115 A zebra has four possible locations to cross the Zambezi

1.15 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 is, the chance it will catch the zebra) is 1. The payoff to the crocodile is 1/2 if they choose adjacent locations, and O 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 x 2 game? (c) Find the value of the game (to the crocodile) and optimal strategies for both.

Solution

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The matrix cannot be reduced as the rank of the matrix = no of rows = 4

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c) value of the game = For crocodile = 1.5a+2b+2c+1.5d

   For zebra = 1.5a+2b+2c+1.5d

As both have the same value of game, strategies for both are selecting c and d instead of a and b

	Pay off matrix
Location	a	b	c	d
zebra a	1	0.5	0	0
b	0.5	1	0.5	0
c	0	0.5	1	0.5
d	0	0	0.5	1