Marco Serena

A Belief-based Theory for Private Information Games1

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).

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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 . 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-, where the1 player’s entire hierarchy of beliefs is correct, and a B0 plays the Nash equilibrium of the symmetric-type 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. However, the type-projection of a B0 neglects the prior F and its features, including the correlation across types. Therefore, in the extreme case of perfect negative

3 1 2 of player 1 (2) is distributed according to F1 (F2) over the domain D (D ). In the symmetrized version of the game the domain is extended to Dext = D1 D2, the ext ext ext [1 distribution F1 de…ned over D is as F1 with 0 mass for all types D =D , and the ext ext ext 2 distribution F2 de…ned over D is as F2 with 0 mass for all types D =D . The de…nition of a Bk player is then applicable to such a symmetrized game. Consider, for instance, a game where the state of the world is A; B; or C, all equally likely. Player 1 knows whether the state is C or not-C (in short, AB), and player 2 knows whether the state is A or not-A (in short, BC).4 In the symmetrized version of the ext 1 2 ext ext game, D = D D = AB; C; A; BC , F1 has mass 2=3; 1=3; 0; 0 over D , ext [ f gext f g and F2 has mass 0; 0; 1=3; 2=3 over D . If player 1 is a B1 of type AB, then she correctly understandsf that playerg 2 can tell apart A from BC, but she fails to understand that player 2 understands that she can tell apart C from AB.5 A comparison with the level-k theory is useful to clarify the many distinctive features of the belief-based theory.6 According to the level-k theory, typically applied to complete information games,7 players have in mind a simpli…ed model of the game which is characterized by a level of reasoning k; an L1 player best responds to an L0 player, an L2 best responds to an L1, and so on. Thus, the level-k theory has two main ingredients: the iterative de…nition of Lk as a function of k 1, and the anchoring L0 speci…cation. Similarly, the belief-based theory provides a non- equilibrium concept characterized by a single-parameter k capturing how many steps of “mental process” are computed by the player. In the level-k theory the mental steps are the iteration of best responses, while in the belief-based theory the mental steps are the iteration of reciprocal beliefs. Despite this paper aiming to neither extend nor generalize the level-k theory, it remains useful to discuss the di¤erences between Lk and Bk, and between L0 and B0. Lk and Bk. Applying the standard level-k model to private information games, an Lk player best responds to a “50-50 rival”(i.e., a rival who is equally likely to be a high- or a low-type), who in turn best responds to a 50-50 rival, and so on exactly k times. Thus, in private information games, when an Lk iterates the exactly k times, she is also forming up to the kth-order of beliefs according to the distribution of types (in this case, the “50-50 distribution”). So far, the literature on private information level-k models has stopped both processes (iterations of best

(positive) correlation across types, the heuristic beliefs of a B0 turn out to be incorrect (correct) with certainty. 4 This game is in Brocas et al. (2014). 5 The symmetrization approach works also in case the information partitions are identical, but the type distributions are di¤erent across players, and in case of one-sided private information. For details, see Section 4, where we apply the belief-based theory to the games analysed in Eyster and Rabin (2005) and we show the predictions of the symmetrization approach. 6 The standard level-k theory has been pioneered, among other, by Stahl and Wilson (1994, 1995), Nagel (1995), and Costa-Gomes, Crawford and Broseta (2001). See also Costa-Gomes and Crawford (2006) and Costa-Gomes, Crawford and Iriberri (2009). For a survey on theory and evidence on level-k models, see, Crawford et al. (2013). 7 A few papers analyze private information level-k models. For instance, the level-k study of auctions in Crawford and Iriberri (2007a), Crawford et al. (2009) and Crawford (2016), focuses on models of auctions with private information. Brocas et al. (2014) test subjects’strategic thinking in a private information setting. Brown et al. (2012) propose a private information level-k model to explain why moviegoers fail to infer the low quality of movies produced by studios which withhold the movie from critics before release.

4 responses and formation of beliefs) for an Lk player after exactly k iterations, and from there a behavioral L0 action which ignores both best responses and formation of beliefs is assumed. The belief-based theory, instead, is entirely focused on beliefs; a Bk stops only the belief formation process after k iterations, and from there onward the Bk has heuristic beliefs, but she remains capable of computing the best response to her beliefs. In fact, after k iterations, the Bk expects the B0 action to be played, which corresponds to in…nite iterations of heuristic reciprocal beliefs. L0 and B0. There are two key di¤erences between L0 and B0. First, what di¤erentiates an L0 from the other Lk’sis that an L0 does not maximize her payo¤. Contrarily, in the belief-based theory all Bk’s, including B0, maximize payo¤s given their beliefs, and thus what tells a B0 apart from a Bk is exclusively the belief formation process. In a nutshell, rather than through a behavioral action as for L0, we specify the B0 through behavioral beliefs; a B0 can be seen as “informationally naive,”but fully capable of maximizing her payo¤given her heuristic beliefs. Second, the L0 speci…cation is typically an impulsive reaction that the speci…c game being played evokes in the players’minds. In this sense, the L0 speci…cation is necessarily tailored to the speci…c game being played; examples are the randomization over a domain in p-beauty contests (Nagel, 1995), the truth-telling action in sender-receiver games (Crawford, 2003; Cai and Wang, 2006; Tao-Yi, Spezio and Camerer, 2010), and the salient action in salience games (Rubinstein, Tversky and Heller, 1996).8 Yet, in other games the domain of randomization or the salient action could be arbitrary, and there might not be a truth-telling action. Rather than playing a game- plausible action as an L0, a B0 has heuristic beliefs (“my rival’stype is the same as mine”) and thus a B0 plays the Nash equilibrium of the symmetric-type version of the game with complete information. Therefore, the belief-based approach enables the speci…c game being played to speak for the B0 action, rather than handcu¢ ng the analysis to a game-speci…c and game-plausible L0 speci…cation. Finally, the appropriateness of alternative anchoring L0 and B0 speci…cations remains a purely empirical question which we do not address here despite our acknowledgment of its importance (see e.g., Crawford et al., 2008; Hargreaves-Heap et al., 2014; Crawford, 2014). Another closely related literature is that of self-similarity, developed by Rubin- stein and Salant (2016). They also adopt the idea that a player might project herself into her rival. According to self-similarity, however, the projection is of one own’s ac- tion, rather than of one own’stype; “Self-similarity means that a player who chooses some action X tends to believe, to a greater extent than a player who chooses a dif- ferent action, that other players will also choose action X.” Thus, the concept of

8 Several other speci…cations of L0 have been suggested by the literature. In salience games, besides choosing the salient action either with certainty or with greater probability than the other actions (e.g., Rubinstein, Tversky and Heller, 1996), an L0 could alternatively be role-asymmetric (e.g., in the hide-and-seek games of Crawford and Iriberri, 2007b, L0 seekers favor salience and hiders avoid it). In the asymmetric X-Y game, an L0 has a “payo¤ bias” in that she chooses the action that might yield her the maximum attainable payo¤ more than any other action (e.g., Crawford et al., 2008). In the coordinate attack game, Kneeland (2016) proposes a broad family of L0 behaviors which satis…es a set of intuitive assumptions, such as L0 reacting monotonically to the information on the private and public signal that they receive. In independent private value …rst- and second-price auctions, an L0 bids the valuation that her own private information suggests (see Crawford and Iriberri, 2007a, and the naive model of Lind and Plott, 1991).

