A choice prediction competition for a social preferences in simple extensive form games: An introduction

Ert, Eyal; Erev,Ido; Roth, Alvin E. Article A choice prediction competition for a social preferences in simple extensive form games: An introduction Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Ert, Eyal; Erev,Ido; Roth, Alvin E. (2011) : A choice prediction competition for a social preferences in simple extensive form games: An introduction, Games, ISSN 2073-4336, MDPI, Basel, Vol. 2, Iss. 3, pp. 257-276, https://doi.org/10.3390/g2030257 This Version is available at: https://hdl.handle.net/10419/98537 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Games 2011, 2, 257-276; doi:10.3390/g2030257 games ISSN 2073-4336 www.mdpi.com/journal/games Article A Choice Prediction Competition for Social Preferences in Simple Extensive Form Games: An Introduction Eyal Ert 1,*, Ido Erev 2 and Alvin E. Roth 3,4 1 Agricultural Economics and Management, The Hebrew University of Jerusalem, P.O. Box 12, Rehovot 76100, Israel 2 Max Wertheimer Minerva Center for Cognitive Studies, Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel; E-Mail: [email protected] 3 Department of Economics, 308 Littauer, Harvard University, Cambridge, MA 02138, USA 4 Harvard Business School, 441 Baker Library, Boston, MA 02163, USA; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]. Received: 14 March 2011 / Accepted: 5 July 2011 / Published: 25 July 2011 Abstract: Two independent, but related, choice prediction competitions are organized that focus on behavior in simple two-person extensive form games (http://sites.google.com/site/extformpredcomp/): one focuses on predicting the choices of the first mover and the other on predicting the choices of the second mover. The competitions are based on an estimation experiment and a competition experiment. The two experiments use the same methods and subject pool, and examine games randomly selected from the same distribution. The current introductory paper presents the results of the estimation experiment, and clarifies the descriptive value of some baseline models. The best baseline model assumes that each choice is made based on one of several rules. The rules include: rational choice, level-1 reasoning, an attempt to maximize joint payoff, and an attempt to increase fairness. The probability of using the different rules is assumed to be stable over games. The estimated parameters imply that the most popular rule is rational choice; it is used in about half the cases. To participate in the competitions, researchers are asked to email the organizers models (implemented in computer programs) that read the incentive structure as input, and derive the predicted behavior as an output. The submission deadline is 1 December 2011, the results of the competition experiment will not be revealed until that date. The submitted models will be ranked based on their prediction OPEN ACCESS Games 2011, 2 258 error. The winners of the competitions will be invited to write a paper that describes their model. Keywords: social preferences; fairness; reciprocity; social welfare; trust; altruism 1. Introduction Experimental studies of simple social interactions reveal robust behavioral deviations from the predictions of the rational economic model. People appear to be less selfish, and less sophisticated than the assumed Homo Economicus. The main deviations from rational choice (see a summary in Table 1) can be described as the product of a small set of psychological factors. Those factors include (i) altruism or warm glow [1]; (ii) envy or spitefulness [2]; (iii) inequality aversion [3,4]; (iv) reciprocity [5,6]; (v) maximizing joint payoff [7]; (vi) competitiveness [7]; and (vii) level-k reasoning [8]. Table 1.Sequential two-player games that show deviations from rationality.1 Game Description Rational prediction Main findings Ultimatum [9] A Proposer offers an allocation of a pie (e.g., $10) between herself and a responder. If the responder accepts the offer, the money is allocated. If she rejects, both get nothing The responder maximizes own payoff thus agrees to any allocation. The proposer, anticipating that, offers the lowest amount possible to the responder Most proposers suggest equal split when such split is possible. Low offers (below 30% of the pie) are typically rejected. [9] Dictator [10] A “Dictator” determines an allocation of an endowment (e.g., $10) between herself and a recipient The dictator, maximizing their own payoff, gives $0 to the recipient Dictators, on average, give 30% of the endowment. [10] Trust [11] A sender receives an endowment (e.g., $10) and can send any proportion of it to the responder. The amount sent is multiplied (e.g., by 3). The responder then decides how much to send back. Responder maximizes their own payoff and thus sends back $0. The sender, anticipating that, sends $0. Most senders send half or more of their endowment. Many responders (e.g., 44% in [11]) return at least the amount sent. Gift exchange [12] A “manager” (M) determines a wage (w) to “hire” an “employee”(E). The employee then choose an effort level e which is costly (c(e)). The profit functions are: The employee chooses minimum effort. Anticipating that, the manager chooses the minimum wage. The minority of transactions (less than 9%) involve minimal wages and effort. About 2/3 of the managers offers are higher than 50. [12] Recent research demonstrates the potential of simple models that capture these psychological factors e.g., [3,5,13]. However, there is little agreement concerning the best abstraction and the relative importance of the distinct psychological factors. The main goal of the current project is to address the 1 Each of those games has been studied extensively with different variations. Table 1 is focused on the motivating experiments that introduced those games. Games 2011, 2 259 quantitative “best abstractions and relative importance” questions with the organization of a choice prediction competition. It focuses on simple response games, similar by structure to the games studied by Charness and Rabin [14]. The game structure, presented in Figure 1, involves two players acting sequentially. The first mover (Player 1) chooses between action Out, which enforces an “outside option” payoffs on the two players (f1 and s1 for Players 1 and 2 respectively), and action In. If In is chosen then the responder (Player 2) determines the allocation of payoffs by choosing between actions Left that yields the payoffs f2, s2, and action Right that yields the payoffs f3, s3. Figure 1.The structure of the basic game. Player 1 (P1) selects between Out and In, and Player 2 (P2) selects between Left and Right. The selected structure has two main advantages: the first is that, depending on the relation between the different payoffs, this structure allows for studying games that are similar to the famous examples considered in Table 1 (ultimatum, trust etc.), and at the same time allows for expanding the set of game-types. Secondly, although this structure is simple, it is sufficient to reproduce the main deviations from rational choice considered by previous studies. For example, both players can exhibit their preferences over different outcome distributions between themselves and the other player. In addition, the “opting out” feature of the game can color action In by Player 1 as a selfish, altruistic, or neutral act (depending on the payoffs), allowing for reciprocal behavior by Player 2 (punishing or rewarding Player 1’s decision). These properties make the current set of games a natural test case for models of social preferences. Figure 2 presents a classification of the space of games implied by Figure 1’s game structure. The classification is based on the relation between the different payoffs for each player in a specific class of games. It provides a closer look into the potential conflicts that may be involved in each game, as well as clarifying the game’s likelihood of occurrence under a random selection from that space. The classification reveals that only a small proportion of the games in Figure 2’s space are similar to the games that have been extensively used by previous studies. For example, ultimatum-like games, special cases of the costly punishment (c.p.) type of games in which f1 = s1, capture less than 1% of the entire space of possible games. Similarly, the likelihood that a trust-like game would be randomly sampled from the space is only about 4%. The most likely games are typically less interesting than the famous examples: a large proportion of games include a “common interest” option that maximizes both player’s payoffs, and many other games are “safe shot” games for Player 1 as the payoff from In is higher than that of Out independently of Player 2’s choice. Games 2011, 2 260 Figure 2.The structure