Robust Equilibria under Non-Common Priors

Abstract

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.

Coordination-free equilibria in cheap talk games

“…Online Appendix C shows that if the robustness concept were relaxed to require only that some "nearby" strategy pro…le be nearly optimal, then the results would still hold if heterogeneous priors about noise are allowed.3 For example,Oyama and Tercieux (2010) show, in …nite complete information games, that generically, an equilibrium is robust only if it is the unique rationalizable action pro…le Weinstein and Yildiz (2007). show a similar result when, instead, interim beliefs are concentrated around the complete information payo¤s.…”

Robustness to incomplete information in repeated games

“…A.1.2 General noncommon priors In the case where players have significantly different priors (P i ) i∈N , in the sense that m(P P i ) is large, robustness to such perturbations is a much more stringent requirement than robustness to common-prior perturbations. In a generic static game, Oyama and Tercieux (2010) show that a Nash equilibrium is robust to incomplete information with noncommon priors if and only if it is iteratively dominant. One can extend their result to dynamic settings and show that a PPE is strongly robust to incomplete information with noncommon priors if and only if it is iteratively stage-dominant.…”

“…Note also that KM maintain the common-prior assumption. Oyama and Tercieux (2010) and Izmalkov and Yildiz (2010) consider incomplete-information perturbations that do not satisfy the common-prior assumption. We relax the common-prior assumption in Appendix A.…”

Robust Rationalizability Under Almost Common Certainty of Payoffs

“…In this case, they showed that with high probability, there is common p-belief about payoffs with p < 1/|I|, where |I| is the number of players. Oyama and Tercieux (2010) discuss how much their result relies on the common prior assumption and in particular, how much common p-belief about payoffs must be weakened when players may have ex ante heterogeneous beliefs. 2 One can ask the converse of Monderer and Samet's result: what is the weakest topology over higher-order beliefs that guarantees the continuity of equilibria or rationalizable actions?…”

“…We do not, however, take this route in this paper, for we are afraid of introducing ex ante perspectives in a setup with otherwise exclusively interim perspectives. See Oyama and Tercieux (2010), who discuss the relation between robustness notions from ex ante and interim perspectives.…”