Lecture12 - Mixed Strategies Matching Pennies We denote the...

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3/3/2009 1 Mixed Strategies – Matching Pennies We denote the strategies on the game bi-matrix Guildenstern q 1-q Heads Tails Guildenstern’s payoff to H : (-1)p + 1(1-p) = 1-2p T : (1)p + (-1)(1-p) = 2p-1 Rosencrantz p 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 probability 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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3/3/2009 2 Preferences Involving Gambles Desirable Property #1: Since outcomes themselves can be thought of as gambles, v should agree with u on such “degenerate gambles”.
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