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Chimpanzee game theory

PNAS MANUSCRIPT 4/4/12

CLASSIFICATION: Biological Sciences, Social Sciences (Psychology, Economics)

TITLE: Experienced chimpanzees behave more game-‐theoretically than humans in simple competitive interactions

AUTHORS: Christopher Flynn Martin1, Rahul Bhui2, Peter Bossaerts2, 3, Tetsuro Matsuzawa1, Colin Camerer2, 3

1Department of Brain and Behavioral Sciences, Kyoto University Primate Research Institute, Inuyama, Aichi 484-‐8506 Japan, 2HSS Division, Caltech, Pasadena CA 91125 USA, 3Computational & Neural Systems, Caltech

CORRESPONDING AUTHOR:

Colin Camerer

Caltech

1200 E. Calif. Blvd.

Pasadena CA 91106

Email: [email protected]

MANUSCRIPT INFORMATION:

NUMBER OF TEXT PAGES (incl refs + fig. legends): 13

NUMBER OF FIGS.: 4 NUMBER OF TABLES: 0

ABBREVIATIONS: Nash equilibrium (NE)


ABSTRACT WORD COUNT: 245 words

CHARACTER COUNT: (49,000 limit, incl. text, spaces, Figs., tables, equations)

TEXT: 30,497 including acknowledgements, main text, footnotes, Fig. captions, and refs

FIGS. & EQUATIONS: 18,500

TOTAL: 48,997

AUTHOR CONTRIBUTIONS:

Designed research: CM, CC, PB, TM

Performed research: CM

Contributed new analyses: RB, PB

Analyzed data: RB, CM, PB, CC

Wrote paper: CC, CM, TM, RB


ABSTRACT: [245 words]

The capacity of humans and other animal species to think strategically about the likely payoff-‐relevant actions of conspecifics is not thoroughly understood. Games are mathematical descriptions of canonical ways in which joint choices determine interdependent rewards. Game theory is a collection of ideas about how strategic thinking and learning determine choice. We test predictions of game theory in three simple competitive abstract games with chimpanzee and human participants. Subjects make choices on a dual touch-‐screen panel and earn food or coin rewards. The chimpanzee and human protocols are closely matched on experimental procedures. The results show that aggregated frequencies of chimpanzee choices are very close to equilibrium points; and choices shift with reward changes almost exactly as predicted by equilibrium theory. Remarkably, chimpanzee choices are closer to the equilibrium prediction than human choices are. Chimpanzee and human choices also exhibit unpredictability on average from trial-‐to-‐trial (a property which is adaptive in competitive games), but individual subject-‐sessions show substantial predictability of choices from past choices and rewards. The results are generally consistent with the cognitive tradeoff hypothesis, which conjectures that some human cognitive ability inherited from chimpanzee kin may have been displaced by dramatic growth in the human neural capacity for language (and perhaps associated skills). As a result, chimpanzees retained the ability, slightly superior to humans, to adjust strategy competitively and in unpredictable ways, conforming remarkably closely to equilibrium predictions from game theory.


ACKNOWLEDGMENTS: The Ministry of Education, Sports, Technology, and Culture (MEXT) No. 16002001 and No. 20002001, JSPS-‐GCOE (A06, Biodiversity) (TM, CM) Tamagawa GCOE (CC), and the Gordon and Betty Moore Foundation. (CC, PB). Thanks to D. Biro for help in task design and M. Tanaka for help with building the touchpanel setup.


Humans are very social. Most of our closest great ape relatives are social too. However, the evolutionary origin and extent of sociality in all these species in ecologically important situations is still not well understood (even for humans). We take a step forward by observing and comparing behavior of humans and our closest extant relatives, chimpanzees (Pan troglodytes) as they make choices in incentivized interactions with similar experimental protocols (1).

Our experimental design is guided by the formal structure of game theory. Games are mathematical distillations of the basic action-‐reward structures of ecologically valid situations. Interactive experimental strategic games with reward payoffs have been used with primates and apes to assess prosociality and coordination of mutually beneficial actions (2-‐4), and in hundreds of human studies (5). A recent study (1) introduced a standardized experimental method for ‘uplinking’ game-‐playing protocols used in monkeys to humans.

We extend this experimental method to interactions that are direct and competitive: Joint actions always create one winner and one loser. Both players have two possible actions, pressing Left or Right touch-‐screen buttons. A Matcher player earns rewards if their choices match (e.g., Left-‐Left). A Mismatcher earns rewards if their choices mismatch (e.g., Right-‐Left). The interactions are also direct because the joint actions of both players immediately determine their reward through the shared touch-‐screen software, with no subject-‐experimenter interaction (cf. (1)) . Ours is the first experiment in which chimpanzees compete directly with other chimpanzees for competitively-‐determined rewards. We compare their outcomes to those from humans.

