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pr9ab.zerosum

# pr9ab.zerosum - Zero-Sum Games Part A Pure Strategies...

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Zero-Sum Games Part A: Pure Strategies 2 ECON 461 Spring 2010, H. Ofek Zerosum Games A Zero-Sum Game in Bimatrix Form (Including Redundancy) 3 t 1 t 2 t 3 t 4 s 1 3, -3 1 , -1 4 , -4 6 , -6 s 2 5 , -5 3 , -3 5 , -5 6 , -6 s 3 4 , -4 2 , -2 2 , -2 1 , -1 s 4 3 , -3 1 , -1 1 , -1 0 , 0 Since U II (s, t) = - U I (s, t), it is redundant to write down the payoffs for player II.

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3 ECON 461 Spring 2010, H. Ofek Zerosum Games Player I ’s payoff matrix is enough 3 A zero-sum game is ordinarily represented by the payoff matrix of player I alone. For any practical purpose, these payoffs can be viewed as payments from players II to player I. Player I seeks to maximize these payoffs, and player II seeks to minimize them. t 1 t 2 t 3 t 4 s 1 3 1 4 6 s 2 5 3 5 6 s 3 4 2 2 1 s 4 3 1 1 0 4 Zerosum Games The minimax method t 1 t 2 t 3 t 4 s 1 3 1 4 6 1 s 2 5 3 5 6 3 s 3 4 2 2 1 1 s 4 3 1 1 0 0 max U I 5356 min U I First step : The smallest value in each row (s) and the largest value in each column (t) are listed in an auxiliary column and auxiliary row (respectively appended to the game table). Recall : This is the payoff table for player I. Player I favors larger numbers in this table and player II favors smaller ones.
5 ECON 461 Spring 2010, H. Ofek Zerosum Games maximin & minimax, obtained t 1 t 2 t 3 t 4 s 1 3 1 4 6 1 s 2 5 3 5 6 3 maximin s 3 4 2 2 1 1 s 4 3 1 1 0 0 max U I 53 minimax 56 maximin =3 minimax =3 min U I In this example maxmin = minimax = 3 Hence, the game is said to have a value in pure strategies: v = 3 This is also the security level for both players. Nash equilibrium is obtained at (s 2 ,t 2 ), (e.g., by best response). This point is a saddle point of the payoff matrix. 6 Zerosum Games minimax & maximin (in pure strategies ) Interpretation (maximin): The pure-strategy maximin is the largest payoff a player can guarantee in pure strategies whatever the other player may do. o As such, the maximin value is the player’s security level in pure strategies but not quite yet the player’s security level in the game as a whole (because there is also a maximin in mixed strategies that may guarantee a better payoff).

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pr9ab.zerosum - Zero-Sum Games Part A Pure Strategies...

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