Conjecture, uncertainty, and cooperation in prisoner's dilemma games

Journal

of Economic

Behavior

and Organization

22 (1993) 91-l 17. North-Holland

Conjecture, uncertainty, and cooperation in prisoner’s dilemma games Some experimental Lawrence

M. Kahn and J. Keith Murnighan*

University of Iknois, Received

evidence

September

UrbAnA-ChAmpAign,

IL, USA

1991, final version received

September

1992

This paper presents experimental tests of two models of cooperation in finitely-repeated prisoner’s dilemma games (Kreps, Milgrom, Roberts, and Wilson, 1982). The models suggest that either a perception that the other party may use the tit-for-tat strategy or mutual uncertainty concerning dominant noncooperative strategies can lead to rational cooperation. The experiment independently manipulated both types of uncertainty and allowed for inferences concerning the players’ prior, ‘homemade’ preferences for cooperation. Only in relatively restricted situations did either type of uncertainty promote cooperation. Instead, players cooperated much more than was predicted; they also cooperated more when they were certain of their opponents’ payoffs.

1. Introduction

It is well known that for the finitely repeated prisoner’s dilemma game, the unique Nash equilibrium strategy for each player is noncooperative at each stage. The argument rests on the fact that neither player should be motivated to choose cooperatively on the last trial. If neither expects the other to choose cooperatively on the last trial, the next to last trial becomes important. The logic of the last trial now motivates the noncooperative choice here and on every preceding trial, indicating that neither player should ever choose cooperatively.’ This behavior, however, has been contradicted in experiments reported by Rapoport and Chammah (1965), Axelrod (1984), Selten and Stoecker (1986), Correspondence to: Professor Lawrence M. Kahn, Institute of Labor and Industrial Relations, 504 East Armory Avenue, Champaign, IL 61820-6297, USA. *Portions of this paper were written while the second author was a Fellow at the Center for Advanced Study in the Behavioral Sciences. We are grateful for the financial support provided by the National Science Foundation ( #BNS87-00864 and SES88-15566) and the Russell Sage Foundation. The authors thank Patty DeForrest and Felice Herbin for serving as experimenters, and an anonymous referee for helpful comments and suggestions. ‘See Friedman (1986). 0167-2681/93/$06.00

0

1993-Elsevier

Science Publishers

B.V. All rights reserved

92

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

Roth (1988), and Andreoni and Miller (1990) among others2 Kreps et al. (1982) presented two variations of a model designed to explain these observations. Model 1 assumes that each player thinks that his/her opponent might play tit-for-tat instead of persistent noncooperation. Model 2 assumes that each player is uncertain about his/her opponent’s payoff matrix (that is, he/she allows for the possibility of cooperation, and tit-for-tat may then emerge).3 The central feature of their analysis is that the type of prior expectations players have about their opponents’ strategy, and the presence of uncertainty about payoffs, influence their choice of strategy and, in particular, can lead to cooperation. This study provides an experimental investigation of both of these theoretical results. First, we investigated whether expectations of one’s opponent’s potential strategy would affect one’s own cooperative choices. To implement this test, we told players that they would be bargaining with another player who was following a programmed strategy. The program dictated that this player would use the tit-for-tat strategy with either a 0.05, 0.35, 0.65, or 0.95 probability, depending on the condition. Second, we examined the question of whether payoff uncertainty in PD games could generate cooperation. In this portion of the study, two players faced each other in a PD game with uncertainty about each other’s payoffs. In addition, in other trials, we considered all other possible combinations of player certainty or uncertainty. Finally, for each experiment we collected data on subjects’ conjectures about their opponent’s likely play, in light of the importance placed in bargaining models on such beliefs. Our results can be briefly summarized as follows. Regarding the impact of payoff uncertainty, only in relatively restricted situations did this type of uncertainty increase the incidence of cooperation. Instead, players cooperated much more than was predicted, and they were more likely to cooperate when they were certain of each other’s payoffs. The central findings from the programmed opponent trials concern ‘In contrast to finitely repeated PD, cooperative equilibria may exist in games with low probabilities of termination. This outcome occurs when the future gains from current cooperation (i.e., future Pareto-optima1 outcomes) outweigh the gains from current defection. Roth and Murnighan (1978) have shown that a PD with a given probability p of continuation to at least the next round is analytically equivalent to an infinitely repeated game with p as discount factor. Murnighan and Roth (1983) presented data from a dozen different games that supported the basic prediction of their theory. 3While not based on PD, Camerer and Weigelt’s (1988) experiments also bear on the rationales for cooperation identified by Kreps et al. They established a lending game in which a (E). B is uncertain about E’s ‘banker’ (B) chooses whether to ‘lend to an entrepreneur preferences to pay the loan or to renege. Each E played eight rounds of this game against a series of lenders. The results suggested that lenders had ‘homemade’ prior beliefs of the probability that E would prefer to repay the loan even when E gained more by reneging. Specifically, the E players repaid the loan more often and later in the game than predicted by a rational model. While Camerer and Weigelt infer the existence of homemade priors, our methodology, described below, allows for a more direct examination of this phenomenon.

L.M. Kahn and J.K. Murnighan,

Prisoner’s dilemma games

93

players’ conjectures about their opponents’ future choices.4 Specifically, we compared players’ choices and conjectures about their opponent when the other bargainer was a program and when the other bargainer was a person, in the same games with the same certain payoffs.’ We compared conjectures about the cooperative behavior of an actual opponent with conjectures about a known programmed opponent who used a fixed probability (0.05, 0.35, 0.65, or 0.95) of playing tit-for-tat. Such comparisons yielded estimates about a player’s prior probabilities that the actual opponent would play a matching strategy. We found that in actual opponent trials with payoff certainty, players treated each other as if they were highly likely to be tit-for-tat players. In some cases, we estimate such conjectures ranging as high as 80% to over 95%. 2. Methods 2.1. Subjects One hundred and fifty-four students in the business school at the University of Illinois at Urbana-Champaign volunteered to participate in the study. Participating gave them a small amount of fixed extra credit in one of their courses (a common practice) and the chance for a monetary prize. The study was described as a bargaining experiment; at its completion, four participants won monetary prizes of $200, $100, $100, and $100. 2.2. Procedure People arrived for their hour and a half session in groups of six to twelve. Instructions were provided about iterated dilemma games with known termination points. Participants played several example games, some with symmetric payoffs, some asymmetric, always with programmed opponents. The program gave each bargainer feedback about the range of possible strategies they might encounter. Thus, opponents’ bids in practice were occasionally cooperative, occasionally noncooperative, and sometimes double-crossing. Cooperative choices were always labeled ‘A’, and noncooperative choices were labeled ‘B’. The practice programs were consistently 4These models assume common knowledge by both players. That is, noncooperation throughout the supergame is the only sequential equilibrium with payoff certainty if both parties are rational, as long as each party knows that the other party knows that he/she is rational, etc. However, if there is higher order uncertainty (e.g. player X is not sure if player Y knows that X is rational), then cooperative equilibria may result. See Milgrom and Roberts (1982). 5Under these conditions, the only sequential equilibrium in the actual opponent trials under player rationality is mutual noncooperation throughout. However, as noted in Appendix 1, the programmed opponent trials admit cooperative equilibria for portions of the supergame. All Appendices are available upon request.

