Unformatted text preview: with probability 0.25*0.8 = 0.2. This is Game 3! • If choose Game 2 for 2nd stage, win nothing with probability 0.75 and win $3000 with probability 0.25. This is Game 4! Ilya Pollak Overall Winnings and Their ProbabiliRes for the Two Stage Game • If choose Game 1 for 2nd stage, win nothing with probability 0.75 + 0.25*0.2 = 0.8, and win $4000 with probability 0.25*0.8 = 0.2. This is Game 3! • If choose Game 2 for 2nd stage, win nothing with probability 0.75 and win $3000 with probability 0.25. This is Game 4! • In “standard formulaRon”, Game 3 is preferred to Game 4. • In “sequenRal formulaRon”, Game 4 is preferred to Game 3. This is called “isolaRon eﬀect”. Ilya Pollak Framing • More generally, the phenomenon that people’s preferences can change depending on the formulaRon of the quesRon   even though the underlying choices are idenRcal   is called framing. Ilya Pollak Prospect Theory • D. Kahneman and A. Tversky. Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47(2):263 291, March, 1979. • Overcomes a number of experimentally observed inconsistencies of uRlity theory. Ilya Pollak ExplanaRon 2: What about risk? • Since the expected proﬁt is inﬁnite, the variance of the proﬁt is undeﬁned. • Need some other way of reasoning about the risk. • Note that the E[proﬁt] is inﬁnite only because of humongous proﬁts associated with extremely unlikely events. • Suppose the casino charges $10,000. Then – P(win) = P(1st H on 14th toss or later) = 1/214 + 1/215 + … = 1/214(1+1/2 + … ) = 1/213 ≈ 0.0001    i.e., you would expect to lose in 9999 out of every 10,000 games! – The most likely outcome is a single toss, which happens with probability 1/2 and leads to a loss of $9,998. Ilya Pollak More on the risk • What is the expected number of tosses per game? • The number of tosses is geometric with parameter p=1/2. • Hence, its expected value is 2. • Thus, in an “average” game, you will win $4 minus the fee. Ilya Pollak Any other risks for the player? • Note that the inﬁnite expectaRon is conRngent upon the nonzero probabiliRes of arbitrarily large payoﬀs. • But, if the ﬁrst tail comes on the 40th toss, will the casino actually be able to pay you one trillion dollars? This is called credit risk. • Suppose the original condiRons only apply to the ﬁrst 30 tosses. If 30
n>30 then you only win $2 . • Then the expected winnings are • The same result is obtained by assuming that, beyond some very 30 large number ($2 in our example), it makes no diﬀerence to people what the winnings are   i.e., by assuming that the uRlity funcRon is perfectly ﬂat for large winnings (G. Cramer, 1728.) Ilya Pollak...
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 Fall '08
 GELFAND
 Electrical Engineering, Game Theory, Ilya, Pollak, Ilya 00

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