225 prove that linear programs 215 and 216 are dual

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2.25 Prove that linear programs (2.15) and (2.16) are dual to each other. 2.9 Solved Exercises 2.9.1 Another Betting Game Consider the betting game with the following payo ff matrix: player II L R player I T 0 2 B 5 1 Draw graphs for this game analogous to those shown in Figure 2.1. Solution: Suppose player I plays T with probability x 1 and B with probability 1 x 1 , and player II plays L with probability y 1 and R with probability 1 y 1 . (We note that in this game, there is no saddle point.) Reasoning from player I’s perspective, her expected gain is 2(1 y 2 ) for playing the pure strategy T , and 4 y 2 + 1 for playing the pure strategy B . Thus, if she knows y 2 , she will pick the strategy corresponding to the max- imum of 2(1 y 2 ) and 4 y 2 + 1. Player II can choose y 2 = 1 / 6 so as to mini- mize this maximum, and the expected amount player II will pay player I is 5 / 3. This is the player II strategy that minimizes his worst-case loss. See Figure 2.11 for an illustration. From player II’s perspective, his expected loss is 5(1 x 1 ) if he plays the pure strategy L and 1 + x 1 if he plays the pure strategy R , and he will
48 Two-person zero-sum games 0 1 2 5 1 1/6 Player II’s mixed strategy Expected loss of player II : when player I plays B : when player I plays T Worst-case loss 0 1 2 5 1 Player I’s mixed strategy Expected gain of player I : when player II plays L Worst-case gain 2/3 : when player II plays R Fig. 2.11. The left side of the figure shows the worst-case expected gain of player I as a function of her mixed strategy (where she plays T with probability x 1 and B with probability 1 x 1 ). This worst case expected gain is maximized when she plays T with probability 2/3 and B with probability 1/3. The right side of the figure shows the worst-case expected loss of player II as a function of his mixed strategy (where he plays L with probability y 1 and R with probability 1 y 1 . The worst case expected loss is minimized when he plays L with probability 1/6 and R with probability 5/6. aim to minimize this expected payout. In order to maximize this minimum, player I will choose x 1 = 2 / 3, which again yields an expected gain of 5 / 3.
3 General-sum games We now turn to the theory of general-sum games . Such a game is given by two matrices A and B , whose entries give the payo ff s to the two players for each pair of pure strategies that they might play. Usually there is no joint optimal strategy for the players, but the notion of Nash equilibrium remains relevant. These equilibria give the strategies that “rational” players might choose. However, there are often several Nash equilibria, and in choosing one of them, some degree of cooperation between the players may be desirable. Moreover, a pair of strategies based on cooperation might be better for both players than any of the Nash equilibria. We begin with two examples.

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