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Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
24, 10]24 Ž1998.
GA980638
Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining Kevin A. McCabe, Stephen J. Rassenti, and Vernon L. Smith* Uni¨ ersity of Arizona, Tucson, Arizona 85721-0108 Received February 13, 1996
We report decision making in two-person extensive form game trees, using six treatments that vary matching protocol, payoffs, and payoff information. Our objective is to examine game theoretic hypotheses of decision making based on dominance and backward induction in comparison with the culturally or biologically derived hypothesis that reciprocity supports more cooperation than predicted by game theory. We find strong support for cooperation under complete information, even in single-play treatments and in games of trust, unreinforced by the prospect of punishment for defection from reciprocity. Only under private information do we observe strong support for noncooperative game theory. Journal of Economic Literature Classification Numbers: C78, C92. Q 1998 Academic Press
1. INTRODUCTION A number of simple experimental games like the ultimatum and dictator game ŽForsythe et al., 1994; Hoffman et al., 1994; Eckel and Grossman, 1996. and investment trust game ŽBerg et al., 1995. show that many people do not follow self-interested dominant strategies, nor do they expect such behavior from their counterparts in these two-person anonymous interactions. But these are cases of boundary social interactions, which provide extreme conditions in the payoff environment under which the elementary game theoretic principles of dominance and backward induction fail to predict behavior in the one-shot anonymity condition}a condition thought to be most favorable for eliciting noncooperative play. Evolutionary psychologists have argued that the human mind is evolutionarily adapted to help make social exchange work in family and tribal groups, and they have conducted reasoning experiments showing that subjects have a pronounced tendency to punish cheaters in social exchange contexts ŽCosmides, 1985; Cosmides and Tooby, 1992.. Behavior in ultimatum games can be interpreted as an expression of this tendency ŽHoffman *E-mail: [email protected]. 10 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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et al., 1994.. But in the dictator game no such interpretation is possible, since the decision maker’s offer of a share of the pie to her counterpart cannot be rejected. Although positive offers in the dictator game can be explained by models in which the players have a taste for ‘‘fairness’’ ŽRabin, 1993., such models do not explain, at least as currently formulated, why utility varies with treatments designed to operate on people’s expectations ŽHoffman et al., 1996a, b.. An alternative hypothesis for explaining these anomalous results is the concept of reciprocal ‘‘altruism,’’ or simply reciprocity Žno utilitarian altruism is involved, because it is an exchange., developed in the evolutionary biology and psychology literatures ŽCosmides and Tooby, 1992, pp. 170]179.. In this literature, people are assumed to possess a specialized mental algorithm in which their long-term self interest is best served by an unconscious willingness to punish cheating on cooperative social exchanges Žnegative reciprocity.. We also postulate a willingness to reward the initiation of cooperative social exchanges Žpositive reciprocity.. This idea is suggested in Hoffman et al. Ž1996c., where it is proposed that subjects are culturally, if not biologically, disposed to reputation building across games; i.e., the game of life is a repeated sequence of constituent games that may differ markedly. Hence, if you want to have a reputation for reciprocity, you do not engage in noncooperativerdefectionist strategies just because the experimenter thinks he has presented you with a one-shot game. If you exploit situations in that you think you cannot be ‘‘found out,’’ a condition which is uncertain, then how can you be sure that you will not slip into exploitative tactics at other times, and compromise your reputation? It is well known that borderline sociopaths encounter costly negative reactions from others. This explains the rejection of ‘‘unfair’’ offers in the ultimatum game, the ‘‘unreasonable’’ generosity of people in anonymous investment and dictator games, and the considerable reduction in this generosity when you run them double blind ŽHoffman et al., 1996b., so that it is clear that no one can know the subject’s decision. It also has an informal basis in game theory via The Folk theorem, and goes to the core of the question of how people perceive the games they play. We economists have accepted the maintained hypothesis of the theory that people perceive one-shot games to be isolated interactions, while seeking to resolve anomalies by inserting ‘‘fairness’’ considerations into the utility function Žas in Rabin, 1993..
2. THE CONSTITUENT GAMES In this paper we explore these issues in a somewhat richer two-person extensive form bargaining game environment than in the above games, but
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MCCABE, RASSENTI, AND SMITH
FIG. 1. Extensive form game tree, Games 1 and 2. Game 1 payoffs are closest to the tree. Game 2 differs from Game 1 by interchanging the payoffs Ž50, 50. and Ž60, 30. in the left branch.
one that is still elementary from the perspective of theory. In constituent Game 1 ŽFig. 1., at the initial decision node x 1 , player 1 has an outside option yielding the asymmetric payoff Ž35, 70. respectively to Žplayer 1, player 2., which for player 1 is somewhat less attractive than the symmetric subgame perfect ŽSP. outcome Ž40, 40. available by player 2 moving right at node x 6 . If player 1 moves down at x 1 , then player 2 chooses between subsequent play in the right subgame containing the unique SP outcome, or in the left subgame containing the symmetric joint maximum ŽSJM. outcome available at node x 3 . To achieve the SJM outcome, player 2 must move left at x 2 , but player 1 can defect on this signal by moving down at x 3 forcing player 2 at x 5 to choose between left, yielding the outcome Ž60, 30., and down to incur a costly choice that directly punishes player 1. But this direct punishment strategy is subgame dominated by the move left at x 5 . If player 2 always chooses undominated strategies, and believes that player 1 will do the same, then player 2 will not move left at x 2 in a single play of the constituent game, and will expect to achieve SP by moving right at x 2 . Player 1, using the same reasoning, will move down at x 1 , since 40 at SP is better for player 1 than 35, the outside option.
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Game 2 is the same as Game 1, except that the payoffs Ž60, 30. and Ž50, 50. have been interchanged in the left branch of the decision tree, thereby removing the threat of direct punishment. In both games the use of reciprocity is restricted to the left subgame, starting at node x 3 , allowing us to study backward induction in the right subgame, starting at x 4 , where reciprocity is not an issue. Furthermore, in both games the cooperative SJM outcome Ž50, 50. and the SP outcome Ž40, 40. are symmetric, allowing control over any payoff equity motivations that might influence subject choice.
3. REPEAT PLAY OF GAME 1 AND GAME 2 In the supergame treatment below, the same pairs repeat the constituent Game 1 Žor Game 2. for 20 repetitions Žsee the procedures below.. In Game 1, player 2 can avoid playing dominated punishing strategies Žand in Game 2 can introduce indirect punishment., by following an individually rational trigger strategy: in Game 1, if on repetition t, player 1 defects, then player 2 moves left at x 5 , but right at x 2 in repetition t q 1, punishing player 1 with at least the SP outcome, then in repetition t q 2 returns to moving left at x 2 . If this strategy is invoked each time player 1 defects, player 2 avoids the more costly outcome Ž20, 20.. But if player 1 is to be made strictly worse off, player 2 must punish, or credibly threaten to punish, with a right move at x 2 more than once for each defection, since just one punishment gives player 1 Ž60 q 40.r2 s 50, the same payoff as at SJM. The same trigger strategy in Game 2 allows player 2 to punish player 1 on any t where player 1 moves left at x 3 . Hence, in repeated play, if subjects use the trigger strategy for indirect punishment, Game 1 and Game 2 are equivalent, but if they tend to use the direct punishment option, then Game 1 will yield more outcomes at SJM than Game 2.