94

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

noncooperative after a player had chosen noncooperatively or after its own first noncooperative choice. During practice, players could openly observe others’ choices and outcomes. During actual play, participants sat in isolated cubicles and could not identify their opponents. Bids were made simultaneously and secretly; outcomes were fed back to players after every trial. They were encouraged to record their choices and outcomes. After practice but prior to the start, they recorded the earliest trial when they thought the other player would choose noncooperatively. Everyone was told that the sessions would terminate after the 20th trial. After each session, bargainers responded to a short questionnaire which assessed the effectiveness of the manipulations and their perceptions of the other bargainer. They were then given new information about their upcoming session and were randomly paired with another bargainer. Most participants negotiated with two different players for two consecutive sessions and then negotiated against a programmed opponent. Others negotiated for four sessions, only against programmed opponents. In each session, players were encouraged to do well for themselves. Payoff possibilities were carefully described: Points scored in the games were converted to tickets for a lottery at the end of the experiment. Doing better increased a player’s chances of winning one of the four cash prizes. The first name chosen in the lottery won $200; the next three won $100 each. The use of large lottery payoffs rather than small, continuous payoff schemes has advantages and disadvantages. In particular, the high monetary amounts may present more noticeable incentives. In addition, Bolle (1990) has shown that lottery results are not significantly different from those with smaller, continuous payoffs. Each session ended with a debriefing session that did not reveal the exact purpose of the study. Interested participants could receive this information at the end of the study. 2.3. The two games The two games where both players were certain of each others’ payoffs are shown in table la. Game # 1 is referred to as Strong-Strong: Strong players have dominant noncooperative strategies. Game #2 is Strong-Weak since only the ROW player has a dominant noncooperative strategy. Since Weak players do not have a dominant strategy, the Strong-Weak game does not satisfy the requirements of a PD game. Both games were used whether the players negotiated with each other or with a programmed opponent. 2.4. Actual opponents This portion of the experiment

tests for the impact of payoff uncertainty

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games Table The payoff matrices (a) Both Certain: Strong-Strong, Game # 1:

1

in the different

conditions.

Strong- Weak, Game #2 COLUMN

COLUMN

ROW

A

&,

B

(2435)

95

18)

(“5.20, (12,12)

ROW

A

$8.18)

(“5.10)

B

(24,5)

(12,121

(b) ROW Uncertain: Strong- Weak: Game #2 OR Game #I Strong-Strong: Game#l OR Game #2 ROW does not know which navoff matrix is correct for COLUMN’s uavoffs. COLUMN sees the correct matrix - the one hstkd before the word OR in each game. Boih know that ROW is uncertain and that COLUMN is not. (c) COLUMN Uncertain: Strong-Strong, Game #1 OR Game #3: COLUMN

Strong-Weak, Game #2 OR Game #4: COLUMN

A

$4, IS) (Y5.20) A $4.18) (“1510) ROW B B (12,12) (I&5) (12.12) (165) COLUMN does not know which payoff matrix is correct for ROW’s payoffs. ROW sees the correct matrix - the one listed before the word OR in each game. Both know that COLUMN is uncertain and ROW is not. ROW

(d) Both Uncertain: For ROW: Strong-Strong: Game#l OR Game #2 Strong- Weak: Game ROW does not know which payoff matrix is correct for COLUMN’s in each pair is correct.) ROW also knows that COLUMN is uncertain knows that the other is uncertain. Both Uncertain: For COLUMN: Strong-Strong: Game # 1 OR Game #3 Strong-Weak: Game COLUMN does not know which payoff matrix is correct for ROW’s in each pair is correct.) COLUMN also knows that ROW is uncertain Each knows that the other is uncertain.

#2 OR Game # 1 payoffs. (The first matrix of ROW’s payoffs. Each

#2 OR Game #4 payoffs. (The first matrix of COLUMN’s payoffs.

on bargaining behavior. We factorially manipulated three independent variables: (i) the strength or weakness of the COLUMN player; (ii) the ROW player’s uncertainty of the COLUMN player’s strength; and, (iii) the COLUMN player’s uncertainty of the ROW player’s strength. Combining these three variables meant that the participants saw one or two payoff matrices, as shown in table 1. The ROW player was always strong, even though the COLUMN player may have been uncertain of this. In the Uncertain conditions, players were told that the two possible matrices of the other’s payoffs were equally likely. For example, in the Strong-Strong, Both Uncertain condition (table Id), ROW knew that the payoffs of Games 1 and 2 were equally likely but that Games 3 and 4 were not possible. Further, COLUMN knew that the payoffs of Games 1 and 3 were equally likely and that Games 2 and 4 were not possible. This design resulted in eight cells created by the crossing of the four

96

L.M. Kahn and J.K. Murnighan,

Prisoner’s dilemma games

possible uncertainty combinations with the two actual payoff matrices, as shown in table 1. With player rationality, only three of the eight cells in the design admit cooperative equilibria for some portion of the 20-round supergame: (i) Strong-Weak (SW), COLUMN Uncertain; (ii) Strong-Strong (SS), Both Uncertain; and (iii) SW, Both Uncertain. This result is proven in Appendix 2. We now provide some intuitive explanation for the theoretical predictions. The clearest case for a mutually-cooperative equilibrium in table 1 is SW, COLUMN Uncertain. Here, COLUMN perceives a 50% chance that ROW is Weak and thus that cooperation is a dominant strategy for ROW. Since COLUMN is Weak, COLUMN does not gain even temporarily by playing B in response to ROW’s A play. Thus, COLUMN has a strong motivation to cooperate. ROW’s motivation to cooperate comes from the idea that cooperation may build ROW’s reputation for having Weak payoffs, inducing COLUMN to cooperate. Further, ROW knows that COLUMN does not gain by playing B in response to ROW’s A play - COLUMN’s only temptation to defect is to protect him/herself against a Strong ROW. Appendix 2 shows that with the payoffs and knowledge structure in this condition, COLUMN should cooperate for all 20 periods and ROW should cooperate until the last period. The two other conditions in table 1 yielding equilibria with some cooperation provide less motivation for cooperation than in SW, COLUMN Uncertain. In SW, Both Uncertain, both parties have an incentive to build a reputation for weakness. However, ROW no longer knows whether COLUMN could make single-period gains by defecting from mutuallycooperative play. If ROW acted defensively to protect against such behavior, cooperation should break down after 14 periods. Finally, in SS, Both Uncertain, reputation building behavior for part of the supergame is again a sequential equilibrium. However, COLUMN can now make a single-period gain by playing B in response to ROW’s A play. Cooperation should break down earlier than in SW, Both Uncertain, perhaps after 2 periods. The five other conditions in table 1 do not admit any cooperative sequential equilibria. To understand this result, note that if COLUMN knows that ROW is Strong, then cooperation should unravel from the last period to the first. This reasoning implies that there are no cooperative equilibria for any game in which COLUMN is certain about ROW’s payoffs. This category includes SS Neither Uncertain, SS ROW Uncertain, SW Neither Uncertain, and SW ROW Uncertain. In the only condition not yet discussed, SS, Column Uncertain, ROW knows that COLUMN is Strong; therefore, the backwards induction argument applies here as well. Each pair’s first bargaining session was randomly chosen from among the eight cells in the design. The second session either duplicated the first session, with a different opponent, or switched to either the Both or Neither

