonzero values of a_ij*L
H*Initialize B matrix so that all elements are zero*L
cr
Print@"biga3 = ", MatrixForm@biga3DD; Print@" "D;
ip t
biga3 = ReplacePart@biga3, 881, 3< -> 1, 82, 3< ® 2 11, 82, 6< ® 9 11, 83, 3< -> 1 55, 83, 6< ® 27 55, 83, 9< ® 27 55, 84, 6< ® 2 11, 84, 9< ® 36 55, 84, 12< ® 9 55, 85, 6< -> 1 22, 85, 9< ® 6 11, 85, 12< ® 9 22, 86, 6< -> 1 154, 86, 9< ® 27 77, 86, 12< ® 9 14, 87, 9< ® 2 11, 87, 12< ® 9 11, 88, 9< ® 4 55, 88, 12< ® 51 55, 89, 9< -> 1 55, 89, 12< ® 54 55, 810, 12< -> 1, 811, 12< -> 1, 812, 12< -> 1
us
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, bigb3@@i, jDD = 0;D;D; H*Now insert the correct nonzero values of b_ij*L
d
M
an
bigb3 = ReplacePart@bigb3, 881, 1< -> 1, 82, 3< ® 2 11, 82, 4< ® 9 11, 83, 3< -> 1 55, 83, 4< ® 9 55, 83, 6< ® 18 55, 83, 7< ® 27 55, 84, 4< ® 1 55, 84, 6< ® 9 55, 84, 7< ® 18 55, 84, 9< ® 18 55, 84, 10< ® 9 55, 85, 6< -> 1 22, 85, 7< ® 3 22, 85, 9< ® 9 22, 85, 10< ® 27 110, 85, 12< ® 9 55, 86, 6< -> 1 154, 86, 7< ® 3 77, 86, 9< ® 24 77, 86, 10< ® 18 77, 86, 12< ® 9 22, 87, 7< ® 1 154, 87, 9< ® 27 154, 87, 10< ® 27 154, 87, 12< ® 9 14, 88, 9< ® 4 55, 88, 10< ® 6 55, 88, 12< ® 9 11, 89, 9< -> 1 55, 89, 10< ® 3 55, 89, 12< ® 51 55, 810, 10< -> 1 55, 810, 12< ® 54 55, 811, 12< -> 1, 812, 12< -> 1
te
Print@"bigb3 = ", MatrixForm@bigb3DD; Print@" "D;
Ac ce p
H*Matrix H*L Print@"bigh3 = ", MatrixForm@bigb3 - biga3DD; Print@" "D; H*Below is the calculation for 2-person prisoner's dilemma with 6 groups*L
Print@ "BELOW IS THE TRANSITION MATRIX FOR 2-PERSON PRISONER'S DILEMMA WITH 6 GROUPS"D Print@" "D H*define s function*L
s@num_D := Product@2 * i - 1, 8i, 1, num 2
Print@"s@nD = ", s@nDD; Print@" "D
Page 44 of 50
TransitionMatrices.nb
biga2 = Array@a, 8n, n
H*Initialize A matrix so that all elements are zero*L
H*Now insert the correct nonzero values of a_ij*L
cr
Print@"Matrix A initialized."D;
ip t
For@i = 1, i £ n, i ++, For@j = 1, j £ n, j ++, biga2@@i, jDD = 0;D;D;
Print@"biga2 = ", MatrixForm@biga2DD; Print@" "D
an
us
For@i = 1, i £ n, i ++, If@Mod@i, 2D 0, H*if i is even*LFor@j = i, j £ Min@2 i, nD, j += 2, biga2@@i, jDD = HBinomial@i, j - iD * Binomial@n - i, j - iD * HHj - iL !L * s@2 i - jD * s@n - jDL denom;D;, H*if i is odd*L For@j = i + 1, j £ Min@2 i, nD, j += 2, biga2@@i, jDD = HBinomial@i, j - iD * Binomial@n - i, j - iD * HHj - iL !L * s@2 i - jD * s@n - jDL denom;D;D;D;
M
H*Initialize H matrix so that all elements are zero*L
d
bigh2 = Array@h, 8n, n
te
Print@"Matrix H initialized."D;
H*Now insert the correct nonzero values of h_ij*L
