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Chance he will receive a ¿100 ticket on route a and

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Unformatted text preview: chance he will receive a ¿100 ticket on route A and only a 30% chance he will receive a ¿100 ticket on route B. The police force has three options; namely, full police force on route A, full police force on route B and a 50-50 split of the force on routes A and B. Determine the police payo matrix and determine the optimal mixed strategy for Alan and the police. (13 marks) AMA405 1 Turn Over AMA405 2 (i) Given the Minimax Theorem, min i n X j =1 a ij q j = max j m X i =1 a ij p i = ! ; prove the following statements: (a) There exists at least one pure strategy for each player which, if used against his opponent's optimal mixed strategy (M), yields the value of the game ( ! ) . (b) Against an opponent's optimal mixed strategy (M) a pure strategy cannot yield a higher expected payo than M. (c) A player's optimal mixed strategy (M) contains no pure strategy which yields less than the value of the game, when that pure strat- egy is used against an opponent's optimal mixed strategy. (11 marks) (ii) Consider the IP problem max z = 3 x 1 + x 2 subject to 2 x 1 x 2 6 3 x 1 + 9 x 2 45 where x 1 and x 2 are non-negative integers. The optimal tableau for the relaxed continuous LP problem is x 1 x 2 y 1 y 2 Solution z 8 = 7 5 = 21 123 = 7 x 1 1 6 = 14 1 = 21 33 = 7 x 2 1-1/7 2 = 21 24 = 7 where y 1 and y 2 are the slack variables of the rst and second constraints....
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