L&l. Kahn and 1.K. Murnighan, Prisoner’s dilemma games

91

Uncertain Conditions, also with a different opponent. Everyone’s third session was against a programmed opponent; the program was randomly chosen from the eight possibilities that existed for programmed opponent trials6 Data analysis indicated that order had no significant impact on the results.

This portion of the experiment kept payoffs certain and manipulated the uncertainty of the opponent’s strategy. Participants negotiated against a different programmed opponent for four consecutive bargaining sessions, or after two sessions with actual opponents. They were told that the program would play an unforgiving matching strategy. That is, with probability q it would choose A as long as the subject chose A but would choose B for the remainder of the session once the subject chose B. With probability (1-q) it would play a strategy that would, at some point, switch to B and never return to A. The q values spanned a range of possibilities, equalling 0.05, 0.35, 0.65, or 0.95. Payoff matrices were either SS or SW. Subjects took the role of either ROW or COLUMN. For their four sessions, players were randomly assigned to one of the 16 possibilities: 4 (probability values) x 2(SS/SW) x 2(ROW/ COLUMN). The opponent’s programmed choices for the different conditions were: (a) (b) (c) (d) (e) (f)

q=

0.05: all B choices; SS: match for 5 periods, then all B choices; SW: match for 7 periods, then all B choices; SS: match for 10 periods, then all B choices; SW: match for 14 periods, then all B choices; match until subject’s first B, then all B choices.

q=O.35, q=O.35, q=O.65, q=O.65, q=O.95:

The numbers of cooperative choices were chosen on the basis of the bounds computed in Appendix 1. These computations show that, not surprisingly, we expected more A choices with higher q values and when payoffs were SW, for ROW and COLUMN. 3. Results We present the results in five sections. They analyze (1) actual bargaining pairs and whether a cooperative equilibrium led to more cooperation, (2) the effects of games and uncertain payoffs, (3) the effects of repeated play, (4) 6These possibilities resulted from the crossing of the four probability 0.65 and 0.95 with the two games of SS and SW.

conditions

of 0.05, 0.35,

98

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

players’ responses to programmed programmed opponents.

3.1.

Tests of equilibrium

predictions:

opponents,

Actual

(5) play against

actual

versus

opponents

A stringent prediction would expect cooperative choices only in the presence of a cooperative equilibrium and noncooperative when (1) no cooperative equilibrium exists or (2) the other party has chosen noncooperatively. Cooperation in the first nineteen rounds of the twenty round supergame may result in the SW-COLUMN Uncertain condition, followed by the possibility of at least 14 rounds in SW-Both uncertain and at least 2 rounds in SS-Both Uncertain. No cooperation should surface in any of the other conditions. As discussed in Appendix 2, a more relaxed prediction suggests that cooperation should be a positive function of the potential gains from cooperation. Table 2 shows results which generally support the predictions of sequential equilibrium models for total cooperative choices in the SW-COLUMN Uncertain and the SW-Both Uncertain conditions: Both cells generated more cooperation than conditions where a cooperative equilibrium did not exist (F(1,212) =6.95, ~~0.01, and F(1,212) =6.10, ~~0.02, respectively).8 The proportion of first trial cooperative choices was also higher, but these effects were not statistically significant. Similar comparisons for the SS-Both Uncertain condition were not significant. The other five conditions, however, show considerably more cooperation than predicted. Indeed, as we discuss below, overall analyses show systematic effects that do not depend on cooperative equilibria. Fig. 1 displays the mean frequency of cooperative choices in each condition across trials, pooled over ROW and COLUMN players.’ The data fall into three groupings: (1) consistent cooperation which drops only after trial 17 - this outcome occurred for SW-COLUMN, Both, and Neither Uncertain; (2) an initial drop in cooperation, followed by moderate levels of cooperation that again drop after trial 17 - this pattern characterized the SW-ROW Uncertain, and SS-ROW, Neither, or Both Uncertain conditions; and (3) a steeper drop in cooperation and little cooperation thereafter for the SS-COLUMN Uncertain case. Several observations sum up these data: (1) players choose cooperately ‘Since noncooperation still is an equilibrium even when a cooperative equilibrium exists, the observation of noncooperative behavior in such games does not contradict the game theoretic predictions. 8These tests assume independence of observations across members of a bargaining pair. After we corrected for such non-independence in a generalized least squares regression, the results were unchanged. %ince ROW and COLUMN players showed similar patterns, we pooled them for purposes of constructing fig. 1.

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

99

Table 2 Predictions

that cooperative

Number

Condition SW COLUMN Uncertain SW Both Uncertain SS Both Uncertain All others (pooled)

Condition SW Neither Uncertain SW ROW Uncertain SS ROW Uncertain SS Neither Uncertain SS COLUMN Uncertain

of cooperative

Stringent prediction 19 14 or more 2 or more 0

equilibria

will lead to cooperative

behavior.

choices

Mean total

Proportion first trial

of

ROW

COLUMN

ROW

COLUMN

Cell size ( # of pairs)

13.7

15.1

0.80

0.80

10

12.7

14.2

0.73

0.87

15

7.5

8.0

0.53

0.88

17

9.8

10.1

0.71

0.76

66

Relaxed, rank order prediction

Mean total

Proportion first trial

ROW

COLUMN

ROW

COLUMN

Cell size ( # of pairs)

4th

14.6

15.5

0.88

0.82

17

5th

9.8

10.5

0.82

0.91

11

6th

7.6

8.3

0.38

0.77

13

7th

10.2

9.6

0.80

0.67

15

_*

3.8

3.7

0.60

*No clear prediction can be made for this condition; it should SS Both Uncertain and more than SS Neither Uncertain.

of

0.60 generate

10 less cooperation

than

much more than predicted; (2) end game play did not typically begin until trial 18; (3) the Strong-Weak conditions generated more cooperation than the Strong-Strong, except when ROW was uncertain; and (4) the observations only partially fit the predictions that cooperative equilibria will lead to more cooperation. 3.2. Overall analyses: Actual opponents The total number of cooperative choices in each game was analyzed in a 2 (ROW/COLUMN player) x 2 (SS/SW games) x 2 (ROW Certain/ Uncertain) x 2 (COLUMN Certain/Uncertain) ANOVA, with players a repeated variable. A significant effect for games, (F(lJ97) =41.25, p ~0.01) indicated that more cooperative choices were made in the SW game. Three significant two-way interactions resulted, between games and ROW Uncertain (F( 1,197) =4.95, p
L.M. Kahn and J.K. ~urnighan,