and space of the games. The games are classified according to the relations between their outcomes for each player separately, and their main properties are described below the graph. Cells marked with gray are defined as “trivial games.” The lower panel shows the proportion of games under random sampling from the space, from the space excluding the “trivial games,” and under the quasi random sampling algorithm used in the estimation study. The current study focuses on games that were sampled from Figure 2’s space using the quasi random sampling algorithm described in Appendix 1. The algorithm excludes the sampling from the trivial class (games in which action Out yields the best payoff for both players or in which all payoffs are equal for a player—the dark gray cells) and implies slight oversampling of interesting games. The implied sampling proportions by this algorithm are presented on the rightmost column in the lower panel of Figure 2. The Structure of the Competition and the Problem Selection Algorithm The current paper introduces two independent but related competitions: one for predicting the proportion of In choices by Player 1, and the other for predicting the proportion of Right choices by Player 2. Both competitions focus on the same games, and are based on two experiments: An “estimation” and a “competition” experiment.2 We describe the first experiment (and several baseline models) below, and challenge other researchers to predict the results of the second experiment. The 2This structure follows the structure of previous competitions we organized on other research questions [15,16]. Prop. f1=f2=f3 f1>f2>f3 f1>f2=f3 f1>f3>f2 f1=f2>f3 f2>f1>f3 f2>f1=f3 f2>f3>f1 f2=f3>f1 f3>f2>f1 f3>f1=f2 f3>f1>f2 f3=f1>f2 in space: s1=s2=s3 0.34% s1>s2>s3 c.p s.s s.s s.s s.s s.s r.p 13.84% s1>s2=s3 f.p s.s s.s s.s & s.d s.s s.s f.p 2.77% s1>s3>s2 r.p s.s s.s s.s s.s s.s c.p 13.84% s1=s2>s3 n.d n.d n.d c.i s.s s.s s.s s.s s.s c.i 2.77% s2>s1>s3 n.d n.d n.d f.h c.i s.s & c.i s.s & c.i s.s & c.i s.s s.s 13.84% s2>s1=s3 n.d n.d n.d f.h c.i s.s & c.i s.s & c.i s.s & c.i s.s s.s c.h c.h 2.77% s2>s3>s1 n.d n.d n.d f.h c.i s.s & c.i s.s & c.i s.s & c.i s.s s.s c.h. & tr c.h 13.84% s2=s3>s1 n.d n.d & s.d n.d f.h s.s s.s s.s & s.d s.s s.s f.h 2.77% s3>s2>s1 n.d n.d n.d c.h c.h. & tr s.s s.s s.s s.s & c.i s.s & c.i c.i f.h 13.84% s3>s1=s2 n.d n.d n.d c.h c.h s.s s.s s.s s.s & c.i s.s & c.i c.i f.h 2.77% s3>s1>s2 n.d n.d n.d s.s s.s s.s s.s & c.i s.s & c.i c.i f.h 13.84% s3=s1>s2 n.d n.d n.d c.i s.s s.s s.s s.s s.s c.i 2.77% proportion in space: 0.34% 13.84% 2.77% 13.84% 2.77% 13.84% 2.77% 13.84% 2.77% 13.84% 2.77% 13.84% 2.77% 100.00% notation class Main Properties Prop. in space Prop. by algorithm c.i common interest 19.7% 19.0% s.s safe shot 35.9% 38.0% s.d strategic dummy Player 2 cannot affect the payoffs: dictator is private case 0.2% 3.5% n.d near dictator Best payoff for player 1 is independent of player 2's choice 21.1% 32.0% c.p costly punish Punishing player 1's In choice is costly 3.8% 4.0% f.p free punish Player 2 can punish player 1's In choice with no cost 0.7% 1.2% r.p rational punish Punishing player 1's In choice maximizes player 2's payoff 3.8% 5.3% c.h costly help Improving other's payoff is costly for the helper 5.5% 7.4% f.h free help Improving others payoff is not costly 1.8% 2.8% tr trust game Choosing In improves 2's payoffs but reciprocation is irrational for player 2 3.8% 6.5% trivial games the player's three payoffs are equal or Out maximizes both players' payoffs 11.0% -- Notice: the sum of the proprotions is higher than 100% since some games can be clasified to more than one type 0.8% 4.4% 23.7% 0.2% 40.3% -- 4.3% 2.1% 6.2% 4.4% In is the optimal choice for player 1 f1 best f2 best f3 best s1 best s2 best s3 best There is one option which is best for both players Prop. in space - trivial excluded 22.1% Games 2011, 2 261 competition criteria are described in Section 4. The two experiments examine different games and different participants, but use the same procedure, and sample the games and the participants from the same space. Each experiment includes 120 games. The six parameters that define each game (the payoffs f1–f3, and s1–s3, see Figure 1) were selected using the algorithm presented in Appendix 1. The algorithm implies a nearly uniform distribution of each parameter in the range between −$8 and + $8. The 120 games studied in the estimation experiment are presented in Table 1. 