Clancy has $1,800.He plans to bet on a boxing match between Sullivan and Flanagan.He finds that he can buy coupons for $1 each that will pay off $10 each if Sullivan wins.He also finds in another store some coupons that will pay off $10 if Flanagan wins.The Flanagan tickets cost $9 each.Clancy believes that the two fighters each have a probability of 1/2 of winning.Clancy is a risk averter who tries to maximize the expected value of the natural log of his wealth.Which of the following strategies would maximize his expected utility?
A)Don't gamble at all.
B)Buy 450 Sullivan tickets and 50 Flanagan tickets.
C)Buy exactly as many Flanagan tickets as Sullivan tickets.
D)Buy 900 Sullivan tickets and 100 Flanagan tickets.
E)Buy 450 Sullivan tickets and 100 Flanagan tickets.

Question

Quizplus · Accepted Answer

Let X be the amount of money Clancy wins from the bet. 
If Sullivan wins, He wins $10 for each coupon, so he wins $10X coupons. 
If Flanagan wins, He wins $10 for each coupon, but he has to subtract $9 for each Flanagan coupon he buys. So he wins $(10-9)X = $1X coupons.

Let p be the probability of Sullivan winning, which is 1/2. Then the probability of Flanagan winning is also 1/2.

Let Y be Clancy's wealth after buying the coupons. His initial wealth is $1800, and he spends $1 for each Sullivan coupon and $9 for each Flanagan coupon.

Thus, his wealth after buying s Sullivan coupons and f Flanagan coupons is:

Y = 1800 - s - 9f

If Sullivan wins, his wealth becomes:

Y + 10Xs

If Flanagan wins, his wealth becomes:

Y + X

Clancy is a risk averter who tries to maximize the expected value of the natural log of his wealth. Thus, his utility function is:

U = ln(Y + 10Xs) * p + ln(Y + X) * (1-p)

We want to maximize his expected utility:

E[U] = p*ln(Y + 10Xs) + (1-p)*ln(Y + X)

Taking the derivative with respect to s, we get:

dE[U]/ds = (10p/(Y + 10Xs)) - (X(1-p)/(Y + X))

Setting this equal to zero, we get:

(10p/(Y + 10Xs)) = (X(1-p)/(Y + X))

Solving for s, we get:

s = (X/(10* p))*Y + X/(10*(1-p))

Since p=1/2, this simplifies to:

s = (X/Y) * 5 + X/9

We want to choose s and f to maximize E[U], subject to the constraint that Y >= 0.

Since s and f are integers, we can use trial and error to find the combination of s and f that satisfies the constraint and maximizes E[U].

The maximum value of E[U] is attained when Clancy buys 900 Sullivan tickets and 100 Flanagan tickets.

Thus, the best choice is D.

Clancy Has $1,800