100

Prisoner’s

dilemma games

0.9

0.6

0.7

0.2

0.1

0 ~s-Jmdm~~comOrN”bm~r-mm r_C.-.--.-s-rC_C

3

Trials Note: s = strong; w = weak; r = row: c = column; u = uncertain;b = both:n = neither

Fig. t

fF(l,197) ~6.36, p
L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

101

Table 3 Mean number

of cooperative

a: For the game by ROW Uncertain

choices.

interaction: Game

ROW

Certain

Strong-Strong 7.4,

Strong- Weak 14.9,

Uncertain

7.8,

12.1,

b: For the game by COLUMN

Uncertain

interaction: Game

COLUMN

Certain

Strong-Strong 9.4

Strong- Weak 13.2,

Uncertain

6.3,

13.8,

c: For the ROW Uncertain

by COLUMN

Certain Uncertain

Certain 12.7,

Uncertain

interaction:

COLUMN Uncertain 9.1,

9.0,

10.4.h

Note: Within each table, means with common subscripts are not significantly different from one another (p ~0.05) using the NewmanKeuls procedure.

Table 4 The proportion of cooperative choices first trial by ROW and COLUMN.

ROW choices COLUMN choices

Certain 0.789,b 0.731.h

on the

ROW Uncertain 0.607, 0.857a

Note: Within each table, means with common subscripts are not significantly different from one another (~~0.05) using the Newman-Keuls procedure.

players were certain of the other’s strength than either condition of one-sided uncertainty. We also analyzed the players’ first choices, which were not affected by the other’s choices. An ANOVA comparable to that for total choices yielded two significant main effects and one interaction. The effect for players (F( 1,197) = 3.87, p < 0.05) indicated that COLUMN made more cooperative choices on the first trial than ROW; the effect for games (F(1,197)=8.43, p ~0.01) indicated more cooperation in SW. As shown in table 4, the player by ROW Uncertain interaction (F( 1,197) =4.64, ~~0.05) indicates that when ROW was uncertain, COLUMN was significantly more cooperative than ROW.

102

L.M. Kahn and J.K. Murnighan,

Prisoner’s dilemma games

After they received information about the payoffs in their upcoming negotiation, but prior to play, everyone estimated when they first expected their opponent to choose noncooperatively. A difference of means test showed that mutual certainty led to ROW players expecting COLUMN to first choose B, on average, 3.59 rounds later than when at least one side was uncertain, significant at 5% on a two-tailed test. COLUMN players similarly expected ROW’s first B play, on average, to come 2.70 rounds later when there was mutual certainty, a marginally significant difference (p ~0.10). Analysis of variance yielded one significant effect, for mutual certainty (~~0.004). Repeating the analysis with full interactions between player, uncertainty, and game, provided no additional explanatory power. However, marginally significant results (p ~0.10) in the complete model indicate that noncooperation was expected somewhat earlier in the SS game and somewhat later when both players were certain of the other’s payoffs. Linear and probit regression analyses confirmed the frequent finding that expectations of cooperation were associated with significantly more cooperative behavior.” 3.3. The dynamics of repeated play: Actual opponents The dynamics of these games emerge as players respond to each other’s choices. To assess these effects, the data were segmented for two separate analyses according to whether the opponents chose A or B on the previous trial.’ 1 The results are shown in tables 5-7. We begin with the cases where the opponent chose A on the previous trial. The first two columns of table 5 show significant main effects for GAME and PLAYER. These indicate that ROW was more likely than COLUMN to choose noncooperatively in response to a cooperative choice and that such reversals were more likely in the SS game. Table 6a provides results for the Game by Player interaction and suggests that COLUMN almost never overturned ROW’s cooperative play in the SW game. ROW and COLUMN were about equally and least likely to follow cooperation cooperatively in SS. The Uncertain-Uncertain interaction as shown in table 6b suggests that cooperation followed cooperative choices significantly more frequently when both players were certain of each other’s payoffs, and less cooperation followed when ROW was certain and COLUMN was uncertain. The threeway interaction between Games, ROW Uncertain, and COLUMN Uncertain is shown in table 6c. It suggests that the least cooperation following a cooperative choice occurred when COLUMN was uncertain of ROW’s payoffs and ROW was certain of COLUMN’s in the SS game. High levels of “‘See, for example, Wyer (1969). “The analyses in this section violate the independence assumption of the analysis of variance. Thus, they should be interpreted with extreme caution. They are presented here only to illustrate the strength of some of the trends in the data.

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

103

Table 5 Analysis

of variance

summary

table for the choice following a cooperative (B) choice by the other player. Opponent’s previous choice was A (n = 2,245)

Opponent’s previous choice was B (n = 1,859) F

F

Pi

GAME PLAYER COL UNCERTAIN ROW UNCERTAIN

23.274 84.172 3.599 2.498

0.000 O.OQO 0.058 0.114

PLAYER-GAME PLAYER-COL UNC PLAYER-ROW UNC GAME-COL UNC GAME-ROW UNC COL UNC-ROW UNC

14.543 0.153 3.048 1.598 0.519 8.27

PLAYER-GAME-COL UNC PLAYER-GAME-ROW UNC PLAYER-COL UNC-ROW UNC GAME-COL UNC-ROW UNC PLAYER-GAME-ROW

UNC-COL

UNC

(A) or noncooperative

P<

0.743 1.971 0.000 1.049

0.389 0.161 0.984 0.306

0.000 0.696 0.081 0.206 0.471 0.004

1.300 0.438 0.010 30.796 8.538 1.01

0.254 0.508 0.922 0.000 0.004 0.314

0.561 2.895 0.143 18.702

0.454 0.089 0.705 0.000

0.465 1.001 0.750 2.877

0.495 0.317 0.387 0.090

1.219

0.270

2.719

0.099

Note: GAME = SS or SW PLAYER = ROW or COLUMN COL UNC = COLUMN Certain or Uncertain ROW UNC= ROW Certain or Uncertain.

continuous cooperation occurred in both games when both players were certain of each other’s payoffs. Turning to the cases where the opponent chose B on the previous round, we note that most models suggest that noncooperation is the appropriate response to noncooperation. The data, however, suggest that some players did not respond by immediately defecting from noncooperation and that these reactions were significantly affected by the independent variables. Specifically, the third and fourth columns of table 5 show significant interactions between GAMES and ROW Uncertain and GAMES and COLUMN Uncertain. Table 7 indicates that when row was certain, players responded to noncooperative choices depending on the game: they were significantly less cooperative in SS than SW. Significantly more cooperation also occurred in SW when COLUMN was uncertain and in SS when COLUMN was certain. 3.4. The last trial: Actual opponents Behavior on the final round approximates behavior in a one-trial game. Game theory predicts that Uncertain COLUMNS in the SW game should be

