Question

Doug does not follow the expected utility formula. Instead he prefers options that maximize the quantity:

(Pr(C₁))²U(C₁) + ... + (Pr(C_n))²U(C_n)

Doug is indifferent between the following two options.

• He wins $25 with probability 1/2, otherwise he gets $0.
• He wins $100 with probability 1/4, otherwise he gets $0.

If U($0) = 1 and U($100) = 3, what is U($25)?

(Round your answer to three decimal places. Example: if the true answer is 2/3, you should enter 0.667.)

Answer 1

a) Doug is indifferent between the following two options because the formula he uses gives equal output for both the options

That is to say

For the first option, there is a 50% chance of winning $25 and 50% chance of winning $0.

EU for the above case

= (Pr($25))^2U($25) + (Pr($0))^2U($0)

= (1/2)^2 * U($25) + (1/2)^2 * 1

= U($25)/4 + 1/4

For the second option, there is a 25% chance of winning $100 and 75% chance of winning $0.

EU for the above case

= (Pr($100))^2U($100) + (Pr($0))^2U($0)

= (1/4)^2 * U($100) + (3/4)^2 * 1

= 3/16 + 9/16 = 3/4

Now, since the expected utility value for both the above case is same . Hence we can write

U($25)/4 + 1/4 = 3/4

or, U($25)/4 = 3/4 - 1/4

or, U($25) = 2

Hence U($25) = 2 (answer)

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