Question

Let L denote the set of simple lotteries over the set of outcomes C = {c1,c2,c3,c4}. Consider the von Neumann-Morgenstern utility function U : L → R defined by

U(p1,p2,p3,p4)=p2u2 +p3u3 +4p4,
where ui,i = 2,3, is the utility of the lottery which gives outcomes ci, with certainty.

Suppose that the lottery L1 = (0, 1/2, 1/2,0) is indifferent to L2 = (1/2,0,0, 1/2) and that L3 =

(0, 1/3, 1/3, 1/3) is indifferent to L4 = (0, 1/6, 5/6,0). Find the values of u2 and u3.

Answer 1

Let us first calculate the utilities from each of the lotteries.

Utility from L_{1 :}

Utility from L2 _:

Utility from L3 _:

Utility from L4 _:

Indifference in two lotteries implies that the utility achieved from each of those lotteries is the same, hence the indifference.

First we have L1 indifferent to L2, this gives:

Next from L3 indifferent to L4, we have:

Adding equations (1) and (2), we get,