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Is there as-if bargaining?
The Journal of Socio-Economics 36 (2007) 546–560
Is there as-if bargaining? Sven Fischer a,1 , Werner G¨uth a,∗ , Kerstin Pull b,2 a
Max Planck Institute for Research into Economic Systems, Kahlaische Str. 10, D-07745 Jena, Germany b Eberhard Karls Universit¨ at T¨ubingen, Abteilung Personal und Organisation, Nauklerstr. 47, D-72074 T¨ubingen, Germany
Abstract “As-if bargaining” claims that ultimatum demands and demands in the Nash demand game are analogously derived and are therefore similar to each other. By comparing experimentally the ultimatum and the Nash demand game with varying conflict payoffs we try to assess the empirical validity of this hypothesis. In addition, we test how the implicit power assessments of the participants depend on their conflict payoffs. Both, average proposer demands and marginal reactions to changing conflict payoffs differ significantly between the two games. Nevertheless a minority of subjects plays game-invariantly, as postulated by asif bargaining. Strikingly, the frequently used generalized Nash bargaining solution neither accounts for ultimatum nor for Nash demand behavior. © 2007 Elsevier Inc. All rights reserved. JEL classification: C78; C91 Keywords: Bargaining behavior; Ultimatum game; Nash demand game; Experiment; Behavioral economics
1. Introduction In the ultimatum game (e.g., G¨uth, 1976; G¨uth et al., 1982) a proposer is asked to divide a given monetary amount P(> 0) between himself and a responder who may then accept or reject the offer. Rejection leaves both with nothing. Game theoretically, the proposer should receive (almost) the whole amount. Numerous ultimatum experiments (see Camerer et al., 2004; Roth, ∗
Corresponding author. Tel.: +49 3641 686 621; fax: +49 3641 686 623. E-mail addresses: [email protected] (W. G¨uth), [email protected] (S. Fischer), [email protected] (K. Pull). 1 Tel.: +49 3641 686 627; fax: +49 3641 686 667. 2 Tel.: +49 7071 29 78186; fax: +49 7071 29 5077. 1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.013
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1995, for surveys) however, do not support the game theoretic prediction: With average offers of 30–40% of the pie and the 50:50 split as the mode, proposers are far from exploiting their strategic advantage. G¨uth et al. (1982) saw these findings as evidence for equity theory (e.g., Homans, 1961) in circumstances where its prerequisites are not granted (G¨uth and Tietz, 1986). More recently this approach has led to a purely outcome-based utility theory which incorporates fairness as an aversion against unequal outcomes (see Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999). Another explanation for the observed behavior is that the proposer in an ultimatum game behaves as-if in a procedurally fair bargaining situation (Pull, 1999, 2003; Selten, 2000). 3 Several conjectures may motivate the as-if bargaining hypothesis: experimental participants may simply be more familiar with a bargaining situation with symmetric roles than with the rather extreme and special ultimatum game. The as-if hypothesis could further be related to a norm of procedural fairness or ‘justice’ (Konow, 2003). While allocative fairness is related to distributive or outcome-based fairness, procedural fairness is distinct in the sense that the procedure is valued in itself (Frey and Stutzer, 2001). Whether a procedure is fair can have enormous effects on behavior. In variations of the mini ultimatum and battle-of-the-sexes games Bolton et al. (2005) are able to show that “(r)esults produced by an unbiased procedure tend to be more acceptable than those produced by a biased procedure” (Bolton et al., 2005, p. 21).4 In the ultimatum game the two roles have very different strategic possibilities. More specifically, one could view the responder as being less powerful since threats to refuse certain offers (above his conflict payoff) become incredible, once the proposer made her decision. Participants may want to neglect this role asymmetry and fulfil norms of procedural fairness by bargaining simultaneously with an imaginary (Selten, 2000) partner. The strict interpretation of the as-if bargaining hypothesis would predict identical demands in the ultimatum and the Nash (1950, 1953) demand game. In the ultimatum game both, proposer and responder would then have to completely ignore their strategic position and to expect the other party to do so as well. A more moderate interpretation therefore rather predicts that proposers implicitly bargain with a partner who has the same right of making an offer, but with the proposer being somewhat stronger. While this allows for different demands in the two games, in how far demands adjust to changes in the conflict points, however, should not depend on the game. Two parties X and Y, engaged in some negotiation, can rely on different assessments of their own relative bargaining power αx and αy (with αX , αY ∈ (0, 1)). Let us first neglect such discrepancies by assuming αx = 1 − αy = α, i.e., that X’s and Y’s assessments of relative bargaining power match. While the generalized Nash bargaining solution relies on power index α, reflecting the relative bargaining power of the proposer, there is no equivalent for α in the ultimatum game. However, we may infer from the offer in the ultimatum game how the proposer perceives his own bargaining position and compare this to his power assessment in Nash demand games. The generalized Nash bargaining solution for a commonly known and accepted relative bargaining power distribution α is given by maximizing: N = (x − cX )α (y − cY )(1−α)
s.t. x + y = P.
(1)
Here x and y denote X’s, or respectively Y’s demand, P(> 0) the given pie, α indicating X’s bargaining power, 1 − α is the relative power of Y, and nonnegative cX , cY represent X’s and Y’s 3
Selten (2000) uses the term “imaginary bargaining”, Pull (1999, 2003) speaks of “as-if” or “ implicit bargaining.” Bolton et al. (2005) induce fair procedures by random draws. In the ultimatum game, for example, an unfair outcome is rejected less often, if the proposal was produced by an unbiased random process rather than a person. Here unbiasedness means that the random process itself did not favor the unequal division of the pie. 4
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conflict payoffs (satisfying cX + cY < P). Maximizing N yields the solution demands: x = α(P − cX − cY ) + cX
and y = (1 − α)(P − cX − cY ) + cY .
(2)
By inverting the equations in (2), we can infer subjectively perceived and therefore possibly inconsistent5 power assessments in the Nash demand game from the proposals (x, y) via: y − cy x − cX . (3) and αY = αX = P −C P −C where C ≡ cX + cY and αi stands for i’s bargaining power as revealed by his demand (x(y) in case of X(Y)). More specifically, we assume that subjects wish to find an efficient agreement (x + y = P) and expect their opponents to have an assessment of relative bargaining power which is matching their own. In the ultimatum game, we derive proposer X’s implicit assessment of α from his offer y or equivalently from his demand x = P − y via: y − cY x − cx =1− . (4) αX = P −C P −C For responder Y, one can use the acceptance threshold to infer Y’s power assessment αY . The solution6 demand of the ultimatum game x = P − cY would yield αX = 1, whereas αX = 0.5 requires y = (P − cX + cY )/2 which implies an equal split only for cX = cY .7 By comparing the participants’ implicit assessments of α for different conflict payoffs vectors and the two games (Nash demand versus ultimatum game), especially by comparing the adjustment of α to changes in c, we test the ‘as-if bargaining’ hypothesis. The generalized Nash bargaining solution in (1) with α indicating relative bargaining power has been widely adopted, e.g., in collective wage bargaining theory (Cahuc and Zylberberg, 2004). It is claimed to “yield ‘common sense’ comparative static results” (Nickel and Andrews, 1983, p. 185) and can be derived as a limit case of Rubinstein’s (1982) sequential bargaining model.8 Note, however, that relative bargaining power α is independent of conflict points. We refer to this as the ‘conflict invariance of α’ hypothesis which can also be tested with our experimental data. For ‘as-if bargaining’ to hold, ‘conflict invariance of α’ is not necessary. The first can also hold under conflict sensitivity of α, namely when in both games (ultimatum and Nash demand game) patterns of α adjustment are identical. 2. The two games In the Nash demand game two parties, X and Y, bargain about how to divide a given positive amount of money, the ‘pie’ P. Each party states two demands nX and gX or respectively nY and gY with 0 ≤ gi ≤ ni ≤ P (for i = X, Y ) simultaneously. We refer to ni as i’s demand and to gi as i’s acceptance border. The pie P and both conflict payoffs cX and cY are common knowledge. The payoff rules are as follows: Here inconsistent means that the individual assessment of own bargaining power differs between parties, i.e., αX + αY = 1. 6 The solution can be justified by once repeated elimination of (weakly) dominated strategies or as a unique (subgame) perfect equilibrium. 7 For α = 1/2, behavior can be described as: “We both keep our own conflict payoffs and split the surplus P−C equally.” 8 Assuming that time intervals between offers and counter offers tend to zero, the solution of the Rubinstein (1982) bargaining model tends to the generalized Nash solution in (2) with α becoming a function of the parties discount rates (see Binmore et al., 1986; Osborne and Rubinstein, 1990). 5
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Fig. 1. Set of strict equilibria in payoff space.