4. TREATMENTS AND PROCEDURE Our treatments are summarized in Table I. We adopt the convention of naming a cell by the matching treatment and the game number. Thus Same 1 refers to experiments in which players keep the same counterpart and play Game 1 repeatedly. Single 1 refers to experiments in which Game 1 is played once. Single 1, with anonymously matched pairs, allows us to give the SP prediction its best shot. If some subjects contrarily play for the SJM outcome, then Single 2 allows us to test whether that outcome is supported primarily by the prospect of direct punishment.
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MCCABE, RASSENTI, AND SMITH
TABLE I Experimental Design: Treatments and Number of Pairs
Designation Single 1 Single 2 Same 1 Same 2 Private Same 1 Private Same 3
Constituent game
Matching Protocol
Number pairs Žobservations.
1 2 1 2 1 3
Single play Single play Repeat same pairs Repeat same pairs Repeat random pairs with same roles Repeat same pairs
30 Ž30. 26 Ž26. 22 Ž440. 23 Ž460. 24 Ž480. a 24Ž480. a
a Sessions consisted of 12 subjects, 6 pairs matched repeatedly. Others consisted of at least 8 subjects.
In the repeat play treatments, subjects are recruited for 2 hours, even though the experiments last only slightly over 1 hour, and players are told nothing about how many trials will be played, making credible the expectation of a long series of repeated plays. The treatments also vary what the subjects know about their counterpart’s payoffs. Under private information, subjects know only their own payoffs. This precludes a subject’s ability to make inferences about her counterpart’s intentions}to read a counterpart’s mind, to determine why a choice might have been made, and to engage in reciprocity. Thus, in Same 1, a left move at x 2 under complete payoff information can be read as a signal to player 1, saying that player 2 wants to reach Ž50, 50. which dominates Ž40, 40., and that if player 1 defects, 2 has the option of punishing him. With private information, such ‘‘mind reading’’ is not possible. ŽSee Baron-Cohen, 1995, for a mental mechanism theory and experiments on the human ability to infer mental states in others from their actions, and the fact that autism, a predisposition to which appears to be inherited, prevents or seriously compromises mind reading.. The failure or inability to read intentions and reciprocate may explain why noncooperative game theory has been fruitfully applied to biological evolution, in which genes are modeled as selfish Žmotivated to increase their reproductive fitness. in an environment of private information, and why in a wide variety of previous bargaining and market experiments, it has been found that private information favors Nashrcompetitive outcomes ŽFouraker and Siegel, 1963; Smith, 1982.. In all experiments, subjects were paid a nonsalient fixed fee of $5.00 for appearing on time for the experiment and, at the end of the experiment, their accumulated earnings. The payoffs are in cents when the games are repeated for 20 trials; in the single trial experiments the payoffs are multiplied by 20.
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5. ALTERNATIVE HYPOTHESES We propose reciprocity as a test alternative to the predictions of noncooperative game theory in the single play experiments. Specifically, we will test the following hypotheses. H1. Based on the principles of dominance and backward induction, right game play will predominate in Single 1 and Single 2 against the null alternative of equal play in the right and left game trees. H2. The Ž50, 50. SJM outcomes in Single 1 will exceed those in Single 2. This is because we hypothesize, from game theory, that reciprocity will be more difficult in Single 2, without the punishment option, than in Single 1. H3. Based on the Folk Theorem, the repeated games Same 1 and Same 2 will result in more cooperation than Single 1 and Single 2, with left play more common in Same 1 than in Same 2. H4. Private 1 will yield a higher proportion of right game play, and of the SP outcome, than any other treatment condition.
6. RESULTS: TESTS OF HYPOTHESES Table II summarizes the outcome frequencies across all treatments. The first data column lists the left game play results, and the sixth lists the right game results. The remaining columns show the outcomes conditional upon being in the left branch or the right branch of the decision tree. The unconditional relative outcome frequencies are shown in parentheses. H1. In the first two rows we see that right game play does not predominate in either Single 1 or Single 2. Using a binomial test, we cannot reject the null hypothesis that left and right game play are equally likely under either treatment Ž p s 0.29; 0.23.. Contingent on right game play, however, we observe strong support for SP in both Single 1 and Single 2, easily rejecting the hypothesis that SP is only 50% likely Ž p 0.001.. H2. The SJM outcome is significantly more frequent in Single 1 than in Single 2 Ž p - 0.001.. H3. The outcomes in Same 1 and Same 2 show significantly more left Žand Same 2 more cooperative. play than in the corresponding Single 1 and Single 2 experiments. Similarly, Same 1 shows more left plays and more cooperative outcomes than Same 2 Ž p - 0.001..
12r26 s 0.46 353r433 s 0.82 261r423 s 0.62 68r428 s 0.16
Single 2 Same 1 Same 2 Private 1
11r13 s 0.85 Ž0.37 . 6r12 s 0.50 Ž0.23 . 312r353 s 0.88 Ž0.72 . 220r261 s 0.84 Ž0.52 . 29r68 s 0.43 Ž0.07 .
50 50 1r13 s 0.08 Ž0.03 . 6r12 s 0.50 Ž0.23 . 14r353 s 0.04 Ž0.03 . 41r261 s 0.16 Ž0.10 . 38r68 s 0.56 Ž0.09 .
60 30
0 1r68 s 0.015 Ž0.
0 22r353 s 0.06 Ž0.05 .
1r13 s 0.08 Ž0.03 .
20 20
0
0
0 5r353 s 0.014 Ž0.01 .
0
0 0
360r428 s 0.84
162r423 s 0.38
80r433 s 0.18
14r26 s 0.54
17r30 s 0.57
Right branch
0 3r80 s 0.038 Ž0.01 . 27r162 s 0.17 Ž0.06 . 51r360 s 0.14 Ž0.12 .
0
30 60 16r17 s 0.94 Ž0.53. 14r14 s1 Ž0.54 . 68r80 s 0.85 Ž0.16. 114r162 s 0.70 Ž0.27. 304r360 s 0.84 Ž0.71.
40 40
0
0 0
0 0 4r80 5r80 s 0.05 s 0.06 Ž0.01 . Ž0.01. 21r162 s 0.13 Ž0.05. 0 5r360 s 0.014 Ž0.01. 0
1r17 s 0.06 Ž0.03 .
15 30
NA
46.9
46.6
40.0
46.2
E Žp 2 < Left .
NA
NA
31.2
}
40.0
E Žp 1 < Defect .
Conditional frequencies are computed contingent on left or right branch play. Frequencies in parentheses are computed unconditionally across all payoff outcomes. NA, Data not applicable.
a
13r30 s 0.43
Left branch
Single 1
Treatment
TABLE II Branch Conditional ŽUnconditional. a Outcome Frequencies by Treatment, All Trials
16 MCCABE, RASSENTI, AND SMITH
RECIPROCITY, TRUST, AND PRIVACY
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H4. Comparing Private 1 and Same 1, noncooperative right game play is very much larger in the former Ž p - 0.001.. Across all treatments, Private 1 yields by far the highest incidence of noncooperative right game play. In Table II we also report the expected profit to player 2 from a left move at x 2 , EŽp 2 < Left., based on the outcome frequencies conditional upon left game play. Similarly, we report the expected profit to player 1 from defection Žmoving down. at x 3 in game 1, EŽp 1 < Defect.. This leads to two important results: Ž1. on average across subject pairs, it pays for subjects in role 2 to move left at x 2 . The gain is strictly better than the SP payoff Ž40. in all treatments except Single 2, where the expected payoff is exactly 40. Ž2. Across both treatments for Game 1, it does not pay for subjects in role 1 to defect on the SJM outcome Ž50.. Hence subjects are not behaving in an irrational manner, in the sense that, given the reciprocity types in the sample, and the subjects’ ability to read each other, on average they are collecting more money than if they behaved in a myopic self-interested way, ignoring the reciprocity characteristics of those with whom they are matched. We note that in Single 1 and Single 2, a little more than half play the right game, and earn Žweakly. less than those who play the left game, but in Same 1 and Same 2, where there is an opportunity for subjects to learn more about their counterparts, many more play the left game, defections decline, and EŽp 2 < Left. increases.