5 self-similarity acquire full-meaning when the action sets are identical across players; otherwise, action X might not exist in the space of some players. Similarly, the belief-based theory acquire full-meaning when the type sets are identical across players; otherwise, the type of a player might not exist in the type space of some other players, thus undermining the type projection. Our favorite way to cope with such asymmetric type sets is the symmetrization approach described above and put to the test in Section 4. The belief-based theory is most closely related to the concept of cursed equilib- rium (Eyster and Rabin, 2005), to which we dedicate Section 4. In Section 2 we ground the heuristic beliefs in the psychology literature. Section 3 proposes a game meant to capture and illustrate the belief-based theory.9 Section 5 concludes.

2 B0 speci…cation as a cognitive bias - Review of the psychological literature

"perch’elli ’ncontra che più volte piega // l’oppinïon corrente in falsa parte, // e poi l’a¤etto l’intelletto lega. " Dante Alighieri, Divina Com- media, Paradiso 13, 118-121. English translation: "opinion— hasty— often can incline to the wrong side, and then a¤ection for one’sown opinion binds, con…nes the mind."

In the Dante passage Thomas Aquinas suggests proceeding with caution when facing uncertain issues, and to abstain from hasty judgments, not only because they are often wrong, but also because the a¤ection for one’s own opinion impedes the reconsideration of wrong ideas. What Dante describes is what modern psychologists would call con…rmation bias; that is, the tendency to stick with one’s own idea and to believe in its universality.10 The con…rmation bias is often endogenous, in the sense that people tend to notice and actively look for what con…rms one’s preexisting ideas, and to neglect and not look for alternative possibilities which might falsify it. In a private information game, this can be seen as placing an excessive weight on one’s own type. For instance, say a person’s type is to believe in a correlation between the department where a journal is edited and the editors’ leniency toward papers submitted by authors a¢ liated with the same department. Then, according to the con…rmation bias, people tend to attach greater importance to evidence that upholds their type and to ignore evidence that challenges their type; in other words, people “notice more” papers published by authors a¢ liated with the department where the journal is edited than papers published by authors with a di¤erent a¢ liation. As a result, people tend to overestimate the number of people who share their same type. The B0 speci…cation is an extreme con…rmation

9 In a typical version of the p-beauty contest players simultaneously pick a number between 0 and 100, and the winner is the player whose number is the closest to p times the average of all numbers submitted, where p is some fraction lower than 1. Anchoring the L0 to pick 50 in expectation, the Lk picks 50pk and the unique Nash equilibrium is to pick 0. 10 For an excellent review of con…rmation biases, see Nickerson (1998).

6 bias, in that a B0 completely ignores the possibility of heterogeneity of types and rather believes that her type is universally valid. An alternative is to specify a B0 as a player who underestimates, rather than overestimates, the probability that the other players are of her same type. Although this B0 speci…cation might be meaningful in some games, we decided not to take this avenue because projecting one’s own type onto others is the most accessible heuristic one could think of; as reported in Kahneman (2003), “a de…ning property of intuitive thoughts is that they come to mind spontaneously, like percepts. The technical term for the ease with which mental contents come to mind is accessibility (E. Tory Higgins, 1996). To understand intuition, we must understand why some thoughts are accessible and others are not.”The exceptionally high accessibility of one’s own observation is hardly debatable; a naive player drawing a red ball from an urn has easy access to the heuristic of the form “my rival will also draw a red ball.” Another strand of the psychological literature in support of the B0 speci…cation is the representativeness heuristic; the estimates of subjects are insensitive to prior probabilities when subjects are given some piece of information. In Kahneman and Tversky (1973)’sexperiment, subjects are given a description of a person named Jack,11 and they are told that Jack was randomly chosen from a population with a proportion of engineers equal to 0:3 in one treatment and 0:7 in another treatment. When asked to assess the probability that Jack is an engineer, subjects of the two treatments essentially estimated the same probability. They relied on whether the description they read was representative of the stereotype of an engineer, rather than on the provided prior probability. A readily accessible assessment of one’sown observation (“Is Jack’sdescription typical of an engineer?”) is often preferred to a more di¢ cult judgment of probabilities. Neglecting the prior probability is the B0 speci…cation, which in this sense is in line with the naivety of subjects who base their choice on representativeness heuristic.12 The hindsight bias is the subjects’inability to accurately remember their prior expectations after observing some new piece of information. In particular, subjects tend to ex-post overestimate the ex-ante probability of the observed piece of in- formation.13 For instance, subjects in Loewenstein, Moore and Weber (2006) have to spot the one di¤erence between two otherwise identical pictures. Some subjects were actually informed about the di¤erence, and others were not. When informed subjects are incentivized to guess the fraction of uninformed subjects who would be able to spot the di¤erence, they guessed 58%, while the true fraction was 20%. Such subjects’ overestimation, in hindsight, of the ex-ante probability of the observed

11 Namely, “Jack is a 45-year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies which include home carpentry, sailing, and mathematical puzzles.” 12 For numerous other examples and applications of the representativeness heuristic see Kahne- man et al. (1982), chapters 2-6. 13 This phenomenon was initially found by Fischho¤ (1975), and the subsequent literature proved its robustness. See Blank et al. (2007) for an excellent survey of the hindsight bias research. A theoretical model on the e¤ect of hindsight bias voters on political elections is Schuett and Wagner (2011). Biais and Weber (2009) characterize the e¤ect of hindsight bias in the …nancial market on investment and trading.