Game theory offers a benchmark of optimal performance: Players should guess accurately what others actually do, and should choose strategies with maximal expected reward given those guesses (a “Nash equilibrium”). Superior performance of this type depends on recognizing patterns in the opponent’s history of play, while hiding patterns in one’s own history.

The behavior of chimpanzees in these games is interesting because many important interactions in the wild have similar competitive reward structures. It has been hypothesized (6-‐7) that primates are well-‐adapted to such games, and may even be cognitively superior to humans (8). This prediction about game-‐theoretic play is interesting for social science because the lab and mixed field evidence suggests that human choices often deviate from game theory predictions in competitive games (5; Supplemental Online Material (SOM)).

Our experiments therefore address two questions: Does game theory accurately predict how chimpanzees play competitive games? And what similarities and differences are there between chimpanzee and human play?

Methods

Players made choices on pairs of computer touch-‐panel screens. Each screen displayed two identical stimuli (45mm light blue square buttons) on the left and


right sides of the screen (Fig. 1a). If both subjects chose the button on the right, or if both subjects chose the button on the left, then the “Matcher” was rewarded. If the subjects chose buttons on different sides, then the “Mismatcher” was rewarded. Payoff structures changed across three kinds of games (Fig. 1c). Pairs played 200 rounds of a game per session. Chimpanzees switched roles between sessions and played game 1 (symmetric matching pennies) for 10 sessions, game 2 (asymmetric matching pennies) for 5 sessions, and game 3 (inspection game) for 4 sessions. Pairs of humans played game 3 (inspection game) for 2 sessions, switching roles once. During the games, players were seated in an experimental booth facing away from each other (Fig. 1b). Universal feeding machines (Biomedica Model BUF-‐310), delivered 8 by 8mm cubes of apple (or coins in the case of humans) on a trial-‐to-‐trial basis. A single PC running a Visual Basic 6 program controlled all experimental events involving the two touch-‐screens and feeders.

Subjects

Six chimpanzees (Pan Troglodytes) at the Kyoto University Primate Research Institute voluntarily participated in the experiment. The subjects were three mother-‐offspring dyads: Ai and son Ayumu (ages 31 and 9); Chloe and daughter Cleo (30 and 9); and Pan and daughter Pal (27 and 9). These dyads were pair-‐matched with each other for all the experimental games. All participants had previously taken part in cognitive studies, including social tasks involving food and token sharing (10,11), and a dual touch-‐panel study in which they observed and copied the behavior of a conspecific model (12). However, the dual touch-‐panel competitive game in this study was novel to the participants. The 6 participants lived with 7 other chimpanzees in a semi-‐natural enriched enclosure (312), and were not food or water deprived during the period of the study. The use of the chimpanzees during the experimental period adhered to the Guide for the Care and Use of Laboratory Primates (2002) of the Primate Research Institute of Kyoto University.

16 human participants (13 female) participated in the experiment. The participants were students of Gifu University and Kyoto University. In a player-‐matched design, pairs of subjects were exposed to 50 training trials in each of the Matcher and Mismatcher roles, to gain familiarity with the task and payoff structure. They then played 200 rounds in each of the two roles. The experimental design and procedure was identical that of the chimpanzee task, except that coins (1 yen pieces) were dispensed from the feeders instead of apple pieces, and an opaque barrier was placed between the stations to prevent collusion. In order to maximize parity between chimpanzee and human conditions, the human subjects were given minimal verbal instructions prior to the task (they were told only “try to gain as many coins as possible”), and were not told that they were to play a competitive game against each other. After completing the task, participants were given 500-‐yen (approximately 6 US dollar) gift-‐cards. The ethical committee of Primate Research Institute of Kyoto University approved the use of human subjects.

Results:


The behavioral results can be summarized using three types of statistics: Frequencies of Left (L) and Right (R) choices, (P(L) and P(R) respectively); statistical tests for dependence of current choices on previous history; and rewards accumulated.

There is little evidence that experience across sessions generally moves behavior closer to the predicted equilibrium play. Data are therefore aggregated across trials. Figs. 2a-‐c plot the frequencies with which the different specific chimpanzee subjects chose each action in the three games, along with the Nash equilibrium (NE) prediction. The theory predicts that for Matchers, P(R)=.50 in all three games. The theory also predicts that Mismatchers will vary P(R) from .50 to .75 to .80 across the three games. Note that these predictions are extremely counterintuitive because they state that behavior will only change across the games for the player whose own rewards do not change.

Fig. 2d plots the cross-‐subject averages from all trials for all three games. This plot highlights whether the chimpanzees’ behavior changed across the three payoff conditions as predicted by equilibrium theory. The Matcher P(R) rates are indeed close to half on average. Even more strikingly, the Mismatchers’ P(R) frequencies do shift in close numerical proximity to those predicted rates (overall frequencies are 0.50 to 0.73 to 0.79, within .01 of the predicted rates on average).

