M Equilibrium A Theory of Beliefs and Choices in Games Jacob K. Goeree and Philippos Louis1 April 20, 2021 Abstract We introduce a set-valued solution concept, M equilibrium, to capture empirical regularities from over half a century of game-theory experiments. We show M equilibrium serves as a meta theory for various models that hitherto were considered unrelated. M equilibrium is empirically robust and, despite being set-valued, falsifiable. We report results from a series of experiments comparing M equilibrium to leading behavioral-game-theory models and demon- strate its virtues in predicting observed choices and stated beliefs. Data from experimental games with a unique pure-strategy Nash equilibrium and multiple M equilibria exhibit coordi- nation problems that could not be anticipated through the lens of existing models. arXiv:1811.05138v2 [econ.TH] 19 Apr 2021 Keywords: M equilibrium, µ equilibrium, profect Nash equilibrium, noisy decisionmaking, imperfect beliefs 1Goeree: AGORA Center for Market Design, UNSW. Louis: Department of Economics, University of Cyprus and AGORA Fellow. We gratefully acknowledge funding from the Australian Research Council (DP190103888). We thank the Co-Editor Jeffrey Ely and three anonymous reviewers for their extremely constructive andin- sightful suggestions and colleagues as well as seminar participants for their helpful feedback. Special thanks to Vai-Lam Mui for stressing the importance of alternatives to fixed-point models and to Lucas Pahl who taught us much of what we know about the geometry of optimization. The \M" terminology was inspired by https://en.wikipedia.org/wiki/M-theory. 1. Introduction The Nash (1950) equilibrium has been the centerpiece of game theory since its introduction more than seventy years ago. Its simple formulation and broad applicability have helped game theory gain the central role originally envisioned by von Neumann and Morgenstern (1944). The Nash equilibrium is one of the most commonly used constructs in economics, with applications in other social sciences, evolutionary biology, computer science, and philosophy. However, the Nash equilibrium fares poorly when confronted with data because of its strong assumptions and stark predictions. Players are subsumed to have perfect foresight, i.e. they anticipate others' choices, and optimize perfectly, i.e. they best respond to their beliefs. Ran- domization occurs only when players are indifferent. As a result, Nash equilibrium implies a high-degree of homogeneity, e.g. in symmetric games with a unique equilibrium, players' choices are identical (and if the equilibrium is in pure strategies, any deviation causes a zero-likelihood problem). Even in games where players' choices are predicted to differ, their beliefs are iden- tical. A survey of over half a century of experimental game theory results reveals that neither the assumptions underlying Nash equilibrium nor its predictions are validated empirically. 1.1. Empirical Motivation for M equilibrium There is overwhelming evidence from the laboratory that subjects do not play Nash equilibrium. Their choices are often far away from Nash predictions.1 Moreover, observed choice behavior is heterogeneous, both within and across individuals. By `within individuals' we mean that a subject may play the same game differently on two separate occasions, a phenomenon labeled \stochastic choice" in decision theory.2 By `across individuals' we mean that subjects produce different choice distributions when playing the same role in the same game.3 Nash (1950) defined equilibrium solely in terms of choices, i.e. as a fixed-point ofthebest reply mappings, implicitly assuming that players' beliefs are correct. One possibility why Nash did not model beliefs explicitly is that he considered them ephemeral. However, by adapting belief-elicitation methods from decision theory, recent work in experimental game theory has produced a wealth of data on the relationship between beliefs and choices.4 If the empirical support for Nash-equilibrium choices is poor, it is worse for Nash-equilibrium 1See e.g. Lieberman (1960, 1962); Brayer (1964); Messick (1967); O'Neill (1987); Brown and Rosenthal (1990); Rapoport and Boebel (1992); Stahl and Wilson (1994, 1995); Nagel (1995); McKelvey and Palfrey (1992, 1995); Goeree and Holt (2001); Crawford et al. (2013); Goeree et al. (2016); Dhami (2016); Eyster (2019). This list is far from exhaustive. 2See e.g. Tversky (1969); Hey and Orme (1994); Ballinger and Wilcox (1997); Hey (2001); Agranov and Ortoleva (2017); Friedman and Ward (2019). 3See Brown and Rosenthal (1990); Rapoport and Boebel (1992); Stahl and Wilson (1994, 1995); Nagel (1995); McKelvey et al. (2000); Costa-Gomes et al. (2001); Arad and Rubinstein (2012); Goeree et al. (2017); DellaVigna (2018) for examples and discussion. 