7 piece of information can be stretched to its extreme of perceiving the prior as a degenerate distribution attributing all the weight to one’sown observation; that is, the B0 speci…cation. When “type”is interpreted as norms, preferences, habits, or opinions, then over- estimating the degree to which one’sown information is shared by other people often takes the name of false consensus e¤ect. Because you think that A is better than B, you assume that the vast majority of other people also feel the same way you do.14 This cognitive bias tends to lead to the perception of a consensus that does not exist, a “false consensus”indeed. An American going to Australia for the …rst time will tend to incorrectly believe that “up” of a toggle switch turns the light on, rather than o¤. The experiment carried out in Ross et al. (1977) reports that college students who preferred brown bread over white bread estimated that 52.5% of all college students preferred brown bread, whereas college students who pre- ferred white bread estimated that only 37.4% of all college students preferred brown bread. This suggests that some people who observe that brown bread is preferred by themselves, tend to project the same preference onto others. In this sense the false consensus e¤ect resembles the B0 speci…cation. In conclusion, the discussion of the B0 speci…cation necessarily depends on the interpretation that one gives to players’types and on the game being played, and in fact the literature strands that we mentioned spanning anthropology, sociology, and mostly psychology, are highly intertwined and often overlap.15

3 The Up-or-Down game

In this section we propose a novel dominance-solvable private information game which captures the main features of the belief-based theory. For the sake of illus- tration and brevity, we present here a simple parametrized version of it, and we generalize and formally analyze it in Appendix A.

The Up-or-Down game. Each of two identical decks contains cards from 1 to M, where M is large.16 Each player draws and privately observes a random card, of value i 1; ::; M , from her deck. Players simultaneously put their cards on the table either2 f face Downg (action D) or face Up (action U). The payo¤ matrix for players is as follows,

14 See Mullen et al. (1985) for a meta-analysis of the false consensus e¤ect. 15 Recent works in economics have highlighted the importance of agents’ limited capacity of processing information. For instance, a sound branch of dynamic programming started with the seminal contribution of Sims (2003), who shows that accounting for the limited information- processing capacity of agents can explain several observed macroeconomic behaviors. One way to intrepret the heuristic beliefs is of a limited capacity to process the information contained in the prior distribution of types. A theoretical model of information projection proposed by Madarasz (2012) and re…ned and tested by Madarasz (2014) and Danz et al. (2017) reports that “people too often think as if others knew what they did, and too often act as if others could guess their private information correctly.”In Madarasz’model, players systematically overestimate the access of other players to their own private information. 16 Formally, in this introductory example, we need M > 6, and the technical reason will be clear in Appendix A; see assumption (A3) and “Remark on M”in Appendix A.

8 2’saction D U 10 if i > j 1’saction D (0; 0) (1;") with = 5 if  =  i 8 i j U ("; 1) (" + 1;" + 2) < 0 if i < j where the …rst (second) number in brackets is the payo¤: of player 1 (2), and " > 0 is arbitrarily small.17 Note that the values of the cards (i.e., types) matter only if both cards are face up, in which case a prize of 10 is given to the player with the highest card (and equally shared in case of ties).

Plausible Line of Thought. Before formally linking the Up-or-Down game to the belief-based theory, we informally spell out a plausible line of thought. In the shoes of player 1, we can ignore column D because if our rival plays D our action does not signi…cantly a¤ect our payo¤, but if our rival plays U then playing D secures a payo¤ of 1 and playing U makes us compete for a fair split of a prize of 10. Thus, just playing action U is arguably a tempting “…rst reaction” to the game.18 However, if we expect our rival to follow the same line of thought and just play U, then when our card is one of the bottom 10% of the cards, our probability of winning the prize of 10 is lower than 0:1, and thus for such “bad cards”we are better o¤ playing D (so as to get a payo¤ of 1). We call this strategy Utop90% (i.e., U for the top 90% of the cards and D otherwise). However, if we expect our rival to follow the same line of thought and play Utop90%, then when our card is one of the bottom 10% of the top 90% cards of the deck, our probability of winning the prize of 10 is lower than 0:1, and we are thus better o¤ playing D. We call this strategy Utop81%. If we iterate this reasoning a su¢ ciently high number of times, we play U only for a very small set of top cards.

Link with the Belief-Based Theory. The above line of thought closely follows the mental steps of a Bk player in the belief-based theory. Consider a B0 player; due to her mental type-projection, a B0 holds the subjective belief that 1 = 2 = 5, and thus she plays U regardless of the card she draws.19 We call this strategy blind-U, which is the tempting “…rst reaction”to the game described above. Consider now a B1; she has correct …rst-order beliefs (i.e., she believes that her rival’stype follows j U 1; ::; M ), but she has heuristic higher-order beliefs (she believes that her rival reasonsf throughg type-projection). In other words, a B1 best responds to a B0, who plays blind-U and whose type is uniformly distributed in

17 The role of " > 0 is to avoid indi¤erence between U and D if the rival plays D, which would yield D;D as a B0 equilibrium. One could equivalently rule out D;D as a B0 equilibrium by settingf " =g 0 and applying to the B0’s action an equilibrium re…nement,f g such as trembling hand, strict, or payo¤-dominance. 18 We ask the reader to keep in mind risk-neutral players for the sake of this introductory expo- sition. We discuss the role of risk attitudes below in the paragraph “Risk neutrality.” 19 In the Up-or-Down game i is a step function in i’s. This makes the B0 expect 5 out of the fair lottery for a prize equal to 10, which we deem plausible. However, identical types is a contingency with probability of 1=M to occur. Instead of such a step function, one can also analyze the more complicated but qualitatively similar case where the game payo¤s in case of actions U; U are f g 1 = 5 + (1 2) and 2 = 5 + (2 1). For ease of exposition we take the simpler path of assuming a step function.

9 1; :::; M . Such a best response is to play U if her card is any of the top 90% of thef cardsg (i.e., strategy Utop90%).20 AB2 has correct …rst- and second-order beliefs, and heuristic higher-order beliefs. In other words, a B2 best responds to a B1, who plays Utop90% and whose type is uniformly distributed in 1; :::; M . Such a best response is to play U if her card is any of the top 90% of thef top 90%g of the cards (i.e., strategy Utop81%). As k increases, the threshold above which a Bk plays U decreases all the way down to the very top cards only. In fact, in the standard game theoretic BNE of the Up-or-Down game U is played only for the six highest cards of the deck, regardless of how big M is; that is, if M the set of cards for which U is played in the BNE is negligible in the set of !M 1cards.21 Such a standard game theoretic BNE might be surprising at …rst, given how tempting it is to play U as a “…rst reaction” of a risk-neutral player. This surprise makes the Up-or-Down game a particularly interesting …eld to study the belief-based theory.