• If nX + nY ≤ P, payoffs are nX for X and nY for Y. • If nX + nY > P but gX + gY ≤ P, party X receives gX and Y receives gY . • If gX + gY > P, conflict results and X receives cX and Y receives cY where cX , cY ≥ 0 and 0 < cX + cY < P. These rules explain our design of the Nash demand game (N-game). An efficient border agreement, gX + gY = P, with X earning gX and Y earning gY , is quite dangerous, especially since parties cannot coordinate on how to consider asymmetric conflict payoffs. To alleviate this problem, parties may strive for efficiency by their demands and play safe when specifying their border. Fig. 1 gives a graphical representation of the game theoretic problem when choosing the borders gX and gY . The conflict payoffs reduce the set of strict equilibrium payoffs to the interior line segment between the two points (cX , P − cX ) and (P − cY , cY ). The symmetric Nash solution (α = 1/2) relies on ( 21 (P + cX − cY ), 21 (P − cX + cY )) so that ni ≥ gi = 21 (P + ci − cj ) for i, j = X, Y and i = j. In our experimental implementation demand ni can be more ambitious, i.e., what one hopes to get, whereas gi is an acceptance threshold, i.e., the minimum one is willing to accept. Game theoretically, only the borders (gi ) are relevant.9 For commonly known values of P, cX and cY the procedure of the ultimatum game is as follows: • First, X suggests how to divide the pie, (dX , dY ) with dX + dY = P. • Then, not knowing (dX , dY ), Y submits an acceptance threshold bY . • If bY ≤ dY , the game ends with payoffs (dX , dY ).
9 To derive more specific predictions for demands, one would have to assume perturbances in the sense of Selten (1975), e.g., uniformly perturbed games (Harsanyi and Selten, 1988).
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• If bY > dY , then Y is informed about dY and decides between ‘accepting’ or ‘rejecting’ (dX , dY ). The payoffs are (dX , dY ) if Y accepts and (cX , cY ) if Y rejects. We employ a mixture of the positional (earlier moves are not announced) and the sequential (with announced earlier moves) order protocol to collect more informative data. However, only a low ratio of last-stage acceptances (i.e., accepting the offer dY in the last stage in spite of bY > dY ) supports the interpretation of bY as the relevant acceptance threshold. Apparently, many subjects preferred to know dY rendering bY data unreliable. 3. Experimental design Altogether four sessions were conducted in the computer laboratory of the Max Planck Institute in Jena with 24 subjects each. Student participants from different fields were invited during lectures at the Friedrich Schiller University in Jena. Ages ranged from 19 to 29 years (22.6 on average). With 56% a slight majority of females participated in our experiments. The experiment was programmed in z-tree.10 Roles were randomly assigned and remained fixed during the entire session. No subject participated in more than one session. Each subject played both the ultimatum (U) and the Nash demand (N) game with a fixed pie of P = 90 ECU (experimental currency units) and confronted in a random order11 11 different conflict vectors i , ci ) = ((i − 1)5, (11 − i)5) for i = 1, . . . , 11, all implying C = 50 ECU and thus the ci = (cX Y constant surplus P − C = 40 ECU. In two sessions subjects first played the Nash demand game, in the other two they started with the ultimatum game. For a given game type (U or N), subjects first encountered all 11 conflict vectors in a random order which was repeated once. Thus, subjects played altogether 44 rounds with feedback between rounds in each session. In each session the 24 subjects were partitioned into three matching groups of eight subjects with four X- and four Y-participants each to be randomly rematched in each round. Note that only matching group data qualify as independent. To discourage effects of repeated interaction, the instructions only mentioned random matching and furthermore stated that “Therefore, it is quite unlikely that you meet the same participant again in the next round.” (See the translated instructions in Appendix A) Subjects were randomly assigned to visually isolated booths where they received written instructions informing them on procedural details and about the first game. Instructions were identical for all participants. Before starting the first round, subjects had to answer three control questions to ensure that they really understood the rules of the game. When all subjects had completed the first part (round 1–22), the instructions for the second part were distributed and subjects were again asked to answer control questions (before the 23rd round). In each round of the Nash demand game, subjects were first informed about (own and their partners’) conflict payoffs. They were then simultaneously asked to state two demands: n and g, with n ≥ g guaranteed by the program. After each round, subjects were informed about their partner’s and their own demands and both payoffs. Similarly, each round of the ultimatum game started with the information about the conflict vector. The first mover (“Max”) then had to suggest a division (dX , dY ) with the program guar-
10 11
Zurich Toolbox for Readymade Economic Experiments (Fischbacher, 1999). The order of conflict vectors was randomly determined by a random number generator at the beginning of a session.
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Table 1 Average demands and conflict rates in the N-game (both roles combined) ci
ni (σni )
ni − ci
rc (n)
gi (σgi )
gi − ci
rc (g)
0 5 10 15 20 25 30 35 40 45 50
35.30(17.07) 39.50(15.27) 42.30(14.53) 45.97(13.53) 46.68(10.76) 49.75(9.36) 54.79(9.87) 57.73(11.50) 62.96(11.91) 66.74(12.38) 73.24(11.62)
35.30 34.50 32.30 30.97 26.68 24.75 24.79 22.73 22.96 21.74 23.24
0.770 0.771 0.729 0.698 0.719 0.667 0.719 0.698 0.729 0.770 0.770
25.10(10.93) 29.45(10.84) 33.34(11.22) 37.19(9.51) 40.18(7.70) 44.73(5.93) 47.29(8.22) 49.74(9.68) 54.81(11.59) 60.42(12.99) 66.35(13.97)
25.10 25.45 23.34 22.19 20.18 19.73 17.29 14.74 14.81 15.42 16.35
0.401 0.443 0.297 0.245 0.214 0.208 0.214 0.245 0.297 0.443 0.401
All
52.27(17.06)
27.27
0.731
44.51(16.08)
19.51
0.310
Note: Standard deviations in brackets. rc stands for the conflict rate and ci for i’s own conflict payoff.
anteeing dX + dY = P. Simultaneously, the second mover (“ Moritz”) was informed that “ Right now Max submits his proposal” and asked “In the meantime, please answer the following hypothetical question: Suppose you were in Max’s place, how would you decide?” The next screen requested an acceptance threshold bY from the second mover and a hypothetical one from the first mover.12 Only in case of bY > dY , a final decision screen appeared informing the second mover about the actual offer dY and asking him to choose between ‘accepting’ and ‘rejecting’. Finally, every subject was informed about all relevant decisions made by both subjects in that round and about their own payoff. Subjects did not receive a show up fee. All rounds were paid according to an exchange rate of 1D = 130 ECU. Each session lasted about 70 min, and subjects earned on average 13.61D (σ = 1.11) for X and 12.73D (σ = 0.73) for Y with maximum (minimum) payoffs of 15.20 (9.90) for X and 14.20 (11.20) for Y. 4. Description of results We have a total of 12 independent (group) observations and 96 data points per role and conflict payoff in each game. A first overview is provided by Tables 1 and 2 and by Figs. 2–4. Figs. 2 and 3 are boxplots of N-game demands n and g and Fig. 4 is a boxplot of proposer demands dX in the U-game. Tables 1 and 2 report major descriptive statistics of experimental behavior for all different levels of c and aggregated over all levels of c (last row of tables). Table 1 lists average demands ni and borders gi for the Nash demand game including the standard deviations in parentheses.13 Furthermore, the differences between demands and own conflict payoff (ni − ci and gi − ci ) are included as well as the resulting conflict rates in the Ngame for n-demands (rc (n)) and the final and much smaller conflict rates (rc (g)) due to g-demands. Table 2 reports for the U-game average proposer demand dX , average proposer demand net of cx (dX − cX ) and overall rejection rates rr (rejection via both threshold bY and final statement) 12 In detail, the first mover (Max) was informed that “Right now Moritz submits his choice” and asked “In the meantime, please answer the following hypothetical question: Suppose you were in Moritz’s place, how would you decide?” However, according to our analysis the answers to the hypothetical questions did not yield any valuable data. 13 Here i stands for the particular subject which can be of role X or Y.
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Fig. 2. Boxplot of demands ni by own conflict payoffs (N-game).
Fig. 3. Boxplot of border demands gi by own conflict payoffs (N-game).
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Table 2 Average proposer demands and responder rejection rate in the U-game cX
dX (σdX )
dX − cX
rr (cY )
0 5 10 15 20 25 30 35 40 45 50
31.98(12.07) 38.61(11.99) 41.64(10.80) 43.11(7.28) 47.77(8.85) 50.51(7.96) 55.84(7.45) 59.56(8.89) 61.53(11.22) 68.01(11.28) 72.26(10.51)
31.98 33.61 31.64 28.11 27.77 25.51 25.84 24.56 21.53 23.01 22.26
0.333 0.323 0.281 0.156 0.260 0.208 0.240 0.146 0.146 0.198 0.167
All
51.89(15.68)
26.89
0.223
Note: Standard deviations in brackets. rr is rate of overall rejection by responders given the proposer’s (X) conflict payoff cX .
by responders in the U-game. Please note that rejection rates rr are listed by the conflict payoffs of the proposer (cX ). For example, the first entry in the last column of Table 2 means that in 33.3% of all cases in which the proposer’s conflict payoff was 0 (i.e. the responder’s conflict payoff was cY = 50), the responder turned down the offer (which on average was equal to about 90 − 31.98 = 58.02). As the sum of the conflict payoffs cX + cY is constant, according to game theory, the difference gi − ci in the N-game should always be equal to 20, whereas in the U-game the difference dX − cX
Fig. 4. Boxplot of proposer demands dX by own conflict payoffs (U-game).