7. EVOLUTION ACROSS REPETITION BLOCKS Tables III]V report conditional outcome frequencies by blocks of five trials over the 20 trials of Same 1, Same 2, and Private 1. In Same 1 ŽTable III., left branch play starts higher than in Single 1, and tends to increase across trial blocks. Knowing that they are matched with the same counterpart for what is expected to be a long game, subjects appear to harbor higher expectations of achieving the SJM outcome, and they are right. Although defections start out at around 18%, they decline to about 7%. Punishment rates begin at 10% of subjects playing the left branch and decline to 3]4%. In Same 2 ŽTable IV., left branch play starts no higher than Single 2, but increases more than in Same 1, as subjects learn to overcome the inherent instability of playing the left branch of Game 2. Defection from SJM by a left move at x 3 starts high Ž36%., but declines steadily to 9% by the last trial block. Right branch play in Same 2 is not only more common than in Same 1, but involves more punishing outcomes. At least some of the moves down at x 6 appear to be trigger strategies designed to induce
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MCCABE, RASSENTI, AND SMITH
TABLE III Branch Conditional a Outcome Frequencies by Repetition Block, Same 1
Repetitions 1]5 6 ] 10 11 ] 15 16 ] 20 a
Left branch
50 50
60 30
20 20
78r109 s 0.716 88r108 s 0.815 94r106 s 0.887 93r110 s 0.845
64r78 s 0.821 76r88 s 0.864 87r94 s 0.926 85r93 s 0.914
6r78 s 0.077 1r88 s 0.011 4r94 s 0.043 3r93 s 0.032
8r78 s 0.103 7r88 s 0.080 3r94 s 0.032 4r93 s 0.043
Right branch
30 60
31r109 2r31 s 0.284 s 0.065 20r108 s 0.185 0 12r106 s 0.113 0 17r110 1r17 s 0.155 s 0.059
40 40
15 30
E Žp 2 < E Žp 1 < Left . Down .
28r31 s 0.966 0 19r20 s 0.95 0 12r12 s1 0 9r17 4r17 s 0.529 s 0.235
45.4
37.2
45.1
16.6
48.2
42.8
47.5
32.5
Conditional frequencies are computed contingent on left or right branch play.
players 1 to move right at x 1 on subsequent plays. Thus from Table I we see that there are 460 observations or 115 per five-trial block in Same 2, but total play in the left or right branch is 100, 110, 108, and 105 by trial block in Table IV. The differences are right moves at x 1 Ž15, 5, 7, 10, respectively.. In Private 1 ŽTable V., left branch play declines steadily as subjects discover its instability without being able to see how their counterpart’s payoffs explain this instability. Although right branch play starts high and goes higher, initially nearly a third of the outcomes are at Ž30, 60.. Apparently some subjects 1, having observed their counterpart move right at x 2 , reason that if they move down at x 4 , the uncertain possible outcomes for player 1 are 40 and 15. The sure thing is 30 by moving right
TABLE IV Branch Conditional a Outcome Frequencies by Repetition Block, Same 2
Repetitions 1]5 6]10 11]15 16]20
Repetitions 1]5 6]10 11]15 16]20 a
Left branch
60 30
50 50
45r100 s 0.450 65r110 s 0.591 73r108 s 0.676 78r105 s 0.743
16r45 s 0.356 10r65 s 0.154 8r73 s 0.110 7r78 s 0.090
29r45 s 0.644 55r65 s 0.846 65r73 s 0.890 71r78 s 0.910
Right branch
30 60
40 40
15 30
EŽp 2 N Left.
55r100 s 0.550 45r110 s 0.409 35r108 s 0.324 27r105 s 0.257
7r55 s 0.127 11r45 s 0.244 6r35 s 0.171 3r27 s 0.111
42r55 s 0.764 26r45 s 0.578 23r35 s 0.657 23r27 s 0.852
6r55 s 0.109 8r45 s 0.178 6r35 s 0.171 1r27 s 0.037
42.9 46.9 47.8 48.2
See Table III for footnote.
a
50 50 13r26 s 0.5 6r17 s 0.353 6r13 s 0.462 4r12 s 0.333
Left branch 26r109 s 0.239 17r107 s 0.159 13r103 s 0.126 12r109 s 0.110
See Table III for footnote.
1]5 6]10 11]15 16]20
Repetitions 13r26 s 0.5 11r17 s 0.647 7r13 s 0.538 7r12 s 0.583
60 30 0 0 0 1r12 s 0.083
20 20 83r109 s 0.761 90r109 s 0.826 90r103 s 0.874 97r109 s 0.890
Right branch
26r83 s 0.313 13r90 s 0.144 5r90 s 0.056 7r97 s 0.072
30 60
40 40 57r83 s 0.687 75r90 s 0.833 83r90 s 0.922 88r97 s 0.907
TABLE V Branch Conditional a Outcome Frequencies by Repetition Block, Private Same 1
0 2r90 s 0.022 2r90 s 0.022 1r97 s 0.010
15 30
40 37.1 39.2 36.7
E Žp 2 < Left .
RECIPROCITY, TRUST, AND PRIVACY
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MCCABE, RASSENTI, AND SMITH
at x 4 , and a great many do. The more hesitant and perhaps risk-averse subjects are reluctant to explore below x 4 , but as repetitions continue, more and more players 1 experiment with a down move at x 4 , almost always finding that it leads to 40. The bottom line is that the great majority of subjects grope for stable self-interested outcomes in this private information environment, and find that SP is a strong point of attraction for this behavior. Since complete payoff information strains credulity outside the laboratory, we think that this is a fruitful way to model SP equilibrium outcomes ŽKalai and Lehrer, 1993..
8. DOES PRIVATE INFORMATION SUPPORT FOR SP WEAKEN IF IT IS MADE STRATEGICALLY MORE COSTLY? In Private 1 we found substantial support for SP. Our explanation of why support required time to develop over successive trials focused on strategic cost: risk averse, timid subjects 1 are reluctant to explore far enough into the tree. If this interpretation is correct, it should be testable, for example, by placing SP at the bottom of the game tree, with low payoff opportunities along the way for both players. This reasoning led us to the new game tree, Game 3, shown in Fig. 2. The SP payoffs are sweetened to Ž95, 95. with player 1 playing down at x 7 . But we place the outcomes Ž75, 15. and Ž15, 75., respectively, on the left at x 3 and x 5 . This, we hypothesize, will discourage timid players from playing the left game by providing transparent indications of the risk in achieving SP. Players 2 should be reluctant to move left at x 2 , giving up a sure 55 on the right, for an uncertain mixture of 15, 75, 15, and 95; players 1 should see this and be more likely to move down at x 1 than in Private 1. But many players 1, if they reach x 3 , are likely to be tempted to go left, settling for the sure thing, 75, in place of the uncertain 95 at the bottom. This, in turn, will discourage players 2 from moving left at x 2 . In particular, at x 2 , if player 2 considers the payoff boxes in the left game to be equally likely, then his expected payoff is Ž15 q 75 q 15 q 95.r4 s 50, compared with a sure thing of 55 on the right. So left at x 2 is not for the timid or risk-averse, although it is certainly appropriate in a long game for those with an exploratory bent. Similarly, at x 3 , if player 1 considers the payoffs in the subgame below x 3 to be equally likely, then her expected payoff is Ž15 q 95.r2 s 55 compared with the sure thing, 75, by moving left at x 3 . If this prospect is part of player 2’s reasoning, then the prospect of him moving left at x 2 is further diminished.