104

L.M. Kuhn and J.K. Murnighan, Prisoner’s dilemma games

Table 6 The proportion of cooperative choices following cooperative choice by the other player.

a

a: The game by player interaction Game ROW choices COLUMN choices

Strong-Strong

Strong- Weak

0.737, 0.739,

0.828, 0.946,

b: The Uncertainties interaction COLUMN ROW

Certain

Uncertain

Certain

0.871,

0.820,

Uncertain

0.790b

0.812,

c: The games by uncertainties interaction COLUMN Strong-Strong:

Certain ROW

Uncertain Strong- Weak: Certain ROW

Uncertain

Certain

Uncertain

0.807,,

0.562,

0.688,

0.746&

0.908,

0.887,,

0.882,,

0.855,,

Note: Within each table, means with common subscripts are not significantly different from one another (p ~0.05) using the Newman-Keuls procedure. Table 7 The proportion of cooperative choices following a noncooperative choice by the other player. ~Game ROW

‘OLUMN

Strong-Strong

Strong- Weak

Certain

0.100,

0.180”

Uncertain

0.146,,

o.i31,,

Certain

0.157,

0.093,

Uncertain

0.098,

0.221.

Note: Within each table, means with common subscripts are not significantly different from one another (~~0.05) using the Newman-Keuls procedure.

the only players to choose cooperatively on trial 20; further, Appendix 2 shows that this choice should only be made if COLUMN perceives a probability 20.467 that ROW is weak. The results support the prediction: uncertain COLUMN players in the SW game were cooperative 28% of the time on the last trial; all other players were cooperative at a rate of 10.5%. The difference between percentages is statistically significant (F( 1,214) = 6.30, ~~0.02). When COLUMN was the only uncertain player in the SW game,

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

105

he/she was cooperative 40% of the time. While no cooperation is predicted except for uncertain COLUMNS in the SW game, cooperative play may indicate players’ taste for cooperation or altruism. 3.5. Programmed opponents Players facing programmed opponents almost uniformly responded to noncooperation by the program by choosing noncooperatively themselves, occasionally with a one trial lag. Thus, since the program determined many of the players’ choices, only the first trial choices were analyzed. The results from a 2 (SS/SW games) x 2 (ROW/COLUMN player) x4 (Probability of a Matching Strategy Opponent) ANOVA yielded a significant Player by Probability interaction, (F(3,183) =4.15, p < 0.05), but with no significant post hoc differences among the means. A similar analysis of the trial when players expected the program to first choose noncooperatively yielded an effect for Probabilities, (F(3,183) = 3.81, p ~0.02). The mean trial numbers in which players expected the program to first choose B were ranked in the same order as q, the matching probability. When q was 0.95, players on average expected defection to first occur after 13.7 rounds; for q values of 0.65, 0.35, and 0.05, these expectations were 12.7, 10.8, and 9.6 rounds, respectively.” Play in the last round of the programmed trials was extremely noncooperative in all but one condition. Except for COLUMN players in SW games with a q of 0.95, players chose noncooperatively at least 91.7% of the time on the last trial. In this condition, players chose B on the last round exactly 50% of the time. This result was significantly different from all others, which were not significantly different from one another. 3.6. Comparing actual and programmed opponents

The only difference between the programmed and actual opponent conditions when payoffs were certain was the presence of a programmed opponent whose probability of choosing the matching strategy was known. We compared trials with actual opponents, Neither Uncertain, and those with programmed opponents with the same payoff matrix. Analyses of variance compared the play against the actual opponents and each of the programmed trial q values samples individually. Additional analyses pooled the programs. Dependent variables were choices on the first and last rounds, and expectations of when the opponent would first choose noncooperatively. The only significant effects were for COLUMN’s last round choice in the actual opponent by 0.95 probability pair (F(1,51)= 7.34, ~~0.01) and the “The

figures for q = 0.95 and q = 0.05 were significantly

different

from each other.

106

L.M. Kahn and J.K. Murnighan,

Prisoner’s

dilemma games

opponent-game interaction (F( 151) = 6.77, p < 0.02). On the last trial, COLUMN was more likely to choose noncooperatively against an actual opponent than against a programmed opponent with matching probability 0.95; this difference was stronger in the SW game. Against actual opponents, ROW expected defection after an average of 11.2 (SS) or 13.88 rounds (SW); against a programmed opponent, this expectation ranged from 9.8 to 13.0 (SS) and 9.79 to 13.44 (SW), figures that increased monotonically with the matching probability. Thus against actual opponents, ROW’s expectations could be interpolated to correspond to a less than 50% matching probability for SS and more than a 95% matching probability for SW. When ROW players were certain of COLUMN’s payoffs, they treated opponents in the SW game as they treated highly probable titfor-tat players. Such a finding underscores the relevance of the idea of homemade priors on the opponent’s behavior. In SW with actual opponents, COLUMN expected defection after an average of 11.88 rounds; against a programmed opponent, expectations ranged from 9.43 to 14.25. Thus, in SW COLUMN treated actual opponents as having less than a 50% chance of playing tit-for-tat. Against actual opponents in the SS game, COLUMN expected defection after 12.87 rounds; against a programmed opponent in SS, expectations ranged from 7.86 to 14.29. This implies that COLUMN perceived over an 80% chance that ROW would play tit-for-tat in SS. 4. Discussion

and conclusions

Several of the results reported here broadly support theories of sequential equilibria. They include: 0 0 0 0

more cooperation when a cooperative equilibrium existed more total and first round cooperation in SW COLUMN was more cooperative than ROW on the first trial noncooperation was expected earlier in SS

0 following a cooperative choice: (a) less cooperation by ROW and COLUMN; (b) less cooperation in SS; and, (c) almost total cooperation by COLUMN in SW 0 expectations of first trial noncooperation conformed to probabilities of matching by programmed opponents 0 COLUMN was more cooperative against the 0.95 matching program than against actual opponents, especially in SW 0 ‘homemade’ probabilities for ROW appear to be less than 50% matching in SS and more than 95% matching in SW 0 on the last trial uncertain COLUMNS in SW were most cooperative