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should always be equal to 40. In Table 1, one can see that, while the differences ni − ci and gi − ci decrease with the own conflict payoff, actual demands increase. Furthermore, deviances of gi − ci from 20 are considerable and prove to be significant (p < 1%) in a controlled estimation14 of means for all ci ’s except ci = 20 and 25. The standard deviations of demands increase with the asymmetry of conflict payoffs what accounts for the considerable increase in the conflict rates rc (·) with increasing asymmetry. As subjects submit a higher (‘ambitious’) demand n in 69.4% of all cases, conflict rates are higher for n than for g demands. Although subjects could reach an agreement via two demands (ni and gi ), the overall conflict rate in the N-game of about 31% is quite high. These findings are supported by the boxplots in Figs. 2 and 3, showing wider distributions of demands for asymmetric conflict vectors and much more concentrated ones for more symmetric conflict payoffs. In the U-game, proposer demands tend to be higher than the g-demands in the N-game. Patterns in distribution are, however, similar (see Table 2 and Fig. 4). Deviances of dX − cX from 40 are considerable, and, in a controlled estimation (see footnote 14) of means prove to be significant (p < 1%) for all ci ≥ 10. Contrary to conflict rates in the N-game, rejection rates in the U-game increase with the conflict payoff of the responder and not with asymmetry per se as in the N-game. Furthermore, the overall rejection rate in the U-game is much smaller than the overall conflict rate in the N-game. The procedurally unfair ultimatum game is thus more efficient (compared to the demand game). In the following section, we present a regression analysis of gX and dX data concerning our two hypotheses. As bY data proved to be rather uninformative15 no analysis concerning this data is presented. 5. Regression analysis In order to test the two hypotheses we performed a mixed effects regression analysis. If not mentioned otherwise, significance level is set to 1% throughout the following analysis. The first column of (Table 3) reports results of a mixed effects estimation16 of αX (as inferred by demands gX and dX ) by own conflict payoff cx and several shift and slope dummies. The estimated model is specified as αit = β0 + δU DU + δH DH + βc cit + βU DU cit + βH DH cit + ζ0i + ζ1i cit + ζUi DU + uit . (5) Here αit is subject i’s assessment of his own bargaining power αX (note that only X data is included in this regression) at period t of the session, DH a dummy with 1 if the particular game (U- or N-game) is being repeated (rounds 12–22 and 34–44), and DU a dummy with 1 for the U-game. Finally, vit = ζ0i + ζ1i cit + ζUi DU + uit is a composite error term consisting of three different subject specific random effects (ζ(·) ) and a standard white noise term uit .
14 For a controlled comparison of means, we ran mixed effects regressions estimating average demands per conflict payoff whilst controlling for matching group and subject effects via (nested) random effects. 15 In altogether 821 cases (77.7%) the stated b was larger than the offer. In 585 of those 821 cases (71.3%), this rejection Y was revised in the subsequent final stage, i.e., the offer was nevertheless accepted. Furthermore, in 165 of all 1056 cases subjects stated a bY of exactly 90 or 89. Thus, subjects had a clear preference to know how much they were actually offered and bY data cannot conclusively be interpreted as acceptance thresholds. 16 For a detailed description of mixed effects estimations, see, e.g., Pinheiro and Bates (2000).
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The model explaining the data best17 proved to be the one in (5). Note first that the order of play (whether the U- or N-game was played first) was insignificant and is therefore not included.18 Concerning the main effect for the N-game (βc ) and U-game (βc + βU ), a coefficient of 0 corresponds to a marginal increase in demands by 1 with respect to changes in cit , whereas a coefficient equal to −0.025 (= −1/40) implies that demands do not react to changes in cit at all, i.e. demands are identical for all levels of cX . Observe that αX decreases significantly with increasing cX in both games (βc < 0 and βc + βU < 0). This is in line with the observed patterns of demands seen in Tables 1 and 2. As the effects for both, DU and DU cit , are significant and positive, there is a systematic difference between demands in the N- and the U-game, with proposers in the U-game systematically demanding more and reactions of demands to cit being stronger in the U-game.19 Hence on average, proposers in the U-game do not simply ignore their strategic advantage, although they do not exploit it as radically as predicted. As both coefficients for DH and DH cit are significant, but combinations of both with dummy DU are insignificant, experience affects behavior similarly in both games. Average demands become smaller with experience20 and as the DH cit effect is significant and positive (given βc < 0), subjects’ α’s become less responsive to changes in c (indicating that demands become more reactive to c). This already allows us to summarize our main results concerning our two hypotheses. First, as α varies systematically between games, the as-if hypothesis must be rejected. Second, our observation that αi varies within the N-game (depending on the value of ci ) also challenges the ‘conflict invariance of α’ hypothesis. The results of separate, similar mixed effects regressions on demands gi (this time including data of both roles) and dX themselves, listed in the second and last column of Table 3, mirror our previous results. Column 3 in Table 3 shows the results of a corresponding regression of ni demands in the N-game, which by and large have the same properties as demands gi . For both, gi and ni data it is important to add that the role had no effect on demands. To obtain a more detailed picture, we ran simple maximum likelihood estimations including only the fixed effects of model (5) individually for each of the 48 proposers, this time relying on a significance level of 5%. Concerning differences in behavior between the two games, we observe that for altogether 10 subjects (20.8%), we are not able to reject the hypothesis of both game effects (δU and βU ) being jointly insignificant. Put differently, the as-if bargaining hypothesis holds for altogether 20.8% of players whose behavior in the two games does not differ systematically. Furthermore, “conflict invariance of α” within both games (coefficient βc = 0 in N-game and βc + βU = 0 in U-game) is for altogether 15 subjects (31.25%) at least partly supported.
17
For model selection the Schwarz and Akaike information criteria were used. Also other interaction effects proved to be insignificant (at the level of 1%). Other time-series effects than the one included did not explain our data better, or were insignificant. Tested time-series effects were time trend in session period or game period, ARMA processes and effects of lagged variables, e.g., a lagged dummy indicating whether a subject experienced conflict in the previous round. Also variable ait = |cit /5 − 5| indicating the asymmetry of the conflict vector, was tested and proved to be insignificant. 19 Note that β + β is still significantly smaller than 0 (p < 0.001). C U 20 As the lines of the fitted functions intersect, this is not obvious. However, rearranging the fitted functions to functions of demands rather than α, it is easy to prove that the integral of the fitted function for the second half of each game is smaller than for the first one. This is also true for the fitted functions in the second and last column of Table 3. 18
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Table 3 Fixed effects of mixed effects estimations Coeffficient (var.)
β0 δU (DU ) δH (DH ) βc (cit ) βU (DU cit ) βH (DH cit ) Pseudo R2
Regressor αX
gi
ni
dX
0.7087∗∗ (28.86) 0.1522∗∗ (7.61) −0.1029∗∗ (−6.54) −0.0086∗∗ (−9.63) 0.0015∗ (2.75) 0.0032∗∗ (5.95) n.a.
27.081∗∗ (32.55) – −3.4310∗∗ (−5.12) 0.7017∗∗ (23.59) – 0.1248∗∗ (5.50) 0.1359
36.306∗∗ (30.96) – −4.3408∗∗ (−5.62) 0.6495∗∗ (20.03) – 0.1348∗∗ (5.16) 0.1058
34.082∗∗ (36.97) – −3.8112∗∗ (−4.88) 0.7406∗∗ (19.70) – 0.0910∗ (3.44) 0.1547
Note: t-Values in parentheses. ∗ p-value < 1%. ∗∗ p-value < 0.1%. All regressions are mixed effects ML estimations. Table 4 Nested mixed effects regression of expected dividends (both roles) Coefficient
β0
δY (DY )
δU (DU )
δH (DH )
δYU (DY DU )
Estimate t-Value
10.82∗∗ 27.15
1.046 2.01
1.371∗ 2.80
0.991∗∗ 4.92
−3.968∗∗ −5.74
Note: ∗ p-values < 1%, ∗∗ p-value < 0.1%. F -value = 268. One-sided test: reject H0 : δU + δYU = 0 with p < 0.01%.
