Journal of Comparative Psychology
Manuscript version of
Like Chimpanzees (Pan troglodytes), Pigeons (Columba livia domestica) Match and
Nash Equilibrate Where Humans (Homo sapiens) Do Not
Yosuke Hachiga, Lindsay P. Schwartz, Christopher Tripoli, Samuel Michaels, David Kearns, Alan Silberberg
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• National Institutes of Health
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Like Chimpanzees (Pan troglodytes), Pigeons (Columba livia domestica) Match and
Nash Equilibrate Where Humans (Homo sapiens) Do Not
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Yosuke Hachiga, Lindsay P. Schwartz, Christopher Tripoli,
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Samuel Michaels, David Kearns, and Alan Silberberg
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American University
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Author Note
Yosuke Hachiga, Lindsay P. Schwartz, Christopher Tripoli, Samuel Michaels, David
Kearns, and Alan Silberberg, Department of Psychology, American University
Correspondence concerning this article should be addressed to Yosuke Hachiga,
Department of Psychology, American University, Washington, DC 20016.
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Contact: hachi@american.edu
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This work was funded by NIH 1 R15 MH109922 to American University and by JSPS
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Overseas Research Fellowships to Yosuke Hachiga.
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Abstract
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Martin, Bhui, Bossaerts, Matsuzawa and Camerer (2014) found that chimpanzee pairs competing
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in matching-pennies games achieved the Nash Equilibrium whereas human pairs did not. They
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hypothesized this outcome may be due to: (a) chimpanzee ecology producing evolutionary
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changes that give them a cognitive advantage over humans in these games; and (b) humans being
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disadvantaged because the cognition necessary for optimal game play was traded off in evolution
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to support language. We provide data relevant to their hypotheses by exposing pairs of pigeons
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to the same games. Pigeons also achieved the Nash Equilibrium, but did so while also
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conforming with the Matching-Law prediction on concurrent schedules that choice ratios covary
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with reinforcer ratios. The cumulative-effects model, which produces matching on concurrent
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schedules, also achieved the Nash Equilibrium when it was simulated on matching-pennies
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games. The empirical and simulated compatibility between Matching-Law and Nash-
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Equilibrium predictions can be explained in two ways. Choice to concurrent schedules, where
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matching obtains, and choice in game play, where the Nash Equilibrium is achieved, may reflect
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the operation of a common process in choice (e.g., reinforcer maximization) for which matching
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and achieving the Nash Equilibrium are derivative. Alternatively, if matching in choice is innate
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as some accounts argue, then achieving the Nash Equilibrium may be an epiphenomenon of
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matching. Regardless, the wide species generality of matching relations in nonhuman choice
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suggests game play in chimpanzees would not prove advantaged relative to most species in the
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animal kingdom.
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Keywords: Nash Equilibrium, matching pennies, matching law, chimpanzees, humans,
pigeons
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Like Chimpanzees (Pan troglodytes), Pigeons (Columba livia domestica) Match and
Nash Equilibrate Where Humans (Homo sapiens) Do Not
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Dennett (1983) has suggested that theorizing about mental-function levels in humans and
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other animals is done by two types of researchers, killjoys and romantics. Killjoys place humans
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at the apex of any ranking of mental function, viewing their mental capacities as unrivalled in the
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animal kingdom; romantics, on the other hand, often see equivalences in mental function
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between humans and some other animals.
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We have no quarrel with this dichotomy and see the literature on comparative cognition
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as offering cases where one or the other view seems sensible and useful. However, there is a
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small literature that falls outside the Dennett dichotomy. This literature claims to show mental
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function in chimpanzees that is superior to that seen in humans in terms of short-term memory
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(Inoue & Matsuzawa, 2007), self-control in intertemporal preferences (Rosati, Stevens, Hare, &
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Hauser, 2007) and rational maximizing (Jensen, Call, & Tomasello, 2007).
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We have opposed these claims, arguing that, in one way or another, they overreach
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(Genty, Karpel, & Silberberg, 2012; Silberberg & Kearns, 2009; Smith & Silberberg 2010).
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These references once challenged all published claims of mental function outside the Dennett
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dichotomy of which we were aware; however, matters have recently changed. Martin, Bhui,
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Bossaerts, Matsuzawa, and Camerer (2014) offer credible demonstrations that when pairs of
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chimpanzees or humans make repeated binary choices in competition for reinforcement,
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chimpanzees come closer to achieving the so-called Nash Equilibrium (NE)—a point where each
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participant in a competitive game cannot gain by a unilateral change in choice strategy if the
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strategy of the opponent remains unchanged. To familiarize the reader with this work, we
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summarize their report below.