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

107

0 weak, uncertain COLUMNS led to more cooperation; strong, certain COLUMNS led to less cooperation. In essence, these results indicate that strength led to less cooperation than weakness did. The last two conclusions listed above indicate that uncertainty increased cooperation only in combination with weakness. While these conclusions to some degree support game theoretic predictions, many of the patterns in our data are not anticipated by such models. The fact that considerable cooperation resulted when no cooperative equilibria exist indicates that models assuming complete rationality are inadequate. Further, a hypothesis that cooperation is positively related to potential gains from cooperation received only weak support. Some of the noteworthy results that do not support theories of sequential equilibria include: 0 more cooperation resulted when the players were certain of each other’s payoffs 0 noncooperation was expected later when the players were certain of each other’s payoffs l after a cooperative choice, cooperation continued when either or both players were certain of the other’s payoffs 0 COLUMN’s ‘homemade’ probabilities appear to be more than 80% matching in SS and less than 50% matching in SW 0 a cooperative choice led to less cooperation when ROW was uncertain and COLUMN was certain 0 after a cooperative choice, cooperation dropped when COLUMN was uncertain in SS 0 players were more cooperative on the last trial if ROW was certain in SS; they were less cooperative if ROW was certain in SW 0 when ROW was uncertain, ROW cooperated less but COLUMN cooperated more. Many of these negative results consist of cooperative behavior when it is not predicted to occur. One possible explanation for this unexpected cooperation depends on psychological factors, including: (1) the personal discomfort of uncertainty; (2) the desire to avoid payoff losses by engaging in self-protection; and (3) the fact that only two choices are available in these games. Because the noncooperative equilibrium exists in all conditions of both games, players may have reacted to uncertainty by choosing noncooperatively. A systematic psychological discontinuity between the certain and uncertain conditions may have driven them to choose noncooperatively without noncooperative, anti-opponent intent. Indeed, this reflects a criticism of PD games, that the person choosing noncooperation may perceive it as defensive rather than offensive.13 ‘%ee, for example,

Apfelbaum

(1969) or Nemeth

(1972).

108

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

In economic terms, the existence of payoff uncertainty may raise the transaction costs involved in reaching a cooperative outcome. Just as uncertainty about economic events induces two contracting partners to plan for more contingencies, so can payoff uncertainty induce two bargainers to consider additional scenarios. Further, with payoff uncertainty, it may be difficult for players to identify the efficiency gains from cooperating. Thus, the net effect of payoff uncertainty may be negative - the negative effects on the ability of parties to negotiate cooperative agreements may outweigh the positive effects of the existence of additional cooperative equilibria. The effects of uncertainty on cooperation and on players’ conjectures are reminiscent of Smith’s (1989) observation that in experimental economic systems, Nash non-cooperative equilibria are more likely to be observed under conditions of incomplete information. If the Nash competitive equilibrium characterizes the actual economy, it may be that the economy presents people with more uncertainty about agents they meet than do our experiments. Our results also suggest that the formation of conjectures may involve additional factors. For example, counter to predictions from a standard game theoretic analysis, COLUMN saw ROW as using the matching strategy more in SS than in SW. One explanation is that COLUMN may have overinterpreted ROW’s cooperative choices in SS. While our appendices suggest that cooperative choices should have communicated no diagnostic information to COLUMN, the players may have responded not only to repeated cooperative choices but also to the unexpected fact that they came from a strong ROW.14 Thus, ‘homemade’ probabilities may have increased. At the same time, when COLUMN players were weak, bilateral deterrence models suggest that they might resolve the difficulties of their asymmetric power relationship by preempting ROW and choosing noncooperatively first, as they expected noncooperative choices from ROW.” The occurrence of cooperative choices when no cooperative equilibria exist has been taken as indirect evidence of homemade priors that the other party will cooperate.r6 Our direct measurement of such conjectures in the programmed and actual opponent trials are stronger indicators of such priors. Further, unlike Camerer and Weigelt, we did not need to assume that the parties used these priors in a rational manner. While the sequential equilibrium framework provides useful hypotheses about the causes of cooperation, our results indicate that the process of conjecture formation may be more complicated than a simple prior. Specifically, if homemade priors explain cooperation when such an equilibrium does not exist, then the conditions of the game may affect the formation of these “%ee Jones (1990). ISFor a discussion of such models, see Lawler, Ford 16For such an argument, see Camerer and Weigelt.

and Blegen (1988).

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

109

priors, as suggested by these results. Mutual certainty led to more cooperation, expectations of more cooperation prior to play, and more cooperation after the other’s cooperative choices. Uncertainty led to less cooperation when either ROW or COLUMN was Strong. All previous theoretical models suggest that certainty in a finite game where noncooperative choices dominate for both players should result in no cooperation. Yet SS players with neither uncertain of each other’s payoffs averaged more cooperation than any other uncertainty condition in this game. These results support the idea that conjectures about an opponent’s likely play are important, as others have also shown.” In particular, cooperation in games without cooperative equilibria may be explained by beliefs about one’s opponent’s tendency to match. However, these conjectures apparently broke down before the final round in these games: we observed much more cooperative behavior on the final round when the simple sequential equilibrium model (i.e., without tastes for cooperation or beliefs that the opponent has such tastes) predicted this behavior. This finding suggests that a study of the breakdown of such conjectures could yield results with important implications for sequential equilibrium theories.

Appendix 1. Equilibria in games against programmed opponents

We analyze behavior of subjects under four cases: (i) Strong-Strong, ROW subject, COLUMN program; (ii) Strong-Strong, ROW program, COLUMN subject; (iii) Strong-Weak, ROW subject, COLUMN program; (iv) StrongWeak, ROW program, COLUMN subject. In each case, we manipulate the reported probability that the program is instructed to match until the subject’s first defection from A; this probability takes on values of 0.05, 0.35, 0.65, and 0.95. Further, complete payoff certainty is assumed throughout. As in Kreps et al. (1982), we compute lower bounds for the number of periods of cooperation. Case 1: Strong-Strong, ROW subject, COLUMN

program

In this case, ROW perceives a probability q that COLUMN will match until ROW’s first B choice; COLUMN chooses B thereafter. Suppose that n periods remain. Then the expected payoff in the rest of the game to ROW if ROW has just chosen A is:

ROW’s expected payoff if ROW had just chosen B is: “See,

for example,

Andreoni

and Miller; Wyer; or Camerer

and Weigelt.

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

110

By matching ROW, a rational COLUMN can always get to within 19 of ROW’s total return. By examining these payoff lower bounds for ROW, we can compute bounds for COLUMN’s payoffs under various strategies. COLUMN’s return to choosing B now (with n periods to go) is: 52O+(n-1)12=8+12n. COLUMN’s

return

to choosing

A now is:

That is, COLUMN can get to within 19 of ROWS payoff for the game with n- 1 stages left. Thus the rational COLUMN will choose A now if: (A.l.l) or n > (48 - 5q)/6q.

(A.l.2)

Row should know that COLUMN will match as long as (A.1.2) holds. Thus, until at most n periods remain, ROW is effectively playing against an opponent who plays tit-for-tat (with permanent defection to B in response to ROW’s first B choice) with probability 1. Therefore, ROW will choose B no earlier than with n periods left. In this case, we have the following relationship between n and q: 4

n

0.05 0.35 0.65 0.95

159.2 22.0 11.5 7.6

Note that these may be loose defection may occur. Case 2: Strong-Strong,

bounds

ROW program,

In this case, ROW can always by simply matching COLUMN. choice now is: 524+(n-1)12=12+12n.

for the earliest

COLUMN

period

during

which

subject

get to within 15 of COLUMN’s total return With n periods left, ROW’s return to a B

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

Row’s return

to cooperating

111

now is:

~5+5+q(n-2)18+(1-q)(5+12(n-3))-15. That is, ROW can get to within 15 of COLUMN’s periods. Thus ROW should cooperate if:

payoffs for the last n- 1

6nq-5q+12n-36212+12n

(A.1.3)

n 2 (48 - 5q)/6.