6. Payoffs According to the data of both games (i) demanded dividends (surplus above own ci ) decrease with increasing own conflict payoffs and (ii) conflict rates in the N-game increase with asymmetry, whereas in the U-game rejection rates increase with proposer’s conflict payoffs. How do these observed effects influence resulting payoffs? Tables 4 and 5 list mixed effects regressions of expected dividends which were calculated by assuming that in each round a subject is equally likely to meet any of his four opponents within his matching group. The mixed effects regression in Table 4 includes data of both roles and nested random effects in all 11 levels of ci . Dummy DY indicates the role of the responder. The estimation therefore resembles a controlled comparison of average expected dividends per role and game with expected proposer dividends in the N-game as the basis for comparisons. Note that, while δY is insignificant, the sum δU + δYU is significantly smaller than zero. The result shows that responders earn significantly less and proposers earn significantly more in the U-game than in the N-game, i.e., the higher rates of conflict in the N-game do not render responder payoffs in the Ngame equal (or smaller than) those in the U-game. Furthermore, differences of expected dividends between roles are insignificant in the N-game, but in the U-game responders earn significantly less than proposers. The two rows in Table 5 report regressions including slope effects in ci separately for proposers (X) and responders (Y). For proposers the shift effect between U- and N-game (δU ) is insignificant; however, the slope in ci is significantly greater in the U-game (βcU ). The slope in the U-game remains significantly negative.21 For responders the picture is different. Here, the smaller average payoffs in the U-game result from a shift of the expected dividends curve (δU ). Therefore responder behavior in the U-game renders their expected payoffs to have the same reactions towards changes in cY in both games. 21
Reject one-sided H0 : βc + βcU = 0 at p < 0.1%.
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Table 5 Mixed effects regression of expected dividends β0
δU (DU )
δH (DH )
βc (ci )
βcU (DU ci )
βcH (DH ci )
Estimate t
15.81∗∗ 24.83
−0.345 −0.42
−1.755∗ −3.18
−0.194∗∗ −10.22
0.067∗ 2.70
0.102∗∗ 5.45
Estimate t
17.38∗∗ 25.81
−3.742∗∗ −5.44
−0.931 −1.66
−0.227∗∗ −12.54
0.047 2.26
0.085∗∗ 4.80
X
Y
Note: ∗ p-values < 1%, ∗∗ p-value < 0.1%. F -valueX = 309, F -valueY = 274.
7. Conclusions Introducing asymmetry via asymmetric conflict payoffs in both, the Nash demand and the ultimatum game, allowed us to reject the ‘as-if bargaining’ as well as the ‘conflict invariance of α’ hypothesis. The significant differences in behavior between the U- and the N-game are not restricted to average demands but also to how one reacts to changes in the conflict payoffs (namely more strongly in the U- than in the N-game). Nevertheless, for a minority share of participants (about 20%), the “as-if bargaining” hypothesis seems to hold: their power assessments do not depend significantly on the game. There is also a small share of participants for whom we cannot reject ‘conflict invariance of α’ since their power assessments do not react significantly to changes in conflict payoffs. Overall, we observe considerable heterogeneity in behavior, especially in asymmetric situations. We find a curse of strength in the sense that with increasing higher own conflict payoff subjects not only demand but also obtain a lower share of the total dividend (of 40). The curse of strength is also reflected in the fact that rejection rates are increasing in cX in the U-game and that in the N-game conflict rates are increasing with asymmetry. The overall failure of the Nash bargaining solution to account for U-game as well as for N-game behavior questions its frequent use in different fields of economics as, for example, labor economics and family economics, especially when conflict payoffs of the parties differ considerably. Acknowledgements We gratefully acknowledge comments by Manfred Stadler and the very helpful and constructive advice by our two referees. Appendix A. Translated instructions A.1. General instructions Welcome and thank you very much for participating in this experiment. Please read the following instructions carefully and cease any communication with other participants. If you have any questions or if there is something unclear, please raise your hand. A supervisor will come to you and answer your questions. The instructions are identical for all participants. All data will be treated anonymously. How much you will earn in the following experiment depends on your and other participant’s decisions and randomized events. During the experiment all amounts will be denoted in ECU
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(experimental currency unit). Thereby 130 ECU equal exactly 1D . Your earnings will be paid to you in private in EURO at the end of the experiment. No other participant will learn from us what decisions you made and how much you earned. At the beginning of the experiment you will be randomly assigned one of the two roles (Max or Moritz). You will keep this role during the entire experiment. One half of the subjects will decide in the role of Max, the other half in the role of Moritz. A.2. Instructions22 The experiment consists of several rounds. In each round a participant in the other role will be randomly assigned to you. Therefore, it is quite unlikely that you meet the same participant again in the next round. A.2.1. Course of one round In each round, Max and Moritz can allocate a total of 90 ECU to each other. Both choose two amounts: N and G with 90 ≥ N ≥ G ≥ 0, i.e., Max determines NMax and GMax and Moritz determines NMoritz and GMoritz . • If the sum (NMax + NMoritz ) ≤ 90, then Max receives NMax and Moritz receives NMoritz . • If the sum (NMax + NMoritz ) > 90, but the sum (GMax + GMoritz ) ≤ 90, then Max receives GMax and Moritz receives GMoritz . • If both (NMax + NMoritz ) > 90 and (GMax + GMoritz ) > 90, then Max receives CMax and Moritz receives CMoritz . You will be informed about the values of CMax and CMoritz at the beginning of each round. A.3. Instructions 223 The experiment consists of several rounds. In each round a participant in the other role will be randomly assigned to you. Therefore, it is very unlikely that you meet the same participant again in the next round. A.3.1. Course of one round In each round, Max and Moritz can allocate a total of 90 ECU to each other. First, Max submits a suggestion (DMax , DMoritz ) with DMax + DMoritz = 90 and DMax , DMoritz ≥ 0. Subsequently, but still uninformed about Max’s suggestion, Moritz submits a number BMoritz : • Whenever DMoritz ≥ BMoritz , then Max receives DMax and Moritz receives DMoritz . • If DMoritz < BMoritz , then Moritz is being informed about DMoritz and must decide between ‘accepting’ and ‘rejecting’. • If Moritz accepts, then Max receives DMax and Moritz receives DMoritz . • If Moritz rejects, then Max receives CMax and Moritz receives CMoritz . You will be informed about the values of CMax and CMoritz at the beginning of each round. 22 In sessions where subjects first played the ultimatum game, this sheet was titled “Instructions 2” and not distributed before the end of the first part of the experiment. This part was printed on a separate sheet. 23 In sessions where subjects first played the Nash demand game, this sheet was titled “Instructions”. This part was printed on a separate sheet.
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Appendix B. Translated control questions B.1. Control questions before the ultimatum game24 Suppose that Max has already submitted his suggestion (DMax , DMoritz ) and Moritz his number BMoritz . Assume that DMoritz > BMoritz . Please click the correct answers: Question 1: How much do Moritz and Max get?
... Assume that DMoritz < BMoritz . Please click the correct answers: Question 2: Moritz accepts. How much do Moritz and Max subsequently receive? ... ... Assume that DMoritz < BMoritz . Please click the correct answers: Question 3: Moritz rejects. How much do Moritz and Max subsequently receive? ...
B.2. Control questions before the Nash demand game25 Suppose Max has already submitted his numbers NMax and GMax and Moritz his numbers NMoritz and GMoritz . Please click the correct answers: Question 1: Assume (NMax + NMoritz ) > 90 and (GMax + GMoritz ) > 90. What do Moritz and Max subsequently receive?
24 Text identical to the previous question is replaced by dots (. . .). The questions were presented on the PC screen. Subjects could not proceed before clicking the correct answer. 25 Text identical to the previous question is replaced by dots (. . .). The questions were presented on the PC screen. Subjects could not proceed before clicking the correct answer.
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... Question 2: Assume (NMax + NMoritz ) > 90 and (GMax + GMoritz ) ≤ 90. What do Moritz and Max subsequently receive? ... ... Question 3: Assume (NMax + NMoritz ) ≤ 90 and (GMax + GMoritz ) ≤ 90. What do Moritz and Max subsequently receive? ...
