Marco Serena A Belief-based Theory for 1 Private Information Games Max Planck Institute for Tax Law and Public Finance Working Paper 2018 – 12 August 2017 Max Planck Institute for Tax Law and Public Finance Department of Business and Tax Law Department of Public Economics http://www.tax.mpg.de 1 A previous version of this paper was entitled “A Level-k Theory for Private Information Games” Working papers of the Max Planck Institute for Tax Law and Public Finance Research Paper Series serve to disseminate the research results of work in progress prior to publication to encourage the exchange of ideas and academic debate. Inclusion of a paper in the Research Paper Series does not constitute publication and should not limit publication in any other venue. The preprints published by the Max Planck Institute for Tax Law and Public Finance represent the views of the respective author(s) and not of the Institute as a whole. Copyright remains with the author(s). Max Planck Institute for Tax Law and Public Finance Marstallplatz 1 D-80539 Munich Tel: +49 89 24246 – 0 Fax: +49 89 24246 – 501 E-mail: [email protected] http://www.tax.mpg.de A belief-based theory for private information games Marco Serenay May 22, 2019 Abstract We propose a belief-based theory for private information games. A Bk player forms correct beliefs up to the kth-order, and heuristic beliefs from the (k + 1)th-order onwards. Correct beliefs follow the prior distribution of types, as in standard game theory. Heuristic beliefs ignore the distribution of types and are rather heuristic projections of one own’s type onto the rival, of the form “my rival is of my type.”A Bk best responds to those partially correct and partially heuristic beliefs. As a result, a B plays the standard game theoretic Bayesian-Nash equilibrium, where the1 player’s entire hierarchy of beliefs is correct, and a B0 plays the Nash equilibrium of the symmetric-type complete information version of the game, where the entire hierarchy of beliefs is heuristic. We ground the belief-based theory on the psychological literature, we illustrate it through a simple yet novel game, we apply it to standard games and we compare its predictions with those of cursed equilibrium (Eyster and Rabin, 2005), which is another single-parameter generalization of the standard game theoretic Bayesian-Nash equilibrium. Despite the two theories are conceptually di¤erent, predictions often overlap. Keywords: common-knowledge, hierarchy of beliefs, private information games. C72, D83 Max Planck Institute for Tax Law and Public Finance. Email: [email protected] yA previous version circulated under the title “Level-k reasoning in reciprocal beliefs.”I would especially like to thank Amanda Friedenberg and Tilman Klumpp for many substantial comments. I am also grateful to Vincent Crawford and Mario Gilli for useful suggestions. I would like to thank participants at the Ce2 Workshop (Warsaw), the 13th European Meeting on Game Theory SING (Paris), the 18th meeting of the Association for Public Economic Theory (Paris), and the V Informal Research Meeting at the Max Planck Institute for Tax Law and Public Finance (Munich). Yichun Dong provided excellent research assistance. All errors are my own. 1 1 Introduction A growing body of experimental evidence supports models where players have a distorted perception of the game being played. In games of private information, Eyster and Rabin (2005) postulate that players underestimate the degree to which other players’actions are correlated with the information they possess. We propose an alternative theory entirely based on beliefs over types, which is to the best of our knowledge an unexplored avenue. Consider the in…nite reciprocal beliefs over types (“I believe that your type follows the prior, and I believe that you believe that my type follows the prior,...”)of standard game theory. In a nutshell, we propose that a player might correctly iterate this process of belief formation only up to some order of belief k, which de…nes a Bk player. We ground the belief-based theory in the psychological literature, and we test it in various models. More in details, we propose that a Bk player forms correct beliefs up to the kth-order, and heuristic beliefs from the (k + 1)th-order onwards. Correct beliefs are probability distributions over types which follow the prior distribution of types, as standard game theory predicts in a Bayesian Nash Equilibrium (BNE). Heuristic beliefs, conversely, ignore the distribution of types; in particular, we argue that in private information games it is natural to model “ignoring the distribution of types”as projecting the type that a player observes for herself onto her