2. Experimental Method The estimation experiment was run in the CLER lab at Harvard. One hundred and sixteen students participated in the study, which was run in four independent sessions, each of which included between 26 and 30 participants. Each session focused on 60 of the 120 extensive form games presented in Table 2, and each subset of 60 games was run twice, counterbalancing the order of problems.3 The experiment was computerized using Z-Tree [17]. After the instructions were read by the experimenter, each participant was randomly matched with a different partner in each of the 60 games, and played each of the 60 games using the strategy-method.4 That is, participants marked their choices without knowing what the other player had chosen. Moreover, they did not receive any feedback during the experiment. At the end of the session one game was selected at random to determine the players’ payoffs and the participants were reminded of the game’s payoffs, and their choices, and were informed about the choice of the other player in that game and consequently their payoffs (see a copy of the instructions below). The whole procedure took about 30 minutes on average. Participants’ final payoffs were composed from the sum of a $20 show-up fee, and their payoff (gain/loss) in one randomly selected trial. Final payoffs ranged between $14 and $28. 3. Experimental Results Table 2 presents the 120 games, and the proportion of In choices (by Player 1) and Right choices (by Player 2). The rightmost columns of this table present the predicted choices under the subgame perfect equilibrium (SPE), 5 and the average Mean Squared Deviation (MSD) of the observed proportions from the two predictions. The results reveal high correlations between the players’ behavior and the equilibrium predictions especially for Player 2’s choices (r = 0.88, and r = 0.98 for Players 1 and 2 respectively).The ENO (equivalent value of observations6) score of the equilibrium prediction is 0.87 for Player 1, and 7.5 for Player 2. 3We checked the data for potential “order of game” effect but no such effect was found. 4Previous research that compares the strategy method to a sequential-decision method shows little difference between the two [18,19], albeit levels of punishment seem to be lower with the strategy method [19]. 5The SPE prediction for Player 2 is 0 (Left) if S2 > S3, 1 (Right) if S2 < S3 and 0.5 (random choice) otherwise. To define the SPE prediction for Player 1 let EIN be the expected payoff for Player 1 from In assuming that Player 2 follows the SPE predictions. The SPE prediction for Player 1 is 0 (Out) if F1 > EIN, 1 (In) if F1 < EIN and 0.5 (random choice) otherwise. 6In order to clarify this measure, consider the task of predicting the entry rate in a particular game. Assume that you can base your estimate on the observed rate in the first m cohorts that plays this game, and on a point prediction made by a specific model. It is easy to see that the value of the observed rate increases with m. ENO of 7.5 means that the prediction of the model is expected to be more accurate than the observed rate in an experiment with 6 cohorts. The exact computation is explained in [20]. Games 2011, 2 262 Table 2. The 120 games studied in the estimation experiment ranked by the Mean Squared Deviation from the Subgame Perfect Equilibrium prediction. The left-hand columns present the payoffs of the 120 games, the right-hand columns present the experimental results (proportions of In and Right choices), the subgame perfect equilibrium predictions for these choices (SPE), and the average MSD (over the two players) between the predictions and the results. Rank Game f1 s1 f2 s2 f3 s3 P(In) P(Right) P(In) SPE P(Right) SPE MSD 1 16 0 8 1 −6 −6 3 0 1 0 1 0 2 18 3 −4 −2 8 3 −5 0 0 0 0 0 3 31 −6 −6 2 −3 1 2 1 1 1 1 0 4 43 −7 1 6 4 0 −6 1 0 1 0 0 5 49 5 −2 4 −5 −3 5 0 1 0 1 0 6 56 0 −2 8 3 7 −7 1 0 1 0 0 7 63 5 3 −3 6 0 8 0 1 0 1 0 8 70 −6 −6 0 8 −1 −6 1 0 1 0 0 9 84 −4 1 1 −8 8 0 1 1 1 1 0 10 86 −5 1 0 −6 −4 −4 1 1 1 1 0 11 90 −4 −4 0 −7 7 −4 1 1 1 1 0 12 93 2 4 −7 7 3 −5 0 0 0 0 0 13 97 0 6 3 −7 6 −2 1 1 1 1 0 14 105 8 −5 −6 −5 −3 6 0 1 0 1 0 15 111 −5 6 4 −4 5 4 1 1 1 1 0 16 117 −3 −6 8 0 7 3 1 1 1 1 0 17 35 0 1 6 1 8 −6 1 0.03 1 0 0 18 39 5 7 7 −7 0 4 0.03 1 0 1 0 19 46 −4 −2 7 6 −4 3 1 0.03 1 0 0 20 3 −7 −4 −5 8 −5 8 1 0.47 1 0.5 0 21 4 −3 −4 −1 4 3 7 1 0.97 1 1 0 22 30 2 −8 