Mixed strategy game

In a mixed strategy game the players will select between different strategies.

		B1	B2
Game player A	A1	15	20
	A2	30	10

Player a will select strategy A1 as this is the best of the minimum gains (15). Player B will select strategt B2 as this is the smallest of the maxiumum losses. There is no saddle point in this game.

		B1	B2	Row minimum
Game player A	A1	15	20	15
	A2	30	10	10
		30	20

Player 'A' can decide which portion of the time it should play strategy A1 and which portion of the time strategy A2 to maximize pay out. This can be though of as a probability. Let p be the proportion of the games that player A selects A1. (1-p) then will represent the percentage of the time that player A selects strategy A2.

If B selects B1 the expected gain is: 15p + 30 (1-p) = 30 - 15p If B selects B2 the expected gain is: 20p + 10 (1-p) = 10 +10p Solve this: {{{ 30 - 15p = 10 + 10p 20 = 25p p = 0.8 }}} The expected gain for A (assuming B1) is: {{{ 15 * 0.8 + 30 * 0.2 = 18 }}} And assuming B2: {{{ 20 * 0.8 + 10 * 0.2 = 18 }}}

Same calculations for player B. Let q be the proportion of the time player B selects strategy B1 and (q-1) the proportion of the time he selects B2. If A selects A1 the expected loss is: 15 * q + 20 (1-q) If A selects A2 the expected loss is: 30 * q + 10 (1-q) Solve this: {{{ 15 * q + 20 (1-q) = 30 * q + 10 (1-q) -5q + 20 = 20q + 10 10 = 25q q = 0.4 }}} The expected value for B in this case is also 18.

Part of M7-Quantitative Decision Methods-S5 - Game Theory