Abstract

Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. The analyst’s model approximately describes each player’s hierarchies of beliefs over payoff-relevant states, but the true incomplete information game may have correlated duplicated belief hierarchies, and the analyst has no information about the correlation. Under these circumstances, a natural candidate for the analyst’s prediction is the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s incomplete information game. We introduce the concept of robustness for BIBCE: a subset of BIBCE is robust if every nearby incomplete information game has a BIBCE that is close to some BIBCE in this set. Our main result provides a sufficient condition for robustness by introducing a generalized potential function of an incomplete information game. A generalized potential function is a function on the Cartesian product of the set of states and a covering of the action space which incorporates some information about players’ preferences. It is associated with a belief-invariant correlating device such that a signal sent to a player is a subset of the player’s actions, which can be interpreted as a vague prescription to choose some action from this subset. We show that, for every belief-invariant correlating device that maximizes the expected value of a generalized potential function, there exists a BIBCE in which every player chooses an action from a subset of actions prescribed by the device, and that the set of such BIBCE is robust, which can differ from the set of potential maximizing BNE.

Introduction

Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. He believes that his model correctly describes the probability distribution over the players’ Mertens-Zamir hierarchies of beliefs over payoff-relevant states (Mertens and Zamir, 1985). However, players may have observed signals generated by an individually uninformative correlating device (Liu, 2015), which allows the players to correlate their behavior. In other words, the true incomplete information game may have correlated duplicated belief hierarchies (Ely and Peski, 2006; Dekel et al., 2007). Then, a natural candidate for the analyst’s prediction is the set of outcomes that can arise in some Bayes Nash equilibrium (BNE) of some incomplete information game with the same distribution over belief hierarchies. Liu (2015) shows that this set of outcomes can be characterized as the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s model. A BIBCE is a Bayes correlated equilibrium (BCE) in which a prescribed action does not reveal any additional information to the player about the opponents’ types and the payoff-relevant state, thus preserving the player’s belief hierarchy.

WP019

2020.03

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The abstract of our 1997 survey paper Kajii and Morris (1997b) on "Refinements and Higher Order Beliefs" reads:

This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher order beliefs literature” (Rubinstein 1989; Monderer and Samet 1989; Morris, Rob and Shin 1995; Kajii and Morris 1997a) and the “payoff uncertainty approach” (Fudenberg, Kreps and Levine 1988; Dekel and Fudenberg 1990).

In particular, this paper provided a unified framework to relate the notion of equilibria robust to incomplete information introduced in Kajii and Morris (1997a) [Hereafter, KM1997] to the classic refinements literature. It followed Fudenberg, Kreps and Levine (1988) and Kreps (1990) in relating refinements of Nash equilibria to a "complete theory" where behavior was rationalized by explicit incomplete information about payoffs, rather than depending on action trembles or other exogenous perturbations. It followed Fudenberg and Tirole (1991), chapter 14, in providing a unified treatment of refinements and a literature on higher-order beliefs rather than proposing a particular solution concept.

The primary purpose of the survey paper was to promote the idea of robust equilibria in KM1997 and we did not try to publish it as an independent paper. Since we wrote this paper, there have been many developments in the literature on robust equilibria, fortunately. But there has been little work emphasizing a unified perspective, and consequently this paper seems more relevant than ever. We are therefore very happy to publish it twenty years later. We provide some notes in the following on relevant developments in the literature and how they relate to the survey. These notes assume familiarity with the basic concepts introduced in the survey paper and KM1997.

WP006

2019.03

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