Epistemic Game Theory (2018)
Prepared for Handbook of Rationality
Abstract: In this chapter we review some of the most important ideas, concepts and results in epistemic game theory, with a focus on the central idea of common belief in rationality. We start by showing how belief hierarchies can be encoded by means of epistemic models with types, and how this encoding can be used to formally define common belief in rationality. We next indicate how the induced choices can be characterized by a recursive elimination procedure, and how the concept relates to Nash equilibrium. Finally, we investigate how the idea of common belief in rationality can be extended to dynamic games, by looking at several plausible ways in which players may revise their beliefs.
Previous version appeared as EPICENTER Working Paper No. 13
Abstract: This paper investigates static games with unawareness, where players may be unaware of some of the choices that can be made by other players. That is, different players may have different views on the game. We propose an epistemic model that encodes players' belief hierarchies on choices and views, and use it to formulate the basic reasoning concept of common belief in rationality. We do so for two scenarios: one in which we only limit the possible views that may enter the players' belief hierarchies, and one in which we fix the players' belief hierarchies on views. For both scenarios we design a recursive elimination procedure that yields for every possible view the choices that can rationally be made under common belief in rationality.
with Arkadi Predtetchinski
Abstract: In this paper we focus on stochastic games with finitely many states and actions. For this setting we study the epistemic concept of common belief in future rationality, which is based on the condition that players always believe that their opponents will choose rationally in the future. We distinguish two different versions of the concept -- one for the discounted case with a fixed discount factor δ, and one for the case of uniform optimality, where optimality is required for "all discount factors close enough to 1".
We show that both versions of common belief in future rationality are always possible in every stochastic game, and always allow for stationary optimal strategies. That is, for both versions we can always find belief hierarchies that express common belief in future rationality, and that have stationary optimal strategies. We also provide an epistemic characterization of subgame perfect equilibrium for two-player stochastic games, showing that it is equivalent to common belief in future rationality together with some "correct beliefs assumption".
EPICENTER Working Paper No. 1
Abstract: Harsanyi (1967--1968) showed that belief hierarchies can be encoded by means of epistemic models with types. Indeed, for every type within an epistemic model we can derive the full belief hierarchy it induces. But for one particular belief hierarchy, there are in general many different ways of encoding it within an epistemic model. In this paper we give necessary and sufficient conditions such that two types, from two possibly different epistemic models, induce exactly the same belief hierarchy. The conditions are relatively easy to check, and seem relevant both for practical and theoretical purposes.
with Geir Asheim
Abstract: We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility, proper rationalizability and full permissibility by means of the concepts of preference restrictions and likelihood orderings. We apply the algorithms for comparing iterated admissibility, proper rationalizability and full permissibility, and provide a sufficient condition under which iterated admissibility does not rule out properly rationalizable strategies. Finally, we use the algorithms to examine an economically relevant strategic situation, namely a bilateral commitment bargaining game.
with Elias Tsakas
Abstract: In this paper we introduce a novel framework for modeling bounded reasoning in dynamic games, based on the idea that at each history of the game each player pays attention to some -- not necessarily all -- histories. We refer to this phenomenon as local reasoning, and we show that several extensively studied types of bounded rationality can be studied within this framework, such as for instance limited memory or limited foresight. Then, we proceed to study a standard form of reasoning within our framework, according to which each player tries to rationalize her opponents' past actions at the histories that she reasons about. As a result we obtain a generalized solution concept, which we call common strong belief in rationality. We characterize the strategy profiles that can rationally be played under our concept by means of a simple iterative elimination procedure. Finally, we show that standard existing solution concepts -- such as extensive-form rationalizability or the backward dominance procedure -- are special cases of rationality and common strong belief in rationality.
with Christian Nauerz
EPICENTER Working Paper No. 6
Abstract: Abstract models of decision-making under ambiguity are widely used in economics. One stream of such models results from weakening the independence axiom in Anscombe et al. (1963). We identify necessary assumptions on independence to represent the decision maker's preferences such that he acts as if he maximizes expected utility with respect to a possibly local prior. We call the resulting representation Local Prior Expected Utility, and show that the prior used to evaluate a certain act can be obtained by computing the gradient of some appropriately defined utility mapping. The numbers in the gradient, moreover, can naturally be interpreted as the subjective likelihoods the decision maker assigns to the various states. Building on this result we provide a unified approach to the representation results of Maximin Expected Utility and Choquet Expected Utility and characterize the respective sets of priors.
