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Stat155 Game Theory Lecture 21: Midterm 2 Review

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Stat155 Game Theory Lecture 21: Midterm 2 Review Midterm 2 Stat155 Game Theory Lecture 21: Midterm 2 Review “This is an open book exam: you can use any printed or written material, but you cannot use a laptop, tablet, or phone (or any device that can communicate). There are three questions, each consisting of three parts. Peter Bartlett Each part of each question carries equal weight. Answer each question in the space provided.” November 8, 2016 1 / 59 2 / 59 Topics Series and parallel games Series and parallel games. A series game: Both players first play G1, then both play G2. Two-player general-sum games Value is V1 + V2. Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games A parallel game: Both players simultaneously decide which game to Utility functions, Nash equilibria. Nash’s Theorem play, and an action in that game. Congestion games and potential games If they choose the same game, they get the payoff from that game. Congestion games: Every congestion game has a pure Nash equilibrium If they choose different games, the payoff is 0. Potential games: Every congestion game is a potential game Value is1 /(1/V1 + 1/V2). Criticisms of Nash equilibria Evolutionarily stable strategies Electrical networks ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks 3 / 59 4 / 59 Topics Cooperative versus noncooperative games Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Noncooperative games Congestion games and potential games Players play their strategies simultaneously. Congestion games: Every congestion game has a pure Nash equilibrium Potential games: Every congestion game is a potential game They might communicate (or see a common signal; e.g., a traffic Criticisms of Nash equilibria signal), but there’s no enforced agreement. Evolutionarily stable strategies Natural solution concepts: ESS and Nash equilibria Nash equilibrium, correlated equilibrium. Multiplayer ESS No improvement from unilaterally deviating. Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks 5 / 59 6 / 59 General-sum games General-sum games Dominated pure strategies Notation A pure strategy e for Player I is dominated by e in payoff matrix A if, for A two-person general-sum game is specified by two payoff matrices, i i 0 m n all j 1,..., n , A, B R × . ∈ { } ∈ aij ai j . Simultaneously, Player I chooses i 1,..., m and the Player II ≤ 0 ∈ { } chooses j 1,..., n . Similarly, a pure strategy ej for Player II is dominated by ej0 in payoff ∈ { } matrix B if, for all i 1,..., m , Player I receives payoff aij . ∈ { } Player II receives payoff bij . b b . ij ≤ ij0 7 / 59 8 / 59 General-sum games General-sum games Nash equilibria Safety strategies A pair( x , y ) ∆m ∆n is a Nash equilibrium for payoff matrices ∗m n∗ ∈ × A safety strategy for Player I is an x ∆m that satisfies A, B R × if ∗ ∈ ∈ min x>Ay = max min x>Ay. max x>Ay = x>Ay , y ∆n ∗ x ∆m y ∆n x ∆m ∗ ∗ ∗ ∈ ∈ ∈ ∈ max x>By = x>By . x maximizes the worst case expected gain for Player I. y ∆n ∗ ∗ ∗ ∗ ∈ Similarly, a safety strategy for Player II is a y ∆n that satisfies ∗ ∈ If Player I plays x and Player II plays y , neither player has an min x>By = max min x>By. ∗ ∗ x ∆m ∗ y ∆n x ∆m incentive to unilaterally deviate. ∈ ∈ ∈ x is a best response to y , y is a best response to x . y maximizes the worst case expected gain for Player II. ∗ ∗ ∗ ∗ ∗ In general-sum games, there might be many Nash equilibria, with different payoff vectors. 9 / 59 10 / 59 Comparing two-player general-sum and zero-sum games Comparing two-player general-sum and zero-sum games Zero-sum games Zero-sum games 1 A pair of safety strategies is a Nash equilibrium (minimax theorem) 4 If each player has an equalizing mixed strategy 2 Hence, there is always a Nash equilibrium. (that is, x>A = v1> and Ay = v1), 3 If there are multiple Nash equilibria, they form a convex set, and the then this pair of strategies is a Nash equilibrium. expected payoff is identical within that set. (from the principle of indifference) Thus, any two Nash equilibria give the same payoff. General-sum games General-sum games 4 If each player has an equalizing mixed strategy 1 A pair of safety strategies might be unstable. for their opponent’s payoff matrix (Opponent aims to maximize their payoff, not minimize mine.) (that is, x>B = v21> and Ay = v11), 2 There is always a Nash equilibrium (Nash’s Theorem). then this pair of strategies is a Nash equilibrium. 