An Impossibility Theorem on Beliefs in Games

Authors: Brandenburger, Adam¹; Keisler, H.²

Source: Studia Logica, Volume 84, Number 2, November 2006 , pp. 211-240(30)

Publisher: Springer

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Abstract:

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell's Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:

Ann believes that Bob assumes that

Ann believes that Bob's assumption is wrong

This is formalized to show that any belief model of a certain kind must have a `hole.' An interpretation of the result is that if the analyst's tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.

Keywords: belief model; complete belief model; game; first order logic; modal logic; paradox

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s11225-006-9011-z

Affiliations: 1: Email: adam.brandenburger@stern.nyu.edu 2: Email: keisler@math.wisc.edu

Publication date: 2006-11-01