References Binmore, K., Rubinstein, A., Wolinsky, A., 1986. The nash bargaining solution in economic modelling. The Rand Journal of Economics 17 (2), 176–188. Bolton, G.E., Brandts, J., Ockenfels, Axel, 2005. Fair procedures: evidence from games involving lotteries. The Economic Journal 115 (506), 1054–1076. Bolton, G.E., Ockenfels, A., 2000. ERC: a theory of equity, reciprocity, and competition. The American Economic Review 90 (1), 166–193. Cahuc, P., Zylberberg, A., 2004. Labor Economics. MIT Press, Cambridge and London. Camerer, C.F., Weber, R.A., Knez, M., 2004. Timing and virtual observability in ultimatum bargaining and ‘weak link’ coordination games. Experimental Economics 7 (1), 25–48. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics 114 (3), 817–868. Frey, B.S., Stutzer, A., 2001. Beyond bentham—measuring procedural utility. CESifo Working Paper 492, Center Econ. Studies and Institute for Econ. Research, Munich. G¨uth, W., 1976. Towards a more general study of v. Stackelberg-situations. Journal of Institutional & Theoretical Economics 132 (4), 592–608. G¨uth, W., Schmittberger, R., Schwarze, B., 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behavior & Organization 3 (4), 367–388. G¨uth, W., Tietz, R., 1986. Auctioning ultimatum bargaining positions—how to act if rational decisions are unacceptable?. In: Scholz, R.W. (Ed.), Current Issues in West German Decision Research. Peter Lang. Homans, G.C., 1961. Social Behavior: Its Elementary Forms. Harcourt, Brace & World, New York. Konow, J., 2003. Which is the fairest one of all? A positive analysis of justice theories. Journal of Economic Literature 41 (4), 1188–1239. Nash, J., 1950. The bargaining problem. Econometrica 18 (2), 155–162. Nash, J., 1953. Two-person cooperative games. Econometrica 21 (1), 128–140. Nickel, S.J., Andrews, M.J., 1983. Unions, real wages and employment in Britain 1951–1979. Oxford Economic Papers, 35:183–206 (New Series, Supplement: The Cause of Unemployment). Osborne, M.J., Rubinstein, A., 1990. Bargaining and markets. Harcourt Brace Jovanovich. Academic Press, San Diego, London, Sydney and Toronto. Pull, K., 1999. What is the fair wage? A model of as-if-co-operation. Quint-Essenzen No. 58, Trier. Pull, K., 2003. Ultimatum games and wages: evidence of an ‘implicit bargain’?. Schmalenbach Business Review 55, 161–171. Roth, A.E., 1995. Bargaining experiments. In: Kagel, J.H., Roth, A.E. (Eds.), The Handbook of Experimental Economics. Princeton University Press, pp. 253–348. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50 (1), 97–109. ¨ Selten, R., 2000. Th¨unen-Vorlesung: Eingeschr¨ankte Rationalit¨at und Okonomische Motivation. Zeitschrift f¨ur Wirtschafts- und Sozialwissenschaften 9 (Special issue), 129–157.
Is there as-if bargaining? Sven Fischer a,1 , Werner G¨uth a,∗ , Kerstin Pull b,2 a
Max Planck Institute for Research into Economic Systems, Kahlaische Str. 10, D-07745 Jena, Germany b Eberhard Karls Universit¨ at T¨ubingen, Abteilung Personal und Organisation, Nauklerstr. 47, D-72074 T¨ubingen, Germany
Abstract “As-if bargaining” claims that ultimatum demands and demands in the Nash demand game are analogously derived and are therefore similar to each other. By comparing experimentally the ultimatum and the Nash demand game with varying conflict payoffs we try to assess the empirical validity of this hypothesis. In addition, we test how the implicit power assessments of the participants depend on their conflict payoffs. Both, average proposer demands and marginal reactions to changing conflict payoffs differ significantly between the two games. Nevertheless a minority of subjects plays game-invariantly, as postulated by asif bargaining. Strikingly, the frequently used generalized Nash bargaining solution neither accounts for ultimatum nor for Nash demand behavior. © 2007 Elsevier Inc. All rights reserved. JEL classification: C78; C91 Keywords: Bargaining behavior; Ultimatum game; Nash demand game; Experiment; Behavioral economics
1. Introduction In the ultimatum game (e.g., G¨uth, 1976; G¨uth et al., 1982) a proposer is asked to divide a given monetary amount P(> 0) between himself and a responder who may then accept or reject the offer. Rejection leaves both with nothing. Game theoretically, the proposer should receive (almost) the whole amount. Numerous ultimatum experiments (see Camerer et al., 2004; Roth, ∗
Corresponding author. Tel.: +49 3641 686 621; fax: +49 3641 686 623. E-mail addresses: [email protected] (W. G¨uth), [email protected] (S. Fischer), [email protected] (K. Pull). 1 Tel.: +49 3641 686 627; fax: +49 3641 686 667. 2 Tel.: +49 7071 29 78186; fax: +49 7071 29 5077. 1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.013
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1995, for surveys) however, do not support the game theoretic prediction: With average offers of 30–40% of the pie and the 50:50 split as the mode, proposers are far from exploiting their strategic advantage. G¨uth et al. (1982) saw these findings as evidence for equity theory (e.g., Homans, 1961) in circumstances where its prerequisites are not granted (G¨uth and Tietz, 1986). More recently this approach has led to a purely outcome-based utility theory which incorporates fairness as an aversion against unequal outcomes (see Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999). Another explanation for the observed behavior is that the proposer in an ultimatum game behaves as-if in a procedurally fair bargaining situation (Pull, 1999, 2003; Selten, 2000). 3 Several conjectures may motivate the as-if bargaining hypothesis: experimental participants may simply be more familiar with a bargaining situation with symmetric roles than with the rather extreme and special ultimatum game. The as-if hypothesis could further be related to a norm of procedural fairness or ‘justice’ (Konow, 2003). While allocative fairness is related to distributive or outcome-based fairness, procedural fairness is distinct in the sense that the procedure is valued in itself (Frey and Stutzer, 2001). Whether a procedure is fair can have enormous effects on behavior. In variations of the mini ultimatum and battle-of-the-sexes games Bolton et al. (2005) are able to show that “(r)esults produced by an unbiased procedure tend to be more acceptable than those produced by a biased procedure” (Bolton et al., 2005, p. 21).4 In the ultimatum game the two roles have very different strategic possibilities. More specifically, one could view the responder as being less powerful since threats to refuse certain offers (above his conflict payoff) become incredible, once the proposer made her decision. Participants may want to neglect this role asymmetry and fulfil norms of procedural fairness by bargaining simultaneously with an imaginary (Selten, 2000) partner. The strict interpretation of the as-if bargaining hypothesis would predict identical demands in the ultimatum and the Nash (1950, 1953) demand game. In the ultimatum game both, proposer and responder would then have to completely ignore their strategic position and to expect the other party to do so as well. A more moderate interpretation therefore rather predicts that proposers implicitly bargain with a partner who has the same right of making an offer, but with the proposer being somewhat stronger. While this allows for different demands in the two games, in how far demands adjust to changes in the conflict points, however, should not depend on the game. Two parties X and Y, engaged in some negotiation, can rely on different assessments of their own relative bargaining power αx and αy (with αX , αY ∈ (0, 1)). Let us first neglect such discrepancies by assuming αx = 1 − αy = α, i.e., that X’s and Y’s assessments of relative bargaining power match. While the generalized Nash bargaining solution relies on power index α, reflecting the relative bargaining power of the proposer, there is no equivalent for α in the ultimatum game. However, we may infer from the offer in the ultimatum game how the proposer perceives his own bargaining position and compare this to his power assessment in Nash demand games. The generalized Nash bargaining solution for a commonly known and accepted relative bargaining power distribution α is given by maximizing: N = (x − cX )α (y − cY )(1−α)
s.t. x + y = P.
(1)
Here x and y denote X’s, or respectively Y’s demand, P(> 0) the given pie, α indicating X’s bargaining power, 1 − α is the relative power of Y, and nonnegative cX , cY represent X’s and Y’s 3
Selten (2000) uses the term “imaginary bargaining”, Pull (1999, 2003) speaks of “as-if” or “ implicit bargaining.” Bolton et al. (2005) induce fair procedures by random draws. In the ultimatum game, for example, an unfair outcome is rejected less often, if the proposal was produced by an unbiased random process rather than a person. Here unbiasedness means that the random process itself did not favor the unequal division of the pie. 4
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conflict payoffs (satisfying cX + cY < P). Maximizing N yields the solution demands: x = α(P − cX − cY ) + cX
and y = (1 − α)(P − cX − cY ) + cY .
(2)
By inverting the equations in (2), we can infer subjectively perceived and therefore possibly inconsistent5 power assessments in the Nash demand game from the proposals (x, y) via: y − cy x − cX . (3) and αY = αX = P −C P −C where C ≡ cX + cY and αi stands for i’s bargaining power as revealed by his demand (x(y) in case of X(Y)). More specifically, we assume that subjects wish to find an efficient agreement (x + y = P) and expect their opponents to have an assessment of relative bargaining power which is matching their own. In the ultimatum game, we derive proposer X’s implicit assessment of α from his offer y or equivalently from his demand x = P − y via: y − cY x − cx =1− . (4) αX = P −C P −C For responder Y, one can use the acceptance threshold to infer Y’s power assessment αY . The solution6 demand of the ultimatum game x = P − cY would yield αX = 1, whereas αX = 0.5 requires y = (P − cX + cY )/2 which implies an equal split only for cX = cY .7 By comparing the participants’ implicit assessments of α for different conflict payoffs vectors and the two games (Nash demand versus ultimatum game), especially by comparing the adjustment of α to changes in c, we test the ‘as-if bargaining’ hypothesis. The generalized Nash bargaining solution in (1) with α indicating relative bargaining power has been widely adopted, e.g., in collective wage bargaining theory (Cahuc and Zylberberg, 2004). It is claimed to “yield ‘common sense’ comparative static results” (Nickel and Andrews, 1983, p. 185) and can be derived as a limit case of Rubinstein’s (1982) sequential bargaining model.8 Note, however, that relative bargaining power α is independent of conflict points. We refer to this as the ‘conflict invariance of α’ hypothesis which can also be tested with our experimental data. For ‘as-if bargaining’ to hold, ‘conflict invariance of α’ is not necessary. The first can also hold under conflict sensitivity of α, namely when in both games (ultimatum and Nash demand game) patterns of α adjustment are identical. 2. The two games In the Nash demand game two parties, X and Y, bargain about how to divide a given positive amount of money, the ‘pie’ P. Each party states two demands nX and gX or respectively nY and gY with 0 ≤ gi ≤ ni ≤ P (for i = X, Y ) simultaneously. We refer to ni as i’s demand and to gi as i’s acceptance border. The pie P and both conflict payoffs cX and cY are common knowledge. The payoff rules are as follows: Here inconsistent means that the individual assessment of own bargaining power differs between parties, i.e., αX + αY = 1. 6 The solution can be justified by once repeated elimination of (weakly) dominated strategies or as a unique (subgame) perfect equilibrium. 7 For α = 1/2, behavior can be described as: “We both keep our own conflict payoffs and split the surplus P−C equally.” 8 Assuming that time intervals between offers and counter offers tend to zero, the solution of the Rubinstein (1982) bargaining model tends to the generalized Nash solution in (2) with α becoming a function of the parties discount rates (see Binmore et al., 1986; Osborne and Rubinstein, 1990). 5
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Fig. 1. Set of strict equilibria in payoff space.