RECIPROCITY, TRUST, AND PRIVACY
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FIG. 2. Extensive form game tree, Game 3.
As in Private 1, Private 3 uses subjects who, after random assignment to a pair and to the role of player 1 or 2, maintain the same counterpart and role across all trials. Four sessions with six pairs in each session entered decisions for 20 trials, yielding a total of 480 observations. Comparing Table VI with Table V, we see that support for SP builds steadily across repetitions in Game 3 as in Game 1, but at lower levels. Almost all of this increasing support is coming from increased downward exploration in the left game; right game play changes very little across the four trial blocks. In the final trial block, 16]20, we have right game, 44 observations Ž36.7%., SP, 64 observations Ž84.2%., with the payoff Ž75, 15. at x 3 accounting for the remaining 12 observations Ž15.8%.. We conclude that support for SP under private information is robust in moving from Game 1 to Game 3, but importantly compromised by elementary considerations of strategic risk in achieving SP. When we lengthen the chain of moves to achieve SP, displaying obvious possible traps along the way, this reduces support for SP. In fact, it should not be too difficult to invent other environments that would make SP even more difficult to achieve but not less rational, given the strategic cost of attaining it. Yet it is clear from a wide variety of experiments over the decades that noncoop-
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TABLE VI Branch Outcome Frequencies by Repetition Block, Private Same 3
Repetitions 1]5 6]10 11]15 16]20 1]20
Left branch
75 15
15 75
95 95
Right branch
70r119 s 0.588 77r120 s 0.642 77r120 s 0.642 76r120 s 0.633 300r479 s 0.626
39r70 s 0.557 20r77 s 0.260 19r77 s 0.247 12r76 s 0.158 90r300 s 0.300
11r70 s 0.157 6r77 s 0.078 2r77 s 0.026 0
20r70 s 0.286 51r77 s 0.662 56r77 s 0.727 64r76 s 0.842 191r300 s 0.637
49r119 s 0.412 43r120 s 0.358 43r120 s 0.358 44r120 s 0.367 179r479 s 0.374
19r300 s 0.063
erative equilibria fare better under private, than under complete, payoff information.
9. DISCUSSION Contrary to noncooperative game theory, but consistent with the reciprocity hypothesis, many subjects achieve the symmetric joint maximum under the single play anonymous interaction conditions that are expected to give the subgame perfect equilibrium its best shot. Reciprocity Žas a hypothesis., on average, enables many subjects to achieve individually rational states whose rewards exceed that attainable if each person in a trial behaves on the assumption that his counterpart will follow her own myopic self-interest. Furthermore, these achievements are to a considerable extent driven by trust, and do not depend critically on the prospect of punishment. Although the resulting rewards are larger, subjects exhibit differing behaviors Žas is common in biological models and phenomena generally.. In single play treatments, about half go for the SP, but in repeat interaction, consistent with the game theoretic Folk Theorem, support for the symmetric joint maximum is increased substantially. Also consistent with game theory is the very strong support for SP contingent on right branch play in both Games 1 and 2. Reciprocity and its origins in trust andror punishment is in need of being modeled to account for a variety of behavioral reputational types: those who offer cooperation on the basis of pure trust, those who require the prospect of punishment, those who defect when cooperation is offered, and those who, faced with defection, tend to respond with punishment.
RECIPROCITY, TRUST, AND PRIVACY
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These types are not necessarily independent. Thus it is reasonable to postulate that a person who as Player 2 moves left at x 2 to signal cooperation is also likely to reciprocate as a Player 1 by moving left at x 3 . Similarly, noncooperative Player 2s are expected to be more likely to defect as Player 1s. Any such modeling of player types is complicated by the strong likelihood that behavior will not be independent of opportunity costs and, in view of the strong support here and elsewhere for the Folk Theorem, will certainly not be independent of counterpart signals in a supergame. Simpler recent models, however, have dealt with some of these features, and the results are promising; e.g., punishment that is directed solely at noncooperation can lead to the evolution of cooperation, Žsee Boyd and Richerson, 1992, and the references therein .. Similarly, reciprocity has been modeled by Kranton Ž1996. as a self-interested gift exchange, sustained through the rewards of exchange and the punishment of partners who renege on their exchange response, whereas, consistent with our trust results, Carmichael and MacLeod Ž1995. show that a stable gift-giving custom may emerge that does not depend upon punishment.
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Hoffman, E., McCabe, K., and Smith, V. Ž1996b.. ‘‘Social Distance and Other-Regarding Behavior in Dictator Games,’’ Amer. Econ. Re¨ . 86, 653]660. Hoffman, E., McCabe, K., and Smith, V. Ž1996c.. ‘‘Behavioral Foundations of Reciprocity: Experimental Economics and Evolutionary Psychology,’’ Econ. Inquiry Žto appear.. Kalai, E., and Lehrer, E. Ž1993.. ‘‘Rational Learning Leads to Nash Equilibrium,’’ Econometrica 61, 1019]1045. Kranton, R. E. Ž1996.. ‘‘Reciprocal Exchange: A Self Sustaining System,’’ Amer. Econ. Re¨ . 86, 830]851. Rabin, M. Ž1993.. ‘‘Incorporating Fairness into Game Theory and Economics,’’ Amer. Econ. Re¨ . 81, 1281]1302. Smith, V. L. Ž1982.. ‘‘Microeconomic Systems as an Experimental Science,’’ Amer. Econ. Re¨ . 72, 923]955.
24, 10]24 Ž1998.
