Bayesian game theorists and non-Bayesian players

Abstract

Bayesian game theorists claim to represent players as Bayes rational agents, maximising their expected utility given their beliefs about the choices of other players. I argue that this narrative is inconsistent with the formal structure of Bayesian game theory. This is because (i) the assumption of common belief in rationality is equivalent to equilibrium play, as in classical game theory, and (ii) the players' prior beliefs are a mere mathematical artefact and not actual beliefs held by the players. Bayesian game theory is thus a Bayesian representation of the choice of players who are committed to play equilibrium strategy profiles.

Keywords:

• Bayesianism
• common belief in rationality
• epistemic game theory
• interactive epistemology
• prior beliefs

JEL CODES:

• B21
• C72
• D81

Acknowledgements

Several ideas presented in this article originated from an ongoing collaboration with Lauren Larrouy on coordination in game theory. I would like to thank her for various discussions on this subject. I am also grateful to participants to the 21st ESHET annual conference, and to two anonymous reviewers for their comments and suggestions.

Notes

1 Throughout the rest of this paper, I will use the terms ‘Bayesian’ and ‘epistemic’ game theory interchangeably.

2 According to Giocoli, this evolution results from the combination of Harsanyi’s influence on the development of the refinement literature, on models of imperfect competition (in particular as a reaction against the Chicago antitrust arguments in US Courts), and on mechanism design theory (see Schotter and Schwödiauer (1980) for a literature review of the period).

3 Games of imperfect information, on the other hand, correspond to games of complete information in which players are unaware of the actions chosen by others, for instance in sequential games.

4 Harsanyi argues in Part I that an uncertainty about the set of strategies can be reduced to a strategy about the payoff function. If a strategy x_i is not available to player i, then it is as if i always receives an extremely low payoff when she chooses this strategy (we therefore know that i will never choose this strategy if it were available).

5 Note that we assume that all the relevant parameters of the model can be summarised in a single variable, without explicitly defining those parameters. Mertens and Zamir (1985) will give a precise mathematical definition of this idea. They offer in particular a proof of the existence of a space of infinite hierarchies of beliefs.

6 Harsanyi originally used the terms ‘information vector’, ‘attribute vector’ and ‘type’. Following Aumann and Maschler’s (1967, 1968 [reprinted in Aumann and Maschler 1995]) suggestion, Harsanyi and later authors eventually adopted the term ‘type’ (see Myerson, 2004, 1820).

7 Spohn (1982) raises a similar point, and argues that Harsanyi’s idea of ‘Bayesian’ game theory is ‘still more game theoretic than decision theoretic in spirit and hence criticisable on similar grounds as the standard accounts’ (Spohn 1982, 240).

8 The condition stated in this extract only implies a 2nd-order belief in rationality (we all believe that we all believe that we are rational). Harsanyi seems however to interpret the condition of mutually expected rationality as common belief in rationality: this is clearly the case in his defence of the rationality postulates of game theory (Harsanyi 1961, 180), where he argues that the postulates of ‘utility-maximisation’ and ‘mutually expected rationality’ implies the iterated elimination of dominated strategies (which however requires CBR).

9 ‘Simulation’ refers to a specific theory of mindreading, according to which the individual forms her beliefs about the other’s actions by imagining what she would do if she were in the others’ shoes. See Guala (2016, 89–101) and Larrouy and Lecouteux (2017) for a game-theoretic analysis of simulation thinking.

10 See Vanderschraaf and Sillari (2014) for a history of the concept and a review of the different formalisations.

11 The basic intuition of the theorem is the following. If we share a common prior and receive some private information, then our posteriors may differ if the information was different. However, if we both know our ‘initial’ posteriors, then we must revise them to take into account the fact that the other seems to have received an information that led her to revise her beliefs in a different way. The theorem states that this process of belief revision will necessarily lead the players to hold the same posterior belief.

12 The ‘deeming possible’ operator was introduced by Kripke (1963), and is slightly more general than Aumann’s model. Samet (1990) characterises the restrictions that should be put on the ‘deeming possible’ operator to obtain Aumann’s model. See also Bacharach (1985).

13 I will not distinguish between common belief and common knowledge in this paper: the main reason is that the notion of common knowledge introduced in most of the papers that will be discussed below correspond to a notion of common belief with probability 1.