• If nX + nY ≤ P, payoffs are nX for X and nY for Y. • If nX + nY > P but gX + gY ≤ P, party X receives gX and Y receives gY . • If gX + gY > P, conflict results and X receives cX and Y receives cY where cX , cY ≥ 0 and 0 < cX + cY < P. These rules explain our design of the Nash demand game (N-game). An efficient border agreement, gX + gY = P, with X earning gX and Y earning gY , is quite dangerous, especially since parties cannot coordinate on how to consider asymmetric conflict payoffs. To alleviate this problem, parties may strive for efficiency by their demands and play safe when specifying their border. Fig. 1 gives a graphical representation of the game theoretic problem when choosing the borders gX and gY . The conflict payoffs reduce the set of strict equilibrium payoffs to the interior line segment between the two points (cX , P − cX ) and (P − cY , cY ). The symmetric Nash solution (α = 1/2) relies on ( 21 (P + cX − cY ), 21 (P − cX + cY )) so that ni ≥ gi = 21 (P + ci − cj ) for i, j = X, Y and i = j. In our experimental implementation demand ni can be more ambitious, i.e., what one hopes to get, whereas gi is an acceptance threshold, i.e., the minimum one is willing to accept. Game theoretically, only the borders (gi ) are relevant.9 For commonly known values of P, cX and cY the procedure of the ultimatum game is as follows: • First, X suggests how to divide the pie, (dX , dY ) with dX + dY = P. • Then, not knowing (dX , dY ), Y submits an acceptance threshold bY . • If bY ≤ dY , the game ends with payoffs (dX , dY ).
9 To derive more specific predictions for demands, one would have to assume perturbances in the sense of Selten (1975), e.g., uniformly perturbed games (Harsanyi and Selten, 1988).
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• If bY > dY , then Y is informed about dY and decides between ‘accepting’ or ‘rejecting’ (dX , dY ). The payoffs are (dX , dY ) if Y accepts and (cX , cY ) if Y rejects. We employ a mixture of the positional (earlier moves are not announced) and the sequential (with announced earlier moves) order protocol to collect more informative data. However, only a low ratio of last-stage acceptances (i.e., accepting the offer dY in the last stage in spite of bY > dY ) supports the interpretation of bY as the relevant acceptance threshold. Apparently, many subjects preferred to know dY rendering bY data unreliable. 3. Experimental design Altogether four sessions were conducted in the computer laboratory of the Max Planck Institute in Jena with 24 subjects each. Student participants from different fields were invited during lectures at the Friedrich Schiller University in Jena. Ages ranged from 19 to 29 years (22.6 on average). With 56% a slight majority of females participated in our experiments. The experiment was programmed in z-tree.10 Roles were randomly assigned and remained fixed during the entire session. No subject participated in more than one session. Each subject played both the ultimatum (U) and the Nash demand (N) game with a fixed pie of P = 90 ECU (experimental currency units) and confronted in a random order11 11 different conflict vectors i , ci ) = ((i − 1)5, (11 − i)5) for i = 1, . . . , 11, all implying C = 50 ECU and thus the ci = (cX Y constant surplus P − C = 40 ECU. In two sessions subjects first played the Nash demand game, in the other two they started with the ultimatum game. For a given game type (U or N), subjects first encountered all 11 conflict vectors in a random order which was repeated once. Thus, subjects played altogether 44 rounds with feedback between rounds in each session. In each session the 24 subjects were partitioned into three matching groups of eight subjects with four X- and four Y-participants each to be randomly rematched in each round. Note that only matching group data qualify as independent. To discourage effects of repeated interaction, the instructions only mentioned random matching and furthermore stated that “Therefore, it is quite unlikely that you meet the same participant again in the next round.” (See the translated instructions in Appendix A) Subjects were randomly assigned to visually isolated booths where they received written instructions informing them on procedural details and about the first game. Instructions were identical for all participants. Before starting the first round, subjects had to answer three control questions to ensure that they really understood the rules of the game. When all subjects had completed the first part (round 1–22), the instructions for the second part were distributed and subjects were again asked to answer control questions (before the 23rd round). In each round of the Nash demand game, subjects were first informed about (own and their partners’) conflict payoffs. They were then simultaneously asked to state two demands: n and g, with n ≥ g guaranteed by the program. After each round, subjects were informed about their partner’s and their own demands and both payoffs. Similarly, each round of the ultimatum game started with the information about the conflict vector. The first mover (“Max”) then had to suggest a division (dX , dY ) with the program guar-
10 11
Zurich Toolbox for Readymade Economic Experiments (Fischbacher, 1999). The order of conflict vectors was randomly determined by a random number generator at the beginning of a session.
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Table 1 Average demands and conflict rates in the N-game (both roles combined) ci
ni (σni )
ni − ci
rc (n)
gi (σgi )
gi − ci
rc (g)
0 5 10 15 20 25 30 35 40 45 50
35.30(17.07) 39.50(15.27) 42.30(14.53) 45.97(13.53) 46.68(10.76) 49.75(9.36) 54.79(9.87) 57.73(11.50) 62.96(11.91) 66.74(12.38) 73.24(11.62)
35.30 34.50 32.30 30.97 26.68 24.75 24.79 22.73 22.96 21.74 23.24
0.770 0.771 0.729 0.698 0.719 0.667 0.719 0.698 0.729 0.770 0.770
25.10(10.93) 29.45(10.84) 33.34(11.22) 37.19(9.51) 40.18(7.70) 44.73(5.93) 47.29(8.22) 49.74(9.68) 54.81(11.59) 60.42(12.99) 66.35(13.97)
25.10 25.45 23.34 22.19 20.18 19.73 17.29 14.74 14.81 15.42 16.35
0.401 0.443 0.297 0.245 0.214 0.208 0.214 0.245 0.297 0.443 0.401
All
52.27(17.06)
27.27
0.731
44.51(16.08)
19.51
0.310
Note: Standard deviations in brackets. rc stands for the conflict rate and ci for i’s own conflict payoff.
anteeing dX + dY = P. Simultaneously, the second mover (“ Moritz”) was informed that “ Right now Max submits his proposal” and asked “In the meantime, please answer the following hypothetical question: Suppose you were in Max’s place, how would you decide?” The next screen requested an acceptance threshold bY from the second mover and a hypothetical one from the first mover.12 Only in case of bY > dY , a final decision screen appeared informing the second mover about the actual offer dY and asking him to choose between ‘accepting’ and ‘rejecting’. Finally, every subject was informed about all relevant decisions made by both subjects in that round and about their own payoff. Subjects did not receive a show up fee. All rounds were paid according to an exchange rate of 1D = 130 ECU. Each session lasted about 70 min, and subjects earned on average 13.61D (σ = 1.11) for X and 12.73D (σ = 0.73) for Y with maximum (minimum) payoffs of 15.20 (9.90) for X and 14.20 (11.20) for Y. 4. Description of results We have a total of 12 independent (group) observations and 96 data points per role and conflict payoff in each game. A first overview is provided by Tables 1 and 2 and by Figs. 2–4. Figs. 2 and 3 are boxplots of N-game demands n and g and Fig. 4 is a boxplot of proposer demands dX in the U-game. Tables 1 and 2 report major descriptive statistics of experimental behavior for all different levels of c and aggregated over all levels of c (last row of tables). Table 1 lists average demands ni and borders gi for the Nash demand game including the standard deviations in parentheses.13 Furthermore, the differences between demands and own conflict payoff (ni − ci and gi − ci ) are included as well as the resulting conflict rates in the Ngame for n-demands (rc (n)) and the final and much smaller conflict rates (rc (g)) due to g-demands. Table 2 reports for the U-game average proposer demand dX , average proposer demand net of cx (dX − cX ) and overall rejection rates rr (rejection via both threshold bY and final statement) 12 In detail, the first mover (Max) was informed that “Right now Moritz submits his choice” and asked “In the meantime, please answer the following hypothetical question: Suppose you were in Moritz’s place, how would you decide?” However, according to our analysis the answers to the hypothetical questions did not yield any valuable data. 13 Here i stands for the particular subject which can be of role X or Y.