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Martin et al. (2014) procedure and results
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Martin et al. (2014) exposed pairs of chimpanzees, sitting in proximity, to three
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matching-pennies games in which each subject chose on its own touch screen between
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illuminated left- and right-side squares. In all games, one member of the pair, the matcher, was
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rewarded with apple only when it chose the same square on its touch screen as the other member;
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the mismatcher, on the other hand, was rewarded only when it chose the square opposite that
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chosen by the matcher.
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The payoff structure differed in each game for the matcher: In the 2,000-trial symmetric
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game, matches resulted in a single apple cube; in the 1,000-trial asymmetric game that followed,
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left-side matches yielded three apple pieces and right-side matches yielded one; and finally, in
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the 800-trial inspection game, left-side matches produced four apple pieces vs. one apple piece
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for right-side matches. For mismatchers in these games, correct mismatches yielded one apple
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piece in the symmetric game and two apple pieces in the other two games regardless of the side
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chosen.
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Martin et al. (2014) also exposed pairs of Japanese and African humans to the inspection
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game for 200 trials as a matcher and 200 trials as a mismatcher, replacing food with money
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amounts. The Japanese responded to the touch screens used with chimpanzees whereas the
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Africans chose between bottle tops that were facing up or down. When bottle tops or side
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preferences matched, the matcher was paid; when not, the mismatcher was paid. Payoffs on left
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and right sides or up and down bottle tops mirrored those of the chimpanzees described above for
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the inspection game.
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Martin et al. (2014) calculated for each game the relative frequency of right-side or top-
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up choices for chimpanzees and humans and compared it to the equilibrium point where neither
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competitor could gain by a change in strategy if the competitor’s strategy remained unchanged.
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This point, the NE, was approximated by the across-trials average performance of each
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chimpanzee pair; however, the mismatching humans playing the inspection game chose the right
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side or top up roughly 60% of the time, a value well short of the NE prediction of 80%.
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Martin et al. (2014) speculated that the chimpanzee-human performance difference they
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obtained was due to two factors. Regarding humans, evolutionary specializations such as
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language and categorization skills required that other cognitive capacities, such as those used in
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playing the inspection game, be compromised. They called this notion the “cognitive-tradeoff
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hypothesis”, and it may account for why humans failed to achieve the NE on this task.
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Regarding chimpanzees, their dominance-mediated social environment may have created a
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fitness value for strength and speed in chimpanzees that translates into “a possible parallel
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cognitive advantage… (p. 4)”. This added fitness value also may have aided chimpanzee
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performance.
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As noted above, our goal is to test psychological accounts that fall outside the Dennett
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(1983) dichotomy. Martin et al.’s (2014) theorizing about the causes of chimpanzee superiority
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in matching pennies certainly qualifies. Unfortunately, we see no way to test these accounts
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individually: Are chimpanzees better at matching pennies than humans because humans have
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been dumbed down by trading off task-specific cognitive function for language? Or are
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chimpanzees smarter than humans because chimpanzees have a task-specific parallel cognitive
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advantage? Either hypothesis rationalizes chimpanzee outperformance of humans.
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Nevertheless, we do see a way these hypotheses can be tested jointly. The key is to choose a
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species to match pennies so cognitively inferior to humans that readers will doubt its superior
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performance, assuming it obtains, could plausibly be explained by either of Martin et al.’s
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accounts.
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Toward this end, we have selected pigeons to match pennies. Our argument is that even
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if human mental facility specific to matching pennies has been compromised by the advent of
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language, the consequences of that compromise could not credibly be so severe as to rationalize
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why pigeons’ performance is superior to humans’. If pigeons are superior to humans in
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matching pennies, this conceptual damage also extends to the notion of chimpanzees’ parallel
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cognitive advantage because theorists that could believe chimpanzees have a task-specific
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cognitive advantage would be less likely to extend this claim to pigeons.
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This report compares chimpanzee approximations to the NE from Martin et al. (2014)
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with those from pigeons on procedures like the ones they used with chimpanzees. As it happens,
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pigeons have already been tested on a matching-pennies task by Sanabria and Thrailkill (2009).
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They played a variety of matching-pennies games, but only two were also played in Martin et al.
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(2014), the symmetric and asymmetric game. Based on the last session of performance in each
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game from Sanabria and Thrailkill, pigeons appear to have matched chimpanzees in
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approximating the NE in the symmetric game, but not in the asymmetric game.
