Question

I. Zoe, who satisfies the axioms of expected utility, faces the following decision problem, where the numbers are her von Neumann-Morgenstern utilities probability tt state-, s act (a) What action will she choose? Suppose that, before making her choice, Zoe is offered the chance to consult, for free, an expert. The expert will perform a test, which will consist in dipping a piece of paper in a special liquid. After a few minutes the paper will turn either red (which happens if the state is s,) or blue (which happens if the state is s) or yellow (which happens if the state is either s2 or s,). (b) Should Zoe consult the expert and make her choice after the consultation, or should she make her choice without consulting the expert? Your answer must be supported by appropriate calculations. In particular, you must calculate the updated beliefs and the expected utility of consulting the expert (c) Suppose now that consulting the expert involves an expense that Zoe considers to be equivalent to a utility loss ofx in every state. For what value of x would she be indifferent between consulting the expert and not consulting the expert?

Answer 1

1.a. expected utilities for each of the actions (a,b or c)

FORMULA : expected utility = E [U] = $\sum$ P_i U_i

where P_i is probability of a state to occur and U_i is utility derived from that state.

for the 3 potential actions to take, expected utilities are as follows.

E[U] for action a = (1/8) 5+(1/4) 2+(1/8) 5+(1/2) 6 = 4.75

E[U] for action b = (1/8) 5+(1/4) 2+(1/8) 5+(1/2) 6 = 2.375

E[U] for action c = (1/8) 5+(1/4) 2+(1/8) 5+(1/2) 6 = 5.25

expected utility is maximum for action c. that is what zoe will choose in case of no expert consultation

1.b free consultancy

here, the consultant will do litmus test for which piece of paper will turn red for s₁ (probability 1/8), blue for s₄ (probability 1/2) and yellow for s₂ and s₃ ( probability (1/4)+(1/8)= 3/8 )

if it turns red (s1), action a will be chosen as it has highest utility ---- 5

if it turns blue (s4), action 3 will be chosen as it has highest utility ---- 8

if it turns yellow (s2 or s3), action c will be chosen as it has highest expected utility ---- 3.333 (explanation below)

if it turns yellow (s2 or s3), choice must be made between action b and action c because c gives highest utility for s2 and b gives highest utility for s3

expected utilities are calculated for case when consultant strip turns yellow - given its yellow, probability of it being s2 is [(1/4) / (1/4 +1/8)] = 2/3 and probability of it being s3 is [(1/8) / (1/4 +1/8)] = 1/3

E[U] for action b = (2/3)1+(1/3)6 = 2.666

E[U] for action c = (2/3)3+(1/3)4 = 3.333

since expected utility of action c is higher, zoe must choose action c in case litmus test shows yellow where expected utility is 3.333

Expected utility when expert is consulted is = (1/8)5 + (1/2)8 + (3/8)3.333 = 5.875

consulting expert gives better expected utility, so zoe will opt to consult an expert before taking decision

1.c if x utility is lost when consultation is taken,

zoe becomes indifferent to consulting or not consulting if expected utilities become equal in case of consulting and not consulting

i.e 5.875 - x = 5.25

solving above equation, we get x = 0.625

when x is 0.625, zoe is indifferent between consulting and not consulting an expert.