GA980638
Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining Kevin A. McCabe, Stephen J. Rassenti, and Vernon L. Smith* Uni¨ ersity of Arizona, Tucson, Arizona 85721-0108 Received February 13, 1996
We report decision making in two-person extensive form game trees, using six treatments that vary matching protocol, payoffs, and payoff information. Our objective is to examine game theoretic hypotheses of decision making based on dominance and backward induction in comparison with the culturally or biologically derived hypothesis that reciprocity supports more cooperation than predicted by game theory. We find strong support for cooperation under complete information, even in single-play treatments and in games of trust, unreinforced by the prospect of punishment for defection from reciprocity. Only under private information do we observe strong support for noncooperative game theory. Journal of Economic Literature Classification Numbers: C78, C92. Q 1998 Academic Press
1. INTRODUCTION A number of simple experimental games like the ultimatum and dictator game ŽForsythe et al., 1994; Hoffman et al., 1994; Eckel and Grossman, 1996. and investment trust game ŽBerg et al., 1995. show that many people do not follow self-interested dominant strategies, nor do they expect such behavior from their counterparts in these two-person anonymous interactions. But these are cases of boundary social interactions, which provide extreme conditions in the payoff environment under which the elementary game theoretic principles of dominance and backward induction fail to predict behavior in the one-shot anonymity condition}a condition thought to be most favorable for eliciting noncooperative play. Evolutionary psychologists have argued that the human mind is evolutionarily adapted to help make social exchange work in family and tribal groups, and they have conducted reasoning experiments showing that subjects have a pronounced tendency to punish cheaters in social exchange contexts ŽCosmides, 1985; Cosmides and Tooby, 1992.. Behavior in ultimatum games can be interpreted as an expression of this tendency ŽHoffman *E-mail: [email protected]. 10 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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et al., 1994.. But in the dictator game no such interpretation is possible, since the decision maker’s offer of a share of the pie to her counterpart cannot be rejected. Although positive offers in the dictator game can be explained by models in which the players have a taste for ‘‘fairness’’ ŽRabin, 1993., such models do not explain, at least as currently formulated, why utility varies with treatments designed to operate on people’s expectations ŽHoffman et al., 1996a, b.. An alternative hypothesis for explaining these anomalous results is the concept of reciprocal ‘‘altruism,’’ or simply reciprocity Žno utilitarian altruism is involved, because it is an exchange., developed in the evolutionary biology and psychology literatures ŽCosmides and Tooby, 1992, pp. 170]179.. In this literature, people are assumed to possess a specialized mental algorithm in which their long-term self interest is best served by an unconscious willingness to punish cheating on cooperative social exchanges Žnegative reciprocity.. We also postulate a willingness to reward the initiation of cooperative social exchanges Žpositive reciprocity.. This idea is suggested in Hoffman et al. Ž1996c., where it is proposed that subjects are culturally, if not biologically, disposed to reputation building across games; i.e., the game of life is a repeated sequence of constituent games that may differ markedly. Hence, if you want to have a reputation for reciprocity, you do not engage in noncooperativerdefectionist strategies just because the experimenter thinks he has presented you with a one-shot game. If you exploit situations in that you think you cannot be ‘‘found out,’’ a condition which is uncertain, then how can you be sure that you will not slip into exploitative tactics at other times, and compromise your reputation? It is well known that borderline sociopaths encounter costly negative reactions from others. This explains the rejection of ‘‘unfair’’ offers in the ultimatum game, the ‘‘unreasonable’’ generosity of people in anonymous investment and dictator games, and the considerable reduction in this generosity when you run them double blind ŽHoffman et al., 1996b., so that it is clear that no one can know the subject’s decision. It also has an informal basis in game theory via The Folk theorem, and goes to the core of the question of how people perceive the games they play. We economists have accepted the maintained hypothesis of the theory that people perceive one-shot games to be isolated interactions, while seeking to resolve anomalies by inserting ‘‘fairness’’ considerations into the utility function Žas in Rabin, 1993..
2. THE CONSTITUENT GAMES In this paper we explore these issues in a somewhat richer two-person extensive form bargaining game environment than in the above games, but
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MCCABE, RASSENTI, AND SMITH
FIG. 1. Extensive form game tree, Games 1 and 2. Game 1 payoffs are closest to the tree. Game 2 differs from Game 1 by interchanging the payoffs Ž50, 50. and Ž60, 30. in the left branch.
one that is still elementary from the perspective of theory. In constituent Game 1 ŽFig. 1., at the initial decision node x 1 , player 1 has an outside option yielding the asymmetric payoff Ž35, 70. respectively to Žplayer 1, player 2., which for player 1 is somewhat less attractive than the symmetric subgame perfect ŽSP. outcome Ž40, 40. available by player 2 moving right at node x 6 . If player 1 moves down at x 1 , then player 2 chooses between subsequent play in the right subgame containing the unique SP outcome, or in the left subgame containing the symmetric joint maximum ŽSJM. outcome available at node x 3 . To achieve the SJM outcome, player 2 must move left at x 2 , but player 1 can defect on this signal by moving down at x 3 forcing player 2 at x 5 to choose between left, yielding the outcome Ž60, 30., and down to incur a costly choice that directly punishes player 1. But this direct punishment strategy is subgame dominated by the move left at x 5 . If player 2 always chooses undominated strategies, and believes that player 1 will do the same, then player 2 will not move left at x 2 in a single play of the constituent game, and will expect to achieve SP by moving right at x 2 . Player 1, using the same reasoning, will move down at x 1 , since 40 at SP is better for player 1 than 35, the outside option.
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Game 2 is the same as Game 1, except that the payoffs Ž60, 30. and Ž50, 50. have been interchanged in the left branch of the decision tree, thereby removing the threat of direct punishment. In both games the use of reciprocity is restricted to the left subgame, starting at node x 3 , allowing us to study backward induction in the right subgame, starting at x 4 , where reciprocity is not an issue. Furthermore, in both games the cooperative SJM outcome Ž50, 50. and the SP outcome Ž40, 40. are symmetric, allowing control over any payoff equity motivations that might influence subject choice.
3. REPEAT PLAY OF GAME 1 AND GAME 2 In the supergame treatment below, the same pairs repeat the constituent Game 1 Žor Game 2. for 20 repetitions Žsee the procedures below.. In Game 1, player 2 can avoid playing dominated punishing strategies Žand in Game 2 can introduce indirect punishment., by following an individually rational trigger strategy: in Game 1, if on repetition t, player 1 defects, then player 2 moves left at x 5 , but right at x 2 in repetition t q 1, punishing player 1 with at least the SP outcome, then in repetition t q 2 returns to moving left at x 2 . If this strategy is invoked each time player 1 defects, player 2 avoids the more costly outcome Ž20, 20.. But if player 1 is to be made strictly worse off, player 2 must punish, or credibly threaten to punish, with a right move at x 2 more than once for each defection, since just one punishment gives player 1 Ž60 q 40.r2 s 50, the same payoff as at SJM. The same trigger strategy in Game 2 allows player 2 to punish player 1 on any t where player 1 moves left at x 3 . Hence, in repeated play, if subjects use the trigger strategy for indirect punishment, Game 1 and Game 2 are equivalent, but if they tend to use the direct punishment option, then Game 1 will yield more outcomes at SJM than Game 2.
4. TREATMENTS AND PROCEDURE Our treatments are summarized in Table I. We adopt the convention of naming a cell by the matching treatment and the game number. Thus Same 1 refers to experiments in which players keep the same counterpart and play Game 1 repeatedly. Single 1 refers to experiments in which Game 1 is played once. Single 1, with anonymously matched pairs, allows us to give the SP prediction its best shot. If some subjects contrarily play for the SJM outcome, then Single 2 allows us to test whether that outcome is supported primarily by the prospect of direct punishment.
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MCCABE, RASSENTI, AND SMITH
TABLE I Experimental Design: Treatments and Number of Pairs
Designation Single 1 Single 2 Same 1 Same 2 Private Same 1 Private Same 3
Constituent game
Matching Protocol
Number pairs Žobservations.
1 2 1 2 1 3
Single play Single play Repeat same pairs Repeat same pairs Repeat random pairs with same roles Repeat same pairs
30 Ž30. 26 Ž26. 22 Ž440. 23 Ž460. 24 Ž480. a 24Ž480. a
a Sessions consisted of 12 subjects, 6 pairs matched repeatedly. Others consisted of at least 8 subjects.