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Obviously, these data suggest that challenging Martin et al.’s (2014) two-theory account
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of task-specific cognitive superiority in chimpanzees by showing pigeons are also superior to
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humans may be empirically ill considered. However, we persist because we speculate that
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pigeons’ failure to approximate the NE as well as chimpanzees may have been due to Sanabria
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and Thrailkill’s (2009) procedure differing in a critical way from Martin et al.’s: Sanabria and
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Thrailkill had pigeons play the role of matcher and mismatcher in the same session by splitting
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each session into halves and reversing matcher-mismatcher roles between each half of the
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session; Martin et al. did not do this. By creating a multiple schedule of games, Sanabria and
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Thrailkill may have produced interaction between components. Such an outcome is often seen in
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pigeon experiments where multiple schedules are used (e.g., Buck, Rothstein, & Williams,
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1975). Moreover, we note that Martin et al. were scrupulous in ensuring procedural identity,
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insofar as was practical, between the games played by chimpanzees and humans. Given our
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speculation about multiple-schedule interaction and given that there are other procedural
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differences between how Martin et al. and Sanabria and Thrailkill arranged their games, we
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repeat pigeon game play in this report, this time mimicking Martin et al.’s design closely.
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An expected result: Matching
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Given the results of Sanabria and Thrailkill (2009), we are unsure pigeons will achieve
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the NE in our experiment. However, there is another result we do expect: a high correlation
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between changes in reinforcement ratios induced by various games and the choice ratios they
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produce when these ratios are plotted in logarithmic space (Baum, 1974). This outcome defines
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the Matching Law.
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This is not a bold prediction. Give any animal, save humans (Lowe & Horne, 1985), two
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manipulanda, each associated with a schedule of reinforcement, matching nearly always obtains
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(Davison & McCarthy, 2016). In fact, the rapidity and regularity of matching relations in choice
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has led some investigators to argue that matching in choice is innate (e.g., Gallistel, King,
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Gottlieb, Balci, Papachristos, Szalecki, & Carbone, 2007). If our expectation of matching is
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realized, and if pigeons also achieve the NE, interesting questions emerge depending on how the
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reader explains matching.
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One popular explanation is that matching is due to maximizing some dimension of
reinforcement such as its rate or immediacy (Rachlin, Green, Kagel & Battalio, 1976; Shimp,
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1966). To us, such a notion seems broadly compatible with the definition of the NE because the
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NE defines a choice ratio where prospects for reinforcement cannot be improved upon unless
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there is a change in a competitor’s choice allocation. In this case, achieving the NE and
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matching likely share a common maximization process. Alternatively, some accounts of
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matching attribute a matching outcome literally to a matching process (Herrnstein, 1970;
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Gallistel et al., 2007). If this is the case, achieving the NE becomes an epiphenomenal correlate
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of adherence to a matching process.
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There is a long, contentious and unresolved literature on whether matching as outcome is
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due to maximizing or a matching process. Regardless of which account is superior, explaining
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achieving the NE in terms of matching seems advantaged because only this account could
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accommodate choice in simple concurrent schedules as well as in game competition because
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choice in concurrent schedules subsumes choice in competitive play. The converse is not the
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case. This perspective raises the possibility that matching pennies in nonhuman animals may be
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a competition in name only. Indeed, what they may perceive from a cognitive perspective may
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not be a competition, but a standard concurrent schedule such as those seen in other matching
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studies.
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Experiment 1: Matching pennies in pigeons
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Method
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Subjects
Four, one-year-old male Silver King pigeons that were used in a prior version of this
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experiment served as subjects. This earlier experiment presented single-hopper presentations
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that differed in their durations. Unfortunately, the design of the grain hopper failed to ensure
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long hopper durations provided continuous access to grain. For this reason, multiple hopper
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presentations of briefer duration were in the present version of the experiment. This permitted
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grain to flow to the hopper opening at a constant rate.
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Pigeons were maintained at 80% of their free-feeding weights and received supplemental
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feeding of mixed grains when their weights fell below this level. They lived in an animal colony
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where a 14-h light, 10-h dark cycle was maintained (lights on at 8 AM). Water was freely
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available in their home cages. Sessions were conducted five days per week, usually in the
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morning.