Robustness of B0. In the Up-or-Down game, blind-U is the B0 strategy for a wide range of B0’s behavioral decision rules or beliefs over her rival’s action. Examples of the former are as follows. Under the maximin decision rule blind-U is the unique strategy in that it secures the highest certain prize. Also, action U can be seen as the “salient action”(see Schelling, 1960), since U is the only action for which private information plays a role and players e¤ectively compete. Examples of the latter are as follows. A B0 plays blind-U if she believes that her rival plays any randomization between D and U which does not depend on the value of the card. This mirrors the standard L0 speci…cation of the level-k theory to uniformly mix over the action space. If instead the randomization between D and U does depend on the value of the card, a B0 plays blind-U if she believes that her rival’s choice between D and U does not depend “too increasingly”on the value of the card, such as, for instance, strategy Utop90%.22 In general, B0 plays blind-U for any subjective

20 In fact, a player plays U if and only if she holds a subjective belief such that Ei[ i] 1 ". M 1 In Appendix A we compute the exact threshold above which a B1 plays U, which is 10 + 2 . That is, a B1 plays U when drawing any of the top-90% of the card. We are neglecting the integer problem. See Appendix A for a discussion.   21 In the Up-or-Down game, the standard game theoretic BNE coincides with the Bk strategy as k . A sketch of the proof is as follows. A player’saction does not signi…cantly a¤ect her payo¤ if! her 1 rival plays D, and thus we focus on the contingency when her rival plays U. Additionally, a player’sbelief over her i increases in her i, and thus any equilibrium must be of the form “U if my card is above a threshold value”(the threshold might be 1 or M, too). Thus, say that your rival plays U for all the best x 1; :::; M cards; that is, for cards of value at least M x + 1. 2 f g Then, if your card is i = M x + 1, by playing U at best you tie, so that your expected payo¤ of playing U is 5=x+", and since a deviation to D guarantees 1, such a deviation is pro…table x 6. In other words, the condition of indi¤erence between U and D occurs between the 6th and 87thtop card, if the rival also plays according to the same threshold. Thus, a standard game theoretic BNE is of the form U i¤ i M 5 (i.e., only for the six top cards of the deck). Appendix A formally analyzes the Up-or-Down game with a more general matrix of payo¤. 22 Let us de…ne one’srival’s(pure) strategy that depends on the value of the card as a mapping M M 1; :::; M U; D , thus there are 2 behavioral actions. Out of these 2 , given my card i, whatf is theg ! percentage f g of cases in which U is preferred? It is su¢ cient that the mass of my rival’s strategy of choosing U that lies on the right of my i is no more than 10 times as much as the mass on the left of my i. If i, j and j’s strategy are uniformly random, the probability of

10 Figure 2: Evolution of Bk strategy in the Up-or-Down game from k = 0 to k . ! 1

11 belief such that Ei[ i] 1 ", which is a broad set of subjective beliefs since i is the of a fair mechanism of splitting a prize of 10 between two players.23 In conclusion, we are con…dent that blind-U is a suitable anchoring B0 behavior in the Up-or-Down game.

Best responding. Admittedly, best responding in the Up-or-Down game is less straightforward than in many complete information games proposed to illustrate and test the level-k theories (such as the 11-20 money request game of Arad and Rubinstein, 2012), and the reason is the existence of private information, which makes players bear the extra cognitive burden of understanding the role of infor- mation in a¤ecting actions and beliefs. Best responding in the Up-or-Down game is less straightforward because it requires the understanding that playing U is better than D for su¢ ciently good cards only. Nevertheless, we believe that players either act regardless of the value of their card (e.g., playing U for the sake of competing for the prize) which gives rise to the B0 speci…cation, or if they do take into account the value of their cards they understand that higher cards foster the appeal of action U as opposed to action D.

Spectrum of Bk’s action. The threshold above which U is played moves monotonically in the order of correct beliefs k. This monotonicity of the strategy in k is present in many games analyzed by the level-k literature (e.g., in the p-beauty contest, in the 11-20 money request game, in the , in the traveller’s dilemma) since it is a compelling property so as to have a clear identi…cation of subjects’ k. The Up-or-Down game is designed to have such monotonicity, and furthermore the actions of the two extreme hierarchies of beliefs (i.e., k = 0 and k ) lie on the antipodes of the strategy space; that is, the B0 plays blind-U and the! B 1 (standard game theoretic BNE) plays “almost blind-D.”The fact that with di¤erent1 values of k we cover the entire space of (threshold) strategies facilitates the identi…cation of subjects’k. Obviously, the belief-based k cannot be carelessly projected onto the k of level-k theory, since the cognitive burden to push one’s own reasoning to a certain k is of di¤erent nature in the two theories. A judicious approach is to give to k an ordinal, rather than cardinal, interpretation.

Risk neutrality. In the Up-or-Down game the behavior naturally depends on the players’ attitude toward risk. However, we do not regard this as a weakness for two reasons. First, this is in some sense inevitable because payo¤ uncertainty is an inherent feature of games of private information where, in contrast to complete information games, beliefs over types are distributions, and thus they naturally involve risk. Second, and more importantly, if the players’ risk attitude is not neutral, instead of trimming the bottom 10% of the cards for which a B1 plays U, players trim more (if risk averse) or less (if risk loving) than 10%, but this trimming percentage remains constant across all k’s; that is, at any order of correct beliefs k players face the same amount of risk. Therefore, the qualitative results do not depend on risk attitude. this contingency goes to 0 very fast as M increases (for instance, for M = 10, such probability is 0:0002). A MatLab code with numerical simulations is available upon request. 23 Similar arguments are widely present in the literature, and one of the pioneers is Rosenthal (1981).