In the repeat play treatments, subjects are recruited for 2 hours, even though the experiments last only slightly over 1 hour, and players are told nothing about how many trials will be played, making credible the expectation of a long series of repeated plays. The treatments also vary what the subjects know about their counterpart’s payoffs. Under private information, subjects know only their own payoffs. This precludes a subject’s ability to make inferences about her counterpart’s intentions}to read a counterpart’s mind, to determine why a choice might have been made, and to engage in reciprocity. Thus, in Same 1, a left move at x 2 under complete payoff information can be read as a signal to player 1, saying that player 2 wants to reach Ž50, 50. which dominates Ž40, 40., and that if player 1 defects, 2 has the option of punishing him. With private information, such ‘‘mind reading’’ is not possible. ŽSee Baron-Cohen, 1995, for a mental mechanism theory and experiments on the human ability to infer mental states in others from their actions, and the fact that autism, a predisposition to which appears to be inherited, prevents or seriously compromises mind reading.. The failure or inability to read intentions and reciprocate may explain why noncooperative game theory has been fruitfully applied to biological evolution, in which genes are modeled as selfish Žmotivated to increase their reproductive fitness. in an environment of private information, and why in a wide variety of previous bargaining and market experiments, it has been found that private information favors Nashrcompetitive outcomes ŽFouraker and Siegel, 1963; Smith, 1982.. In all experiments, subjects were paid a nonsalient fixed fee of $5.00 for appearing on time for the experiment and, at the end of the experiment, their accumulated earnings. The payoffs are in cents when the games are repeated for 20 trials; in the single trial experiments the payoffs are multiplied by 20.
RECIPROCITY, TRUST, AND PRIVACY
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5. ALTERNATIVE HYPOTHESES We propose reciprocity as a test alternative to the predictions of noncooperative game theory in the single play experiments. Specifically, we will test the following hypotheses. H1. Based on the principles of dominance and backward induction, right game play will predominate in Single 1 and Single 2 against the null alternative of equal play in the right and left game trees. H2. The Ž50, 50. SJM outcomes in Single 1 will exceed those in Single 2. This is because we hypothesize, from game theory, that reciprocity will be more difficult in Single 2, without the punishment option, than in Single 1. H3. Based on the Folk Theorem, the repeated games Same 1 and Same 2 will result in more cooperation than Single 1 and Single 2, with left play more common in Same 1 than in Same 2. H4. Private 1 will yield a higher proportion of right game play, and of the SP outcome, than any other treatment condition.
6. RESULTS: TESTS OF HYPOTHESES Table II summarizes the outcome frequencies across all treatments. The first data column lists the left game play results, and the sixth lists the right game results. The remaining columns show the outcomes conditional upon being in the left branch or the right branch of the decision tree. The unconditional relative outcome frequencies are shown in parentheses. H1. In the first two rows we see that right game play does not predominate in either Single 1 or Single 2. Using a binomial test, we cannot reject the null hypothesis that left and right game play are equally likely under either treatment Ž p s 0.29; 0.23.. Contingent on right game play, however, we observe strong support for SP in both Single 1 and Single 2, easily rejecting the hypothesis that SP is only 50% likely Ž p 0.001.. H2. The SJM outcome is significantly more frequent in Single 1 than in Single 2 Ž p - 0.001.. H3. The outcomes in Same 1 and Same 2 show significantly more left Žand Same 2 more cooperative. play than in the corresponding Single 1 and Single 2 experiments. Similarly, Same 1 shows more left plays and more cooperative outcomes than Same 2 Ž p - 0.001..
12r26 s 0.46 353r433 s 0.82 261r423 s 0.62 68r428 s 0.16
Single 2 Same 1 Same 2 Private 1
11r13 s 0.85 Ž0.37 . 6r12 s 0.50 Ž0.23 . 312r353 s 0.88 Ž0.72 . 220r261 s 0.84 Ž0.52 . 29r68 s 0.43 Ž0.07 .
50 50 1r13 s 0.08 Ž0.03 . 6r12 s 0.50 Ž0.23 . 14r353 s 0.04 Ž0.03 . 41r261 s 0.16 Ž0.10 . 38r68 s 0.56 Ž0.09 .
60 30
0 1r68 s 0.015 Ž0.
0 22r353 s 0.06 Ž0.05 .
1r13 s 0.08 Ž0.03 .
20 20
0
0
0 5r353 s 0.014 Ž0.01 .
0
0 0
360r428 s 0.84
162r423 s 0.38
80r433 s 0.18
14r26 s 0.54
17r30 s 0.57
Right branch
0 3r80 s 0.038 Ž0.01 . 27r162 s 0.17 Ž0.06 . 51r360 s 0.14 Ž0.12 .
0
30 60 16r17 s 0.94 Ž0.53. 14r14 s1 Ž0.54 . 68r80 s 0.85 Ž0.16. 114r162 s 0.70 Ž0.27. 304r360 s 0.84 Ž0.71.
40 40
0
0 0
0 0 4r80 5r80 s 0.05 s 0.06 Ž0.01 . Ž0.01. 21r162 s 0.13 Ž0.05. 0 5r360 s 0.014 Ž0.01. 0
1r17 s 0.06 Ž0.03 .
15 30
NA
46.9
46.6
40.0
46.2
E Žp 2 < Left .
NA
NA
31.2
}
40.0
E Žp 1 < Defect .
Conditional frequencies are computed contingent on left or right branch play. Frequencies in parentheses are computed unconditionally across all payoff outcomes. NA, Data not applicable.
a
13r30 s 0.43
Left branch
Single 1
Treatment
TABLE II Branch Conditional ŽUnconditional. a Outcome Frequencies by Treatment, All Trials
16 MCCABE, RASSENTI, AND SMITH
RECIPROCITY, TRUST, AND PRIVACY
17
H4. Comparing Private 1 and Same 1, noncooperative right game play is very much larger in the former Ž p - 0.001.. Across all treatments, Private 1 yields by far the highest incidence of noncooperative right game play. In Table II we also report the expected profit to player 2 from a left move at x 2 , EŽp 2 < Left., based on the outcome frequencies conditional upon left game play. Similarly, we report the expected profit to player 1 from defection Žmoving down. at x 3 in game 1, EŽp 1 < Defect.. This leads to two important results: Ž1. on average across subject pairs, it pays for subjects in role 2 to move left at x 2 . The gain is strictly better than the SP payoff Ž40. in all treatments except Single 2, where the expected payoff is exactly 40. Ž2. Across both treatments for Game 1, it does not pay for subjects in role 1 to defect on the SJM outcome Ž50.. Hence subjects are not behaving in an irrational manner, in the sense that, given the reciprocity types in the sample, and the subjects’ ability to read each other, on average they are collecting more money than if they behaved in a myopic self-interested way, ignoring the reciprocity characteristics of those with whom they are matched. We note that in Single 1 and Single 2, a little more than half play the right game, and earn Žweakly. less than those who play the left game, but in Same 1 and Same 2, where there is an opportunity for subjects to learn more about their counterparts, many more play the left game, defections decline, and EŽp 2 < Left. increases.