Fig. 2c also plots choice frequencies for the human group in the Inspection game (game 3), using the closely matched low-‐information protocol. Across both roles, the absolute average human deviation is .135, which is consistent with other human data (see SOM Fig. S1). The human deviation is much higher than the chimpanzees’ average deviation of .03.

Figs. 2a-‐c also plot the predictions of two bounded rationality theories that allow parametric imperfections in either response to payoffs (QRE) or in depth of strategic guessing about other players (CH). These theories have been shown to explain when human behavior deviates from NE in a wide variety of games (14-‐17). However, the chimpanzee behavior tends to deviate only slightly from NE in the direction of these alternative theories of bounded rationality (see SOM for details).

The next empirical question is whether choices depend predictably on previous choices. In many domains humans who are motivated to randomize independently actually switch too frequently, as if they regard long streaks as “not random”, and switch to balance recent subsample frequencies (18-‐19).

A simple measure of independence is how often each subject in each session switches choices (from L to R, or R to L on successive trials). Figs. 3a-‐d show histograms of the deviation between the actual numbers of switches in each subject-‐session, and the number switches expected if choices were statistically independent from trial to trial, for the inspection game. The average deviation in switching is close to zero (i.e., overall, choices are roughly temporally independent) but there is a large percentage of individual sequences with too few or too many switches (for


chimpanzees and humans, only 33% and 63% are within their 95% confidence intervals (CI)). In the first two games, played only by chimpanzees, there is a little more variability in switching (see SOM Fig. S2), which means that the chimpanzees’ extensive experience across games is apparently necessary for producing approximate equilibration and partial-‐independence.

Switching rates are only one measure of predictability. A more demanding test uses logit regression to see whether choices in period t depend on any previous choice and reward variables. These regressions do show more predictability; most typically, a subject’s choice depended statistically on their opponent’s previous two choices (SOM). Chimpanzee predictability is somewhat higher than for human choices, but the differences are not reliably significant.

The final analysis is about payoffs won. First, note that Nash equilibrium play is not generally equivalent to joint payoff maximization. Choosing the equilibrium strategy is optimal only if opponents are playing their equilibrium strategies. One group of players can deviate from NE but earn higher payoffs (if others are not optimizing).

To show payoff implications of behavior, Figs. 4a-‐b show histograms of average payoffs for each subject-‐session, for both chimpanzees and humans in the inspection game (see SOM for other games). The expected Nash equilibrium payoffs are also shown. There are no payoff differences between chimpanzee and NE benchmarks. However, for humans payoffs are higher for Matchers. Furthermore, in some previous experiments in a full-‐information protocol with complete information about all possible payoffs, humans deviated even further from Nash equilibrium (and Matchers earned even higher payoffs; see SOM).

The behavioral payoff effect can also be seen in heat maps overlaid on the average choice frequencies for the Matcher (Fig. 4c) and Mismatcher (Fig. 4d). In the vicinity of the actual frequencies, the Matcher payoffs increase as the Mismatcher frequency P(R) decreases. This observation means that the Matcher in the human games are actually earning more than their chimpanzee counterparts on average (1.16 vs. .85 per trial, where NE predicts .80). Their earnings are higher because the human Mismatchers choose R less often than theory predicts, which benefits the Matchers. However, the Mismatcher payoff map (Fig. 4d) exhibits no such effect (the payoffs are 1.03 vs 1.00, where NE predicts 1.00). For Mismatchers, the payoff isoquants are shaped differently and so the difference in the Mismatchers’ P(R) frequencies for chimpanzees and human do not change their own payoffs very much.

Finally, there is an interesting unpredicted difference in response times (RTs): Mismatchers’ RTs are significantly slower than Matcher RTs (see SOM Fig. S3) for all species and all games. In the Inspection game, the median difference within each subject is 78ms (paired Wilcoxon signed-‐rank test, p=0.03) for chimpanzees and 206ms (p=0.003) for humans. The RT difference might be due to a default response to match the behavior of another animal rather than to mismatch (perhaps a byproduct of the capacity for physical imitation).


In an inspection game with the same payoffs as ours, differential neural activation was observed in Mismatcher brains, reflecting extra computational effort (20). The difference was attributed to better reward prospects when the Mismatcher takes into account the influence that her winning could have on future play of the opponent. The matcher has lower influence value and may therefore spend less time computing it. However, this hypothesis cannot account for the role differential in the symmetric matching game.