12 Exogenous i’s. We propose the Up-or-Down game because the choice between U or D depends on the subjective beliefs over ( 1; 2), which in turn depend on the entire hierarchy of reciprocal beliefs over types. If instead ( 1; 2) were exogenous (i.e., the beliefs over ( 1; 2) were commonly known), then the Up-or-Down game would boil down to: 1) a chicken game if 1; 2 [0; 1). In its standard terminology, U corresponds to Swerve, D corresponds to go2 Straight. 2) an Active Samaritan’sDilemma (see Buchanan, 1975) if i [0; 1) and j 1. In its standard terminology, the con…dent player (in Buchanan’s2 story, the donor) has a dominant strategy U (to help the recipient), and since the non-con…dent player (the recipient) knows this, she will choose the action D (low e¤ort) which is not what the con…dent player would like the non-con…dent player to choose. 3) a variation of a prisoner’s dilemma if 1; 2 1, where defection by both players U; U yields a greater payo¤ than cooperation by both players D;D . This gamef is,g however, not interesting per se, since U is a strictly dominant strategyf g and U; U maximizes individual and overall payo¤s. Thus, virtually every game theoreticf predictiong would trivially predict U; U . f g In this section we proposed a parametrized version of the Up-or-Down game. Since to the best of our knowledge this game has not been studied before, it is interesting to study its general properties, which we do in Appendix A.

4 Applications and Cursed Equilibrium24

A seminal concept in games with private information is that of cursed equilibrium (Eyster and Rabin, 2005). In a cursed equilibrium players correctly predict the distribution of other players’ actions, but underestimate the correlation between these actions and other players’ private information. In this Section we discuss similarities and di¤erences between the belief-based theory (henceforth, BB) and the cursed equilibrium theory (henceforth, CE). In a private information game, a player’smental process includes two steps:

1. Step 1; the player forms beliefs about the possible types of her rival (“what is the probability that my rival is a high- or a low-type?”)

2. Step 2; the player forms beliefs about how each such type of her rival turns into an action (“what is the action I expect from a high- and a low-type?”)

BB is de facto a departure from standard game theory in Step 1, while CE is departure in Step 2. In fact, we postulate that players, at some order of beliefs, overestimate the correlation between one own’stype and that of the others; that is, a departure from standard game theory in Step 1. On the contrary, a CE postulates that players underestimate the correlation between players’actions and types; that

24 I thank two anonimous referees for drawing my attention to the similarities with the concept of cursed equilibrium.

13 is, a departure from standard game theory in Step 2. BB and CE are, thus, two com- plementary theories meant to capture diametrically opposed limitations of players’ understanding of the game being played. The limited understanding assumed by CE is that a player incorrectly believes that with some probability her opponent will play the average distribution of actions, irrespective of her type. The limited understanding assumed by BB is that a player incorrectly believes that from some order of belief k onward types stop following the prior distribution. Since both CE with = 0 and BB with k = coincide with the standard game theoretic BNE, we compare the predictions of CE1 and BB on the opposite of the spectrum (namely, = 1 and k = 0) so as to best tell apart the two theories. At the end of this Section we brie‡y discuss intermediate values of and k. In terms of choice of games, we free-ride on the rich and well-selected set of applications provided by Eyster and Rabin to apply BB, and we add the Up-or-Down game.25

One-sided Private Information in Trade. Eyster and Rabin (in the Intro- duction) propose the following stylized variant Akerlof (1970)’slemons model;

“A buyer might purchase a car from a seller at a predetermined price of $1,000 (Canadian). The seller knows whether the car is a lemon, worth $0 (Canadian) to both seller and buyer, or a peach, worth $3,000 (Canadian) to the buyer and $2,000 (Canadian) to the seller. The buyer believes each occurs with probability 1/2 . The parties simultaneously announce whether they wish to trade, and the car is sold if and only if both say they wish to trade.”

In the standard game theoretic BNE, the buyer understands that the seller is only willing to trade lemons, and thus no trade occurs. When instead = 1, a buyer may mistakenly buy a lemon, as Eyster and Rabin show, since she fails to understand that the seller is willing to trade only if the car is a lemon. When applying BB, the sym- metrized domain (see the Introduction) is Dext = P each; Lemon; Uninformed , F seller over Dext is p; 1 p; 0 , and F buyers is 0; 0; 1f . A seller, regardless of her kg, knows the quality off the car,g and thus is alwaysf willingg to trade only lemons. The type-projection of a B0 buyer makes her believe that no one knows the quality of the car, so that trade occurs since both the seller and the buyer expect a non-negative payo¤.26 Thus, a B0 buyer ends up purchasing a lemon, as a cursed buyer. In this stylized model, a B1 buyer is no more fooled into buying a lemon. In fact, a B1 buyer, having correct …rst-order beliefs, understands that the seller knows whether the car is a lemon or a peach, and thus that the seller sells only lemons. Both > 2=3 and k = 0 predict the trade of lemons, and both < 2=3 and k 1 predict no trade.27  25 In some cases we will consider parametrized special cases of Eyster and Rabin (2005)’sappli- cations. 26 In particular, the seller expects $1; 000 0:5 $2; 000 and the buyer expects 0:5 $3; 000 $1; 000. The particular numbers chosen by Eyster  and Rabin create an indi¤erence for the seller when applying BB. Rather than changing their numbers, we innocuously assume that in case of indi¤erence the seller is willing to sell. 27 The fact that a B1 already plays as in a standard game theoretic BNE is due to the binary type space of this stylized Akerlof model. In fact, in the Up-or-Down game instead, the standard game theoretic BNE play requires a higher k.

14 Asymmetric Information Partitions in Trade. Eyster and Rabin (in Sec- tion 3) apply CE to two further trade models; one with two-sided private infor- mation, and the other with one-sided private information, which elaborates on the above stylized trade model.

[Two-sided private information trade model] “Let = !1;!2;!3 be the set of possible payo¤-relevant states of the world, where the two playersf shareg the common prior (!1) = (!2) = (!3) = 1=3. Each trader has private information about the state of the world: Trader A learns when the state is !1, but cannot di¤erentiate between states !2 and !3; Trader B learns when the state is !3, but cannot di¤eren- tiate between states !1 and !2 [...]. A state-contingent trade t : R speci…es a real-valued transfer from Trader A to Trader B in each state of the world.! Consider the trade (1; "; 1) for " > 0, meaning that Trader A pays B 1 in !1, " in !2, and 1 in !3.” A trade is implemented only if both traders agree to it. The standard game theo- retic BNE predicts no trade because Trader A understands that Trader B accepts to trade only if she observes not-!3. When instead = 1 trade occurs in !2, yielding negative payo¤ to Trader A. This is because Trader A accepts to trade when she observes that the state is not-!1, or equivalently !2;!3 , because she expects that f g Trader B may accept trades in !3, while actually Trader B accepts to trade only in !2. Similarly, k = 0 predicts that trade occurs in !2. This is because a B0 who cannot tell apart two !’s fails to understand that her rival is actually capable of telling them apart and thus that her rival’swillingness to trade goes accordingly.28 Both > 2"=(1 + ") and k = 0 predict trade in !2, and both < 2"=(1 + ") and k 1 predict no trade.  [One-sided private information trade model] “A …rm o¤ers itself for sale to a raider; the …rm knows its book value, but the raider does not. The raider has correct priors that the book value of the …rm is uniformly distributed on [0; 1]. Whatever its book value, the …rm values itself at its book value, while the raider values the …rm at 1 times book value. The raider must make the …rm an o¤er, which the …rm then accepts or rejects; without loss of generality we take the raider’s o¤er space to be [0; 1]. The raider seeks to maximize her expected surplus and the …rm accepts any o¤er above its book value. The raider, who has no private information, chooses a price b [0; 1] at which she o¤ers to buy the …rm. The …rm’s optimal strategy is clear: it2 sells at price b if and only if (i¤ ) its book value is less than b.”