For@i = 1, i £ n, i ++, If@Mod@i, 2D 0, H*if i is even*L For@j = i + 2, j £ Min@2 i, nD, j += 2, bigh2@@i, jDD = - HHj - iL iL * biga2@@i, jDD; bigh2@@i, j - 1DD = Abs@bigh2@@i, jDDD;D;, H*if i is odd*L For@j = i + 1, j £ Min@2 i, nD, j += 2, bigh2@@i, jDD = - HHj - iL iL * biga2@@i, jDD; bigh2@@i, j - 1DD = Abs@bigh2@@i, jDDD;D;D;D;
Ac ce p
6
Print@"bigh2 = ", MatrixForm@bigh2DD; Print@" "D H*end of program*L
POPULATION SIZE M=12 MATRIX biga IS THE MATRIX A DEFINED IN THE PAPER MATRIX bigb IS THE MATRIX B DEFINED IN THE PAPER MATRIX bigh IS THE MATRIX H=B-A DEFINED IN THE PAPER
Page 45 of 50
TransitionMatrices.nb
7
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1
bigb12 =
1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
te 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
Ac ce p
bigh12 =
cr
0 0 0 0 0 0 0 0 0 0 0 0
us
0 0 0 0 0 0 0 0 0 0 0 0
an
0 0 0 0 0 0 0 0 0 0 0 0
M
0 0 0 0 0 0 0 0 0 0 0 0
d
biga12 =
0 0 0 0 0 0 0 0 0 0 0 0
ip t
BELOW IS THE TRANSITION MATRIX FOR 12-PERSON PRISONER'S DILEMMA WITH 1 GROUP
0 0 0 0 0 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BELOW IS THE TRANSITION MATRIX FOR 6-PERSON PRISONER'S DILEMMA WITH 2 GROUPS
Page 46 of 50
TransitionMatrices.nb
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1 0 0 0 0
0
0
5 11 2 11 2 33 1 66 1 462
6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bigb6 =
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0 -1 0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
bigh6 =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0
0
0 0 0 0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 0
6 11 9 11 31 33 65 66 461 462
0 0 0 0 0
te
0 0 0 0 0 0
6 11 9 11 31 33 65 66 461 462
ip t
0 0 0 0 0
0
0 0 0 0 0
cr
0 0 0 0 0 biga6 =
0 0 0 0 0
us
0 0 0 0 0
5 11 2 11 2 33 1 66 1 462
an
0 0 0 0 0
1
M
0 0 0 0 0
1 1 1 1 1
d
0 0 0 0 0
0 0 0 0
0 6 11 3 11 4 33 1 22 1 77 1 462
0 0 0 0 -
Ac ce p
8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
BELOW IS THE TRANSITION MATRIX FOR 4-PERSON PRISONER'S DILEMMA WITH 3 GROUPS
Page 47 of 50
TransitionMatrices.nb
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0
0
0
3 11 3 55 1 165
8 11 12 55 8 165 1 165
0 0 0 0
0 0 0 0 0 0
0 0 0 0
0
0 0 0
8 11 36 55 68 165 7 33 1 11 1 33 1 165
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
0 0 0 0
0
0 0
0
0
0
0 0
0
24 55 4 11 34 165 1 11 1 33 1 165
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0 0 0 0 0 0
0 0 0
0
0
0 0
0 0 0
0
0
0 0
0 0 0
0
0
0 0
0
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
Ac ce p
bigh4 =
0 0 0 0 0 0 0 0 0 0 0 0 0 0
d
0
0
1 1 1 1
0
0
0 0 0
0 0 0 0
0 0
0 0 0
8 11 12 55 8 165 1 165
0 0 0 0
0 0
0
0
0 16 55 32 55 26 33 10 11 32 33 164 165