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Apparatus
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Training and testing occurred in two Med Associates (St. Albans, VT) operant pigeon
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chambers (55.8 x 55.8 x 40.6 cm) that were enclosed in sound-attenuation chests. The chambers
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had front and rear aluminum walls, Plexiglas side walls and ceiling, and a floor composed of
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stainless-steel bars. Three evenly spaced cue lights were mounted in the front wall 10 cm above
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the floor. Below the center cue light was a 2- by 3-cm (height by length) oval opening to a
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mixed-grain food dispenser, the lower lip of which was 2.5 cm above the floor. A shielded, 100-
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mA house light was mounted on the ceiling at the front of the chamber. Three 2.5-cm keys that
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could be transilluminated from behind the front wall were arrayed 5.5 cm apart, center to center,
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21.5 cm above the chamber floor.
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Procedure
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Pigeon pairs, one rewarded as a matcher and the other as a mismatcher, were exposed in
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paired chambers to the symmetric, asymmetric and inspection games with reinforcer-amount
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ratios conforming with the values described earlier for each of these games. The individual
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reinforcer amount was 3 s of access to mixed grain. Different reinforcer-amount ratios were
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created by multiple presentations of the hopper with 2 s intervening between successive
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presentations. In the symmetric game, matches resulted in a single hopper presentation; in the
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asymmetric game that followed, left-side matches produced three hopper cycles and right-side
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matches produced one; and in the inspection game, left-side matches produced four hopper
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cycles vs. one for right-side matches. For mismatchers in these games, correct mismatches
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yielded one hopper cycle in the symmetric game and two in the other two games regardless of
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the side chosen.
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At the start of each session, the house light and center key light were illuminated with
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white light. When one pigeon pecked the illuminated center key, it extinguished. When the other
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pigeon pecked its center key, its light extinguished and then the side keys in both chambers were
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illuminated with white light. If both pigeons pecked the same side key, the matcher earned a
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reward. If the pigeons pecked opposite side keys, the mismatcher earned a reward. Once both
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pigeons pecked a side key, the grain hopper provided the reward associated with those choices,
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and the house light for the rewarded pigeon was extinguished for the duration of the hopper
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cycle. A 30-s intertrial interval followed each presentation of a reward cycle. The center key
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light was then illuminated again for the next trial unless the 100-trial, session-ending
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contingency had been met.
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Pigeons were randomly assigned to the roles of matcher and mismatcher. They then
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played the symmetric game, followed by the asymmetric game, and finally the inspection game.
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The pigeons were then switched to the opposite role and played the three games again in the
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same order. Each game lasted no fewer than 12 sessions and no more than 25 sessions. A game
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ended between those numbers if both pigeons in a pair had no data varying more than 15% from
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the moving average of the last five sessions. All data are based on the last 200 trials of a game.
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Results and Discussion
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Pigeons approximate NE in matching pennies
Figure 1 portrays the outcomes during the last 200 trials for the humans and chimpanzees
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in Martin et al. (2014) and for the pigeons in this experiment. The data from Martin et al. were
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digitized from Figures S5 and S6 from their supplemental online data set by using Plot Digitizer
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version 2. The Figure presents the probability of a right choice by the mismatcher as a function
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of the probability of a right by the matcher. The symmetric, asymmetric and inspection games
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are arrayed in three panels from top to bottom. The NE, and preference data for the average
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pigeon, chimpanzee and human are identified by different symbols (see key in bottom panel).
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The error bars through the pigeon data define one standard error of the mean. To a first
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approximation, it looks like humans in the inspection game (bottom panel) are farther from the
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NE than pigeons and chimpanzees, and that averaged across all games, pigeons do as well as
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chimpanzees.
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Figure 2 tests this conclusion by quantifying the Euclidean distance from the NE across
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all games for all species. As is apparent in the inspection game, the only game in which all
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species are compared, humans do worse than pigeons and chimpanzees; and to our eyes, the
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average chimpanzee’s performance is like that of the average pigeon (right pair of bars in Figure
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2). However, it should be noted that averaging masks a possible between-game species
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difference: chimpanzees more closely approximated the NE in the asymmetric game whereas
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pigeons performed better than chimpanzees in the inspection game. These results are not
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amenable to statistical test because Martin et al. (2014) only present population averages.
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Martin et al. (2014) used a runs test to measure how closely chimpanzees and humans
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adhered to the optimal pattern of stochastic response emission. The purpose of this measure was
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to address the possibility that a player achieves the NE in terms of preference level even though
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the player performed poorly because of how response sequences occurred. For example, in the
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symmetric game, optimal performance is to respond stochastically with half of the responses
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being to each alternative. But what if one player responded, say, ten times to the left alternative
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and then ten times to the right alternative to achieve the NE preference-level statistic? In such a
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case, preference level would not properly characterize the inadequacy of this response pattern
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because surely such response sequencing would be co-opted by a competitor.
