Working Papers

Working Papers

A Foundation for Expected Utility in Decision Problems and Games (2021)

For my presentation at the One World Mathematical Game Theory Seminar, click here.

A previous version can be found here:

EPICENTER Working Paper No. 22

Abstract: In a decision problem or game we typically fix the person's utilities but not his beliefs. What, then, do these utilities represent? To explore this question we assume that the decision maker holds a conditional preference relation -- a mapping that assigns to every possible probabilistic belief a preference relation over his choices. We impose a list of axioms on such conditional preference relations, and show that they single out those conditional preference relations that admit an expected utility representation. If there are no weakly dominated choices, the key property is the existence of a uniform preference increase, which states that the decision maker should be able to uniformly increase the preference intensity for one of his choices without contradicting the conditional preference relation. In the presence of weakly dominated choices this condition is strengthened to the existence of coherent uniform preference increases. We also present a procedure that can be used to construct, for a given conditional preference relation satisfying the axioms, a utility function that represents it. If there are no weakly dominated choices, the existence of a uniform preference increase can be replaced by two easily verifiable conditions: strong transitivity and the line property.

Reasoning about Your Own Future Mistakes (2021)

with Martin Meier

A previous version can be found here:

EPICENTER Working Paper No. 21

Abstract: We propose a model of reasoning in dynamic games in which a player, at each information set, holds a conditional belief about his own future choices and the opponents' future choices. These conditional beliefs are assumed to be cautious, that is, the player never completely rules out any feasible future choice by himself or the opponents. We impose the following key conditions: (a) a player always believes that he will choose rationally in the future, (b) a player always believes that his opponents will choose rationally in the future, and (c) a player deems his own mistakes infinitely less likely than the opponents' mistakes. Common belief in these conditions leads to the new concept of strong sequential rationalizability. We show that strongly sequentially rationalizable strategies exist in every finite dynamic game. We prove, moreover, that strong sequential rationalizability constitutes a refinement of both perfect rationalizability (a rationalizability analogue to Selten's (1975) perfect equilibrium) and procedural quasi-perfect rationalizability (a rationalizability analogue to van Damme's (1984) quasi-perfect equilibrium). As a consequence, it avoids both weakly dominated strategies in the normal form and strategies containing weakly dominated actions in the agent normal form.

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.

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.

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.