Zero-sum games of two players. Zero-sum games of two players. Zero-sum games of two players. Zero-sum games of two players

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1 Martin Branda Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization A triplet {X, Y, K} is called a game of two rational players with zero sum, if 1 X is a set of strategies of Player 1 (P1), 2 Y is a set of strategies of Player 2 (P2), 3 K : X Y R is a payoff function of player 1, i.e. if P1 plays x X and P2 plays y Y, then P1 gets K(x, y) and P2 gets K(x, y). Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 For the zero-sum games {X, Y, K} we define upper value of the game uv = inf sup K(x, y), lower value of the game lv = sup inf K(x, y), upper price of the game up = min sup K(x, y), lower price of the game lp = max inf K(x, y). If the lower and upper prices exist and it holds up = lp, then we say that the game has the price p = up = lp. We say that ˆx X is an optimal strategy of P1, if K(ˆx, y) lv for all y Y. ŷ Y is an optimal strategy of P2, if K(x, ŷ) uv for all x X. Upper value can be seen as the lowest payoff of P1, if P1 knows strategy of P2 before his/her move. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24

2 For each zero-sum game {X, Y, K} the upper and lower value exits and it holds lv uv. For each x X and ỹ Y it holds inf K( x, y) K( x, ỹ), sup inf K(x, y) sup K(x, ỹ), lv = sup inf K(x, y) inf sup K(x, y) = uv. (1) For each zero-sum game {X, Y, K} is holds that There is at least one optimal strategy of P1, if and only if the lower price exists. There is at least one optimal strategy of P2, if and only if the upper price exists. : Let ˆx X be an optimal strategy of P1, i.e. K(ˆx, y) lv for all y Y. Then Thus lv inf K(ˆx, y) sup inf K(x, y) = lv. (2) lv = inf K(ˆx, y) = max inf K(x, y) = lp. (3) Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 Let {X, Y, K} be a zero-sum game with X, Y compact and K continuous. Then the upper and lower prices exist. Theorem A zero-sum game {X, Y, K} has a price if and only if the payoff function has a saddle point, i.e. there is a pair a (ˆx, ŷ) such that K(x, ŷ) K(ˆx, ŷ) K(ˆx, y) for all x X and y Y. Then ˆx is an optimal strategy for P1, ŷ is an optimal strategy for P2, and p = K(ˆx, ŷ) is the price of the game. a Such pair can be seen as a Nash equilibrium for two player games. : K(x, ŷ) p K(ˆx, y). Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24

3 John Forbes Nash ( ) Minimax theorem Theorem Let {X, Y, K} be a zero-sum game where X, Y are nonempty convex compact sets and K(x, y) is continuous, concave in x and convex in y. Then, there exists the price of the game, i.e. min max K(x, y) = max min K(x, y). Applicable also out of the game theory, e.g. in robustness. Generalizations: Rockafellar (1970) A Beautiful Mind (2001) Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 Rock paper scissors We say that {X, Y, A} is a matrix game if it a zero sum game (of two players), A R n m is a matrix, and K(x, y) = x T Ay, { X = x R n : Y = y Rm : } n x i = 1, x i 0, i=1 m y j = 1, y j 0. j=1 (4) R P S A = (5) Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24

4 Rock paper scissors lizard Spock Rock paper scissors lizard Spock Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 For a matrix game {X, Y, A}, we define a matrix game with pure strategies {X, Y, A}, where { } n X = x R n : x i = 1, x i {0, 1}, Y = y Rm : i=1 m y j = 1, y j {0, 1}. j=1 (6) Each matrix game has a price and both players have optimal strategies. Matrix game {X, Y, A} has a price in pure strategies if and only if {X, Y, A} has a price. We say that {X, Y, A} has a price in pure strategies if both players have optimal pure strategies. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24

5 Let {X, Y, A} be a matrix game and ˆx X and ŷ Y with price p. Then 1 ˆx is an optimal strategy of P1 if and only if ˆx T A (p,..., p), 2 ŷ is an optimal strategy of P2 if and only if Aŷ (p,..., p) T. ˆx T A (p,..., p) ˆx T Ay p, y Y. ( y & i y i = 1, y = e i ) (Complementarity conditions) Let {X, Y, A} be a matrix game with price p and let ˆx X and ŷ Y be optimal strategies. Then 1 if ˆx i > 0, then (Aŷ) i = p 2 if ŷ j > 0, then (ˆx T A) j = p. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 Example Consider ( ) 5 1 A = 0 7 Let a, b R n. We say that a strictly dominates b (b is strictly dominated by a), if a i > b i for all i = 1,..., n. 5x 1 p, x 1 + 7x 2 p, x 1 + x 2 = 1, x 1 0, x 2 0 and using x 1 + x 2 = 1 max min{5x 1, x 1 + 7x 2 } = p max min{5x 1, 7 6x 1 } = p x 1 0 Let {X, Y, A} be a matrix game. 1 If a row A k, is strictly dominated by a convex combination of other rows, then each optimal strategy of P1 fulfills ˆx k = 0. 2 If a column A,k strictly dominates a convex combination of other columns, then each optimal strategy of P2 fulfills ŷ k = 0. Maximum is attained at ˆx 1 = 7/11, ˆx 2 = 4/11 with the price p = 35/11. Using complementarity conditions, we obtain ŷ 1 = 6/11, ŷ 2 = 5/11. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24

6 Show that (0, 0, 7/11, 4/11) T is optimal strategy for P1, (0, 0, 6/11, 5/11) T for P2, and the price is p = 35/11. Matrix game {X, Y, A} has a price p in pure strategies if and only if matrix A has a saddle point, i.e. there is a pair of indices {k, l} such that A kl = min{a kj : j = 1,..., m} = max{a il : i = 1,..., n}. (minimum in the row, maximum in the column) e k,e l are optimal strategies of P1, P2 (e T k A) j = A kj p, j, (Ae l ) i = A il p, i, (7) A kl = min{a kj : j = 1,..., m} = max{a il : i = 1,..., n}. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24 Example Literature Find the saddle point(s) , , (8) Lachout, P. (2011). Matematické programování. Skripta k (zaniklé) přednášce Optimalizace I (IN CZECH). Rockafellar, R.T. (1970). Convex Analysis. Princeton University Press, Princeton (N.Y.). Webb, J.N. Game Theory (2007). Decisions, Interactions and Evolution. Springer-Verlag, London. Martin Branda (KPMS MFF UK) / 24 Martin Branda (KPMS MFF UK) / 24