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We did not collect runs data in this study, and therefore cannot use it to test for stochastic
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choice allocation in game play. In our view, such a test lacks discriminative power (see Hachiga
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& Sakagami, 2010). To illustrate the problem, imagine a player in the symmetric game achieved
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the NE over 20 choices by responding twice to an alternative before switching. Such a pattern
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would generate ten runs over 20 choices. It would pass the statistical expectation for stochastic
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choice and could also achieve the NE even though choice was obviously patterned. Instead, we
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have created an alternative statistic—payoffs per trial—to evaluate whether successive choices
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were approximately independent. The Appendix shows how this statistic was calculated for all
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three games based on the notion that choice is perfectly stochastic. For all games, Table 1
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compares for each pigeon as a matcher and a mismatcher the expected number of payoffs per
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trial vs. the number obtained. As is apparent, the differences between rewards received and
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rewards expected by probabilistic responding were minor. These results support the view that
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pigeons approximated the NE with successive choices being largely independent of each other.
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This comparison is presented graphically in Figure 3. This figure shows that most of the
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differences are small between the number of payoffs predicted by stochastic responding vs. those
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obtained. Based on a paired t test (df = 47), there is no difference between the mean expected
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and obtained payoffs per trial (-.0004). The 95% confidence interval is -.022 to .021. Despite
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this outcome, we acknowledge that the force of our argument is weakened by our failing to
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record runs data as done by Martin et al. (2014). Had those data been collected, it would have
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been possible to compare directly sequential dependencies as measured by runs between our
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study and theirs and assess whether any differences appeared in this measure between Martin et
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al. and the present report.
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One way to defend Martin et al.’s (2014) parallel cognitive-advantage and cognitive-
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tradeoff hypotheses would be to deny our claim pigeons were in a competitive game analogous
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to that given to Martin et al.’s chimpanzees. In Martin et al., chimpanzees were in the same
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enclosure and each could turn its head to watch how the other chimpanzee responded and, in
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principle, could use that information to make their matching or mismatching choices. Such an
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option was unavailable to the competing pigeons because each was in an opaque enclosure.
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Chimpanzees could see they were competing; pigeons could not. While this difference could
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have mattered, it manifestly did not: Neither matching nor mismatching chimpanzee used the
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choice of its competitor in a trial to guide its choice; had either done so, it would have created a
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different data set, one where the informed subject made few errors and the other made many.
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A final between-study difference warrants mention. Martin et al. (2014) cued the choice
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of each chimpanzee’s opponent after each trial by flashing the opponent’s choice on each
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touchscreen. We viewed such cuing as unnecessary because trial outcome (food or no food)
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provided the same information as the flashing light in Martin et al. Many studies show that when
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a discriminative stimulus is redundant, it does not control behavior (e.g., Reynolds, 1961). For
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this reason, choices of each pigeon’s competitor were not signaled as done in Martin et al.
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Pigeons approximate predictions of Matching Law
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We have shown that pigeons, like chimpanzees, approximately achieve the NE in
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matching pennies. In addition, we predicted pigeon performances, and by implication,
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chimpanzee performances might also be described as due to equating choice ratios to the ratios
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of the food amounts those choices produce (matching). To test this prediction, we need target
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studies from the matching literature where food amounts and payoff likelihoods have been varied
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between two alternatives. We use Schneider (1973) for this purpose. In Schneider’s study,
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pigeons chose between two keys, each of which was associated with a Variable-Interval (VI)
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schedule that, depending on the condition, delivered different ratios of food pellets occasionally
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for a choice. He found that preference depended on both reinforcer rate and reinforcer
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magnitude (number of pellets) delivered for each alternative. He represented choice behavior in
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terms of the following generalized matching equation:
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log (R1/R2) = a log (r1/r2) + b log (q1/q2) + c
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where R, a, r, b, q and c respectively stand for responses (R), a parameter for reinforcer-
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frequency ratios (a), reinforcement frequency (r), a parameter for reinforcer amount ratios (b),
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quantity of food per reinforcer (q), and bias (c). The subscripts 1 and 2 distinguish between the
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choice alternatives. The a and b parameters accommodate slope differences in the plot whereas
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the c parameter accommodates differences in Y-intercept values.