Discussion

We studied strategic interactions between chimpanzees, using a simple touch-‐screen protocol, to measure how well competitive behaviors match the predictions of game theory. In humans, behavior in such games has been used successfully to calibrate different degrees of strategic thinking (5, 17) and prosociality (21,22), correlate prosociality with cultural factors, and to show the strategic effects of psychiatric disorders (23-‐25).

Chimpanzees’ choice frequencies are extremely close on average to those predicted by equilibrium analysis (NE). Behavior varies with changes in payoff structure across three games almost exactly as theory predicts. In addition, the number of switches from one trial to another are, on average, close to the number of switches predicted if choices were independent. However, individual subject-‐sessions often exhibit too few or too many switches, and are often significantly dependent on the previous two choices and outcomes.

In one game we compared chimpanzee and human behavior in a low-‐information protocol designed to match what the two species know as closely as possible. Remarkably, human play is actually further from the NE predicted frequencies than the chimpanzee play is (in the direction of equal choice frequency).1 The average deviation in pooled data is about .033 for chimps and .135 for humans (compared to deviations of .05-‐.10 in other human experimental studies; see SOM).

As a result of this behavioral difference, human players in one of the two player roles actually earn more than their chimpanzee counterparts (because a deviation by one player benefits the opponent). Statistical dependence from trial to trial is a little higher for chimpanzees than for humans, but the difference is not reliably significant.

Since non-‐human subjects cannot verbally report on their decision-‐making processes, we could not directly test whether the chimpanzees comprehended that rewards were contingent on joint actions. Perhaps due to this difficulty, the issue of 1 The SOM reports the first comparison of boundedly rational game theory models, used to study humans, to nonhuman behavioral data. Perhaps surprisingly, the boundedly rational models discussed in SOM offer little general improvement over NE in explaining the chimpanzees’ choices.


non-‐human subjects’ understanding of jointly determined actions in games has been largely avoided in previous studies that have pitted non-‐human animal subjects against computer algorithms (26-‐27 and conspecifics (28).

We do have several reasons to believe that the subjects were aware of the social nature of the task. First, one subject was occasionally observed looking away from her monitor and toward the location of her opponent during delays in the game, suggesting that she was waiting for the other individual to respond. Such behavior is atypical for these chimpanzees, since almost all of their experimental histories are comprised of non-‐social touch-‐panel tasks in which their attention is directed toward only their own monitor. Second, before this experiment, the same subjects did a social imitation task in the same location and using the same apparatus (12), increasing chances that they were aware that the touch-‐panels were interconnected. Third, the chimpanzees might have noticed that feedback sound cues and actual rewards were synchronized between pair members.

But the finding that chimpanzee subjects did play close to the Nash Equilibrium, in itself suggests (implicit or explicit) understanding of joint-‐action contingencies. Adjustment dynamics that do not rely on such understanding typically do not lead precisely to Nash Equilibrium mixtures.

There are two broad hypotheses consistent with the facts that chimpanzee choices are closer to theory than human choices. The first is that there is some species-‐specific confound in the experimental protocols which would bring the chimpanzee and human results closer together if the confound could be eliminated. The second is that chimps actually are better at competitive interaction, and hence learn to better approximate equilibrium choices.

Indeed, there are some obvious confounds. The humans might have been less motivated to work for coins than the chimpanzees are for food. The chimpanzees are also genetically related mother-‐child pairs (though we would therefore expect them to be less competitive and more willing to maximize joint payoffs than the human strangers). And the chimpanzees are highly experienced with working memory tasks and touch-‐screen interaction (their RTs were much faster than the humans’). It is possible that similarly-‐experienced humans would behave like chimpanzees.

Unfortunately, it is extremely difficult to control for species confounds based on reward and experience. However, note that the type of deviations in human play from NE observed in this experiment are quite typical, rather than unusual. Many human lab experiments and some competitive field settings show that human choice frequencies are typically partway between equal mixing and the NE prediction, just as in this experiment (e.g., 20,31, SOM).

It is well known that chimpanzees have physical advantages over humans—they are stronger and faster—which have fitness value in their environment. That chimpanzees are close to NE (in our experiment) and humans are further from NE


(here and elsewhere), suggests that a “cognitive advantage” hypothesis is a potential explanation for the chimp-‐human difference.

One cognitive advantage emerges in lab tasks requiring memory of briefly exposed (210 ms) spatial displays of Arabic numerals. Highly-‐trained young chimpanzees are better at remembering ordered spatial locations than humans (32). Thus, Matsuzawa (9) hypothesizes that chimpanzees are better at such tasks than humans are, because human evolution degraded certain memory skills to make room in the brain for development of human language-‐related skills. Chimpanzees also appear to have cognitive advantages over humans in recognizing upside-‐down faces (33) and voice-‐face matching (34).

