Strategy12Handout - 10/14/2008 Mixed Strategies Matching...

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10/14/2008 1 Mixed Strategies – Matching Pennies ± We denote the strategies on the game bi-matrix Guildenstern q1 - q Heads Tails ± Guildenstern’s payoff to H : (-1)p + 1(1-p) = 1-2p T : (1)p + (-1)(1-p) = 2p-1 Rosencrant zp Heads 1 ± ,-1 -1 , 1 1-p Tails -1 , 1 1 , -1 Preferences Involving Gambles ± We revisit our description of payoffs in the game bimatrix. ± Consider Tom, who is faced with two outcomes ± A: get $1 ± B: get $4 ± Tom likes money, the more the better. So Tom thinks that outcome B is better than A Preferences Involving Gambles ± Tom’s preferences are summarized by ± u1(m)=m, so that u1(1)=1, and u1(4)=4 ± u2(m)=m, so that u2(1)=1, and u2(4)=16 ± u3(m)=m, so that u3(1)=1, and u3(4)=2 any of one these payoff (utility) functions so long as outcomes are “get some money” 2 ½ Preferences Involving Gambles ± We would like to consider preferences over gambles (lotteries) involving outcomes ± Can we come up with a utility function that ranks preferences over gambles (involving outcomes) along with outcomes themselves? ± Example: Suppose G is the gamble ± Get $1 with probability ½, and ± Get $4 with probablity ½ Preferences Involving Gambles ± Note that the expected value of G, E[G] = ½($1) + ½($4) = $2.50 ± Must Tom think that G is just as good as $2.50? ± No ± Maybe, Tom thinks G is better than $2.50, or maybe Tom thinks G is not as good as $2.50 Preferences Involving Gambles ± Suppose we have a utility function over outcomes u:{outcomes} Թ - the real numbers that represents a particular set of outcomes ± It would be nice to have a utility function v:{gambles over outcomes} Թ that ranks gambles over outcomes as well as outcomes
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10/14/2008 2 Preferences Involving Gambles ± Desirable Property #1: Since outcomes
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Strategy12Handout - 10/14/2008 Mixed Strategies Matching...

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