Epistemic game theory: beliefs and types

Abstract

John Harsanyi [19] introduced the formalism of type spaces to provide a simple and parsimonious representation of belief hierarchies. He explicitly noted that his formalism was not limited to modeling a player’s beliefs about payoff-relevant variables: rather, its strength was precisely the ease with wich Ann’s beliefs about Bob’s beliefs about payoff variables, Ann’s beliefs about Bob’s beliefs about Ann’s beliefs about payoff variables, etc. could be represented. This feature plays a prominent role in the epistemic analysis of solution concepts (see the article by Adam Brandenburger elsewhere in this volume), as well as in the literature on global games (Morris and Shin [25]) and on robust mechanism design (Bergemann and Morris [7]). All these applications place particular emphasis on the expressiveness of the type-space formalism. Thus, a natural question arises: just how expressive is Harsanyi’s approach? For instance, solution concepts such as Nash equilibrium or rationalizability can be characterized by means of restrictions on the players’mutual beliefs. In principle, these assumptions could be formulated directly as restrictions on players’ hierarchies of beliefs; but, in practice, the analysis is mostly carried out in the context of a type space a la Harsanyi. This is without loss of generality only if Harsanyi type spaces do not themselves impose restrictions on the belief hierarchies that can be represented. Similar considerations apply in the context of robust mechanism design. A rich literature addresses this issue from different angles, and for a variety of basic representations of beliefs. This article focuses on hierarchies of probabilistic beliefs; however, some extensions are also mentioned. For simplicity, attention is restricted to two players, denoted “1” and “2” or “i” and “−i .” ∗Economics Department, Northwestern University, Evanston, IL 60208-2600. Email: marciano@northwestern.edu. I thank Pierpaolo Battigalli and Adam Brandenburger for helpful discussion.