lec13_slides.pdf - General-Sum Games March 6 2018 Minor canges to class by Prof Peter Bartlett Outline Impartial combinatorial games I Multiplayer

# lec13_slides.pdf - General-Sum Games March 6 2018 Minor...

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General-Sum Games March 6, 2018 Minor canges to class by Prof. Peter Bartlett Outline Impartial combinatorial games I Multiplayer general-sum games 1. Definitions: utility functions, Nash equilibria. 2. Nash’s Theorem I Congestion games and potential games 1. Congestion games 2. Every congestion game has a pure Nash equilibrium 3. Potential games 4. Every congestion game is a potential game Multiplayer general-sum games Notation I A k -person general-sum game is specified by k utility functions, u j : S 1 × S 2 × . . . × S k R . I Player j can choose strategies s j S j . I Simultaneously, each player chooses a strategy. I Player j receives payoff u j ( s 1 , . . . , s k ) . I k = 2 : u 1 ( i , j ) = a ij , u 2 ( i , j ) = b ij . I For s = ( s 1 , . . . , s k ) , let s - i denote the strategies without the ith one: s - i = ( s 1 , . . . , s i - 1 , s i + 1 , . . . , s k ) . I And write ( s i , s - i ) as the full vector. Multiplayer general-sum games Definition A vector ( s * 1 , . . . , s * k ) S 1 × . . . × S k is a pure Nash equilibrium for utility functions u 1 , . . . , u k if, for each player j ∈ { 1 , . . . , k } , max s j S j u j ( s j , s * - j ) = u j ( s * j , s * - j ) .   Multiplayer general-sum games Lemma Lemma Consider a strategy profile x Δ S 1 × . . . × Δ S k . Let T i = { s S i : x i ( s ) > 0 } . The following statements are equivalent. Multiplayer general-sum games Lemma Lemma Consider a strategy profile x Δ S 1 × . . . × Δ S k . Let T i = { s S i : x i ( s ) > 0 } . The following statements are equivalent. I x is a Nash equilibrium. Multiplayer general-sum games LemmaLemmaConsider a strategy profile xΔS1×. . .×ΔSk. LetTi={sSi:xi(s)>0}. The following statements areequivalent.Ix is a Nash equilibrium.IFor each i, there is a cisuch that Multiplayer general-sum games Lemma Lemma Consider a strategy profile x Δ S 1 × . . . × Δ S k . Let T i = { s S i : x i ( s ) > 0 } . The following statements are equivalent. I x is a Nash equilibrium. I For each i, there is a c i such that 1. For s i T i , u i ( s i , x - i ) = c i . Multiplayer general-sum games Lemma Lemma Consider a strategy profile x Δ S 1 × . . . × Δ S k . Let T i = { s S i : x i ( s ) > 0 } . The following statements are equivalent.  • • • 