(1)
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The line of best fit that results from applying Equation 1 to individual pigeon
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performances in Experiment 1 is presented in Figure 4. Do the matching-pennies data from
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Martin et al. (2014) and Sanabria and Thrailkill (2009) look like the data in Figure 4, showing
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matching-like outcomes?
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Figure 4 is based on the actual choice-ratio, reinforcer-frequency and reinforcer-amount
data for Experiment 1. To evaluate Martin et al. (2014) and Sanabria and Thrailkill (2009) in
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terms of the same measures requires the simplifying assumption that all choice is stochastic in
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these two studies. The aggregate amount of reinforcement was estimated from choice
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probabilities. To calculate the expected reinforcement frequency to the left and right keys for the
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matcher, the respective frequency of left and right choices by the matcher was multiplied by the
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respective frequency of left and right choices by the mismatcher. These data were used to
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calculate a left-key/right-key reinforcer ratio. A similar exercise was done in the case of the
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mismatcher, accommodated to reflect that reinforcers for the mismatcher require choice
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mismatches.
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To ensure comparability to our pigeon results, we make the same assumptions for that
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data set. Figure 5 presents data based on these estimates for Experiment 1 (top panel), Martin et
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al. (middle panel) and Thrailkill and Sanabria (bottom panel). All three data sets look like data
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from choice matching studies.
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Experiment 2: Application of the Cumulative-Effects Model to Matching Pennies
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We have argued that chimpanzees and pigeons may view matching pennies not as a
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competitive game, but as a choice procedure akin to those that populate the operant literature on
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concurrent performances. Certainly, the fact that approximating the NE in a matching-pennies
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games also approximates matching relations raises this as a possibility. Arguing that
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chimpanzees in Martin et al. (2014) were simply adhering to the Matching Law does damage to
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their claims of a chimpanzee cognitive advantage in matching pennies because many species
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throughout the animal kingdom match. Presumably, many are less cognitively gifted than
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chimpanzees. And based on what we have seen so far, it may well be the case that animals that
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match also achieve the NE.
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In this second experiment, we demonstrate how a cognitively simple matching model
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from the operant choice literature succeeds in achieving the NE in matching-pennies games.
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This result complements the analytic data from Experiment 1 by now synthesizing the NE
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outcome of chimpanzees by means of the Matching Law.
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There are several models to choose from that predict matching. For example, the copyist
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model (Tanno & Silberberg, 2012) simulates matching even though it posits that an animal
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simply reproduces what was reinforced before. In principle, it could be used to test whether it
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also predicts the NE in matching-pennies games. Nevertheless, we have selected the cumulative-
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effects model for simulation (Davis, Staddon, Machado, & Palmer, 1993) because it produces
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matching in choice, and it is easier to program on a computer playing matching-pennies games.
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The cumulative-effects model posits that an animal calculates the number of reinforcers
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provided by a choice alternative divided by the number of responses made to produce them. The
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alternative which has the higher reinforcers-per-response value garners the next response. Davis
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et al. (1993) demonstrated that this decision rule produces matching. To test whether it also
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achieves the NE in the Martin et al. (2014) games, twelve 400-trial matching-pennies games
358
were played by matcher/mismatcher stat animals that followed the cumulative-effects model in
359
the symmetric, asymmetric and inspection games.
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The stat animals were programmed to begin a simulation by responding with equal
361
probability to each alternative. Once both alternatives had delivered simulated reinforcers to
362
both stat animals according to the requirements of a given game, a reinforcers-per-response
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statistic was calculated for each alternative, and each subsequent choice went to the simulated
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key with the higher value. To illustrate the operation of this simulation, imagine that up to the
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current trial the matcher had obtained 160 reinforcements from making 200 responses on the left
17
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key (reinforcement rate = 160/200 or 0.8), the key that provides a reinforcer four times larger
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than that provided by the right key. If the next left response produced a reward, the left-key
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reinforcement rate was updated to 164/201 or 0.82. On the other hand, if the matcher chose the
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right key and got a four-fold smaller reward, the numerator of the right-key reinforcement rate
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was only increased by one.
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Figure 6 presents the outcomes of these simulations in the format used in Figure 1. As is
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apparent, the cumulative-effects model does an excellent job of achieving the NE in the games
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used by Martin et al. (2014). These outcomes support the view that chimpanzees and pigeons
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may achieve the NE by following a simple matching rule.
