Asked by Connell Maxwell on May 17, 2024
Verified
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.
Risk Averter
An individual or entity that prefers outcomes with less risk and is likely to avoid uncertain prospects in favor of a sure thing.
Expected Value
The weighted average of all possible values of a random variable, considering the probabilities of each outcome.
Natural Log
The logarithm to the base e, where e is an irrational constant approximately equal to 2.71828, often used in mathematics, physics, and engineering.
- Assess the impact of risk aversion on gambling and betting decisions.
- Analyze investment strategies in uncertain contexts to maximize expected utility.
Verified Answer
VJ
Virginia JeffriesMay 22, 2024
Final Answer :
D
Explanation :
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.
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.
Learning Objectives
- Assess the impact of risk aversion on gambling and betting decisions.
- Analyze investment strategies in uncertain contexts to maximize expected utility.