7 −1 6 −7 0.97 0 1 0 0 23 36 −5 3 7 −7 −4 5 1 0.97 1 1 0 24 37 −6 1 −2 −6 1 −3 1 0.97 1 1 0 25 44 −7 6 −5 2 −4 7 1 0.97 1 1 0 26 79 0 0 −5 5 −1 −6 0.04 0 0 0 0 27 89 6 −7 −3 −3 6 −7 0.04 0 0 0 0 28 91 2 −6 −7 −1 0 −6 0.04 0 0 0 0 29 96 2 6 −3 6 3 −3 0 0.04 0 0 0 30 66 −6 −2 1 −4 −4 −4 0.96 0.5 1 0.5 0 31 81 −6 −5 0 −1 −4 −6 0.96 0 1 0 0 32 82 0 8 6 5 8 3 0.96 0 1 0 0 33 83 −6 −6 −1 −1 5 −5 0.96 0 1 0 0 34 103 −7 −5 −5 5 −6 −3 0.96 0 1 0 0 Games 2011, 2 263 Table 2. Cont. Rank Game f1 s1 f2 s2 f3 s3 P(In) P(Right) P(In) SPE P(Right) SPE MSD 35 104 −5 −4 4 −7 2 2 1 0.96 1 1 0 36 26 −1 4 1 0 5 −2 0.97 0.03 1 0 0 37 23 −3 −5 3 −3 6 −1 0.97 0.97 1 1 0 38 53 −6 2 −7 −2 0 2 0.97 0.97 1 1 0 39 67 −3 −1 −5 −3 6 0 0.96 0.96 1 1 0 40 101 2 −6 7 −8 6 −6 0.96 0.96 1 1 0 41 25 −7 1 −6 1 0 3 1 0.93 1 1 0 42 32 −3 6 2 −4 5 −3 1 0.93 1 1 0 43 87 3 −2 2 0 6 3 0.93 1 1 1 0 44 106 4 −1 6 3 0 −2 0.93 0 1 0 0 45 10 −2 5 −4 7 −2 −2 0.07 0 0 0 0 46 24 2 −4 −3 5 3 −2 0 0.07 0 0 0 47 55 6 0 −5 4 0 −2 0 0.07 0 0 0 48 12 2 −7 3 3 4 −6 0.93 0.03 1 0 0 49 15 1 −4 1 2 6 5 0.93 0.97 1 1 0 50 59 −6 5 2 0 7 −8 0.97 0.07 1 0 0 51 120 −2 −7 −3 −6 0 3 0.96 0.93 1 1 0 52 64 6 −2 1 1 4 −3 0.07 0.04 0 0 0 53 102 −1 −6 −4 −2 −4 −3 0.07 0.04 0 0 0 54 62 2 −6 3 −2 −4 5 0.07 0.96 0 1 0 55 99 −5 7 −2 −1 4 −4 0.96 0.07 1 0 0 56 114 −4 1 −3 6 6 1 0.96 0.07 1 0 0 57 118 −5 −6 7 7 4 -6 0.96 0.07 1 0 0 58 11 −2 1 5 −7 5 -7 0.93 0.43 1 0.5 0 59 58 −6 0 −1 −5 −6 −7 0.93 0.07 1 0 0 60 60 −8 3 6 4 −3 −2 0.93 0.07 1 0 0 61 13 5 −6 0 1 1 −7 0.07 0.07 0 0 0 62 29 0 1 −2 5 5 −5 0.1 0.03 0 0 0.01 63 71 7 −1 5 4 −5 7 0.04 0.89 0 1 0.01 64 107 4 −3 5 −7 2 −3 0.11 0.96 0 1 0.01 65 2 7 −6 5 3 −4 0 0.1 0.07 0 0 0.01 66 21 −6 2 0 −7 8 −7 1 0.63 1 0.5 0.01 67 47 7 −7 −6 −6 −4 −6 0 0.63 0 0.5 0.01 68 74 −3 7 0 −5 −3 −6 0.89 0.07 1 0 0.01 69 7 −3 −1 1 −7 −4 6 0.13 0.97 0 1 0.01 70 45 −1 7 4 6 −2 5 0.87 0.07 1 0 0.01 71 14 −6 2 −2 −4 6 −5 1 0.17 1 0 0.01 72 57 3 0 −4 3 −2 −2 0 0.17 0 0 0.01 73 48 7 −7 1 −5 5 2 0.17 0.97 0 1 0.01 74 77 −3 2 0 −5 −6 7 0.18 1 0 1 0.02 75 94 3 −6 3 −6 6 −5 0.82 1 1 1 0.02 Games 2011, 2 264 Table 2. Cont. Rank Game f1 s1 f2 s2 f3 s3 P(In) P(Right) P(In) SPE P(Right) SPE MSD 76 6 −1 4 −7 3 7 0 0.13 0.13 0 0 0.02 77 68 −1 4 2 −4 −3 −2 0.18 0.96 0 1 0.02 78 72 0 0 7 0 −5 7 0.18 0.96 0 1 0.02 79 19 1 −6 4 5 −1 −6 0.8 0.03 1 0 0.02 80 69 −4 8 −3 4 1 −4 0.79 0 1 0 0.02 81 88 −5 6 −4 0 −4 0 0.79 0.46 1 0.5 0.02 82 65 −7 −4 −4 −5 −6 −5 0.96 0.29 1 0.5 0.02 83 119 2 −4 −7 −6 2 −3 0.29 0.93 0.5 1 0.02 84 20 0 −3 −4 6 3 3 0.1 0.2 0 0 0.03 85 95 1 3 5 2 −4 3 0.21 0.89 0 1 0.03 86 112 5 −6 1 −1 6 −1 0.21 0.61 0 0.5 0.03 87 78 6 −1 −5 6 8 8 0.75 1 1 1 0.03 88 100 6 −1 −7 −5 5 4 0.25 1 0 1 0.03 89 116 1 1 2 −1 2 −1 0.75 0.57 1 0.5 0.03 90 51 2 −7 −1 3 7 −6 0.27 0 0 0 0.04 91 76 7 −7 5 6 −7 −5 0.29 0 0 0 0.04 92 108 7 −8 4 0 2 2 0.29 0.93 0 1 0.04 93 52 8 1 0 3 2 2 0 0.3 0 0 0.05 94 1 −3 −7 −7 5 −3 6 0.2 0.93 0.5 1 0.05 95 34 2 2 3 −7 3 −7 0.67 0.43 1 0.5 0.06 96 73 5 −3 2 0 4 5 0.36 1 0 1 0.06 97 92 0 3 5 −2 0 5 0.86 1 0.5 1 0.06 98 22 −5 7 −6 −2 8 −6 0.33 0.17 0 0 0.07 99 38 2 8 6 1 −7 −7 0.63 0.03 1 0 0.07 100 41 1 7 3 2 −4 −6 0.63 0.03 1 0 0.07 101 75 −5 3 −6 5 0 1 0.36 0.11 0 0 0.07 102 61 0 2 −1 6 2 6 0.79 0.82 1 0.5 0.07 103 54 −1 3 7 1 −2 8 0.37 0.87 0 1 0.08 104 98 4 8 6 3 1 2 0.61 0.07 1 0 0.08 105 5 −2 −7 6 5 −2 8 0.83 0.77 0.5 1 0.08 106 80 −1 −2 6 0 −3 1 0.39 0.89 0 1 0.08 107 28 5 −7 −5 −7 8 0 0.57 1 1 1 0.09 108 33 4 0 −8 −8 5 5 0.57 0.97 1 1 0.09 109 113 4 −7 2 −4 7 −4 0.57 0.64 1 0.5 0.1 110 110 1 7 0 2 3 2 0.61 0.79 1 0.5 0.12 111 42 4 −3 −3 7 5 7 0.23 0.93 0 0.5 0.12 112 27 4 1 −6 1 7 5 0.5 1 1 1 0.13 113 50 −1 2 0 1 −5 −3 0.5 0 1 0 0.13 114 9 1 −7 −4 4 3 4 0.4 0.83 0 0.5 0.13 115 109 1 −2 6 6 −3 7 0.5 0.75 0 1 0.16 116 8 0 2 −5 1 2 1 0.47 0.83 0 0.5 0.16 Games 2011, 2 271 (5) Where r = 1 if and r = 0 otherwise; s = 1 if ands = 0 otherwise; q = −1 if P1 “misbehaved” and q = 0 otherwise. and “misbehaved” in the current setting is defined as the case where Player 1 chose In, although choosing Out would have yielded the best joint payoff and the best payoff for Player 2. Modeling the first mover’s behavior requires additional assumptions about his/her beliefs of the responder’s behavior. The current estimation assumes that Player 1 correctly anticipates Player 2’s responses. It further assumes that Player 1 has a similar utility function (excluding the reciprocity parameter q).11The choice function for each player is defined by Equation (2). 