b For an o¤er price b, the raider’ssurplus is b 2 b since the average value of the …rms sold is b . Thus, for < 2 the unique standard game theoretic BNE is 2  b = 0. Therefore, we restrict attention to < 2 . When = 1 Eyster and Rabin

28 A more detailed analysis follows. Trader A observing !1 and Trader B observing !3 expect a payo¤ from trade of 1 regardless of their k and thus reject trade. Consider what happens when instead the state is !2.AB0 Trader A projects her not-!1 type onto her rival, expects (1 ")=2 > 0 and thus trades. Similarly, a B0 Trader B expect (1 + ")=2 > 0 and thus trades. A B1 Trader A (B) knows that her rival trades if observing not-!3 (not-!1) and does not trade if observing !3 (!1); thus, a B1 Trader A (B) knows that trade occurs only in !2, and thus rejects (accepts) trade. Therefore, trade occurs in !2 if Trader A is B0 and Trader B is B0 or B1. Trade never occurs for higher k’s.

15

…nd that b = 4 ; the cursed raider fails to understand that the …rm accepts her o¤er only when its book value is su¢ ciently low. When applying BB, a …rm, regardless of her k, knows the book value and is, thus, willing to sell at any price above that value. AB0 raider believes the …rm does not know the book value, and thus believes that everyone expects a book value of 0:5. Therefore, a B0 raider o¤ers 0:5, which is the lowest price for which the …rm accepts. A B1 raider understands that the …rm knows the book value (correct …rst-order beliefs), thus expects an average value of the …rms b sold to be 2 ; therefore, b = 0. Both = 1 and k = 0 predict a strictly positive o¤er. The former because the raider fails to understand that the …rm accepts her o¤er only for su¢ ciently low book values, and the latter because the raider fails to understand that the …rm knows the value.

Common-Values Auctions. When = 1 in a common-value auction, a bidder fails to understand the link between the other bidders’private signals and their bids, and thus she bids as if the auction were one of private (but uncertain and correlated) values, rather than one of common value. Since the other bidders’ signals in a common-value auction directly a¤ect the value of the object for the cursed bidder too, the bidder incurs in an expected loss. In fact, Eyster and Rabin (in Section 4) show that = 1 captures the winner’scurse. When k = 0, a bidder instead projects her own private signal onto all the other bidders, and thus acts as if everyone received her private signal. Thus, a bidder bids as if the auction were one of public, rather than private, value; the unique is that everyone bids her own signal, and so bidders do when k = 0. This implies that bidders with su¢ ciently high signals incur in an expected loss (winner’scurse). There are two key-common features of = 1 and k = 0. First, in both cases the bidder receiving low signals underbids and the bidder receiving high signals overbids. In this sense, both theories rationalize the winner’scurse. Second, in both cases the bidder bids as if winning the auction conveys no information about the common-value of the object. However, when = 1 winning conveys no information because the bidder fails to link signals and actions of the other bidders, while when k = 0 winning conveys no information because the bidder believes that the others have neither more nor di¤erent information than she already possesses. Both are plausible sources of winner’scurse. The fact that k = 0 predicts that the bidder bids her own private signal provides an interesting parallelism with the work of Crawford and Iriberri (2007a). They apply the standard level-k theory to common-value auctions and rationalize winner’s curse behaviors. In doing so, they consider two anchoring L0 speci…cation; the standard random L0, and truthful L0 who bids the value suggested by its own private information, inspired by the naive model of Lind and Plott (1991). The truthful L0 proposed by Crawford and Iriberri coincides with the B0.

Information Revelation. Besides the two main applications of CE presented by Eyster and Rabin — namely, trade and auctions — , they also present further models (in Section 5). For example, they consider the model of jurors of Feddersen and Pesendorfer (1998). A defendant is guilty (state of the world !G) or innocent (state of the world !I ) with equal probability, and some jurors vote for guilty or

16 innocent after having received a private signal positively correlated with the true state of the world. Consider a unanimous voting rule. In a standard game theoretic BNE jurors with a guilty signal vote guilty with certainty and jurors with an innocent signal vote guilty with positive probability, since a rational juror knows that she is pivotal only when all others vote guilty, and thus when the defendant is most likely guilty. When = 1, Eyster and Rabin show that jurors vote their signals since they fail to understand that guilty votes from the other jurors take place when they receive guilty signals. Similarly, when k = 0 a juror receiving guilty signal is sure that the others received the same signal, thus she believes to be certainly pivotal and vote her signal, as under = 1.

Curse Equilibrium in the Up-or-Down Game. When = 1 a player fails to understand the link between the opponent’s card and action. In Appendix B we show that when = 1 a player plays Utop90%, which is also the strategy of a B1 player. In fact, when = 1 a player believes that her rival plays (the average strategy) regardless of the card she receives, thus she expects the set of cards with which the rival plays U to be also uniformly distributed in 1; :::; M . Therefore, her best reply is to play Utop90%, which is also the strategy off a B1 player.g In fact, the predictions of = 1 and k = 1 coincide even if we change the payo¤ matrix, as long as its values satisfy some regularity conditions (see assumptions (A1)-(A3) in Appendix A) which guarantee that the Up-or-Down game captures BB, and not other games (e.g., a chicken game if both ’sare divided by 10).29

