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Lecture4-addendum

Lecture4-addendum - 6.254 Game Theory Lecture 4 Correlated...

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6.254: Game Theory February 11, 2010 Lecture 4: Correlated Rationalizability Lecturer: Asu Ozdaglar 1 Correlated Rationalizability In this note, we allow a player to believe that the other players’ actions are correlated— in other words, the other players might be in a coalition and thus pick their strategies together instead of individually. To capture this idea, we slightly modify our definition of a belief. Definition 1.1 A belief of player i about the other players’ actions is a probability measure over the set S - i , which we denote as Δ( S - i ) . Note that we allow correlation in our belief. Recall, Δ( S ) denotes a probability distribu- tion over S . One possible type of distribution is the product distribution S 1 × S 2 × ... × S I , which denotes independent mixing between the I players. In general however, the distribu- tion Δ( S ) allows correlation in the strategies of players. Definition 1.2 An action s i S i is a rational action if there exists a belief α - i Δ( S - i ) such that s i is a best response to α - i . To define rationalizable strategies, we eliminate actions that are not best responses to any belief, or in other words, we eliminate actions that are never-best responses . Let us next recall the definitions of “never-best response” strategy and “strictly dominated” strategy. Definition 1.3 1. An action s i is a never-best response if for all beliefs α - i there exists σ i Σ i such that u i ( σ i , α - i ) > u i ( s i , α - i ) . 2. An action s i is strictly dominated if there exists σ i Σ i such that u i ( σ i , s - i ) > u i ( s i , s - i ) for all s - i S - i . 4-1
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It is straightforward to show that a strictly dominated action is a never-best response. Does the other direction hold? We have shown in the previous lecture that it doesn’t hold if beliefs are independent mixings. In this lecture, we will show that this direction holds for correlated beliefs. 2 Strict Dominance & Correlated Rationalizability We first formally define the process of iterative elimination of strictly dominated strategies. Algorithm 2.1 (Strict Dominance Iteration) Let S 0 i = S i and Σ 0 i = Σ i . For each player i ∈ I and for each n 1 , we define S n i as S n i = { s i S n - 1 i | there is no σ i Σ n - 1 i such that u i ( σ i , s - i ) > u i ( s i , s - i ) for all s - i S n - 1 - i } . Independently mix over S n i to get Σ n i . Let D i = n =1 S n i . We refer to the set D i as the set of strategies of player i that survive iterated strict dominance. We next define the process of iterative elimination of never-best response strategies. Re- call our notation that Δ( A ) denotes the set of probability measures over the set A [implying that the set Δ( S - i ) denotes the set of all probability measures over the set S - i , including independent mixings]. Algorithm 2.2 (Correlated Rationalizability Iteration) Start with ˜ S 0 i = S i . Then, for each player i ∈ I and for each n 1 , ˜ S n i = { s i ˜ S n - 1 i | there exists α - i Δ( ˜ S n - 1 - i ) such that u i ( s i , α - i ) u i ( s 0 i , α - i ) for all s 0 i ˜ S n - 1 i } .
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