Abstract
In this paper the concept of belief distorted Nash equilibrium (BDNE) is introduced. It is a new concept of equilibrium for games in which players have incomplete, ambiguous or distorted information about the game they play, especially in a dynamic context. The distortion of information of a player concerns the fact how the other players and/or an external system changing in response to players’ decisions, are going to react to his/her current decision. The concept of BDNE encompasses a broader concept of pre-BDNE, which reflects the fact that players best respond to their beliefs, and self-verification of those beliefs. The relations between BDNE and Nash or subjective equilibria are examined, as well as the existence and properties of BDNE. Examples are presented, including models of a common ecosystem, repeated Cournot oligopoly, a repeated Minority Game or local public good with congestion effect and a repeated Prisoner’s Dilemma.
Highlights
There are many attempts to extend the concept of Nash equilibria to make it work in the case of incomplete information: among others, Bayesian equilibria introduced by Harsanyi (1967), correlated equilibria introduced by Aumann (1974, 1987), -rationalizability introduced by Battigalli and Siniscalchi (2003), self-confirming equilibria introduced by Fudenberg and Levine (1993) and subjective equilibria introduced by Kalai and Lehrer (1993, 1995)
In Example 3, we present a situation when direct comparison between subjective equilibria and pre-belief distorted Nash equilibrium (BDNE) is possible first considered in Kalai and Lehrer (1995)
We start by the continuum of players game and we show that there are beliefs for which the n-player Nash equilibrium can be obtained as a pre-BDNE
Summary
It is claimed that Nash equilibrium is the most important concept in noncooperative game theory. There are many attempts to extend the concept of Nash equilibria to make it work in the case of incomplete information: among others, Bayesian equilibria introduced by Harsanyi (1967), correlated equilibria introduced by Aumann (1974, 1987), -rationalizability introduced by Battigalli and Siniscalchi (2003), self-confirming equilibria introduced by Fudenberg and Levine (1993) and subjective equilibria introduced by Kalai and Lehrer (1993, 1995) These and some other equilibrium concepts tackling the problem of incomplete information are described in a more detailed way in “Other concepts of equilibria taking incomplete, ambiguous or distorted information into account” section in Appendix 3. False knowledge about reality may sustain and people may believe that they play a Nash equilibrium
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