5.5. The Seven Strategies Model The Seven Strategies model is motivated by the regression analysis presented in the results section and the related distinct strategies explanation of the class effect suggested by Table 4. The model does not use the term “utility.” Rather, it assumes that the players consider the seven simple strategies described in Table 3. The probabilities of following the different strategies were estimated using a regression analysis with the restrictions that the sum of the weights equal 1 and the intercept = 0. The estimated parameters imply that the probability of In choice by Player 1 is given by: (6) The parameters and are the estimated probabilities of players choosing according to the distinct strategies. The different strategies are : choosing rationally, : maximizing self payoffs assuming that Player 2 chooses randomly, : maximizing self payoffs assuming that Player 2 chooses the worst payoff for Player 1, : maximizing the payoffs of the player who has the lower payoff of the two players, : maximizing the sum of the two players’ payoffs, and : choosing the option with the minimum payoffs difference. Each of those dummy variables are assigned the value 1 if the strategy implies In choice, 0.5 if the strategy implies random choice, and 0 if the strategy implies Out choice. The probability that Player 2 will choose Right is given by: (7) The parameters and are the estimated proportions of players choosing according to the relevant choice rule. The choice rules variables , and are as described above, and is choosing rationally but maximizing the other player’s payoff if rational choice implies indifference. The results (c.f. Table 6) show that the Seven Strategies model fits both Player 1’s choices and Player 2’s choices better than the other models presented above. 11We also estimated a version that includes individual difference in but this version did not improve the MSD score. Games 2011, 2 272 6. The Equivalent Number of Observations (ENO) In order to evaluate the risk of over fitting the data, we chose to estimate the ENO of the models by using half of the 120 games (the games played by the first cohort) to estimate the parameters, and the other 60 games to compute the ENO. Table 7 shows the results of this estimation. Table 7. The ENO of the baseline models. The parameters of the models were estimated based on the first set of 60 games, and the ENO scores were calculated based on the second set.12 MSD ENO Model Parameters Set player1 player2 player1 player2 Subgame Perfect Eq. (SPE) -- 1 2 0.0535 0.0524 0.0138 0.0073 0.87 0.88 6.16 10.38 Inequality aversion ~U[0, 0.01], ~U[0, 0.05], ~U[0, 0.05], ~U[0, 0.05], = 0.5, = 2.1 1 2 0.0339 0.0267 0.0120 0.0070 4.39 10.70 ERC = 0.40, = 0, = 0.13, = 0.63, = 0.05 1 2 0.0402 0.0333 0.0133 0.0068 3.38 11.39 CR = 0, = 0, = 0.1, = 0.6, = 0.05, = 0.05, = 2.9 1 2 0.0348 0.0236 0.0044 0.0038 5.02 28.56 Seven Strategies = 0.446, = 0.208, = 0.111 = 0.213, = 0.005, = 0.027 = 0.506, = 0.354, = 0.059, = 0.044, = 0.037 1 2 0.0100 0.0142 0.0030 0.0027 9.08 66.84 7. Summary The two choice prediction competitions, presented above, are designed to improve our understanding of the relative I mportance of the distinct psychological factors that affect behavior in extensive form games. The results of the estimation study suggest that the rational model (subgame perfect equilibrium) provides relatively useful predictions of the behavior of Player 2 (ENO = 7.5), and less useful predictions of the behavior of Player 1 (ENO < 1). This observation is in line with those of Engelmann and Strobel [24] who also noticed the significance of efficiency concerns in simple extensive form games. In addition, the results reveal deviations from rational choice that can be attributed to six known behavioral tendencies. These tendencies include: (1) an attempt to be nice (i.e., improve the other player’s payoff) when this behavior is costless; (2) Maxmin: an attempt to maximize the worst possible payoff; (3) Level-1 reasoning: selection of the best option under the assumption that the second agent behaves randomly; (4) Joint max: trying to maximize joint payoff; (5) Helping the weaker player; and (6) an attempt to minimize payoff differences. 