In the wild, chimpanzees also engage in many strategic interactions such as predatory stalking (35), young chimpanzee wrestling (36), border patrolling (which is very much like the Inspection game) (37) and raiding crops from human farms (38). Because competitive reward games are common in chimpanzee life, evolutionary theory predicts that chimpanzees would have developed cognitive adaptations to detect patterns in opponent behavior and to create undetectable predictability in their own behavior. More generally, chimpanzees are capable of strategic thinking in cooperative hunting (39), sneaky copulation (40), future planning (41), and many elements of theory of mind computation (42). Some have argued that the capacity to randomize effectively evolved because primate predatory behavior and routine social interaction selects for unpredictability in counter-‐strategies (6,7). Experiments also show that chimpanzees are better at competitive tasks than at comparable cooperative ones (43).

Finally, the results have implications for what game theory might describe most accurately. Mathematical game theory is usually implicitly thought of as an "eductive" model of human reasoning about strategy. An alternative view is that game theory is "evolutive"—i.e., it describes the limiting the outcome of a long history of evolutionary adaptation or strategic "trial and error" learning by animals (including humans) (29-‐30), or the result of social protean (6) processes. Our results support the evolutive view.


Fig. captions

Fig. 1: The trial progression, touch-‐panel setup, and game payoffs. (A) Two players interacting through touch-‐panel screens are shown a self-‐start key (circle) at the beginning of each trial. After both players press the start key, two choice actions are displayed, represented by squares on the left and right sides of the screens. After both players make a choice, rewards are dispensed to the winner and both players get feedback about their opponent’s choice. (B) Payoff matrices for the 3 games in this study. (C) Subjects sit perpendicular to each other facing touch-‐panel screens that are embedded in the walls of the experimental booth.

Fig. 2: Frequencies of R choices for all pairs in both roles show that chimpanzee behavior is close to game theoretic (NE) predictions. (A) Chimpanzees in the symmetric MP game, (B) chimpanzees in the asymmetric MP game, and (C) chimpanzees and humans in the Inspection game. Predictions of Nash equilibrium (NE), cognitive hierarchy (CH), and Quantal Response Equilibrium (QRE) trajectory (for a range of response sensitivities λ) are marked. (D) Average behavior over all chimpanzees compared to NE for all three games.

Fig. 3: The number of choice switches from trial-‐to-‐trial show approximate statistical independence for both species. Numbers were calculated by taking the observed number of switches in a game and subtracting the number of switches expected from the observed probability mixture in that game assuming independent choices. Chimpanzees had (A) Matcher average = -‐13, sd = 20, and (C) Mismatcher average = -‐3, sd =16. Japanese humans had (B) Matcher average = 6, sd = 15, and (D) Mismatcher average = 1, sd = 17. Chimpanzee and human averages are within 95% confidence intervals (CIs) but only 33% (chimpanzee) and 63% (human) of individual subject-‐sessions are within their session-‐specific CIs (see SOM Fig. S2 for CI construction).

Fig. 4: In the Inspection game, chimpanzees make close to NE payoffs while Matcher humans earn more than NE. (A) Histogram of Matcher payoff averages. (B) Histogram of Mismatcher payoff averages. (Expected average payoffs given NE play shown with dotted vertical lines in A and B). (C) Matcher payoff isoquant heat map for the Inspection game. Lighter regions show higher payoffs; thin lines show isoquants (sets of frequencies that give equal payoffs to Matchers). Empirical relative frequencies of choices are overlaid. The human choice frequencies are closer to a higher-‐value isoquant than chimpanzee frequencies, indicating that human Matchers earn more. (D) Mismatcher payoff isoquant heat map for the Inspection game, overlaid with relative frequencies of choices. The chimpanzee and human choice frequencies are different, but neither is closer to a higher-‐value isoquant, indicating that human Mismatchers do not earn more than chimpanzees.


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Inspection Game

Matcher Mismatcher

Matcher Mismatcher

Matcher Mismatcher

A) Task B) Game Payoffs

Trial start, self-start stimuli presented.

Left/Right choices appear. Players make choice.

Food reward dispensed to winner. Opponent’s choice shown as blinking stimulus for 2000ms.

C) Setup

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Inspection Game

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Nash Equilibrium

All Games (Chimps Only)

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�CH

QRE


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Chimp game theory

Supplemental Online Material

Experienced chimpanzees behave more game-‐theoretically than humans in simple competitive interactions

Chris Martin, Rahul Bhui, Peter Bossaerts, Tetsuro Matsuzawa, Colin Camerer

Contents

I. Discussion of Nash equilibrium and boundedly rational theories

II. Previous lab and field evidence from humans

III. Temporal dependence regression

IV. An interesting difference between Matcher and Mismatcher response times (RTs)

V Histograms of payoffs across subjects and games

I. Discussion of Nash equilibrium and boundedly rational theories

Nash equilibria in simple games like ours satisfy two properties: (a) Players accurately guess what strategies (or mixtures of strategies) others will play; and (b) players choose the optimal strategy with the highest expected payoff—their “best response”—given their (accurate) beliefs from (a).