7. EVOLUTION ACROSS REPETITION BLOCKS Tables III]V report conditional outcome frequencies by blocks of five trials over the 20 trials of Same 1, Same 2, and Private 1. In Same 1 ŽTable III., left branch play starts higher than in Single 1, and tends to increase across trial blocks. Knowing that they are matched with the same counterpart for what is expected to be a long game, subjects appear to harbor higher expectations of achieving the SJM outcome, and they are right. Although defections start out at around 18%, they decline to about 7%. Punishment rates begin at 10% of subjects playing the left branch and decline to 3]4%. In Same 2 ŽTable IV., left branch play starts no higher than Single 2, but increases more than in Same 1, as subjects learn to overcome the inherent instability of playing the left branch of Game 2. Defection from SJM by a left move at x 3 starts high Ž36%., but declines steadily to 9% by the last trial block. Right branch play in Same 2 is not only more common than in Same 1, but involves more punishing outcomes. At least some of the moves down at x 6 appear to be trigger strategies designed to induce
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MCCABE, RASSENTI, AND SMITH
TABLE III Branch Conditional a Outcome Frequencies by Repetition Block, Same 1
Repetitions 1]5 6 ] 10 11 ] 15 16 ] 20 a
Left branch
50 50
60 30
20 20
78r109 s 0.716 88r108 s 0.815 94r106 s 0.887 93r110 s 0.845
64r78 s 0.821 76r88 s 0.864 87r94 s 0.926 85r93 s 0.914
6r78 s 0.077 1r88 s 0.011 4r94 s 0.043 3r93 s 0.032
8r78 s 0.103 7r88 s 0.080 3r94 s 0.032 4r93 s 0.043
Right branch
30 60
31r109 2r31 s 0.284 s 0.065 20r108 s 0.185 0 12r106 s 0.113 0 17r110 1r17 s 0.155 s 0.059
40 40
15 30
E Žp 2 < E Žp 1 < Left . Down .
28r31 s 0.966 0 19r20 s 0.95 0 12r12 s1 0 9r17 4r17 s 0.529 s 0.235
45.4
37.2
45.1
16.6
48.2
42.8
47.5
32.5
Conditional frequencies are computed contingent on left or right branch play.
players 1 to move right at x 1 on subsequent plays. Thus from Table I we see that there are 460 observations or 115 per five-trial block in Same 2, but total play in the left or right branch is 100, 110, 108, and 105 by trial block in Table IV. The differences are right moves at x 1 Ž15, 5, 7, 10, respectively.. In Private 1 ŽTable V., left branch play declines steadily as subjects discover its instability without being able to see how their counterpart’s payoffs explain this instability. Although right branch play starts high and goes higher, initially nearly a third of the outcomes are at Ž30, 60.. Apparently some subjects 1, having observed their counterpart move right at x 2 , reason that if they move down at x 4 , the uncertain possible outcomes for player 1 are 40 and 15. The sure thing is 30 by moving right
TABLE IV Branch Conditional a Outcome Frequencies by Repetition Block, Same 2
Repetitions 1]5 6]10 11]15 16]20
Repetitions 1]5 6]10 11]15 16]20 a
Left branch
60 30
50 50
45r100 s 0.450 65r110 s 0.591 73r108 s 0.676 78r105 s 0.743
16r45 s 0.356 10r65 s 0.154 8r73 s 0.110 7r78 s 0.090
29r45 s 0.644 55r65 s 0.846 65r73 s 0.890 71r78 s 0.910
Right branch
30 60
40 40
15 30
EŽp 2 N Left.
55r100 s 0.550 45r110 s 0.409 35r108 s 0.324 27r105 s 0.257
7r55 s 0.127 11r45 s 0.244 6r35 s 0.171 3r27 s 0.111
42r55 s 0.764 26r45 s 0.578 23r35 s 0.657 23r27 s 0.852
6r55 s 0.109 8r45 s 0.178 6r35 s 0.171 1r27 s 0.037
42.9 46.9 47.8 48.2
See Table III for footnote.
a
50 50 13r26 s 0.5 6r17 s 0.353 6r13 s 0.462 4r12 s 0.333
Left branch 26r109 s 0.239 17r107 s 0.159 13r103 s 0.126 12r109 s 0.110
See Table III for footnote.
1]5 6]10 11]15 16]20
Repetitions 13r26 s 0.5 11r17 s 0.647 7r13 s 0.538 7r12 s 0.583
60 30 0 0 0 1r12 s 0.083
20 20 83r109 s 0.761 90r109 s 0.826 90r103 s 0.874 97r109 s 0.890
Right branch
26r83 s 0.313 13r90 s 0.144 5r90 s 0.056 7r97 s 0.072
30 60
40 40 57r83 s 0.687 75r90 s 0.833 83r90 s 0.922 88r97 s 0.907
TABLE V Branch Conditional a Outcome Frequencies by Repetition Block, Private Same 1
0 2r90 s 0.022 2r90 s 0.022 1r97 s 0.010
15 30
40 37.1 39.2 36.7
E Žp 2 < Left .
RECIPROCITY, TRUST, AND PRIVACY
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MCCABE, RASSENTI, AND SMITH
at x 4 , and a great many do. The more hesitant and perhaps risk-averse subjects are reluctant to explore below x 4 , but as repetitions continue, more and more players 1 experiment with a down move at x 4 , almost always finding that it leads to 40. The bottom line is that the great majority of subjects grope for stable self-interested outcomes in this private information environment, and find that SP is a strong point of attraction for this behavior. Since complete payoff information strains credulity outside the laboratory, we think that this is a fruitful way to model SP equilibrium outcomes ŽKalai and Lehrer, 1993..
8. DOES PRIVATE INFORMATION SUPPORT FOR SP WEAKEN IF IT IS MADE STRATEGICALLY MORE COSTLY? In Private 1 we found substantial support for SP. Our explanation of why support required time to develop over successive trials focused on strategic cost: risk averse, timid subjects 1 are reluctant to explore far enough into the tree. If this interpretation is correct, it should be testable, for example, by placing SP at the bottom of the game tree, with low payoff opportunities along the way for both players. This reasoning led us to the new game tree, Game 3, shown in Fig. 2. The SP payoffs are sweetened to Ž95, 95. with player 1 playing down at x 7 . But we place the outcomes Ž75, 15. and Ž15, 75., respectively, on the left at x 3 and x 5 . This, we hypothesize, will discourage timid players from playing the left game by providing transparent indications of the risk in achieving SP. Players 2 should be reluctant to move left at x 2 , giving up a sure 55 on the right, for an uncertain mixture of 15, 75, 15, and 95; players 1 should see this and be more likely to move down at x 1 than in Private 1. But many players 1, if they reach x 3 , are likely to be tempted to go left, settling for the sure thing, 75, in place of the uncertain 95 at the bottom. This, in turn, will discourage players 2 from moving left at x 2 . In particular, at x 2 , if player 2 considers the payoff boxes in the left game to be equally likely, then his expected payoff is Ž15 q 75 q 15 q 95.r4 s 50, compared with a sure thing of 55 on the right. So left at x 2 is not for the timid or risk-averse, although it is certainly appropriate in a long game for those with an exploratory bent. Similarly, at x 3 , if player 1 considers the payoffs in the subgame below x 3 to be equally likely, then her expected payoff is Ž15 q 95.r2 s 55 compared with the sure thing, 75, by moving left at x 3 . If this prospect is part of player 2’s reasoning, then the prospect of him moving left at x 2 is further diminished.
RECIPROCITY, TRUST, AND PRIVACY
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FIG. 2. Extensive form game tree, Game 3.