12 An exception is the subgame perfect equilibrium predictions. Since this model is free of parameters, ENOs were computed for each of the two sets. Games 2011, 2 273 The results show only weak evidence for negative reciprocity (e.g., punishing unfair actions). Comparison of the current results to previous studies of negative reciprocity suggests that the likelihood of this behavior is sensitive to the context. Strong evidence for negative reciprocity was observed in studies in which the identity of the disadvantaged players remained constant during the experiment. Negative recency appears to be a less important factor when the identity of the disadvantaged players changes between games. We tried to fit the results with two types of behavioral models: Models that abstract the behavioral tendency in the agent's social utility function, and models that assume reliance on several simple strategies. Comparison of the different models leads to the surprising observation that the popular social utility models might be outperformed by a seven-strategy model. We hope that the competition will clarify this observation. Acknowledgements This research was supported by a grant from the U.S.A.–Israel Binational Science Foundation (2008243). References 1. Andreoni, J. Impure altruism and donations to public goods: A theory of warm-glow giving. Econ. J.1990, 100, 464-477. 2. Bolton, G.E.A comparative model of bargaining: Theory and evidence.Am. Econ. Rev.1991, 81, 1096-1136. 3. Fehr, E.; Schmidt, K.M. A theory of fairness, competition, and cooperation.Quart. J. Econ. 1999, 114, 817-868. 4. Loewenstein, G.F.; Thompson, L.; Bazerman, M.H. Social utility and decision making in interpersonal contexts. J. Person. Soc. psychol.1989, 57, 426-441. 5. Falk, A.; Fischbacher, U. A theory of reciprocity.Games Econ. Behav. 2006, 54, 293-315. 6. Rabin, M. Incorporating fairness into game theory and economics. Am. Econ. Rev. 1993, 83, 1281-1302. 7. Messick, D.M.; McClintock, C.G. Motivational bases of choice in experimental games. J. Exper. Soc. Psychol.1968, 4, 1-25. 8. Stahl, D.O.; Wilson, P.W. On players' models of other players: Theory and experimental evidence. Games Econ. Behav. 1995, 10, 218-254. 9. Güth, W.; Schmittberger, R.; Schwarze, B.An experimental analysis of ultimatum bargaining. J. Econ. Behav. Organ. 1982, 3, 367-388. 10. Forsythe, R.; Horowitz, J.L.; Savin, N.E.; Sefton, M. Fairness in simple bargaining experiments. Games Econ. Behav. 1994, 6, 347-369. 11. Berg, J.; Dickhaut, J.; McCabe, K. A comparative model of bargaining: Theory and evidence. Games Econ. Behav. 1995, 10, 122-142. 12. Fehr, E.; Kirchler, E.; Weichbold, A.; Gächter, S. When social norms overpower competition: Gift exchange in experimental labor markets. J. Lab. Econ. 1998, 16, 324-351. Games 2011, 2 274 13. Bolton, G.E.; Ockenfels, A.ERC: A theory of equity, reciprocity, and competition. Am. Econ. Rev. 2000, 90, 166-193. 14. Charness, G.; Rabin, M. Understanding social preferences with simple tests. Quart. J. Econ. 2002, 117, 817-869. 15. Erev, I.; Ert, E.; Roth, A.E.; Haruvy, E.; Herzog, S.M.; Hau, R.; Hertwig, R.; Stewart, T.; West, R.; Lebiere, C. A choice prediction competition: Choices from experience and from description. J. Behav. Dec. Making 2010, 23, 15-47. 16. Erev, I.; Ert, E.; Roth, A.E. A choice prediction competition for market entry games: An introduction. Games 2010, 1, 117-136. 17. Fischbacher, U. Z-Tree: Zurich toolbox for ready-made economic experiments. Exp. Econ. 2007, 10, 171-178. 18. Brandts, J.; Charness, G. Hot vs. cold: Sequential responses and preference stability in experimental games. Exper. Econ. 2000, 2, 227-238. 19. Brandts, J.; Charness, G. The strategy versus direct response method: A first survey of experimental comparisons. Exp. Econ. 2011, doi:10.1007/s10683-011-9272-x. 20. Erev, I.; Roth, A.E.; Slonim, R.; Barron, G. Learning and equilibrium as useful approximations: Accuracy of prediction on randomly selected constant sum games. Econ. Theory 2007, 33, 29-51. 21. Falk, A.; Fehr, E.; Fischbacher, U. On the nature of fair behavior. Econ. Inq. 2003, 41, 20-26. 22. Güth, W.; Huck, S.; Müller, W. The relevance of equal splits in ultimatum games. Games Econ. Behav. 2001, 37, 161-169. 23. De Bruyn, A.; Bolton, G.E. Estimating the influence of fairness on bargaining behavior. Manag. Sci. 2008, 54, 1774-1791. 