0 0 0
0 0 0
1 0 0 -1
0
0 0 0
ip t
0 0 0
0 0 0
0 0 0
bigb4 =
0 0 0
0 0 0
0 0 0 0 0 0
te
biga4 =
0 0 0
cr
0 0 0
3 11 3 55 1 165
0 0 0 0 0 0
0
8 11 12 55 8 165 1 165
0 0 0
0
0 0 0
0
0 0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0
0
0
0 0
0
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0
us
0 0 0
1
an
0 0 0
0
16 55 32 55 26 33 10 11 32 33 164 165
M
0 0 0
1 1 1
0
0 0
0
0
0 0
0
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0
9
16 55 16 55 34 165 4 33 2 33 4 165 1 165
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
BELOW IS THE TRANSITION MATRIX FOR 3-PERSON PRISONER'S DILEMMA WITH 4 GROUPS
Page 48 of 50
TransitionMatrices.nb
9 11 27 55 2 11 1 22 1 154
0 0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0
0 0
0 0
0
0 0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0
0
0 0
0
0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
1 0
0
0
2 11 1 55
9 11 9 55 1 55
0 0 0 0
0 0 0 0 0 0 0 0 0 0
9 55 9 22 9 14 9 11 51 55 54 55
1 1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
18 55 9 55 1 22 1 154
27 55 18 55 3 22 3 77 1 154
0
0
0
18 55 9 22 24 77 27 154 4 55 1 55
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
0
0 0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
0
0 0 0 0
0 0
0
0
9 11 9 55 1 55
0 0 0 -
0
9 11 9 55 1 55
0 0
0 0
9 55 9 22 9 14 9 11 51 55 54 55
M
0
0
0 0
1 0 -1
bigh3 =
0 0
d
0 0
0 0
27 55 36 55 6 11 27 77 2 11 4 55 1 55
ip t
0
0 0
0 0
0 0
bigb3 =
0 0
te
biga3 =
2 11 1 55
cr
0 0
1
us
0 0
an
0 0
Ac ce p
10
0 0 0 0 0
0 0
0 0
1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
27 55 18 55 3 22 3 77 1 154
0 0
27 55 18 55 3 22 3 77 1 154
0 0
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0 0
9 55 27 110 18 77 27 154 6 55 3 55 1 55
0 0 0 0 0 0 0 0 0
0 0
BELOW IS THE TRANSITION MATRIX FOR 2-PERSON PRISONER'S DILEMMA WITH 6 GROUPS
Page 49 of 50
TransitionMatrices.nb
11
s@nD = 10 395
0
1
0
1 11
0 0 0
0 0
0 0 0 0
0 10 11 3 11 1 33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
8 11 16 33 5 33 5 231
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8 33 80 231 20 77 4 33 1 33
8 33 80 231 20 77 4 33 1 33
0
0
0 0
0
0 16 231 8 33 16 33 8 11 10 11
0 0 0 0
0 0
0 0
1 1
0 0 0
0 0 0 0
0 10 11 1 11
0 0
0 -
10 11 1 11
0 0
8 11 8 33 1 33
-
8 11 8 33 1 33
0
0
16 33 4 11 10 77 5 231
16 33 4 11 10 77 5 231
0
0
0
d
0
te
1 -1
M
Matrix H initialized.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Ac ce p
bigh2 =
0
cr
0
8 33 40 77 20 33 16 33 3 11 1 11
us
0
16 33 20 33 30 77 5 33 1 33
an
biga2 =
0
0
ip t
Matrix A initialized.
16 231 40 231 8 33 8 33 2 11 1 11
0
16 231 40 231 8 33 8 33 2 11 1 11
-
0
Page 50 of 50