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value of this zero-sum game and determine optimal strategies forboth players. (Hint: use domination.)2.10A zebra has four possible locations to cross the Zambezi river, callthema,b,c, andd, arranged from north to south. A crocodile canwait (undetected) at one of these locations.If the zebra and thecrocodile choose the same location, the payoff to the crocodile (that
74Two-person zero-sum gamesis, the chance it will catch the zebra) is 1. The payoff to the crocodileis 1/2 if they choose adjacent locations, and 0 in the remaining cases,when the locations chosen are distinct and non-adjacent.(a) Write the payoff matrix for this zero-sum game in normal form.(b) Can you reduce this game to a 2×2 game?(c) Find the value of the game (to the crocodile) and optimal strate-gies for both.2.11A recursive zero-sum game.An inspector can inspect a facilityon just one occasion, on one of the days 1,...,n. The worker at thefacility can cheat or be honest on any given day. The payoff to theinspector is 1 if he inspects while the worker is cheating. The payoffis−1 if the worker cheats and is not caught. The payoff is also−1if the inspector inspects but the worker did not cheat, and there isat least one day left.This leads to the following matrices Γnforthe game withndays: the matrix Γ1is shown on the left, and thematrix Γnis shown on the right.workercheathonestinspectorinspect10wait−10workercheathonestinspectorinspect1−1wait−1Γn−1Find the optimal strategies and the value of Γn.
3General-sum gamesWe now turn to discussing the theory ofgeneral-sum games.Such agame is given instrategic formby two matricesAandB, whose entriesgive the payoffs to the two players for each pair of pure strategies that theymight play. Usually there is no joint optimal strategy for the players, butthere still exists a generalization of the Von Neumann minimax, the so-called Nash equilibrium. These equilibria give the strategies that “rational”players could follow. However, there are often several Nash equilibria, andin choosing one of them, some degree of cooperation between the playersmay be optimal. Moreover, a pair of strategies based on cooperation mightbe better for both players than any of the Nash equilibria. We begin withtwo examples.3.1 Some examplesExample 3.1.1(The prisoner’s dilemma).Two suspects are held andquestioned by police who ask each of them to confess. The charge is serious,but the evidence held by the police is poor. If one confesses and the otheris silent, then the confessor goes free, and the other prisoner is sentencedto ten years. If both confess, they will each spend eight years in prison. Ifboth remain silent, the sentence is one year to each, for some minor crimethat the police are able to prove. Writing the negative payoff as the numberof years spent in prison, we obtain the following payoff matrix:prisoner IIsilentconfessprisoner Isilent(−1,−1)(−10,0)confess(0,−10)(−8,−8)75
76General-sum gamesFig. 3.1. Two prisoners considering considering whether to confess or re-