In competitive games like those we study, the Nash equilibrium mixtures are proportions which lead to (weak) mutual best responses. For the Inspection game, denote NE Matcher P(Left) by p* and Mismatcher P(Left) by q*. The values of p* and q* will satisfy 2p*=2(1-‐p*) and 4q*=1(1-‐q*), or p*=.5, q*=.2.

Note that the mixture probabilities for one player depend only on the payoffs of the other player. For example, if the (Left, Left) Matcher payoff was X (rather than 4), the NE mixtures would be p*=.5 (i.e., the Matcher is not predicted to change

Chimp game theory

behavior at all) and q*=1/(X+1) (i.e., the Mismatcher is predicted to choose Left less often, as if to deny the Matcher the high X payoff). This is a counter-‐intuitive feature. Any learning algorithm that is guided by received payoffs (such as reinforcement learning) will therefore adapt, at least in the short-‐run, in the wrong direction.

Besides learning theories (1), two prominent types of boundedly rational theories have been explored since the mid-‐1990s (and see (2) for newer theories; but cf. (3)).

“Quantal response equilibrium” (QRE) retains assumption (a) but relaxes the optimization condition (b) to allow “softmax” stochastic imperfections in perceiving and responding to payoff differences (4). This can be seen as a biologically plausible hybrid that combines the formal precision of assumption (a) with a reasonable psychophysical constraint on the ability to produce a perfectly optimizing response. QRE typically uses a single parameter (λ) to encode sensitivity of responses; when the parameter is at its maximal value then QRE is equivalent to NE.

Another class of “cognitive hierarchy” (CH) (or level-‐k) theories accounts for limited strategic thinking by maintaining the optimality condition (a) and relaxing the assumption of accurate beliefs (b) (5-‐8). Simple level-‐0 subjects choose using an intuitive heuristic with no cognition about likely choices of others. Higher level subjects guess accurately what lower-‐level subjects are likely to do and optimize. More levels of strategic thinking generally correspond to a more accurate model of the social environment and higher rewards. In the Camerer, Ho, and Chong (7) variant the frequency of subjects at each level corresponds to a Poisson distribution with mean and variance of τ.

Our paper includes the first test of this wide range of rational and boundedly rationality game theory models using nonhuman behavioral data. Text Figure 2 shows the QRE prediction set. It is graphed as a continuous curve spanning values of λ=0 (random play, P(Left)=.5 for both players) to NE (λ→∞). CH predictions are graphed for a single value, τ=1.5 (which fits many experimental and field data sets reasonably well). NE, QRE, and CH all make the same prediction in symmetric matching pennies. For the other two games, the QRE and CH are actually not more accurate than NE for the chimpanzees. However, QRE fits the human Inspection game data more closely.

These results are surprising because QRE and CH typically reliably fit human data as accurately as NE (correcting for their extra degree of freedom, of course). The fact that the chimpanzees are so close to NE in general, and their behavior is not well described by QRE and CH, also supports our conclusion that the experienced chimpanzees seem to have some ability to choose NE mixtures which is apparently superior to that of humans, at least in these simple games.

II. Previous lab and field evidence from humans

Many studies with human subjects have examined how well behavior corresponds to NE predictions. This section is abridged from a longer discussion in Camerer (1)

Chimp game theory

(chapter 3). The empirical background is important for establishing that, for humans, there are typically substantial deviations between NE predicted frequencies and human choices, and that choices are typically not independent over time either.

The earliest studies were conducted in the 1950s, shortly after many important ideas were consolidated and extended in Von Neumann and Morgenstern’s (1944) landmark book. John Nash himself conducted some informal experiments during a famous summer at the RAND Corporation in Santa Monica, California. Nash was reportedly discouraged that subjects playing games repeatedly did not show behavior predicted by theory: “The experiment, which was conducted over a two-‐day period, was designed to test how well different theories of coalitions and bargaining held up when real people were making the decisions. … For the designers of the experiment … the results merely cast doubt on the predictive power of game theory and undermined whatever confidence they still had in the subject.”(9)

In the 1960s similar early experimental results were discouraging. However, subjects were often not financially motivated and sometimes played computerized opponents. One striking result (10) showed that people were capable of mixing game-‐theoretically in a special setting: In their experiment subjects chose first, picked an explicit distribution of strategies (a truly mixed strategy), then the computer observed their mixture and selected a best response. The only way for subjects to win is to choose the equilibrium mixture (since any other choice will be instantly exploited by the computer). In this special setting, they were able to hone in very precisely on NE mixing (65% were playing the exact mixture by the end of a five-‐game sequence).