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Fig. 2. Boxplot of demands ni by own conflict payoffs (N-game).
Fig. 3. Boxplot of border demands gi by own conflict payoffs (N-game).
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Table 2 Average proposer demands and responder rejection rate in the U-game cX
dX (σdX )
dX − cX
rr (cY )
0 5 10 15 20 25 30 35 40 45 50
31.98(12.07) 38.61(11.99) 41.64(10.80) 43.11(7.28) 47.77(8.85) 50.51(7.96) 55.84(7.45) 59.56(8.89) 61.53(11.22) 68.01(11.28) 72.26(10.51)
31.98 33.61 31.64 28.11 27.77 25.51 25.84 24.56 21.53 23.01 22.26
0.333 0.323 0.281 0.156 0.260 0.208 0.240 0.146 0.146 0.198 0.167
All
51.89(15.68)
26.89
0.223
Note: Standard deviations in brackets. rr is rate of overall rejection by responders given the proposer’s (X) conflict payoff cX .
by responders in the U-game. Please note that rejection rates rr are listed by the conflict payoffs of the proposer (cX ). For example, the first entry in the last column of Table 2 means that in 33.3% of all cases in which the proposer’s conflict payoff was 0 (i.e. the responder’s conflict payoff was cY = 50), the responder turned down the offer (which on average was equal to about 90 − 31.98 = 58.02). As the sum of the conflict payoffs cX + cY is constant, according to game theory, the difference gi − ci in the N-game should always be equal to 20, whereas in the U-game the difference dX − cX
Fig. 4. Boxplot of proposer demands dX by own conflict payoffs (U-game).
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should always be equal to 40. In Table 1, one can see that, while the differences ni − ci and gi − ci decrease with the own conflict payoff, actual demands increase. Furthermore, deviances of gi − ci from 20 are considerable and prove to be significant (p < 1%) in a controlled estimation14 of means for all ci ’s except ci = 20 and 25. The standard deviations of demands increase with the asymmetry of conflict payoffs what accounts for the considerable increase in the conflict rates rc (·) with increasing asymmetry. As subjects submit a higher (‘ambitious’) demand n in 69.4% of all cases, conflict rates are higher for n than for g demands. Although subjects could reach an agreement via two demands (ni and gi ), the overall conflict rate in the N-game of about 31% is quite high. These findings are supported by the boxplots in Figs. 2 and 3, showing wider distributions of demands for asymmetric conflict vectors and much more concentrated ones for more symmetric conflict payoffs. In the U-game, proposer demands tend to be higher than the g-demands in the N-game. Patterns in distribution are, however, similar (see Table 2 and Fig. 4). Deviances of dX − cX from 40 are considerable, and, in a controlled estimation (see footnote 14) of means prove to be significant (p < 1%) for all ci ≥ 10. Contrary to conflict rates in the N-game, rejection rates in the U-game increase with the conflict payoff of the responder and not with asymmetry per se as in the N-game. Furthermore, the overall rejection rate in the U-game is much smaller than the overall conflict rate in the N-game. The procedurally unfair ultimatum game is thus more efficient (compared to the demand game). In the following section, we present a regression analysis of gX and dX data concerning our two hypotheses. As bY data proved to be rather uninformative15 no analysis concerning this data is presented. 5. Regression analysis In order to test the two hypotheses we performed a mixed effects regression analysis. If not mentioned otherwise, significance level is set to 1% throughout the following analysis. The first column of (Table 3) reports results of a mixed effects estimation16 of αX (as inferred by demands gX and dX ) by own conflict payoff cx and several shift and slope dummies. The estimated model is specified as αit = β0 + δU DU + δH DH + βc cit + βU DU cit + βH DH cit + ζ0i + ζ1i cit + ζUi DU + uit . (5) Here αit is subject i’s assessment of his own bargaining power αX (note that only X data is included in this regression) at period t of the session, DH a dummy with 1 if the particular game (U- or N-game) is being repeated (rounds 12–22 and 34–44), and DU a dummy with 1 for the U-game. Finally, vit = ζ0i + ζ1i cit + ζUi DU + uit is a composite error term consisting of three different subject specific random effects (ζ(·) ) and a standard white noise term uit .
14 For a controlled comparison of means, we ran mixed effects regressions estimating average demands per conflict payoff whilst controlling for matching group and subject effects via (nested) random effects. 15 In altogether 821 cases (77.7%) the stated b was larger than the offer. In 585 of those 821 cases (71.3%), this rejection Y was revised in the subsequent final stage, i.e., the offer was nevertheless accepted. Furthermore, in 165 of all 1056 cases subjects stated a bY of exactly 90 or 89. Thus, subjects had a clear preference to know how much they were actually offered and bY data cannot conclusively be interpreted as acceptance thresholds. 16 For a detailed description of mixed effects estimations, see, e.g., Pinheiro and Bates (2000).
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The model explaining the data best17 proved to be the one in (5). Note first that the order of play (whether the U- or N-game was played first) was insignificant and is therefore not included.18 Concerning the main effect for the N-game (βc ) and U-game (βc + βU ), a coefficient of 0 corresponds to a marginal increase in demands by 1 with respect to changes in cit , whereas a coefficient equal to −0.025 (= −1/40) implies that demands do not react to changes in cit at all, i.e. demands are identical for all levels of cX . Observe that αX decreases significantly with increasing cX in both games (βc < 0 and βc + βU < 0). This is in line with the observed patterns of demands seen in Tables 1 and 2. As the effects for both, DU and DU cit , are significant and positive, there is a systematic difference between demands in the N- and the U-game, with proposers in the U-game systematically demanding more and reactions of demands to cit being stronger in the U-game.19 Hence on average, proposers in the U-game do not simply ignore their strategic advantage, although they do not exploit it as radically as predicted. As both coefficients for DH and DH cit are significant, but combinations of both with dummy DU are insignificant, experience affects behavior similarly in both games. Average demands become smaller with experience20 and as the DH cit effect is significant and positive (given βc < 0), subjects’ α’s become less responsive to changes in c (indicating that demands become more reactive to c). This already allows us to summarize our main results concerning our two hypotheses. First, as α varies systematically between games, the as-if hypothesis must be rejected. Second, our observation that αi varies within the N-game (depending on the value of ci ) also challenges the ‘conflict invariance of α’ hypothesis. The results of separate, similar mixed effects regressions on demands gi (this time including data of both roles) and dX themselves, listed in the second and last column of Table 3, mirror our previous results. Column 3 in Table 3 shows the results of a corresponding regression of ni demands in the N-game, which by and large have the same properties as demands gi . For both, gi and ni data it is important to add that the role had no effect on demands. To obtain a more detailed picture, we ran simple maximum likelihood estimations including only the fixed effects of model (5) individually for each of the 48 proposers, this time relying on a significance level of 5%. Concerning differences in behavior between the two games, we observe that for altogether 10 subjects (20.8%), we are not able to reject the hypothesis of both game effects (δU and βU ) being jointly insignificant. Put differently, the as-if bargaining hypothesis holds for altogether 20.8% of players whose behavior in the two games does not differ systematically. Furthermore, “conflict invariance of α” within both games (coefficient βc = 0 in N-game and βc + βU = 0 in U-game) is for altogether 15 subjects (31.25%) at least partly supported.
17
For model selection the Schwarz and Akaike information criteria were used. Also other interaction effects proved to be insignificant (at the level of 1%). Other time-series effects than the one included did not explain our data better, or were insignificant. Tested time-series effects were time trend in session period or game period, ARMA processes and effects of lagged variables, e.g., a lagged dummy indicating whether a subject experienced conflict in the previous round. Also variable ait = |cit /5 − 5| indicating the asymmetry of the conflict vector, was tested and proved to be insignificant. 19 Note that β + β is still significantly smaller than 0 (p < 0.001). C U 20 As the lines of the fitted functions intersect, this is not obvious. However, rearranging the fitted functions to functions of demands rather than α, it is easy to prove that the integral of the fitted function for the second half of each game is smaller than for the first one. This is also true for the fitted functions in the second and last column of Table 3. 18
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Table 3 Fixed effects of mixed effects estimations Coeffficient (var.)
β0 δU (DU ) δH (DH ) βc (cit ) βU (DU cit ) βH (DH cit ) Pseudo R2
Regressor αX
gi
ni
dX
0.7087∗∗ (28.86) 0.1522∗∗ (7.61) −0.1029∗∗ (−6.54) −0.0086∗∗ (−9.63) 0.0015∗ (2.75) 0.0032∗∗ (5.95) n.a.