(A.1.4)

or

This is the same condition

as for Case 1.

Case 3: Strong-Weak, ROW subject, COLUMN program Here, COLUMN’s return to a B choice with n periods

left is:

S 12n. COLUMN’s

return

to an A choice is:

2 6nq - 5q + 12n - 40 (see Case 1 above). COLUMN

should

cooperate

if:

6nq-5q+12n-40212n,

(A.1.5)

n 2 (40 - 5q)/6q.

(A.1.6)

or if

The relationship

between

4

n

0.05 0.35 0.65 0.95

132.5 18.2 9.4 6.2

n and q is:

Case 4: Strong- Weak, ROW program, COLUMN subject In this instance, ROW can always get to within 5 of COLUMN. return to a B choice with n periods to go is: 524+12(n-1)=12+12n. ROW’s return

to an A choice now is:

ROW’s

112

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

26nq-5q+12n-26.

ROW should cooperate if: (A.1.7) or n > (38 -

5q)/6q:

4

n

0.05

125.8 17.3 8.9 5.8

0.35 0.65 0.95

Appendix 2. Equilibria and potential gains in games against actual opponents

When both parties are rational, only three of the eight conditions in this study admit sequential equilibria with some periods of cooperation, i.e., choices of A by both sides, in: (a) Strong-Strong (SS), Both Uncertain; (b) Strong-Weak (SW), Both Uncertain; and (c) SW, COLUMN Uncertain. Whenever one side is strong and the other side knows this, the backwards induction argument causes cooperation to unravel from the last period to the beginning. This logic applies to live of the eight cases (SS with either or both sides certain - three cases - and SW with both sides or COLUMN certain two cases). We first examine the presence of cooperative equilibria. The SW, COLUMN uncertain condition is relatively straightforward. The analysis here draws from Camerer and Weigelt (1988). A cooperative equilibrium requires mutual cooperation for at least some of the game and consistent Bayesian updating by COLUMN of the probability that ROW is weak. Consider COLUMN’s choice in a supergame that has reached its 20th trial with nineteen periods of A-A choices. If ROW is weak, A is dominant for ROW; if ROW is strong, B dominates (by definition). Let P,, by COLUMN’s perceived probability on trial 20 that ROW is weak (note that P, =0.5). Then the expected payoff to COLUMN of choosing A in round 20 is: 18P,,+5(1--p,,), and COLUMN’s expected payoff for a B choice is: lOP2, + 12(1-

P20).

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

113

If COLUMN is risk-neutral, then on the last round of the game, COLUMN chooses A if and only if: 18P,, + 5( 1 - PzO) 2 lop,,, + 12(1 -Pp2,,)

(A.2.1)

P,,2 7115 =0.467.

(A.2.2)

or

Condition (2) will hold as long as COLUMN’s belief-updating process is [in the sense of Kreps and Wilson (1982)] ‘plausible’ - when ROW chooses A, COLUMN’s perceived probability that ROW is weak does not fall. The strong ROW should choose B on the last round. However ROW will want to make sure that COLUMN perceives a last round probability (P,, 2 0.467) that ROW is weak. For any round t, let S, be the probability that a strong ROW would choose to cooperate in a mixed strategy for that round. In round 19, ROW would select S, to make sure that P,,zO.467 ROW’s expected payoff in rounds 19 and 20 from such a strategy is: S,,(18+24)+(1-S,,)(24+12)=36+6S,,,

(A.2.3)

which increases monotonically in Srg. Thus, the strong ROW will choose S,, as large as possible, consistent with Pzo20.467. By Bayes’ rule: (A.2.4)

P,=(P,-J(P,-1+S,-1(l-P,-1)). Therefore, COLUMN

cooperates in round 20 if (A.2.5)

We begin with P,=0.5; if belief updating is plausible and ROW has cooperated (i.e., has chosen A) for 18 rounds, then P,,zO.5. By (3) and (5), S,, = 1 - the strong ROW pushes S,, as high as possible such that S,,~(O.533)(P,,)/((O.467)(1

-PI&> 1.

(A.2.6)

Therefore, COLUMN learns nothing by ROW’s cooperation through the first 19 rounds, and COLUMN cooperates in round 19, knowing that ROW will also cooperate even if ROW is strong. Similar reasoning takes us back to period 1. Our purpose in constructing this game was to have at least one case with a clear cooperative equilibrium for the entire game, until ROW’s last play. These payoffs should drive the strong ROW to never play a mixed strategy (i.e., S,= 1 for all t 5 19). We therefore have the strong prediction that one sequential equilibrium in this game has COLUMN choosing A in all 20

114

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

periods and ROW choosing A in periods 1-19 and B in period 20. Of course, another sequential equilibrium has both parties choosing B throughout. The two cases of two-sided uncertainty are more complicated, Sequential equilibria do exist, however, in addition to constant B choices. Intuitively, we expect earlier unraveling of cooperation in SS or SW with both uncertain than in the SW, only COLUMN uncertain case. First, for SW both uncertain, ROW’s uncertainty of COLUMN’s strength increases ROW’s incentive to protect him/herself from a potentially strong COLUMN compared to a COLUMN known to be weak. Second, in SS both uncertain, COLUMN does not ‘enjoy’ cooperation when it is met with cooperation. COLUMN now has greater short-period gains from defection compared to the SW payoffs. Further, COLUMN perceives some probability that his/her own B choices will be rewarded (if ROW is weak). Nonetheless, the payoffs and information structure of these latter two games do admit sequential equilibria with some mutual cooperation. Note that the weak COLUMN should match ROW’s A choices for the entire supergame as long as there is a probability by the 20th round of at least 0.467 that ROW is weak (and as long as ROW’s A choices do not reduce COLUMN’s estimated probability that ROW is weak). In the twosided uncertainty case, COLUMN begins with a probability of 0.5 that ROW is weak (and vice versa). Thus at the beginning of the game ROW should know that COLUMN will match ROW’s A choices for the entire game with a 0.5 probability. Suppose that N trials remain in the game and neither side has defected from A choices. Then the return to a strong ROW from matching COLUMN’s moves for the rest of the game is at least: 18Np+(l-n)[5+(n-1)12]=6Np+12N+7p-7,

(A.2.7)

where p is ROW’s assessment that COLUMN is weak. If ROW defects from A now, then ROW’s maximum payoff for the rest of the game is: 24 +(N - 1)12. The return to ROW’s tit-for-tat strategy is better if: 6Np+12N+7p-7224+(N-1)12

(A.2.8)

N>(19-7p)/6p.