with Christian Bach
Abstract: In games with incomplete information, players face uncertainty about the opponents' utility functions. We follow Harsanyi's (1967-68) one-person perspective approach to modelling incomplete information. Moreover, our formal framework is kept as basic and parsimonious as possible, to render the theory of incomplete information accessible to a broad spectrum of potential applications. In particular, we formalize common belief in rationality and provide an algorithmic characterization of it in terms of decision problems, which gives rise to the non-equilibrium solution concept of generalized iterated strict dominance.
EPICENTER Working Paper No. 8
Abstract: In this paper we investigate the order independence of iterated reduction procedures in dynamic games. We distinguish between two types of order independence: with respect to strategies and with respect to outcomes. The first states that the specific order of elimination chosen should not affect the final set of strategy combinations, whereas the second states that it should not affect the final set of reachable outcomes in the game. We provide sufficient conditions for both types of order independence: monotonicity, and monotonicity on reachable histories, respectively.
We use these sufficient conditions to explore the order independence properties of various reduction procedures in dynamic games: the extensive-form rationalizability procedure (Pearce (1984), Battigalli (1997)), the backward dominance procedure (Perea (2014)) and Battigalli and Siniscalchi's (1999) procedure for jointly rational belief systems (Reny (1993)). We finally exploit these results to prove that every outcome that is reachable under the extensive-form rationalizability procedure is also reachable under the backward dominance procedure.
with Stephan Jagau
Revised version of EPICENTER Working Paper No. 10
Abstract: Belief-dependent motivations and emotional mechanisms such as surprise, anxiety, anger, guilt, and intention-based reciprocity pervade real-life human interaction. At the same time, traditional game theory has experienced huge difficulties trying to capture them adequately. Psychological game theory, initially introduced by Geanakoplos et al. (1989), has proven to be a useful modeling framework for these and many more psychological phenomena. In this paper, we use the epistemic approach to psychological games to systematically study common belief in rationality, also known as correlated rationalizability. We show that common belief in rationality is possible in any game that preserves rationality at infinity, a mild requirement that is considerably weaker than the previously known continuity conditions from Geanakoplos et al. (1989) and Battigalli and Dufwenberg (2009). Also, we provide an example showing that common belief in rationality might be impossible in games where rationality is not preserved at infinity. We then develop an iterative procedure that, for a given psychological game, determines all rationalizable choices. In addition, we explore classes of psychological games that allow for a simplified procedure.
with Christian Bach
EPICENTER Working Paper No. 9
Abstract: In games with incomplete information Bayesian equilibrium constitutes the prevailing solution concept. We show that Bayesian equilibrium generalizes correlated equilibrium from complete to incomplete information. In particular, we provide an epistemic characterization of Bayesian equilibrium as well as of correlated equilibrium in terms of common belief in rationality and a common prior. Bayesian equilibrium is thus not the incomplete information counterpart of Nash equilibrium. To fill the resulting gap, we introduce the solution concept of generalized Nash equilibrium as the incomplete information analogue to Nash equilibrium, and show that it is more restrictive than Bayesian equilibrium. Besides, we propose a simplified tool to compute Bayesian equilibria.
with Christian Bach
EPICENTER Working Paper No. 11
Abstract: This note considers generalized Nash equilibrium as an incomplete information analogue of Nash equilibrium and provides an epistemic characterization of it. It is shown that the epistemic conditions do not imply common belief in rationality. For the special case of complete information, an epistemic characterization of Nash equilibrium ensues as a corollary.
Abstract: Battigalli (1997) has shown that in dynamic games with perfect information and without relevant ties, the forward induction concept of extensive-form rationalizability yields the backward induction outcome. In this paper we provide a new proof for this remarkable result, based on four steps. We first show that extensive-form rationalizability can be characterized by the iterated application of a special reduction operator, the strong belief reduction operator. We next prove that this operator satisfies a mild version of monotonicity, which we call monotonicity on reachable histories. This property is used to show that for this operator, every possible order of elimination leads to the same set of outcomes. We finally show that backward induction yields a possible order of elimination for the strong belief reduction operator. These four properties together imply Battigalli's theorem.