Outline for today. Stat155 Game Theory Lecture 13: General-Sum Games. General-sum games. General-sum games. Dominated pure strategies

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1 Outline for today Stat155 Game Theory Lecture 13: General-Sum Games Peter Bartlett October 11, 2016 Two-player general-sum games Definitions: payoff matrices, dominant strategies, safety strategies, Nash equilibrium. Example: Cheetahs and gazelles Nash equilibrium Example: Polluting factories 1 / 21 2 / 21 Notation A two-person general-sum game is specified by two payoff matrices, A, B R m n. Simultaneously, Player I chooses i {1,..., m} and the Player II chooses j {1,..., n}. Player I receives payoff a ij. Player II receives payoff b ij. Dominated pure strategies A pure strategy e i for Player I is dominated by e i in payoff matrix A if, for all j {1,..., n}, a ij a i j. Similarly, a pure strategy e j for Player II is dominated by e j in payoff matrix B if, for all i {1,..., m}, b ij b ij. 3 / 21 4 / 21

2 Safety strategies A safety strategy for Player I is an x m that satisfies Nash equilibria A pair (x, y ) m n is a Nash equilibrium for payoff matrices A, B R m n if min x Ay = max min x Ay. y n x m y n x maximizes the worst case expected gain for Player I. Similarly, a safety strategy for Player II is a y n that satisfies min x By = max min x By. x m y n x m y maximizes the worst case expected gain for Player II. max x Ay = x Ay, x m max x By = x By. y n If Player I plays x and Player II plays y, neither player has an incentive to unilaterally deviate. x is a best response to y, y is a best response to x. In general-sum games, there might be many Nash equilibria, with different payoff vectors. 5 / 21 6 / 21 Example: Cheetahs and Gazelles Example: Cheetahs and Gazelles Payoff matrices large small large (l/2, l/2) (l, s) small (s, l) (s/2, s/2) (s l). Payoff matrices large small large (l/2, l/2) (l, s) small (s, l) (s/2, s/2) (s l). Dominant strategy? For l 2s, large is a dominant strategy. Suppose l 2s. Pure Nash equilibria? (large, small), (small, large). But who gets the large gazelle? 7 / 21 8 / 21

3 Example: Cheetahs and Gazelles Example: Cheetahs and Gazelles Mixed Nash equilibrium? If Cheetah I plays Pr(large) = x, Cheetah II s payoffs are: large: small: L(x) = l x + l(1 x), 2 S(x) = sx + s (1 x). 2 Equilibrium is when these are equal: x = (2l s)/(l + s). Example: l = 8, s = 6. Equilibrium is when L(x) = S(x): x = (2l s)/(l + s) = 5/7. Think of x as the proportion of a population of cheetahs that would greedily pursue the large gazelle. For a randomly chosen pair of cheetahs, if x > x, S(x) > L(x), and non-greedy cheetahs will do better. And vice versa. Evolution pushes the proportion to x. This is the evolutionarily stable strategy. 9 / / 21 Comparing two-player general-sum and zero-sum games Comparing two-player general-sum and zero-sum games Zero-sum games 1 A pair of safety strategies is a Nash equilibrium (minimax theorem) 2 Hence, there is always a Nash equilibrium. 3 If there are multiple Nash equilibria, they form a convex set, and the expected payoff is identical within that set. Thus, any two Nash equilibria give the same payoff. 1 A pair of safety strategies might be unstable. (Opponent aims to maximize their payoff, not minimize mine.) 2 There is always a Nash equilibrium (Nash s Theorem). 3 There can be multiple Nash equilibria, with different payoff vectors. Zero-sum games 4 If each player has an equalizing mixed strategy (that is, x A = v1 and Ay = v1), then this pair of strategies is a Nash equilibrium. (from the principle of indifference) 4 If each player has an equalizing mixed strategy for their opponent s payoff matrix (that is, x B = v 2 1 and Ay = v 1 1), then this pair of strategies is a Nash equilibrium. 11 / / 21

4 Outline Two-player general-sum games Definitions: payoff matrices, dominant strategies, safety strategies, Nash equilibrium. Example: Cheetahs and gazelles Nash equilibrium Example: Polluting factories Notation A k-person general-sum game is specified by k utility functions, u j : S 1 S 2 S k R. Player j can choose strategies s j S j. Simultaneously, each player chooses a strategy. Player j receives payoff u j (s 1,..., s k ). k = 2: u 1 (i, j) = a ij, u 2 (i, j) = b ij. 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 ). And write (s i, s i ) as the full vector. 13 / / 21 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 u j (s j, s j) = u j (sj, s j). s j S j 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 strategies. Definition A sequence (x1,..., x k ) S 1 Sk (called a strategy profile) is a Nash equilibrium for utility functions u 1,..., u k if, for each player j {1,..., k}, Here, we define max x j Sj u j (x j, x j) = u j (x j, x j). u j (x ) = E s1 x 1,...,s k x k u j (s 1,..., s k ) = x 1 (s 1 ) x k (s k )u j (s 1,..., s k ). s 1 S 1,...,s k S 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 / / 21

5 Example: Polluting Factories Example: Polluting Factories Set x i = (p i, 1 p i ), that is, Pr(Player i plays purify) = p i. For 0 < p i < 1, we have a Nash equilibrium iff u i (purify, x i ) = u i (pollute, x i ). Pure equilibria? (purify, purify, pollute), (purify, pollute, purify), (pollute, purify, purify). Tragedy of the commons : (pollute, pollute, pollute) Solving shows there are two symmetric mixed Nash equilibria: p 1 = p 2 = p 3 = 3 ± / / 21 Example: Polluting Factories Example: Polluting Factories p i = Pr(i plays purify). Plot: cost for p 1 = p 2 = p 3. Blue curve: u i (purify, x i ) = p 2 + 2p(1 p) + 4(1 p) 2 Red curve: u i (pollute, x i ) = 6p(1 p) + 3(1 p) 2. purify pollute p:=p_1=p_2=p_3 Imagine that we draw random factories from a population with proportion p that pollute. What if p is a little less than (3 + 3)/6 0.79? purify has lower cost. What if p is a little more than (3 + 3)/6 0.79? pollute has lower cost. What about near p = (3 3)/6 0.21? Not an attractor! What about near p = 0? pollute has lower cost purify pollute p:=p_1=p_2=p_3 Nash equilibria: (p, p, p) with p = (3 + 3)/ p = (3 3)/ p = 0. (purify, purify, pollute), (purify, pollute, purify), (pollute, purify, purify). 19 / / 21

6 Outline Two-player general-sum games Definitions: payoff matrices, dominant strategies, safety strategies, Nash equilibrium. Example: Cheetahs and gazelles Nash equilibrium Example: Polluting factories 21 / 21