1. Exercise #1 If you have a bankroll of $100, how many tosses could you allow in a finite version of the St. Petersburg game (if you were compelled to pay the winnings from this bankroll)? What are the expected winnings for a game limited to this many tosses? How much do you think your friends might actually pay to play this game? What are the numbers if the bankroll is $100 million ?

2. Answer #1 Bankroll = $100 Max allowable tosses = 14 (see the ‘First tail/Winnings’ table on the slide 11) Liability if all tosses show heads = $81.92 Expected winnings from game limited to 14 tosses = 7 cents Bankroll = $100 million Max allowable tosses = 34 (compute from for i = 1 to 100 do print i, 2^(i-1)/100 ) Liability if all tosses show heads = $85,899,345.92 Expected winnings from game limited to 34 tosses = 17 cents Judging by the popularity of “Powerball” and similar lotteries, a lot of people would likely spend a dollar to potentially win $85M. In the United States, lottery tickets sell for twice to three times their expected value. Interestingly, insurance is typically priced at 4 to 5 times its expected value. First tail Winnings 1 0.01 2 0.02 3 0.04 4 0.08 5 0.16 6 0.32 7 0.64 8 1.28 9 2.56 10 5.12 11 10.24 12 20.48 13 40.96 14 81.92 15 163.84...

3. Exercise #2 Demographic simulations of an endangered marine turtle suggest that conservation strategy “Beachhead” will yield between 3 and 4.25 extra animals per unit area if it’s a normal year but only 3 if it’s a warm year, and that strategy “Longliner” will produce between 1 and 3 extra animals in a normal year and between 2 and 5 extra animals in a warm year. Assume that the probability of next year being warm is between 0.5 and 0.75. Graph the reward functions for these two strategies as a function of the probability of next year being warm. Which strategy would conservationists prefer? Why?

4. Answer #2 // Warm Normal Bw =[3,4.25];Bn =3// reward for Beachhead Lw = [1,3];Ln = [2,5]// reward for Longliner p = [.5,.75] // probability of a warm year for i = 0 to 25 do begin ii = i/25 P = left(p) + ii * width(p) B = P * Bw + (1-P) * Bn L = P * Lw + (1-P) * Ln print P, left(B), right(B), left(L), right(L) end A conservation biologist might favor the “Beachhead” strategy because it has about the same optimistic performance as the “Longliner” strategy but its pessimistic performance would be a lot better. 0 1 2 3 4 0.50.60.70.8 Beachhead Longliner