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Figure 7 produces the matching data (top panel) and expected-payoff data (bottom panel)
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using the axes of Figures 4 and 3, respectively. As is apparent, simulated performances are
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consistent with matching predictions and the expected-payoff data, generated by perfectly
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stochastic choice allocation, look quite like those seen in real pigeons. The mean difference
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between the expected and obtained payoffs/trial was .003. The 95% confidence interval was
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from -.008 to .014.
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General Discussion
Experiment 1 shows that pigeons and chimpanzees are approximately equivalent in their
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capacity to achieve the NE in the competitive games used by Martin et al. (2014), and that both
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species do better in this task than humans. Martin et al.’s theses explaining the superiority of
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chimpanzee-to-human game play are now in question unless the reader extends these notions to
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explain the superiority of pigeon-to-human game play. Moreover, choice allocation in our
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pigeon study was well described by a matching equation. In Experiment 2 we showed by
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simulation that a model that produces matching in concurrent schedules also achieves the NE
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when applied to matching-pennies games. Such an outcome may mean that matching-pennies
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games in nonhumans should not be viewed as competitions to be evaluated by proximity to the
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NE. Instead, they are choice procedures that should be described in terms of the Matching Law.
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These outcomes invite consideration of whether achieving the NE should be viewed as
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due to matching as process or that both matching and achieving the NE are derived from some
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other behavioral principle such as reinforcement maximization. As noted earlier, stochastically
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allocated choice produces matching in standard concurrent schedules when the choice ratio
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approximates matching relations (Rachlin et. al., 1976). This outcome seems compatible with
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the definition of the NE as the choice ratio that optimizes income unless competitor preferences
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change. A second possibility is that matching itself is responsible for achieving the NE, and the
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kinds of measures used by NE theorists fail to evidence this fact. This argument gains credence
400
if the reader accepts the arguments made by Gallistel et al. (2007) that matching as process is
401
innate. In either case, the third possibility to consider—that pursuit of the NE accommodates
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concurrent-schedule performances and competitive game play—does not seem credible. The
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obvious problem is that on conventional concurrent schedules, subject choice does not face a
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competitor.
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Summary, implications and conclusions
406
In this report, we provide data relevant to Martin et al.’s (2014) hypotheses regarding
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superiority in matching pennies by chimpanzees by showing: (a) chimpanzees are likely hardly
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unique in achieving the NE in matching-pennies games; (b) what is viewed as a competitive
409
game by Martin et al. (2014) may, in terms of its psychology, be closer to conventional
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concurrent schedules as used in behavior analysis; and (c) chimpanzees and pigeons achieving
411
the NE in matching pennies may be an epiphenomenal correlate of the operation of a matching
19
412
rule such as that provided by the cumulative-effects model (alternatively both outcomes may be
413
due to another process such as reinforcement maximization). In any case, the finding of
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matching in pigeons in matching pennies (Experiment 1) that can be synthesized by simulation
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(Experiment 2) supports the expectation that many species should achieve the NE in matching
416
pennies because virtually all species tested to date match choice ratios to reinforcer ratios.
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If the reader accepts our view, the findings in this report weaken confidence in the notion
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of Martin et al. (2014) that chimpanzees have task-specific cognitive advantages over humans
419
and possibly many other species in playing matching-pennies games. In our view, Martin et al.’s
420
account is one-sided for advancing unusual explanations for superior game play in chimpanzees
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in the face of an obvious alternative explanation for their data: Humans perform more poorly not
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because chimpanzees are cognitively advantaged in matching pennies or because humans have
423
traded off cognitive capacity for language, but because of the inherent greater complexity of
424
human mental function.
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Evidence of human mental complexity compromising human performances abounds in
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prospect theory where we see many examples of cognitive errors unlikely to be found in other
427
animals (Kahneman & Tversky, 1979). Prospect theorists do not advance these errors as
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examples of how humans are more limited than other animals. Although they are typically silent
429
about nonhumans, we think it likely they would attribute the results of Martin et al. (2014) not to
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the limits of human mental function when compared to other animals, but to their expanse.