4See Schotter and Trevino (2014) and Schlag et al. (2015) for excellent surveys. For tests of the state-of-the- art methods see Holt and Smith (2016) and Danz et al. (2020). 1 beliefs. A main finding from experiments that elicit beliefs and choices is that a significant proportion of choices are not best responses to stated beliefs. In 3 × 3 games, for instance, the percentage of best responses is typically between 45% and 75%. Furthermore, deviations from best responses are sensitive to expected payoffs, i.e. more costly mistakes occur less frequently.5 A second finding is that beliefs are generally not correct. Subjects do not have perfect foresight { not in one-shot games nor in repeated settings where learning is possible. Nonetheless, reported beliefs are not arbitrary { they predict opponents' choices better than chance and reflect strategic thinking.6 A final finding is that beliefs differ across individuals in the same strategic role, and the same individual often reports different beliefs when facing the same strategic situation. If anything, belief heterogeneity appears more pronounced than choice heterogeneity both within and across individuals.7 We look for minimal behavioral assumptions that produce a falsifiable theory consistent with these empirical observations. (i) Monotonicity: choice frequencies are positively but imperfectly related to expected payoffs based on beliefs, i.e. players \better respond" but do not necessarily best respond. (ii) Consequential Unbiasedness: expected payoffs based on beliefs generate choice dis- tributions that do not differ systematically from observed choices. (iii) Set Valued: belief and choice predictions are set valued with predicted choices being a subset of predicted beliefs. Monotonicity captures the idea that unbiased mistakes, or \noise," dilute choice probability from better to worse options but not to the extent that they overturn their ordering. Con- sequential unbiasedness relaxes the rational-expectations assumption of correct beliefs { the intuition is that there is no need to change beliefs if doing so leaves choices unaffected.8 These behavioral postulates have empirical support per se, and the requirement that M equilibrium captures any behavior that abides by them naturally results in a set-valued solution concept. Being set valued, M equilibrium imposes weak restrictions on the data. Hence, when its predictions are not violated, its contribution lies in setting a new baseline from which further restrictions can be added, resulting in sharper prediction that are also not violated. Importantly, M equilibrium is falsifiable. As we show in Section 2.5, the size ofany M-choice set falls 5See Manski (2004); Croson (2000); Nyarko and Schotter (2002); Costa-Gomes and Weizs¨acker (2008); Palfrey and Wang (2009); Rey-Biel (2009); Terracol and Vaksmann (2009); Hyndman et al. (2009, 2013); Wang (2011); Hyndman et al. (2012); Danz et al. (2012); Sutter et al. (2013); Hoffmann (2014); Trautmann and van de Kuilen (2014); Goeree et al. (2017); Alempaki et al. (2019); Esteban-Casanelles and Gon¸calves (2020). 6Huck and Weizs¨acker (2002); Weizs¨acker (2003); Ehrblatt et al. (2005); Hyndman et al. (2010, 2013); Schlag et al. (2015); Goeree et al. (2017); Alempaki et al. (2019); Polonio and Coricelli (2019); Friedman and Ward (2019); Esteban-Casanelles and Gon¸calves (2020); Foroughifar (2020); Wolff and Bauer (2021). 7This increased heterogeneity possibly reflects the different nature of belief elicitation, i.e. a distribution over choices is elicited rather than a single discrete choice (pure strategy). But it may also reflect a different cognitive process when thinking about others' choices versus thinking about own choices (e.g. Bhatt and Camerer, 2005). 8For instance, in a Prisoner's Dilemma game, all beliefs result in the same ranking of payoffs and, hence, choices. There is no need to correctly anticipate the other's behavior. 2 factorially fast with the number of players and strategies. For instance, in a three-player game where each player has three strategies, the size of any M-choice set is less than half a percent of the set of all strategy profiles, and even if there are multiple M equilibria their total size is less than one-sixth. As such, we do not expect M equilibrium to always be correct. When its predictions fail, the weak assumptions underlying M equilibrium offer important insights about behavior: is monotonicity violated or are beliefs biased in terms of payoff consequences? 1.2. Theoretical Motivation for M equilibrium The idea that players may make suboptimal choices, or \tremble," has been part of game theory since its early days. The refinement literature started with Selten (1975)