rival. Given this de…nition of a Bk player, the B0 speci…cation automatically follows; the entire hierarchy of beliefs of a B0 is made of heuristic beliefs. Thus, a B0 believes she is playing a symmetric-type game with complete information. A general Bk player chooses her action to maximize her expected payo¤, given her type and given the structure of correct and heuristic beliefs just described, and pinned-down by the single parameter k. The belief-based theory can be described in a compact logic sentence as follows. A Bk player {believes that her rival’s type follows the distribution of types, and that her rival in turn}k {believes that her rival is of her same type rather than 1 following the distribution of types, and that her rival in turn}1. We exemplify the belief-based theory through a two-player private information game with types drawn from a “50-50 distribution”— i.e., both high- and low-types have the same probability and draws are independent. A high-type (low-type) B0 believes that her rival’s type is also a high-type (low-type), who also believes she is up against a high-type (low-type), and so on, ad in…nitum; that is, the whole hierarchy of beliefs of a B0 is heuristic. The beliefs of a B1 and a B2 in such a game are illustrated in Figure 1, and are described in what follows. A B1 believes she is with probability 0:5 up against a high-type who believes she is up against another high-type (ad in…nitum), and with probability 0:5 up against a low-type who believes she is up against another low-type (ad in…nitum); in other words, a B1 computes correct …rst-order beliefs (“my rival’stype follows the 50-50 distribution”), but heuristic second- and higher-order beliefs (“my rival believes that my type is the same as hers,” rather than “my rival believes that my type follows the 50-50 1 The parts in the curly brackets are to be repeated respectively k and times, so that a Bk forms correct beliefs up to the kth-order and all her subsequent beliefs are1 heuristic. 2 Figure 1: Hierarchy of beliefs over rival’s type of a B1 and a B2 in the two-player private information game with a “50-50 distribution”of types. distribution”). A Bk best responds to her structure of beliefs.2 Thus, a B0 plays the symmetric-type Nash equilibrium of the game with complete information; a B1 best responds to facing a high-B0 and a low-B0 with equal probability; and so on. AB has a correct hierarchy of beliefs at all orders; thus, a B perceives the model1 as standard game theory dictates — i.e., under common-knowledge1 of the prior. Therefore, the belief-based theory postulates that players engage only in a limited number of iterations of reciprocal expectations according to the distribution of types; among the interpretations of this limited computation, compelling ones are the inherent cognitive di¢ culty of the in…nite iterations of reciprocal expectations to be fully accounted in a player’s mental process, and the fact that fully rational players simply do not believe that other players are capable of facing such inherent cognitive di¢ culty. Thus far, we presented the belief-based theory in a setting where two players play a two-sided private information game with a common prior distribution of types F over a common domain D. When instead F and/or D are asymmetric across players, the belief-based theory suggests to …rst symmetrize the type sets.3 Say that the type 2 Despite the entire hierarchy of beliefs playing a crucial role in the belief-based theory, beliefs do not directly enter the utility function of players as in psychological games (Geanakoplos, Pearce and Stacchetti, 1989). The belief-based theory sticks to the assumption that payo¤s depend only on actions, and the beliefs of a player are consistent with her own actions, but might not be consistent with the true beliefs and actions of the others. In this sense, players can be ex-post surprised by noticing that the actions they believed would be played are eventually not played. Thus, the belief-based theory shares the relevance of hierarchy of beliefs with the psychological game literature, and the possibility of being ex-post surprised with the level-k literature. 3 Other extensions of our simple setting are more trivial to see. For instance, with n-player it is natural to assume that a B0 believes that all of her opponents are of her same type, and a Bk forms: i) up to the kth order, correct beliefs over all of her rivals, and ii) from the (k + 1)th-order, heuristic beliefs over all of her rivals. Additionally, when types are correlated, a Bk’s belief over her rival’s type takes into account such correlation, and thus the Bk’s type itself plays a role in Bk’sbeliefs over her rival’stype.