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The literature on probability matching is a classic illustration of where humans do more
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poorly than other animals. In a probability-matching task, a reward is assigned with unequal
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probability to each of two choices. Humans often behave sub-optimally by matching their
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relative choices to the relative outcomes they produce. For example, if a random 70% of the
20
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rewards are given for the choice “left” and the rest are given for the choice “right”, humans will
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tend to choose “left” 70% of the time. Few other species tested on this task make this error—
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virtually all their choices are “left” (Fantino & Esfaniari 2002; Graf, Bullock, & Bitterman,
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1964; Wilson, 1960). To our knowledge, no probability-matching theorist advances the idea that
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humans do more poorly than other animals because of their cognitive limitations. Instead, they
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point to added features of human mental function such as the gambler’s fallacy (Gold, 1997)
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which cause human performance to be inferior (e.g., Tversky & Kahneman, 1974). Data from
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Derks and Paclisanu (1967) support this interpretation. They showed through age five, children
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maximize on probability-matching tasks, just like nonhuman animals; only when older do they
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probability match. If one accepts as plausible that human adults are more cognitively advanced
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than children, it seems reasonable to interpret maximizing to simpler mental processes evident in
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children and probability matching to a cognitive error such as the gambler’s fallacy that is
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present in adults.
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If we attribute inferior matching-pennies performance to the greater complexity of human
449
mental function, predictive parsimony obtains. For example, no longer is explanation burdened
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by the fact that accepting chimpanzees have a parallel cognitive advantage leads to the
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implication that pigeons also have a parallel cognitive advantage. Instead, this perplexing
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outcome for theory disappears: Human mental sophistication causes humans to perform more
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poorly in matching pennies than pigeons. It is also resolves the singular status of Martin et al.’s
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(2014) data being outside the Dennett (1983) dichotomy. They are not.
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457
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Appendix
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Suppose p is the left-choice probability of the mismatcher and q is the left-choice probability of
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the matcher. The expected number of reinforcers per trial in each game was calculated as
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follows:
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Symmetric Game
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For the matcher, p * 1 + (1 - p) * 0 = p for Left, and p * 0 + (1 – p) * 1 = 1 – p for Right.
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For the mismatcher, q * 0 + (1 – q) * 1 = 1 – q for Left, and q * 1 + (1 – q) * 0 = q for Right.
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Asymmetric Game
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For the matcher, p * 3 + (1 – p) * 0 = 3p for Left and p * 0 + (1 – p) * 1 = 1 – p for Right.
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For the mismatcher, q * 0 + (1 – q) * 2 = 2 – 2q for Left and q * 2 + (1 – q) * 0 = 2q for Right.
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Inspection Game
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For the matcher, p * 4 + (1 – p) * 0 = 4p for Left and p * 0 + (1 – p) * 1 = 1 – p for Right.
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For the mismatcher, q * 0 + (1 – q) * 2 = 2 – 2q for Left and q * 2 + (1 – q) * 0 = 2q for Right.
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Table 1
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Figure Captions
Figure 1. Probability of a right in mismatchers as a function of the probability of a right
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in matchers in three different competitive games. Pigeon data are from this experiment. All
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other data are from Martin et al. (2014). See text for other details.
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Figure 2. Distance from the NE for pigeons, chimpanzees and humans in symmetric
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game, asymmetric game and inspection game. Chimpanzee and human data are from Martin et
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al. (2014). Bars on far, right side of panel are all-game averages for pigeons and chimpanzees.
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Figure 3. A graphic presentation of the results in Table 1. This figure presents the
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relative frequency of the differences between the expected payoff per trial that would be
562
produced by stochastic response emission vs. the results obtained based on pigeon choice
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frequencies.
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Figure 4. The panel’s X and Y axes respectively present log choice ratios as a function of
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the log ratio of the total amount of food each choice alternative provided in a session. Individual
566
performances of pigeons in each role (matcher or mismatcher) in each type of matching-pennies
567
game from Experiment 1 populate the graph. The best-fit function is based on Equation 1. The
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R2 statistic in the panel defines the proportion of the individual-subject data variance
569
accommodated by the best-fit function.
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Figure 5. Matching analysis of the results of the current experiment and those for
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chimpanzees from Martin et al. and for pigeons from Sanabria and Thrailkill based on Equation
572
1. See text for other details.
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Figure 6. The probability of a right in mismatchers as a function of the probability of a
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right in matchers in three different matching-pennies games based on the cumulative-effects
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model. In all games, the closed circles representing simulated animal performances overlap
27
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with the NE (open squares). Error bars through these data points define one standard error of the
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mean. Mean performances for chimps from Martin et al. (2014) are also presented.
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Figure 7. Log ratio of choices to each key in the cumulative-effects model simulations
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for all games from Martin et al. for matchers and mismatchers (top panel). The bottom panel
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presents the relative frequency distribution of the differences between expected and obtained
581
payoffs per trial in these simulations.
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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7