These discouraging results turned attention away from mixed-‐strategy games. Game theorists began to activity research games with private information, and repeated games. A revival of interest in competitive games began with O’Neill’s (11) elegant design, a 4x4 game played 500 periods. He reported overall frequencies of play that were much closer to those predicted by NE than the early c. 1960s studies.

However, O’Neill’s data were reanalyzed by Brown and Rosenthal (12). They used more careful tests to show that choices often depend strongly on previous choices and previous outcomes (i.e., independence is violated). (The tests they used are the same ones we conducted, reported below in this Supplemental material section III). Others closely replicated these results in games similar in structure.

While there are clearly reliable deviations between NE and human choice, it is notable that the deviations are often small in magnitude, and across different strategies and games there is a substantial correlation between NE predictions and actual choice frequencies. Intuitively, if one strategy X is predicted to be chosen more often than another strategy Y, then X is almost always chosen more often. A glimpse of several studies illustrating the accuracy of this theory-‐behavior

Chimp game theory

correspondence comes from a figure in Camerer (1), reprinted with our human data added as Fig. S1 below.

The correlation between predictions and behavior is .84. The mean absolute deviation between predictions and data is .067. Furthermore, keep in mind that predictions usually depend on auxiliary assumptions like neutrality toward risk; if those assumptions are violated then the behavior should be a little different than predicted. These results are therefore quite positive in establishing some predictive value of Nash equilibrium predictions. A notable set of experiments with a similarly positive conclusion is Binmore et al. (2001). One lesson from these data, then, is that under some experimental conditions behavior close to Nash equilibrium choice can occur.

Figure S1: A cross-‐study comparison of actual strategy choices frequencies with predictions from Nash mixed-‐strategy equilibrium (MSE). Each data point is one strategy from one study. The light blue circles represent human data reported in this study. See Camerer (2003, chapter 3) for details.

The next important wave of research sought to test whether typical findings in highly-‐controlled lab settings were also evident in naturally-‐occurring settings

Chimp game theory

where randomization is expected. (The quality of field-‐lab correspondence is often of interest, since economists hope to discover theories which work equally well in highly-‐controlled, artificial lab settings and in field settings with similar features. Camerer (13) discusses the ideas and debate about this topic within experimental economics (see also Heckman and Falk (14)). He also surveys the best available studies. Those studies generally show good correspondence between patterns in field data and patterns in closely-‐matched lab settings.) Most of the studies use zero-‐sum competitive sporting events, in which repeatedly playing the same strategy predictably—such as always serving to the same side of the service box in tennis—would typically be noticed and exploited by an opponent.

Early studies of tennis (15, 16) and soccer (17-‐21) found that players’ frequencies corresponded fairly closely to those predicted by an NE analysis, and that choices were also roughly independent. The Palacios-‐Huerta and Volij study is particularly impressive because they are able to match data from actual play on the field from one group of players (in European teams) with laboratory behavior of some players from that group (although not matching the same players’ field and lab data). Importantly, PHV also found that high school students as a group behaved less game theoretically than the soccer pros, except that high school students with substantial experience playing soccer were much closer to game-‐theoretic. However, a reanalysis by Wooders (22) later showed a higher degree of statistical dependence than shown by PHV.

Levitt, List, and Reiley (23) compared behavior of poker, bridge and soccer players (from US teams) in abstract games conducted off the field. They find substantial deviation from NE and violations of independence. However, the soccer players playing for US teams in the sample might have been less likely than their counterparts in PHV to actually randomize independently in the field, so it is not clear their players are randomizing less in the lab than in the field. (The key point here is that the best players, and perhaps the best randomizers, play in soccer-‐crazy Europe rather than in the US.)

Another field study used a simple lottery played in Sweden by about 50,000 people per day, over seven weeks (24). Participants in the “LUPI” lottery paid 1 euro to pick an integer from 1 to 99,999. The lowest unique positive integer won 10,000 euros. The symmetric NE has a dramatic shape, with numbers from 1 to 5513 being chosen almost equally often, but with slightly declining probability (i.e., 1 is most probable, 2 is slightly less probable, etc.). A bold prediction is that numbers above 5000—a range that includes 95% of all available numbers-‐-‐ should rarely be chosen. The actual behavior is not far from the NE prediction and converges over the seven weeks toward the statistical prediction of the NE prediction (e.g., the mean, variance, and other statistics all move toward NE). In a scaled-‐down laboratory replication behavior is even closer to NE, even before there is much feedback to learn from.

