65%(17)11 out of 17 people found this document helpful
This preview shows page 55 - 58 out of 241 pages.
2.25Prove that linear programs (2.15) and (2.16) are dual to each other.2.9 Solved Exercises2.9.1 Another Betting GameConsider the betting game with the following payoffmatrix:player IILRplayer IT02B51Draw graphs for this game analogous to those shown in Figure 2.1.Solution:Suppose player I playsTwith probabilityx1andBwith probability 1−x1,and player II playsLwith probabilityy1andRwith probability 1−y1. (Wenote that in this game, there is no saddle point.)Reasoning from player I’s perspective, her expected gain is 2(1−y2) forplaying the pure strategyT, and 4y2+ 1 for playing the pure strategyB.Thus, if she knowsy2, she will pick the strategy corresponding to the max-imum of 2(1−y2) and 4y2+ 1. Player II can choosey2= 1/6 so as to mini-mize this maximum, and the expected amount player II will pay player I is5/3. This is the player II strategy that minimizes his worst-case loss. SeeFigure 2.11 for an illustration.From player II’s perspective, his expected loss is 5(1−x1) if he plays thepure strategyLand 1 +x1if he plays the pure strategyR, and he will
48Two-person zero-sum games012511/6Player II’s mixed strategyExpectedlossof player II: when player I plays B : when player Iplays TWorst-caseloss01251Player I’s mixed strategyExpectedgainof player I: when player II plays LWorst-casegain2/3: when player IIplays RFig. 2.11. The left side of the figure shows the worst-case expected gainof player I as a function of her mixed strategy (where she playsTwithprobabilityx1and B with probability 1−x1). This worst case expectedgain is maximized when she playsTwith probability 2/3 andBwithprobability 1/3. The right side of the figure shows the worst-case expectedloss of player II as a function of his mixed strategy (where he plays L withprobabilityy1and R with probability 1−y1. The worst case expected lossis minimized when he playsLwith probability 1/6 andRwith probability5/6.aim to minimize this expected payout. In order to maximize this minimum,player I will choosex1= 2/3, which again yields an expected gain of 5/3.
3General-sum gamesWe now turn to the theory ofgeneral-sum games. Such a game is givenby two matricesAandB, whose entries give the payoffs to the two playersfor each pair of pure strategies that they might play. Usually there is no jointoptimal strategy for the players, but the notion of Nash equilibrium remainsrelevant. These equilibria give the strategies that “rational” players mightchoose. However, there are often several Nash equilibria, and in choosing oneof them, some degree of cooperation between the players may be desirable.Moreover, a pair of strategies based on cooperation might be better for bothplayers than any of the Nash equilibria. We begin with two examples.