Unformatted text preview: log2, and if his iniRal wealth is zero, he will derive the same uRlity from playing this game as from being paid $4, because log24 = 2. Ilya Pollak What if prior wealth is not zero? • If the prior wealth is not zero, we can ask what amount a person would be willing to pay for playing the game. • For prior wealth w and fee f, the expected uRlity of the wealth aner the game will be E[log2(w + W – f)]. – This expectaRon is ﬁnite. – A person with iniRal wealth w will be willing to pay any fee f that produces expected uRlity larger than log2w. Ilya Pollak URlity funcRon • U(0) = 0 • U is strictly increasing • U’ is strictly decreasing – risk aversion: we would decline a bet that pays +1 or −1 with equal probability, or in fact any bet that is symmetrically distributed around zero. Ilya Pollak A problem with uRlity theory Game 1: 0.8 1 Game 2: Most people prefer Game 2 to Game 1: 0.8u(4000) < u(3000) (assuming u(0)=0) 0.2 $0 $4000 Game 3: 0.8 $3000 Game 4: 0.75 0.25 0.2 $0 $4000 $0 Most people prefer Game 3 to Game 4: 0.2u(4000) > 0.25u(3000) 0.8u(4000) > u(3000) $3000 Ilya Pollak Certainty Eﬀect • Example is due to Maurice Allais (1953), and is someRmes called “Allais Paradox” • Called “certainty eﬀect” by Kahneman and Tversky (1979): – people overweight outcomes that are considered certain, rela7ve to outcomes which are merely probable Ilya Pollak Two Stage Game • 1st stage: prob = 0.75 to win nothing and end the game; prob = 0.25 to move onto 2nd stage. • 2nd stage: you have a choice between Game 1 and Game 2, i.e., between winning $4000 with probability 0.8 and winning $3000 for sure. • The choice for the 2nd stage must be made before the beginning of the 1st stage. • Most people choose winning $3000 for sure for the 2nd stage. Ilya Pollak Overall Winnings and Their ProbabiliRes for the Two Stage Game • 1st stage: prob = 0.75 to win nothing and end the game; prob = 0.25 to move onto 2nd stage. • 2nd stage: choose between Game 1 ($4000 with p=0.8) and Game 2 ($3000 for certain). • If choose Game 1 for 2nd stage, win nothing with probability 0.75 + 0.25*0.2 = 0.8, and win $4000...
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 Fall '08
 GELFAND
 Electrical Engineering, Game Theory, Ilya, Pollak, Ilya 00

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