The general picture from these decades of field and lab studies is that people are capable of approximating Nash mixtures (and certainly of moving in their direction with learning), but that substantial deviations are to be expected. For simple matrix

Chimp game theory

games like those we study, an absolute deviation of 0.05-‐0.10 between NE prediction and actual frequency is to be expected for human subjects. The average absolute deviations in the Inspection Game 3 are 0.05 and 0.22 , which are comparable to these guesses from many other studies. For chimpanzees, the average across all roles and games is 0.033 (compared to 0.135 for humans).

Table S1: Absolute deviations between NE predictions and average overall frequencies by role, game and species

Deviation (to 2 digits) (All are negative dev’ns)

Matcher Mismatcher

Sym Chimps 0 0

Asym Chimps 0.14 0.02

Insp Chimps 0.03 0.01

Insp Humans 0.05 0.22

III. Temporal dependence regression

The histograms in the text (Figure 2) show the results of a simple test comparing the number of switches in each subject’s time series of L-‐R responses to the number of expected assuming statistical independence. The switching rate histograms for the game-‐role pairs from the symmetric and asymmetric matching pennies payoff games are shown in Figure S2 below. They show a little more deviation from random independent play.

Individual 95% confidence intervals for each subject-‐session uses the mixture probabilities for that subject-‐session, which imply the mean and variance of the number of runs under the hypothesis of independence (the basis for a Wald-‐Wolfowitz runs test: Let the number of L choices = n, R choices = m. Then the mean = 2nm/(n+m) + 1 and variance = 2nm(2nm – n – m)/((n + m)2 (n + m – 1))) . The number of runs is asymptotically normal, providing a 95% confidence interval for each subject-‐session with that mean and variance. These 95% confidence intervals were averaged to produce the confidence intervals shown in Figures 3a-‐d and S2a-‐d.

Our version of the Brown-‐Rosenthal (BR) equation is

!!!! = !! + !!!! + !!!!!! + !!!!!!∗ + !!!!∗ + !!!!!!∗ + !!!! + !!!!!!

where !! is the player’s choice, !!∗ is the opponent’s choice, and !! denotes the winner in period t. This logit regression tests for a variety of temporal dependence effects.

Table S2 shows the percentage of role-‐subject session time series which yield BR coefficients that are significant at the 5% level, for each group of coefficients. For

Chimp game theory

example, for human matchers (role m), 50% of the 16 subjects’ regressions indicated significant dependence of a player’s choices on the previous two opponent choices. A joint test for all effects of previous outcomes and choices (the bottom row of the Table) indicates that in almost all cases some of the coefficients are significantly nonzero, when tested together jointly.

Fig. S2a-‐d: Switch deviations for chimpanzees in symmetric matching pennies (top row) and asymmetric matching pennies (bottom row). Matchers (left) and Mismatchers(right) are plotted separately.

Importantly, however, the human and chimp percentage differences in significant dependence rates (shown in the right-‐hand columns) are generally close together. Z-‐tests of the difference in percentages across chimps and humans do not indicate any strong differences which persist for both roles.

Table S2: Percentage of significant temporal dependence effects for chimp and human inspection games

Chimp game theory

Table S2 shows corresponding percentages (averaging across both Matcher and Mismatcher roles) for the symmetric and asymmetric matching pennies games, and for Brown and Rosenthal’s (1990) original analysis of human data (playing 500 trials). Both human data results, and the chimp inspection game, are comparable in the rates of significant dependence.

Table S3: Percentage of significant temporal dependence effects for all chimp and human games, and original Brown-‐Rosenthal (1990) human data

IV. An interesting difference between Matcher and Mismatcher response times (RTs)

There are some interesting patterns in response times (RTs). Each point Figure S3 shows the pair of averaged RT for each subject, when playing as both Matcher (x-‐axis) and Mismatcher (y-‐axis). One evident result is that Mismatcher RTs are always longer (i.e., slower) than Matcher reactions. One theory to account for this difference is that, in equilibrium, the Mismatchers have to choose unequal portions of L and R responses. However, the slight RT difference is even evident in the symmetric games, where L and R responses are predicted to be equally common (and actually are, empirically) for both Matcher and Mismatcher.

Chimp game theory

Figure S3: Average reaction times for individual subjects when playing as Matcher (x-‐axis) and Mismatcher (y-‐axis).

We speculate that the RT differential might indicate some kind of highly evolved (and conserved across species) speed for physical imitation of movements, compared to anti-‐imitation.

A paired sign test for differences in medians for intra-‐subject RTs rejected the hypothesis of equal medians (median difference 127ms, p<10-‐8) even when including an outlier (Matcher RT 1748ms, Mismatcher RT 1086ms).

V Histograms of payoffs across subjects and games

Chimp game theory

Figure S4: Average payoff distributions across all chimpanzee-‐sessions in Matching Pennies and Asymmetric Matching Pennies games. (These correspond to Fig. 4a-‐b in the main text.)

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