As in Private 1, Private 3 uses subjects who, after random assignment to a pair and to the role of player 1 or 2, maintain the same counterpart and role across all trials. Four sessions with six pairs in each session entered decisions for 20 trials, yielding a total of 480 observations. Comparing Table VI with Table V, we see that support for SP builds steadily across repetitions in Game 3 as in Game 1, but at lower levels. Almost all of this increasing support is coming from increased downward exploration in the left game; right game play changes very little across the four trial blocks. In the final trial block, 16]20, we have right game, 44 observations Ž36.7%., SP, 64 observations Ž84.2%., with the payoff Ž75, 15. at x 3 accounting for the remaining 12 observations Ž15.8%.. We conclude that support for SP under private information is robust in moving from Game 1 to Game 3, but importantly compromised by elementary considerations of strategic risk in achieving SP. When we lengthen the chain of moves to achieve SP, displaying obvious possible traps along the way, this reduces support for SP. In fact, it should not be too difficult to invent other environments that would make SP even more difficult to achieve but not less rational, given the strategic cost of attaining it. Yet it is clear from a wide variety of experiments over the decades that noncoop-
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MCCABE, RASSENTI, AND SMITH
TABLE VI Branch Outcome Frequencies by Repetition Block, Private Same 3
Repetitions 1]5 6]10 11]15 16]20 1]20
Left branch
75 15
15 75
95 95
Right branch
70r119 s 0.588 77r120 s 0.642 77r120 s 0.642 76r120 s 0.633 300r479 s 0.626
39r70 s 0.557 20r77 s 0.260 19r77 s 0.247 12r76 s 0.158 90r300 s 0.300
11r70 s 0.157 6r77 s 0.078 2r77 s 0.026 0
20r70 s 0.286 51r77 s 0.662 56r77 s 0.727 64r76 s 0.842 191r300 s 0.637
49r119 s 0.412 43r120 s 0.358 43r120 s 0.358 44r120 s 0.367 179r479 s 0.374
19r300 s 0.063
erative equilibria fare better under private, than under complete, payoff information.
9. DISCUSSION Contrary to noncooperative game theory, but consistent with the reciprocity hypothesis, many subjects achieve the symmetric joint maximum under the single play anonymous interaction conditions that are expected to give the subgame perfect equilibrium its best shot. Reciprocity Žas a hypothesis., on average, enables many subjects to achieve individually rational states whose rewards exceed that attainable if each person in a trial behaves on the assumption that his counterpart will follow her own myopic self-interest. Furthermore, these achievements are to a considerable extent driven by trust, and do not depend critically on the prospect of punishment. Although the resulting rewards are larger, subjects exhibit differing behaviors Žas is common in biological models and phenomena generally.. In single play treatments, about half go for the SP, but in repeat interaction, consistent with the game theoretic Folk Theorem, support for the symmetric joint maximum is increased substantially. Also consistent with game theory is the very strong support for SP contingent on right branch play in both Games 1 and 2. Reciprocity and its origins in trust andror punishment is in need of being modeled to account for a variety of behavioral reputational types: those who offer cooperation on the basis of pure trust, those who require the prospect of punishment, those who defect when cooperation is offered, and those who, faced with defection, tend to respond with punishment.
RECIPROCITY, TRUST, AND PRIVACY
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These types are not necessarily independent. Thus it is reasonable to postulate that a person who as Player 2 moves left at x 2 to signal cooperation is also likely to reciprocate as a Player 1 by moving left at x 3 . Similarly, noncooperative Player 2s are expected to be more likely to defect as Player 1s. Any such modeling of player types is complicated by the strong likelihood that behavior will not be independent of opportunity costs and, in view of the strong support here and elsewhere for the Folk Theorem, will certainly not be independent of counterpart signals in a supergame. Simpler recent models, however, have dealt with some of these features, and the results are promising; e.g., punishment that is directed solely at noncooperation can lead to the evolution of cooperation, Žsee Boyd and Richerson, 1992, and the references therein .. Similarly, reciprocity has been modeled by Kranton Ž1996. as a self-interested gift exchange, sustained through the rewards of exchange and the punishment of partners who renege on their exchange response, whereas, consistent with our trust results, Carmichael and MacLeod Ž1995. show that a stable gift-giving custom may emerge that does not depend upon punishment.
REFERENCES Baron-Cohen, S. Ž1995.. Mindblindness: An Essay on Autism and Theory of Mind. Cambridge, MA: MIT Press. Berg, J., Dickhaut, J., and McCabe, K. Ž1995.. ‘‘Trust, Reciprocity and Social History,’’ Games Econ. Beha¨ . 10, 122]142. Boyd, R., and Richerson, P. J. Ž1992.. ‘‘Punishment Allows the Evolution of Cooperation Žor Anything Else. in Sizable Groups,’’ Ethology Sociol. 13, 171]195. Carmichael, H. L., and MacLeod, W. B. Ž1995.. ‘‘Gift Giving and the Evolution of Cooperation,’’ mimeo, Queens University and Boston College. Cosmides, L. Ž1985.. ‘‘The Logic of Social Exchange: Has Natural Selection Shaped How Humans Reason? Studies with the Wason Selection Task,’’ Cognition 31, 187]276. Cosmides, L., and Tooby, J. Ž1992.. ‘‘Cognitive Adaptations for Social Exchange,’’ in The Adapted Mind ŽJ. Barkow, L. Cosmides, and J. Tooby, Eds... New York: Oxford University Press. Eckel, C., and Grossman, P. Ž1996.. ‘‘Altruism in Anonymous Dictator Games,’’ Games Econ. Beha¨ . 16, 181]191. Forsythe, R., Horowitz, J. L., Savin, N. S., and Sefton, M. Ž1994.. ‘‘Fairness in Simple Bargaining Experiments,’’ Games Econ. Beha¨ . 6, 347]367. Fouraker, L., and Siegel, S. Ž1963.. Bargaining Beha¨ ior. New York: McGraw-Hill. Hoffman, E., McCabe, K., Shachat, K., and Smith, V. Ž1994.. ‘‘Preferences, Property Rights, and Anonymity in Bargaining Games,’’ Games Econ. Beha¨ . 7, 346]380. Hoffman, E., McCabe, K., and Smith, V. Ž1996a.. ‘‘On Expectations and the Monetary Stakes in Ultimatum Games,’’ Internat. J. Game Theory 3, 1]13.
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Hoffman, E., McCabe, K., and Smith, V. Ž1996b.. ‘‘Social Distance and Other-Regarding Behavior in Dictator Games,’’ Amer. Econ. Re¨ . 86, 653]660. Hoffman, E., McCabe, K., and Smith, V. Ž1996c.. ‘‘Behavioral Foundations of Reciprocity: Experimental Economics and Evolutionary Psychology,’’ Econ. Inquiry Žto appear.. Kalai, E., and Lehrer, E. Ž1993.. ‘‘Rational Learning Leads to Nash Equilibrium,’’ Econometrica 61, 1019]1045. Kranton, R. E. Ž1996.. ‘‘Reciprocal Exchange: A Self Sustaining System,’’ Amer. Econ. Re¨ . 86, 830]851. Rabin, M. Ž1993.. ‘‘Incorporating Fairness into Game Theory and Economics,’’ Amer. Econ. Re¨ . 81, 1281]1302. Smith, V. L. Ž1982.. ‘‘Microeconomic Systems as an Experimental Science,’’ Amer. Econ. Re¨ . 72, 923]955.