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Belief system μ specifies a probability distribution over nodes in each information set. Write μ (z|h) for the probability of node z given information set h.
Strategy profile σ is sequentially rational with respect to belief system μ if each player’s strategy maximizes his expected utility at every information set, given the strategies of other players and the beliefs over nodes at that information set.
Belief system μ is consistent with strategy profile σ if beliefs at each information set reached by σ are generated by σ and Bayes rule.
Definition: A strategy profile and beliefs system, (σ, μ), is a weak perfect Bayesian equilibrium if σ is sequentially rational with respect to μ and μ is consistent with σ. Strategy profile σ is a weak perfect Bayesian equilibrium strategy profile if (σ, μ) is a weak perfect Bayesian equilibrium for some belief system μ.
If strategy profile σ0 has “full support,” all information sets are reached and thus there is a unique belief system μ0 such that μ0 is consistent with σ0. Beliefs μ are said to be fully consistent with σ if there exists a full support sequence σk, with σk → σ, such that unique beliefs μk consistent with σk satisfy μk → μ.
Definition: A strategy profile and beliefs system, (σ, μ), is a sequential equilibrium if σ is sequentially rational w.r.t. μ and μ is fully consistent given σ. Strategy profile σ is sequential equilibrium strategy profile if (σ, μ) is a sequential equilibrium for some belief system μ.
Reference:
Andrew Postlewaite, 2005, Notes on "Game Theory, Choice under Uncertainty and Information Economics"
Kreps and Wilson (1982)
Myerson, Game Theory: Analysis of Conflict, chapter 4
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