Intermediate values of and k. As said, we focused on predictions under = 1 and k = 0 because on the opposite of the spectrum (i.e., when = 0 and k ) both theories coincide with the standard game theoretic BNE. The speci…c form! 1 of transition from = 1 and k = 0 to the standard game theoretic BNE is therefore less informative to tell apart the two theories. For instance, in Appendix B we show that CE in the Up-or-Down game when (0; 1) is not well-behaved; some values of yield multiple equilibria, and equilibrium2 convergence from = 1 to = 0 (i.e., the standard game theoretic BNE) is not monotone. On the other hand, in most of the above applications of BB to the games selected by Eyster and Rabin, the standard game theoretic BNE is already played by a B1 — and thus played by all Bk’swith k 1 — either because one of the two players as a dominant strategy regardless of beliefs (a seller sells if and only if the value is su¢ ciently high), or because the type- and action-space are not “rich”enough.30 Thus, interior values of and k yield well- or badly-behaved predictions depending on technical assumptions of the speci…c game being played, rather than on conceptual di¤erences between belief-based equilibrium and cursed-equilibrium as for the comparison of = 1 or

29 In Appendix B we also provide a characterization of the cursed equilibrium when (0; 1). 30 Consider a with binary type-space (H and L) and binary action-space2 (actions a and b). A (pure) strategy maps types into actions, and thus it can take one of four forms (namely, always a, always b, a if H and b if L, or viceversa). Thus, convergence to the standard game theoretic BNE should occur for a k 4, unless cycles occur. Cycles typically occur when the BNE is in mixed-strategies, as for instance in the following symmetric market entry game. A …rm who does not enter obtains 0. If both …rms enter, each obtains 1. If only one …rm enters, she obtains i. The prior is that H and L are equally likely, and H > L > 1.AB0 of type i enters with probability i= (i + 1).AB1 of type H (L) enters (does not enter). A Bk with k 2 enters if k is even and does not enter if k is odd. 

17 k = 0. This further explains the focus of this Section on comparing BB and CE based on their predictions when = 1 or k = 0.

Conclusions of the BB-CE comparison. We found a prevalence of overlap- ping between the predictions under = 1 and those under k = 0 in the games chosen by Eyster and Rabin to illustrate CE. This is surprising, given that the de…nitions of CE and BB are grounded in sharply di¤erent concepts; the former postulates an (extreme) underestimation the correlation between players’ actions and types ( = 1), while the latter postulates an (extreme) overestimation of the correlation between one own’stype and that of the others (k = 0). The ideal family of games to apply CE is one where the private information of a player directly a¤ects the payo¤ of the other player. This is the case in the above trade models and common-value auctions. The ideal family of games to apply BB is one where the private informa- tion of a player directly a¤ects only the payo¤ of that player, and the other player’s type matters only in that it implies a certain action by that player. This is the case in the Up-or-Down game. Thus, we deem a horse-race between CE and BB too sensitive to the choice of game (whether closer to the former or to the latter family of games) to be generally conclusive and worth being attempted. Furthermore, each of these theories complies with standard game theory where the other theory departs from standard game theory. That is, a cursed player correctly understands that her rival’stype is distributed according to the prior, and a Bk player correctly understands that her rival links her action to her information. This highlights once again the complementarity of the two theories, and points at an hybrid model combining these two layers of “naiveté”as future research project, so as to gain predictive power regardless of which of the two family of games is being played.

5 Conclusions

Standard equilibrium concepts for private information games cling on not only player’s beliefs about each other’s types, but also on beliefs about each other’s beliefs, beliefs about each other’sbeliefs about each other’sbeliefs, and so on. This paper takes advantage of such hierarchy of beliefs to propose a departure from the standard assumption that the prior distribution of types is consistent with every player’sbelief of every order; namely, a Bk player has correct beliefs (i.e., following the prior) only up to the kth-order, and heuristic beliefs at all higher orders. We model heuristic beliefs as type-projections, of the form “my rival is of my same type,”which is an assumption well-grounded in a broad body of the psychological literature. While correct beliefs fully take into account the prior, heuristic beliefs fully ignore the prior. This is a knife-edge speci…cation of beliefs, being the …rst cut into a belief-based theory. However, one could think of a less knife-edge speci…ca- tion; for instance, drawing inspiration from the cursed equilibrium concept, where a player assigns a probability (1 ) to her rival playing type-incontingent (type- contingent) strategy, one could similarly de…ne a Bk with k [0; 1] as a player who assigns a probability k (1 k) to her rival being of her identical2 type (being of 18 type distributed according to the prior). Both our theory and this variant are single- parameter theories which embed the standard game theoretic BNE as a special case. Finally, we analyze a simple dominance-solvable game (the Up-or-Down game) to illustrate and clarify the reasoning of a Bk player, and we apply the belief-based theory to the array of games adopted by Eyster and Rabin to test the predictions of cursed equilibrium. Although the belief-based theory focuses exclusively on one procedural aspect of decision-making in private information games, namely the players’“informational cognition,”by no means we claim that this is the unique or the most relevant aspect a¤ecting players’thinking; thus, the belief-based theory complements the burgeoning behavioral literature (including, e.g., level-k models and cursed equilibrium), as a further step towards a robust hybrid model of players’thinking with a satisfactory degree of portability across games.

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23 Appendix

A The generalized Up-or-Down-game

In this section we generalize the payo¤ matrix of the Up-or-Down game so as to build a better understanding since, to the best of our knowledge, this game has not been studied before. Two players privately observe their types i with i = 1; 2, which is 31 drawn from a commonly known i U 1; :::; M with M “big”enough. We call u (i.e., " arbitrarily small in the main text)f theg certain payo¤ of action U, and we call d the payo¤ that a player obtains if her action is D and her opponent’saction is U. The overall prize in case of actions U; U equals 2. Thus, the generalized payo¤ matrix is f g

D U D (0; 0) (d; u) U (u; d) (u + 1; u + 2)

where,

2 if i > j =  if  =  i 8 i j < 0 if i < j : Assumptions. In what follows we will make three assumptions (A1)-(A3) on the quadruple (u; d; ; M), which are satis…ed by the numerical example in the main text and which make the Up-or-Down game non-trivial.

(A1): 0 < u < d

Namely, 0 < u rules out equilibrium D;D , and u < d avoids action U being strictly dominant regardless of beliefs andf levelg of reasoning. Similarly, in order to avoid action D being dominant we assume that

(A2): d < min u + ; 2 f g In particular, condition d < u +  is necessary to guarantee that the complete information game that a B0 believes she is playing (where 1 = 2 = ) has the unique equilibrium U; U , instead of two equilibria; namely U; D and D;U . Thus, under (A2), actionf gU is tempting enough so as to unambiguouslyf g makef a Bg0 play blind-U. Condition d < 2 avoids action D being dominant regardless of beliefs and level of reasoning when u is arbitrarily small (like in the numerical example of the main text). The last assumption, (A3), is rather technical and will be explained below.

