11_DiscreteRVs-3_StPetersburg_Paradox_packed

Between game 1 and game 2 ie between

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 09/11/2013 for the course ECE 302 taught by Professor Gelfand during the Fall '08 term at Purdue University-West Lafayette.

Ask a homework question - tutors are online