MIT14_126S10_lec01 - Review of Basic Concepts Normal form...

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Review of Basic Concepts: Normal form 14.126 Game Theory Muhamet Yildiz Road Map • Normal-form Games • Dominance & Rationalizability • Nash Equilibrium – Existence and continuity properties • Bayesian Games – Normal-form/agent-normal-form representations – Bayesian Nash equilibrium—equivalence to Nash equilibrium, existence and continuity 1
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Normal-form games A (normal form) game is a triplet ( N , S , u ): N = {1, . . . , n } is a (finite) set of players . S = S 1 × × S n where S i is the set of pure strategies of player i . u = ( u 1 ,…, u n ) where u i : S R is player i ’s vNM utility function . A normal form game is finite if S and N are finite. The game is common knowledge. Mixed Strategies, beliefs Δ ( X ) = Probability distributions on X . Δ ( S i ) = Mixed strategies of player i . Independent strategy profile: σ = σ 1 × ... × σ n ∈ Δ ( S 1 ) × ... × Δ ( S n ) correlated strategy profile: σ ∈ Δ ( S ) Δ ( S -i ) = possible conjectures of player i (beliefs about the other players’ strategies). [ σ -i ∈ Δ ( S -i )] – A player may believe that the other players’ strategies are correlated! Expected payoffs: u i ( σ ) = E σ ( u i ) = Σ s S σ ( s ) u i ( s ) 2
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Rationality & Dominance Player i is rational if he maximizes his expected payoff given his belief. s i * is a best reply to a belief σ -i iff s i S i : u i ( s i *, σ -i ) u i ( s i , σ -i ). B i ( σ -i ) = best replies to σ -i . σ i strictly dominates s i iff s -i S -i : u i ( σ i , s -i ) > u i ( s i , s -i ). σ i weakly dominates s i iff s -i S -i : u i ( σ i , s -i ) u i ( s i , s -i ) with a strict inequality. Theorem: In a finite game, s i * is never a best reply to a ( possibly correlated ) conjecture σ -i iff s i * is strictly dominated (by a possibly mixed strategy). Proof of Theorem Let S -i = { s -i 1 ,…, s -i m }, u i ( s i ,.) = ( u i ( s i , s -i 1 ),…, u i ( s i , s -i m )) U = { u i ( s i ,.)| s i S i } – Co( U ) = convex hull of U = { u i ( σ i ,.)| σ i ∈ Δ ( S i )} (=>) Assume s i * B i ( σ -i ). ⇒∀ s i , u i ( s i *, σ -i ) u i ( s i , σ -i ) ⇒ ∀σ i , u i ( s i *, σ -i ) u i ( σ i , σ -i ) No σ i strictly dominates s i *.
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