31 The reader would have to wait until “Remark on M”below for the meaning of “big.”

24 Analysis of B1.AB1 of a certain type i best responds to a rival whose: i) type is j U 1; :::; M , and ii) action is U. Then, the B1 plays U rather than D i¤  f g u + Ei[ i i; ki = 1] d j 

1 i 1 u +  + 2 d () M M    Pr i=j Pr i>j f g f g M|{z}(d u) 1 i + | {z } (1) ()  2 2

M(d u) 1 De…ne  = + , where x is the smallest integer greater than or B1 2 2 d e equal to x. Then al B1 plays32m

U if i   B1 (2) D if i  1   B1 In the argument given in the main text for simplicity we neglected the probability 1 of ties, which yields term “+ 2 ”in (1). In the numerical example of the main text, 1 " M B 1 = M 10 + 0:5 and, if M is for instance a multiple of 10, then B 1 = 10 + 1 which implies that a B1 plays Utop90%. Thus, neglecting the probability of ties a¤ects the B1 threshold only marginally and for some values of M.

Analysis of Bk. Since from B2 onwards players best respond to a threshold strategy, we already provide the general analysis of Bk with k 2.ABk of type  i best responds to a rival j = i whose: i) type is j U 1; :::; M , and ii) action 6  f g is U if and only if j Bk 1.ABk 1, as in (2), plays U with probability  (M Bk 1 + 1)=M. Thus, the Bk plays U rather than D i¤ M Bk 1 + 1 M Bk 1 u + Ei[ i i; ki = k] d M j  M

M Bk 1 + 1 1 i Bk 1 M Bk 1 u + 2  + 23 d () M M Bk 1 + 1 M Bk 1 + 1  M 6 7 6 Pr i=j Pr i>j 7 6 f g f g 7 4 5  2 | {zM Bk} 1 | {z } u + + i Bk 1 d () M M  M M(d u)   dBk 1 i Bk 1 (3) ()  2

M(d u)  d Bk 1 And if we de…ne Bk = Bk 1 + 2 , then a Bk plays l m U if i   Bk D if i  1   Bk 32 Leaving the threshold as a possibly non-integer number would also yield to an unambiguously de…ned strategy, but we round the threshold in order to be able to use B 1 to compute the exact probability that a B1 plays D when computing the B2’sbest response.

25 Fixed Point. The …xed point at k requires the right-hand side of (3) to equal 0,33 or equivalently, ! 1 d u  B = M (4) 1 d d For …nite M, in the numerical example of the main text, (4) dictates to play U only for the six highest cards of the deck. As M , (4) outlines strategy d u 34 ! 1 Utop d %, so that the set of cards for which U is played is negligible compared to the size of the deck when u is arbitrarily small.  Well-behaved Bk’s. The process determining the subsequent thresholds is a linear …rst-order di¤erence equation with constant coe¢ cient of the form xt = 2 d M(d u)  axt 1 + b with a = 2 and b = 2 , as can be easily seen in (3). Under (A2), a (0; 1) and thus the process converges monotonically. To further guarantee a 2 well-behaved problem, we assume what guarantees that B > B 1, namely, 1 d u  M (d u) 1 M > + d d 2 2 (2 d)(d u)M > (d + 2) () which is equivalent to the following assumption 2 d (A3): (d u)M >  2 + d

2 d Note that by (A2), (0; 1), and thus (A3) implies (d u)M > , which 2+d 2 guarantees B 1 > 1 , see (1). Since B < M — see (4) — assumptions (A2)-(A3) 1 _ together guarantee that 1 < B 1 < ::: < Bk < ::: < B < M; that is, the Up-or-Down game is well-behaved. 1

Remark on M. In the main text we assumed that u is arbitrarily small, d = 1 and  = 5, but we also assumed that the deck is su¢ ciently “big”so as to neglect for simplicity the probability of ties, which are rather crucial for small decks.35 While (A1) and (A2) do not depend on M, (A3) requires the deck to be composed of more than six cards. In particular, for any triple (u; d; ) satisfying (A1) and (A2) there is always an M big enough so that (A3) holds. Finally, note that the limiting case M and the case of a continuous distribution of types are equivalent, and that the! entire 1 analysis can be easily replicated with any distribution of types over a compact space, where the thresholds of the strategies are speci…ed in terms of quantiles of the distribution.

B Cursed Equilibrium in the Up-or-Down-game

A player believes that with probability the rival plays the type-incontingent strat- egy — i.e., regardless of the cards she gets, she plays D with probability (x 1)=M 33 Note that the rounding issue does not a¤ect the …xed point argument. 34 d u Note that d (0; 1) by (A1). 35 2 For instance, if M = 1 then i = j =  k and thus k does not a¤ect the strategies. 8 26 Figure 3: Cursed Equilibria in the Up-or-Down game in the ( ; M)-space. and she plays U with probability (M x + 1)=M, where x has to be determined in equilibrium — , while with probability 1 the rival plays the type-contingent strategy — i.e., a threshold strategy with threshold   . Thus, we look for the in- di¤erence card   yielding indi¤erence between playing U and D, and x =   so as achieve consistency. Equality of payo¤ between playing U and playing D at the indi¤erence card reads

1 M  + 1 1  1 M  + 1 (1 ) 5 + 5 + 10 +" = (1 )1+ 1 X M M M M      (5)

To understand the …rst term, recall that when i =   and i plays U, at best she can tie (with probability 1=X ). 1 M When = 1, as " 0, the two solutions of (5) are   = 1 + M and   = 2 + 10 . In the former (blind-D!) it is pro…table to deviate to U and thus it is inconsistent with the cursed-equilibrium correct anticipation of the average action. Therefore, 1 M the unique equilibrium is i = 2 + 10 , which coincides with the strategy of a B1. When (0; 1), as " 0, (5) can be seen as a polynomial of degree 4 in  , 2 ! which turns out to often have more than one real roots satisfying X [1;M + 1]. If, 1 2 for instance, = 2 and M = 100, then X 3; 20; 89 are all solutions of (5) (after rounding), as can be seen in the …gure below,2 f where weg plot in the ( ; M)-space the

X solving (5). Notice that, besides the multiplicity issue, convergence to standard game theoretic BNE is not monotone.

27