Working Papers

Working Papers

Expected utility as an expression of linear preference intensity (2022)

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 that is both necessary and sufficient for admitting an expected utility representation. Most of these axioms express the idea that the decision maker's preference intensity between two choices changes linearly with the belief. Finally, we show that under certain conditions the relative utility differences are unique across the different expected utility representations.

Structure-Preserving Transformations of Epistemic Models (2021)

with Christian W. Bach

EPICENTER Working Paper No. 24

Abstract:  The prevailing approaches to modelling interactive uncertainty with epistemic models in economics are state-based and type-based. We explicitly formulate two general procedures that transform state models into type models and vice versa. Both transformation procedures preserve the belief hierarchies as well as the common prior assumption. By means of counterexamples it is shown that the two procedures are not inverse to each other. However, if attention is restricted to maximally reduced epistemic models, then isomorphisms can be constructed and an inverse relationship emerges.

Rational Updating at the Crossroads (2021)

with Silvia Milano

EPICENTER Working Paper No. 23

Abstract: In this paper we explore the absentminded driver problem using two di erent scenarios. In the rst scenario we assume that the driver is capable of reasoning about his degree of absentmindedness before he hits the highway. This leads to a Savage-style model where the states are mutually exclusive and the act-state independence is in place. In the second we employ centred possibilities, by modelling the states (i.e. the events about which the driver is uncertain) as the possible nal destinations indexed by a time period. The optimal probability we nd for continuing at an exit is di erent from almost all papers in the literature. In this scenario, act-state independence is still violated, but states are mutually exclusive and the driver arrives at his optimal choice probability via Bayesian updating. We show that our solution is the only one guaranteeing immunity from sure loss via a Dutch strategy, and that { despite initial appearances { it is time consistent

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. 

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.