Abstract
We show that the value of a zero-sum Bayesian game is a Lipschitz continuous function of the players’ common prior belief with respect to the total variation metric on beliefs. This is unlike the case of general Bayesian games where lower semi-continuity of Bayesian equilibrium (BE) payoffs rests on the “almost uniform” convergence of conditional beliefs. We also show upper semi-continuity (USC) and approximate lower semi-continuity (ALSC) of the optimal strategy correspondence, and discuss ALSC of the BE correspondence in the context of zero-sum games. In particular, the interim BE correspondence is shown to be ALSC for some classes of information structures with highly non-uniform convergence of beliefs, that would not give rise to ALSC of BE in non-zero-sum games.
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Einy, E., Haimanko, O. & Tumendemberel, B. Continuity of the value and optimal strategies when common priors change. Int J Game Theory 41, 829–849 (2012). https://doi.org/10.1007/s00182-010-0248-4
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DOI: https://doi.org/10.1007/s00182-010-0248-4
Keywords
- Zero-sum Bayesian games
- Common prior
- Value
- Optimal strategies
- Interim
- Ex-ante
- Bayesian equilibrium
- Upper semi-continuity
- Lower approximate semi-continuity