Working Papers

Two Definitions of Correlated Equilibrium (2018)

with Christian W. Bach

EPICENTER Working Paper No. 18

Abstract: Correlated equilibrium has been introduced by Aumann (1974). Often, in the literature, correlated equilibrium is defined in a simplified as well as more direct way, and sometimes called canonical correlated equilibrium or correlated equilibrium distribution. In fact, we show that the simplified notion of correlated equilibrium is not equivalent – neither doxastically nor behaviourally -- to the original from an ex post perspective. We then compare both solution concepts in terms of reasoning. While correlated equilibrium can be characterized by common belief in rationality and a common prior, the simplified variant additionally requires the one-theory-per-choice condition. Since this condition features a correctness of beliefs property, the latter solution concept exhibits a larger degree of Nash equilibrium flavour than the former.

Order Independence in Dynamic Games (2018)

Previous version appeared as 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.

Common Belief in Future and Restricted Past Rationality (2018)

with Rubén Becerril-Borja

EPICENTER Working Paper No. 17

Abstract: We introduce the idea that a player believes at every stage of a dynamic game that his opponents will choose rationally in the future and have chosen rationally in a restricted way in the past. This is summarized by the concept of common belief in future and restricted past rationality, which is defined epistemically. Moreover, it is shown that every properly rationalizable strategy of the normal form of a dynamic game can be chosen in the dynamic game under common belief in future and restricted past rationality. We also present an algorithm that uses strict dominance, and show that its full implementation selects exactly those strategies that can be chosen under common belief in future and restricted past rationality.

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.

Common Belief in Rationality in Games with Unawareness (2018)

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.

Common Belief in Rationality in Psychological Games (2018)

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.

Incomplete Information and Equilibrium (2017)

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.

Generalized Nash Equilibrium without Common Belief in Rationality (2017)

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.

Incomplete Information and Generalized Iterated Strict Dominance (2017)

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.

Local Prior Expected Utility: A Basis for Utility Representations under Uncertainty (2015)

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.

When do Types Induce the Same Belief Hierarchy? The Case of Finitely Many Types (2014)

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.