Value of this zero sum game and determine optimal

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value of this zero-sum game and determine optimal strategies for both players. (Hint: use domination.) 2.10 A zebra has four possible locations to cross the Zambezi river, call them a , b , c , and d , arranged from north to south. A crocodile can wait (undetected) at one of these locations. If the zebra and the crocodile choose the same location, the payoff to the crocodile (that
74 Two-person zero-sum games is, the chance it will catch the zebra) is 1. The payoff to the crocodile is 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. 2.11 A recursive zero-sum game. An inspector can inspect a facility on just one occasion, on one of the days 1 ,...,n . The worker at the facility can cheat or be honest on any given day. The payoff to the inspector is 1 if he inspects while the worker is cheating. The payoff is 1 if the worker cheats and is not caught. The payoff is also 1 if the inspector inspects but the worker did not cheat, and there is at least one day left. This leads to the following matrices Γ n for the game with n days: the matrix Γ 1 is shown on the left, and the matrix Γ n is shown on the right. worker cheat honest inspector inspect 1 0 wait 1 0 worker cheat honest inspector inspect 1 1 wait 1 Γ n 1 Find the optimal strategies and the value of Γ n .
3 General-sum games We now turn to discussing the theory of general-sum games . Such a game is given in strategic form by two matrices A and B , whose entries give the payoffs 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 there 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, and in choosing one of them, some degree of cooperation between the players may be optimal. 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. 3.1 Some examples Example 3.1.1 ( The prisoner’s dilemma ) . Two suspects are held and questioned 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 other is silent, then the confessor goes free, and the other prisoner is sentenced to ten years. If both confess, they will each spend eight years in prison. If both remain silent, the sentence is one year to each, for some minor crime that the police are able to prove. Writing the negative payoff as the number of years spent in prison, we obtain the following payoff matrix: prisoner II silent confess prisoner I silent ( 1 , 1) ( 10 , 0) confess (0 , 10) ( 8 , 8) 75
76 General-sum games Fig. 3.1. Two prisoners considering considering whether to confess or re-

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