3 There can be multiple Nash equilibria, with different payoff vectors. 11 / 59 12 / 59 Topics Multiplayer general-sum games Series and parallel games. Two-player general-sum games Notation Payoff matrices, dominant strategies, safety strategies, Nash eq. A k-person general-sum game is specified by k utility functions, Multiplayer general-sum games uj : S1 S2 Sk R. Utility functions, Nash equilibria. Nash’s Theorem × × · · · × → Player j can choose strategies s S . Congestion games and potential games j ∈ j Congestion games: Every congestion game has a pure Nash equilibrium Simultaneously, each player chooses a strategy. Potential games: Every congestion game is a potential game Player j receives payoff uj (s1,..., sk ). Criticisms of Nash equilibria Evolutionarily stable strategies ESS and Nash equilibria k = 2: u1(i, j) = aij , u2(i, j) = bij . Multiplayer ESS For s = (s1,..., sk ), let s i denote the strategies without the ith one: Correlated equilibrium − Price of anarchy s i = (s1,..., si 1, si+1,..., sk ). Braess’s paradox − − The impact of adding edges And write( si , s i ) as the full vector. Classes of latencies − Pigou networks 13 / 59 14 / 59 Multiplayer general-sum games Multiplayer general-sum games Definition A sequence( x ,..., x ) ∆ ∆ (called a strategy profile) is a 1∗ k∗ ∈ S1 × · · · × Sk Definition Nash equilibrium for utility functions u1,..., uk if, for each player j 1,..., k , A vector( s ,..., s ) S S is a pure Nash equilibrium for utility ∈ { } 1∗ k∗ ∈ 1 × · · · × k functions u1,..., uk if, for each player j 1,..., k , ∈ { } max uj (xj , x∗ j ) = uj (xj∗, x∗ j ). xj ∆S − − ∈ j max uj (sj , s∗ j ) = uj (sj∗, s∗ j ). sj Sj − − ∈ Here, we define uj (x∗) = Es1 x1,...,sk xk uj (s1,..., sk ) If the players play these sj∗, nobody has an incentive to unilaterally ∼ ∼ deviate: each player’s strategy is a best response to the other players’ = x (s ) x (s )u (s ,..., s ). 1 1 ··· k k j 1 k strategies. s S ,...,s S 1∈ 1X k ∈ k If the players play these mixed strategies xj∗, nobody has an incentive to unilaterally deviate: each player’s mixed strategy is a best response to the other players’ mixed strategies. 15 / 59 16 / 59 Multiplayer general-sum games Multiplayer general-sum games Nash’s Theorem (1951) Every finite general-sum game has a Nash equilibrium. Theorem Consider a strategy profile x ∆ ∆ . Let Proof Idea (two players) ∈ S1 × · · · × Sk T = s S : x (s) > 0 . The following statements are equivalent. We find an “improvement” map M(x, y) = (ˆx, yˆ), so that i { ∈ i i } 1 x is a Nash equilibrium. 1 xˆ>Ay > x>Ay (orˆx>Ay = x>Ay, if x was a best response to y), 2 For each i, there is a ci such that 2 x>Byˆ > x>By (or x>Byˆ = x>By, if y was a best response to x), 1 For all si Si , ui (si , x i ) ci . No better response outside Ti 3 ∈ − ≤ ←− M is continuous. 2 For si Ti , ui (si , x i ) = ci . Indifferent within Ti ∈ − ←− It’s easy to find a map like this. For instance, take a step in the direction of increasing payoff: Compare with the principle of indifference in the zero-sum case. xˆ = (x + ηAy),ˆ y = (y + ηB x). P∆m P∆n > A Nash equilibrium is a fixed point of M. The existence of a Nash equilibrium follows from Brouwer’s fixed-point theorem. 17 / 59 18 / 59 Nash’s Theorem Topics Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Congestion games and potential games Brouwer’s Fixed-Point Theorem Congestion games: Every congestion game has a pure Nash equilibrium A continuous map f : K K from a convex, closed, bounded K Rd Potential games: Every congestion game is a potential game → ⊆ has a fixed point, that is, an x K satisfying f (x) = x. Criticisms of Nash equilibria ∈ Evolutionarily stable strategies ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks 19 / 59 20 / 59 Congestion games: games with a pure Nash equilibrium Congestion games Definition A congestion game has Example k players A m facilities 1,..., m (e.g., edges) { } (1,2,4) (2,3,5) For player i, there is a set Si of strategies that are sets of facilities, s 1,..., m (e.g., paths) ⊆ { } k I, II, III S (1,2,6) T For facility j, there is a cost vector cj R , where cj (n) is the cost of ∈ facility j when it is used by n players. (2,3,5) (2,4,8) For a sequence s = (s1,..., sn), the utilities of the players are defined by B cost (s) = u (s) = c (n (s)), i − i j j j s X∈ i where n (s) = i : j s is the number of players using facility j.
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