24. Engelmann, D.; Strobel, M. Inequality aversion, efficiency, and maximin preferences in simple distribution games. Am. Econ. Rev. 2004, 94, 857-869. Appendix 1 Problem Selection Algorithm: The algorithm generates 60 games in a way that ensures that each of the 10 “game types” from Figure 2 (excluding “trivial games”) is represented at least once in the sample. In addition, it ensures selection of one “ultimatum-like game” (determined as a sub-class of “costly punishment” class in which f1 = s1), and one “dictator-like game” (a sub-class of the “strategic dummy” class in which Player 1’s payoffs are higher than Player 2’s payoffs). The rest of the 48 games in the sample are drawn randomly from the nontrivial games. Notations: f1, f2, f3, s1, s2, s3: The parameters of the game as defined in Figure 1 The basic set: {−8, −7, −6, −5, −4, −3, −2, −1, 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8} fmax = max(f1, f2, f3) smax = max(s1, s2, s3) fbest = 1 if f1 = fmax; 2 if f2 = fmax > f1; 3 if f3 = fmax > f1 and f3 > f2 sbest = max(s1, s2, s3) Games 2011, 2 275 sbest = 1 if s1 = smax; 2 if s2 = smax > s1; 3 if s3 = smax > s1 and s3 > s2 f(x) the payoff for Player 1 in outcome x (x = 1, 2, or 3) s(x) the payoff for Player 2 in outcome x (x = 1, 2, or 3) Trivial game: A game in which (f1 = fmax and s1 = smax) or (f1 = f2 = f3) or (s1 = s2 = s3) Select games: Draw the six payoffs repeatedly from the basic set until: For Game 1 (c.i: “common interest”): there is x such that f(x) = fmax and s(x) = s(max) For Game 2 (s.d: “strategic dummy”): s2 = s3 and f2 = f3 For Game 3(“dictator”): s2 = s3 and f2 = f3 and f2 > s213 For Game 4 (s.s: “safe shot”): f1 ≤ min(f2, f3) For Game 5 (n.d: “near dictator”): f1 = fbest For Game 6 (c.p: “costly punishment”): sbest = 1 and (f2 < f1 < f3 or f3 < f1 < f2) and s(fbest) = max(s2, s3), and s2 ≠ s3 For Game 7 (“ultimatum”): f1 = s1 and sbest = 1 and (f2 < f1 < f3 or f3 < f1 < f2), and s(fbest) = max(s2,s3), and s2 ≠ s3 For Game 8 (f.p: “free punishment”): sbest = 1 and (f2 < f1 < f3 or f3 < f1 < f2) and s(fbest) = max(s2,s3), and s2 = s3 For Game 9 (r.p: “rational punishment”): sbest = 1 and (f2 < f1 < f3 or f3 < f1 < f2) and s(fbest) < max(s2,s3) For Game 10 (f.h: “free help”): sbest > 1 and f1 > min(f2,f3) and f(sbest) = f1 For Game 11 (c.h. “costly help”): s1 < min(s2,s3) and f(sbest) = min(f1,f2,f3) and fbest > 1 For Game 12 (tr: “trust”): s1 < min(s2, s3) and min(f2,f3) < f1 < max(f2,f3) and f(sbest) = min(f1,f2,f3) and s(fbest) < max(s2,s3) For Game = 13 to 60: randomly select (with equal probability, with replacement) six payoffs from the “basic set,” if the game turns out to be “trivial” erase it and search again. Appendix 2 Instructions: In this experiment you will make decisions in several different situations (“games”). Each decision (and outcome) is independent of each of your other decisions, so that your decisions and outcomes in one game will not affect your outcomes in any other game. In every case, you will be anonymously paired with one other participant, so that your decision may affect the payoffs of others, just as the decisions of the other people may affect your payoffs. For every decision task, you will be paired with a new person. 13 Note: Dictator-like game is defined by this algorithm as a private case of strategic dummy in which the first mover’s payoffs are higher than the second mover. Games 2011, 2 276 The graph below shows an example of a game. There are “roles” in each game: “Player 1” and “Player 2”. Player 1 chooses between L and R. If he/she chooses L the game ends, and Player 1’s payoff will be $x dollars, and Player 2’s will be $y. If Player 1 chooses R, then the payoffs are determined by Player 2’s choice. Specifically, if Player 2 selects A then Player 1 receives $z and Player 2 receives $k. Otherwise, if Player 2 selects B then Player 1 gets $i and Player 2 gets $j. player1 Player2 $x $y $z $k $i $j When you make your choice you will not know the choice of the other player. After you make your choice you will presented with the next game without seeing the actual outcomes of the game you just played. The different games will involve the same structure but different payoffs. Before the start of each new game you will receive information about the payoffs in the game. Your final payoff will be composed of a starting fee of $20 plus/minus your payoff in one randomly selected game (each game is equally likely to be selected). Recall that this payoff is determined by your choice and the choice of the person you were matched with in the selected game. Good Luck! © 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/). L R B A