Congratulations, you are a contestant on a game show! You currently have no mone
ID: 1169097 • Letter: C
Question
Congratulations, you are a contestant on a game show! You currently have no money to your name, but now you have a choice: you can take $64 in cash or you can gamble. If you decide to gamble, there are three possible outcomes: door number one, door number two, and door number three. Behind one of the doors is a prize valued at $100, another door has a prize valued at $81, and one door has a prize worth $1. What is the expected value of the gamble? What would a risk neutral person choose? Suppose your utility function is of the form U(M) =.1M0 5 + (M/100)2.5; which option would you choose? Suppose instead your utility function is of the form U(M) =.1M0.5; what is the smallest amount of money that Monty could offer you (with certainty) so that you would just be indifferent between the sure thing and the gamble?Explanation / Answer
a.
Since there are 3 possible outcomes with each having equal chance of occurance. So probability of each event is 1/3.
Expected gain = probability * Money gain from occurance of event
Cash prize from door 1 is $100, expected gain is =100/3
Cash prize from door 2 is $81, expected gain is =81/3
Cash prize from door 3 is $1, expected gain is =1/3
SO Net expected Gain from gambling is (100+81+1)/3 = 182/3
Expected value from gamble is less than the cash reward. So risk neutral person will take the Cash prize.
b.
U=.1M^.5+(M/100)^2.5
Expected utility from Cash reward (M=64) is U=.1*8+.0018=.8018
Expected utility from door 1 is (M=$100), expected utility is =1+1=2
Expected utility from door 2 is (M=$81), expected utility is =1.67
Cash prize from door 3 is $1, expected gain is =.10001
Net expected utility from (2+1.67+.1)/3 =1.2567
Since expected utility from gambling is higher, so person will choose to gamble.
c.
U=.1M^.5
Expected utility from Cash reward (M) is .1M^.5
Expected utility from door 1 is (M=$100), expected utility is =1
Expected utility from door 2 is (M=$81), expected utility is =.9
Cash prize from door 3 is $1, expected gain is =.1
Net expected utility from (1+.9+.1)/3 =2/3
2/3 = .1M^.5 squaring both side and solving the equation
M=44.44
Cash reward of 44.44 will make you indifferent between cash prize and gambling when utlity function is of above form.