27.081∗∗ (32.55) – −3.4310∗∗ (−5.12) 0.7017∗∗ (23.59) – 0.1248∗∗ (5.50) 0.1359
36.306∗∗ (30.96) – −4.3408∗∗ (−5.62) 0.6495∗∗ (20.03) – 0.1348∗∗ (5.16) 0.1058
34.082∗∗ (36.97) – −3.8112∗∗ (−4.88) 0.7406∗∗ (19.70) – 0.0910∗ (3.44) 0.1547
Note: t-Values in parentheses. ∗ p-value < 1%. ∗∗ p-value < 0.1%. All regressions are mixed effects ML estimations. Table 4 Nested mixed effects regression of expected dividends (both roles) Coefficient
β0
δY (DY )
δU (DU )
δH (DH )
δYU (DY DU )
Estimate t-Value
10.82∗∗ 27.15
1.046 2.01
1.371∗ 2.80
0.991∗∗ 4.92
−3.968∗∗ −5.74
Note: ∗ p-values < 1%, ∗∗ p-value < 0.1%. F -value = 268. One-sided test: reject H0 : δU + δYU = 0 with p < 0.01%.
6. Payoffs According to the data of both games (i) demanded dividends (surplus above own ci ) decrease with increasing own conflict payoffs and (ii) conflict rates in the N-game increase with asymmetry, whereas in the U-game rejection rates increase with proposer’s conflict payoffs. How do these observed effects influence resulting payoffs? Tables 4 and 5 list mixed effects regressions of expected dividends which were calculated by assuming that in each round a subject is equally likely to meet any of his four opponents within his matching group. The mixed effects regression in Table 4 includes data of both roles and nested random effects in all 11 levels of ci . Dummy DY indicates the role of the responder. The estimation therefore resembles a controlled comparison of average expected dividends per role and game with expected proposer dividends in the N-game as the basis for comparisons. Note that, while δY is insignificant, the sum δU + δYU is significantly smaller than zero. The result shows that responders earn significantly less and proposers earn significantly more in the U-game than in the N-game, i.e., the higher rates of conflict in the N-game do not render responder payoffs in the Ngame equal (or smaller than) those in the U-game. Furthermore, differences of expected dividends between roles are insignificant in the N-game, but in the U-game responders earn significantly less than proposers. The two rows in Table 5 report regressions including slope effects in ci separately for proposers (X) and responders (Y). For proposers the shift effect between U- and N-game (δU ) is insignificant; however, the slope in ci is significantly greater in the U-game (βcU ). The slope in the U-game remains significantly negative.21 For responders the picture is different. Here, the smaller average payoffs in the U-game result from a shift of the expected dividends curve (δU ). Therefore responder behavior in the U-game renders their expected payoffs to have the same reactions towards changes in cY in both games. 21
Reject one-sided H0 : βc + βcU = 0 at p < 0.1%.
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Table 5 Mixed effects regression of expected dividends β0
δU (DU )
δH (DH )
βc (ci )
βcU (DU ci )
βcH (DH ci )
Estimate t
15.81∗∗ 24.83
−0.345 −0.42
−1.755∗ −3.18
−0.194∗∗ −10.22
0.067∗ 2.70
0.102∗∗ 5.45
Estimate t
17.38∗∗ 25.81
−3.742∗∗ −5.44
−0.931 −1.66
−0.227∗∗ −12.54
0.047 2.26
0.085∗∗ 4.80
X
Y
Note: ∗ p-values < 1%, ∗∗ p-value < 0.1%. F -valueX = 309, F -valueY = 274.
7. Conclusions Introducing asymmetry via asymmetric conflict payoffs in both, the Nash demand and the ultimatum game, allowed us to reject the ‘as-if bargaining’ as well as the ‘conflict invariance of α’ hypothesis. The significant differences in behavior between the U- and the N-game are not restricted to average demands but also to how one reacts to changes in the conflict payoffs (namely more strongly in the U- than in the N-game). Nevertheless, for a minority share of participants (about 20%), the “as-if bargaining” hypothesis seems to hold: their power assessments do not depend significantly on the game. There is also a small share of participants for whom we cannot reject ‘conflict invariance of α’ since their power assessments do not react significantly to changes in conflict payoffs. Overall, we observe considerable heterogeneity in behavior, especially in asymmetric situations. We find a curse of strength in the sense that with increasing higher own conflict payoff subjects not only demand but also obtain a lower share of the total dividend (of 40). The curse of strength is also reflected in the fact that rejection rates are increasing in cX in the U-game and that in the N-game conflict rates are increasing with asymmetry. The overall failure of the Nash bargaining solution to account for U-game as well as for N-game behavior questions its frequent use in different fields of economics as, for example, labor economics and family economics, especially when conflict payoffs of the parties differ considerably. Acknowledgements We gratefully acknowledge comments by Manfred Stadler and the very helpful and constructive advice by our two referees. Appendix A. Translated instructions A.1. General instructions Welcome and thank you very much for participating in this experiment. Please read the following instructions carefully and cease any communication with other participants. If you have any questions or if there is something unclear, please raise your hand. A supervisor will come to you and answer your questions. The instructions are identical for all participants. All data will be treated anonymously. How much you will earn in the following experiment depends on your and other participant’s decisions and randomized events. During the experiment all amounts will be denoted in ECU
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(experimental currency unit). Thereby 130 ECU equal exactly 1D . Your earnings will be paid to you in private in EURO at the end of the experiment. No other participant will learn from us what decisions you made and how much you earned. At the beginning of the experiment you will be randomly assigned one of the two roles (Max or Moritz). You will keep this role during the entire experiment. One half of the subjects will decide in the role of Max, the other half in the role of Moritz. A.2. Instructions22 The experiment consists of several rounds. In each round a participant in the other role will be randomly assigned to you. Therefore, it is quite unlikely that you meet the same participant again in the next round. A.2.1. Course of one round In each round, Max and Moritz can allocate a total of 90 ECU to each other. Both choose two amounts: N and G with 90 ≥ N ≥ G ≥ 0, i.e., Max determines NMax and GMax and Moritz determines NMoritz and GMoritz . • If the sum (NMax + NMoritz ) ≤ 90, then Max receives NMax and Moritz receives NMoritz . • If the sum (NMax + NMoritz ) > 90, but the sum (GMax + GMoritz ) ≤ 90, then Max receives GMax and Moritz receives GMoritz . • If both (NMax + NMoritz ) > 90 and (GMax + GMoritz ) > 90, then Max receives CMax and Moritz receives CMoritz . You will be informed about the values of CMax and CMoritz at the beginning of each round. A.3. Instructions 223 The experiment consists of several rounds. In each round a participant in the other role will be randomly assigned to you. Therefore, it is very unlikely that you meet the same participant again in the next round. A.3.1. Course of one round In each round, Max and Moritz can allocate a total of 90 ECU to each other. First, Max submits a suggestion (DMax , DMoritz ) with DMax + DMoritz = 90 and DMax , DMoritz ≥ 0. Subsequently, but still uninformed about Max’s suggestion, Moritz submits a number BMoritz : • Whenever DMoritz ≥ BMoritz , then Max receives DMax and Moritz receives DMoritz . • If DMoritz < BMoritz , then Moritz is being informed about DMoritz and must decide between ‘accepting’ and ‘rejecting’. • If Moritz accepts, then Max receives DMax and Moritz receives DMoritz . • If Moritz rejects, then Max receives CMax and Moritz receives CMoritz . You will be informed about the values of CMax and CMoritz at the beginning of each round. 22 In sessions where subjects first played the ultimatum game, this sheet was titled “Instructions 2” and not distributed before the end of the first part of the experiment. This part was printed on a separate sheet. 23 In sessions where subjects first played the Nash demand game, this sheet was titled “Instructions”. This part was printed on a separate sheet.
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Appendix B. Translated control questions B.1. Control questions before the ultimatum game24 Suppose that Max has already submitted his suggestion (DMax , DMoritz ) and Moritz his number BMoritz . Assume that DMoritz > BMoritz . Please click the correct answers: Question 1: How much do Moritz and Max get?
... Assume that DMoritz < BMoritz . Please click the correct answers: Question 2: Moritz accepts. How much do Moritz and Max subsequently receive? ... ... Assume that DMoritz < BMoritz . Please click the correct answers: Question 3: Moritz rejects. How much do Moritz and Max subsequently receive? ...
B.2. Control questions before the Nash demand game25 Suppose Max has already submitted his numbers NMax and GMax and Moritz his numbers NMoritz and GMoritz . Please click the correct answers: Question 1: Assume (NMax + NMoritz ) > 90 and (GMax + GMoritz ) > 90. What do Moritz and Max subsequently receive?
24 Text identical to the previous question is replaced by dots (. . .). The questions were presented on the PC screen. Subjects could not proceed before clicking the correct answer. 25 Text identical to the previous question is replaced by dots (. . .). The questions were presented on the PC screen. Subjects could not proceed before clicking the correct answer.
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... Question 2: Assume (NMax + NMoritz ) > 90 and (GMax + GMoritz ) ≤ 90. What do Moritz and Max subsequently receive? ... ... Question 3: Assume (NMax + NMoritz ) ≤ 90 and (GMax + GMoritz ) ≤ 90. What do Moritz and Max subsequently receive? ...
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