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# search-s - Mediators Slides by Sherwin Doroudi Adapted from...

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Mediators Slides by Sherwin Doroudi Adapted from “Mediators in Position Auctions” by Itai Ashlagi, Dov Monderer, and Moshe Tennenholtz

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Bayesian & Pre-Bayesian Games Consider a game where every player has private information regarding his/her “type” A player’s strategy maps types to actions Ex: You are either type A or type B and you have actions “play” and “pass”; one strategy might be A→ “play” and B→ “pass”; we can write this as (A, B) → (“play”, “pass”)
Bayesian & Pre-Bayesian Games These are games of incomplete information In a Bayesian Game there is a commonly known prior probability measure on the profile of types Ex: You are either type A or B, I am either type X or Y, and we know that it is common knowledge that our types are equality likely to be (A, X), (A, Y), (B, X), or (B, Y)

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Bayesian & Pre-Bayesian Games In a pre-Bayesian game, there is no prior probability over they types the players can take Ex: An auction setting in which there is no known distribution with which the players value the goods We will be concerned only with pre- Bayesian games
Equilibria in Pre-Bayesian Games When priors regarding types are not known (i.e. in pre-Bayesian game) we are primarily concerned with ex post equilibrium “A profile of strategies, one for each player, such that no player has a profitable deviation independently of the types of the other players” Requiring dominant strategies is stricter

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Ex: Pre-Bayesian Game Game H : Assume player I has only one type but player II is either type A or type B 5 2 3 0 0 0 4 2 I II α β α β Type(II) = A 2 2 0 0 3 3 5 2 I II α β α β Type(II) = B
Ex: Pre-Bayesian Game Game H : The ex post equilibrium is I plays β and II plays (A, B) →(β, α) 5 2 3 0 0 0 4 2 I II α β α β Type(II) = A 2 2 0 0 3 3 5 2 I II α β α β Type(II) = B

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Question In general, do all pre-Bayesian games have at least one ex post equilibrium?
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search-s - Mediators Slides by Sherwin Doroudi Adapted from...

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