2. Discrete Random Variables Part III: St Petersburg Paradox. ECE 302 Fall 2009 TR 3 4:15pm Purdue University, School of ECE Prof.
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1 2. Discrete Random Variables Part III: St Petersburg Paradox ECE 302 Fall 2009 TR 3 4:15pm Purdue University, School of ECE Prof. Ilya Pollak
2 Problem 2.21: St Petersburg Paradox A coin is flipped repeatedly unpl the first H. If the first H appears on the n th toss, you get paid $2 n. How much should a casino charge you for playing this game?
3 Problem 2.21: St Petersburg Paradox A coin is flipped repeatedly unpl the first H. If the first H appears on the n th toss, you get paid $2 n. How much should a casino charge you for playing this game? Probability mass funcpon of your winnings 1/2 1/4 1/8 1/
4 Problem 2.21: St Petersburg Paradox A coin is flipped repeatedly unpl the first H. If the first H appears on the n th toss, you get paid $2 n. How much should a casino charge you for playing this game? Your expected winnings: Probability mass funcpon of your winnings 1/ = 1/4 1/8 1/
5 How to price the coin flipping game? The customer s expected winnings per game are infinite. The casino should probably not offer this game!
6 What if the casino does offer this game, for a very large price? Will you accept any amount the casino wants to charge in order to play this game? E.g., what if the casino charges $10,000? Your expected winnings are spll 10,000 =
7 What if the casino does offer this game, for a very large price? Will you accept any amount the casino wants to charge in order to play this game? E.g., what if the casino charges $10,000? Your expected winnings are spll 10,000 = But most people people won t pay more than $20 30 to play this.
8 What if the casino does offer this game, for a very large price? Will you accept any amount the casino wants to charge in order to play this game? E.g., what if the casino charges $10,000? Your expected winnings are spll 10,000 = But most people people won t pay more than $20 30 to play this. Paradox: the disconnect between the infinite expected profit and the unwillingness of most people to pay much for this game.
9 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000.
10 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000. Diminishing marginal uplity.
11 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000. Diminishing marginal uplity. Hence, people s uplity funcpons are concave. uplity of wealth diminishing marginal uplity: each addiponal dollar is less valuable than the previous dollar wealth
12 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000. Diminishing marginal uplity. Hence, people s uplity funcpons are concave. uplity of wealth constant marginal uplity diminishing marginal uplity: each addiponal dollar is less valuable than the previous dollar wealth
13 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000. Diminishing marginal uplity. Hence, people s uplity funcpons are concave. uplity of wealth constant marginal uplity diminishing marginal uplity: each addiponal dollar is less valuable than the previous dollar wealth People try to maximize their expected uplity.
14 ExplanaPon 1: Expected UPlity Hypothesis (Gabriel Cramer, 1728 and Daniel Bernoulli, 1738) $1000 is worth more to a person whose total wealth is $1 than to a person whose total wealth is $1,000,000. Diminishing marginal uplity. Hence, people s uplity funcpons are concave. uplity of wealth constant marginal uplity diminishing marginal uplity: each addiponal dollar is less valuable than the previous dollar wealth People try to maximize their expected uplity. E.g., Bernoulli modeled people s uplity funcpon as a log. (Cramer s model was a square root.) I.e., in Bernoulli s model, instead of E[wealth], people try to maximize E[log of wealth].
15 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 PMF of log 2 W 1 2
16 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 1/4 PMF of log 2 W 1 2
17 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 1/4 1/8 1/16 PMF of log 2 W
18 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 1/4 1/8 1/16 PMF of log 2 W log 2 W is a geometric random variable with p=1/2
19 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 1/4 1/8 1/16 PMF of log 2 W log 2 W is a geometric random variable with p=1/2 Hence, E[log 2 W] = 2
20 ExpectaPon of log wealth when prior 1/2 1/4 wealth is zero PMF of winnings W 1/8 1/ /2 1/4 1/8 1/16 PMF of log 2 W log 2 W is a geometric random variable with p=1/2 Hence, E[log 2 W] = 2 If a person s uplity funcpon is log 2, and if his inipal wealth is zero, he will derive the same uplity from playing this game as from being paid $4, because log 2 4 = 2.
21 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.
22 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 uplity of the wealth aler the game will be E[log 2 (w + W f)].
23 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 uplity of the wealth aler the game will be E[log 2 (w + W f)]. This expectapon is finite. A person with inipal wealth w will be willing to pay any fee f that produces expected uplity larger than log 2 w.
24 ExplanaPon 2: What about risk? Since the expected profit is infinite, the variance of the profit is undefined.
25 ExplanaPon 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.
26 ExplanaPon 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.
27 ExplanaPon 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(1 st H on 14 th toss or later) = 1/ / = 1/2 14 (1+1/2 + ) = 1/ i.e., you would expect to lose in 9999 out of every 10,000 games!
28 ExplanaPon 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(1 st H on 14 th toss or later) = 1/ / = 1/2 14 (1+1/2 + ) = 1/ 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.
29 More on the risk What is the expected number of tosses per game?
30 More on the risk What is the expected number of tosses per game? The number of tosses is geometric with parameter p=1/2.
31 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.
32 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.
33 Any other risks for the player? Note that the infinite expectapon is conpngent upon nonzero probabilipes of arbitrarily large payoffs.
34 Any other risks for the player? Note that the infinite expectapon is conpngent upon nonzero probabilipes of arbitrarily large payoffs. But, if the first tail comes on the 40 th toss, will the casino actually be able to pay you one trillion dollars?
35 Any other risks for the player? Note that the infinite expectapon is conpngent upon the nonzero probabilipes of arbitrarily large payoffs. But, if the first tail comes on the 40 th toss, will the casino actually be able to pay you one trillion dollars? Suppose the original condipons only apply to the first 30 tosses. If n>30 then you only win $2 30.
36 Any other risks for the player? Note that the infinite expectapon is conpngent upon the nonzero probabilipes of arbitrarily large payoffs. But, if the first tail comes on the 40 th toss, will the casino actually be able to pay you one trillion dollars? Suppose the original condipons only apply to the first 30 tosses. If n>30 then you only win $2 30. Then the expected winnings are 30 1 n 1 2 n + n 2 30 = n=1 n= 31
37 Any other risks for the player? Note that the infinite expectapon is conpngent upon the nonzero probabilipes of arbitrarily large payoffs. But, if the first tail comes on the 40 th toss, will the casino actually be able to pay you one trillion dollars? Suppose the original condipons only apply to the first 30 tosses. If n>30 then you only win $2 30. Then the expected winnings are 30 1 n 1 2 n + n 2 30 = n=1 The same result is obtained by assuming that, beyond some very large number ($2 30 in our example), it makes no difference to people what the winnings are i.e., by assuming that the uplity funcpon is perfectly flat for large winnings (G. Cramer, 1728.) n= 31
.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
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