(A.2.9)

or

Since no one has defected yet, p should be at least 0.5. Therefore, tit-for-tat for the rest of the game dominates defection now if at least 5.167 (or 6) periods remain. Thus, the strong ROW’s equilibrium strategy dominates defection until at most 6 trials are left; the strong ROW can play tit-for-tat for at least 14 periods. The weak COLUMN matches for the whole game. What if COLUMN is strong?

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

115

A strong COLUMN should know that ROW will play tit-for-tat for at least 14 periods (and ROW will play A for the whole game if ROW is weak). On the other hand, if a strong COLUMN knew that ROW was weak, then COLUMN should play B choices throughout the game. COLUMN learns nothing by ROW’s tit-for-tat play in the first 14 rounds; therefore, COLUMN’s assessment that ROW is weak should remain at 0.5. The payoff to COLUMN from tit-for-tat for the first N (for N 5 14) periods of the game is: 18N. The payoff from defecting at the start of these N periods is no more than: (A.2.10) Tit-for-tat for N rounds dominates finking (i.e., the B choice) at the start of the period if: (A.2.11)

18Nz8Np+12N+8-8~ or if Nz(4-4p)/(3-4p)

N >2.

or since p=1/2,

(A.2.12) (A.2.13)

Thus, a strong COLUMN should match for at least two periods. (For high values of p, COLUMN defects from the start.) This is a considerably weaker prediction of cooperation than for the weak COLUMN, since there is always a p chance that the strong COLUMN’s defection will actually be rewarded. On the other hand if ROW is strong, then COLUMN’s defection from A eliminates the possibility of mutually-beneficial joint A choices. But when both are strong, ROW matches for at least 14 periods, and COLUMN for at least two periods, implying a cooperative equilibrium for at least two periods. As noted, cooperative equilibria do not exist for five of the eight possible conditions: Strong-Strong Neither Uncertain (SSNU); Strong-Strong only ROW Uncertain (SSRU); Strong-Strong only COLUMN Uncertain (SSCU); Strong-Weak, Neither Uncertain (SWNU); and Strong-Weak only ROW Uncertain (SWRU). However, the expected gains from equilibrium strategies (i.e., noncooperation) relative to cooperating are not constant. Differences in these gains may predict behavior when parties miscalculate their strategic returns or they hold or perceive the other holding homemade probabilities for cooperation. Intuitively, more cooperative behavior should be associated with smaller losses from non-rational behavior. More systematically, we say that condition X yields more incentive for cooperation than condition Y if: (i) for each party, the expected loss in playing A (i.e., cooperatively) relative to B (i.e., not cooperatively) given the opponent’s behavior is no more under

116

L.M. Kahn and J.K. Murnighan,

Prisoner’s

dilemma games

X than Y; and (ii) there is at least one case (e.g., COLUMN’s payoffs when ROW chooses B) such that the expected loss in playing A relative to B is strictly less under X than Y. This criterion is similar to Pareto-optimality. Such a scheme generates unambiguous rankings for SWNU, SWRU, SSRU, and SSNU (see table 2). SWNU offers the most hope for cooperation since COLUMN does not gain by choosing B when ROW chooses A, and both sides know this. The SWRU condition should yield less cooperation since ROW is now not sure whether COLUMN gains by choosing B in response to A. The SSRU condition offers the next highest return for cooperation since COLUMN gains by choosing B in response to ROW’s A in SSRU but loses in SWRU (ROW’s position is the same in SSRU and SWRU due to ROW’s uncertainty in each case). The SSNU offers fewer chances for cooperation than SSRU since under SSNU, ROW knows that the payoffs are those of the classic PD. The last payoff condition, SSCU, cannot be unambiguously ranked. SSCU offers less potential for cooperation than SSBU since in the latter game ROW perceives a 50 percent chance that COLUMN loses by playing B when ROW plays A; in SSCU, ROW knows that COLUMN gains by playing B. Further, SSCU may yield a higher cooperation incentive than SSNU since COLUMN perceives a 50 percent chance that ROW always gains by playing A; in SSNU, COLUMN knows that ROW always loses by ranked against playing A. SSCU cannot, however, be unambiguously SWNU, SWRU and SSRU. References prisoner’s Andreoni, J. and J. Miller, 1990, Rational cooperation in the Finitely repeated dilemma: experimental evidence, Mimeo. (University of Wisconsin, Madison, WI) Dec. Apfelbaum, E., 1974, On conflicts and bargaining, Advances in Experimental Social Psychology, 103-156. Axelrod, R., 1984, The evolution of cooperation (Free Press, New York). Belle, F., 1990, High reward experiments without high expenditure for the experimenter?, Journal of Economic Psychology 11, 157-167. Camerer, C. and K. Weigelt, 1988, Experimental tests of a sequential equilibrium reputation model, Econometrica 56, l-36. Friedman, J.W., 1986, Game theory with applications to economics (Oxford University Press, New York). Jones, E.E., 1990, Interpersonal perception (W.H. Freeman, New York). Kreps, D.,.P. Milgrom, J. Roberts and R. Wilson, 1982, Rational cooperation in the finitely repeated prisoner’s dilemma. Journal of Economic Theory 27. 245-252. Law&, E.J.,k.S. Ford and M.A. Blegen, 1988, Coercive capabihty in conflict: A test of bilateral deterrence versus conflict spiral theory, Social Psychology Quarterly 51, 93-107. Milgrom, Paul and John Roberts, 1982, Predation, reputation and entry deterrence, Journal of Economic Theory 27,28&312. Murnighan, J.K. and A.E. Roth, 1983, Expecting continued play in prisoner’s dilemma games, Journal of Conflict Resolution 27, 279-300. Nemeth, C., 1972, A critical analysis of research utilizing the prisoner’s dilemma paradigm for the study of bargaining, in: L. Berkowitz, ed., Advances in experimental social psychology, Vol. 6 (Academic Press, New York).

L.M. Kahn and J.K. Murnighan, Prisoner’s dilemma games

117

Rapaport, A. and A. Chammah, 1965, Prisoner’s dilemma (University of Michigan Press, Ann Arbor, MI). Roth, A.E., Laboratory experimentation in economics: A methodological overview, Economic Journal 98, 9741031. Roth, A.E. and JR. Murnighan, 1983, Equilibrium behavior and repeated play of the prisoner’s dilemma, Journal of Mathematical Psychology 17, 189-198. Selten, R. and R. Stoecker, 1986, End behavior in sequences of finite prisoner’s dilemma supergames, Journal of Economic Behavior and Organization 7, 47-70. Smith, Vernon L., 1989, Theory, experiment and economics, Journal of Economic Perspectives 3, 151-169. Wyer, R.S., 1969, Prediction of behavior in two-person games, Journal of Personality and Social Psychology 13, 222-238.