14 Werner Böge and some of his students in mathematics at Heidelberg developed in the 1970s a formal model of Bayesian games (Böge 1974; Armbruster and Böge 1979; Böge and Eisele 1979). They seem to be the first ones to formally define the notion of common belief in rationality (although they did not use the term, see Perea 2014, 14–15). Armbruster and Böge (1979) and Böge and Eisele (1979) propose a recursive procedure identifying all the choices that rational players could make under an assumption of common belief in rationality, which is essentially the same than in Pearce (1984) – neither Bernheim nor Pearce however refer to those papers.

15 In the first step, we start with the whole set of strategies for every player i. In the second step, we only keep the strategies s_i for which there exists a belief about the choice of j such that s_i is optimal. We continue the same process with the remaining strategies. Bernheim and Pearce’s approaches lead to the exact same set of rationalisable strategies.

16 Their paper distinguishes between ‘independent’ rationalisability (as in Bernheim and Pearce, since the priors are supposed to be distributed independently) and ‘correlated’ rationalisability (which was a case considered by Böge, though his 1979 papers are not mentioned by Brandenburger and Dekel). Correlated rationalisability is equivalent to a posteriori equilibrium, while they introduce the refinement of conditionally independent a posteriori equilibrium for the equivalence with independent rationalisability.

17 The doubled game 2G is defined as the game in which every strategy is available twice (doubling strategies can have a significant impact when priors are correlated, because it allows more subtle beliefs).The difference with Aumann’s 1987 paper is that rational expectations are defined at the interim stage rather than the prior stage (see Aumann and Drèze, 2008, 80). This is why the resulting equilibrium is slightly different between the two papers.

18 Although CBR is routinely used by epistemic game theorists (and considered by Perea (2014, 13) as its ‘central idea’), it is not constitutive of the epistemic program. Aumann and Brandenburger (1995) for instance show that mutual knowledge of rationality combined with common priors and common knowledge of players’ conjectures imply that their beliefs form a Nash equilibrium. The important point here is that EGT looks for a combination of epistemic conditions such that a single ‘equilibrium’ of beliefs remain. CBR is the most efficient condition to generate such consistent beliefs, but other conditions can also work (common priors and the common knowledge of players’ beliefs imply for instance the convergence to a single profile by Aumann’s agreement theorem).

19 If we allow for correlated priors (as in Aumann 1987), then player i’s belief about player j’s action can be conditioned on player i’s own action. This does not however mean that beliefs are action-dependent (see Hammond, 2009), i.e. that i believes that his action could directly influence the action of the others, since the players do not genuinely ‘choose’ in Aumann’s model – their action is indeed given by the state of the world.

20 The important point for the argument is that the prior distribution is a formal tool to encode the beliefs of the players, and not an actual belief held by the player. The argument about the possibility of self-prediction is a disputed topic (see e.g. Joyce 2002, Rabinowicz 2002) but is not of primary importance here.

21 Consider a coordination game in which we can choose either ‘Left’ or ‘Right’. Suppose that I believe you always choose the same side than me (Left when I choose Left, Right when I choose Right), but that you believe that I always choose the opposite side (Left when you choose Right, Right when you choose Left). It turns out that it is impossible to define here a distribution P over the set of possible outcomes from which we could derive our individual beliefs. It means that I cannot define a prior distribution that encodes my beliefs about the game. Larrouy and Lecouteux (2017, 329–332) provide a similar illustration in a prisoner’s dilemma.

22 I thank an anonymous referee for pointing out this reference.

23 A reason why the semantic approach has not been replaced by the ‘hierarchical’ approach of Mertens and Zamir (1985) and Brandenburger and Dekel (1993), which generalises Harsanyi’s type-based approach, and in which the states of the world have a more traditional interpretation, is that the hierarchical approach tends to generate an increasing analytical complexity. Aumann (1999) for instance describes it as ‘cumbersome and far from transparent […] the hierarchy construction is so convoluted that we present it here with some diffidence’ (265–295).

24 We could also consider e.g. team reasoning (see Lecouteux 2018 for a literature review), Stackelberg reasoning (Colman and Bacharach 1997), or the theory of social situations (Greenberg 1990, Mariotti 1997) as alternative approaches.