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  Choice under Uncertainty

Choice under Uncertainty:

In the basic theory about choice under certainty the consumer is assumed to be able to compare and rank all possible consumption bundles. But is this meaningful if the consumer is not certain that she will ever be able to consume any particular bundle? Somehow we must also take into account that with uncertainty different bundles have different probabilities of actually being available for consumption. A way of incorporating uncertainty is to imagine (just for the sake of the argument) that the objects of choice are not consumption bundles, but “lottery tickets” whose prizes are bundles of consumption goods, or more commonly, money prizes. Different lotteries have different probabilities of winning the various possible prizes, and individuals are assumed to know these probabilities and make their choices based on their preferences for consumption and attitudes toward risk taking.

Assume that we have three “lotteries” whose prizes are three sums of money: Y1 = $100, Y2 = $200, and Y3 = $500. If you won these prizes each time you played each lottery (and the cost of the lottery ticket is below $100) you would always prefer the third lottery, to the second, to the first, provided that you prefer more to less. But imagine that the probability of winning in the first lottery is 1/2 , while it is 1/4 in the second and only 1/10 in the third. Is it now obvious how you would choose among these three lotteries? (Assume that you are given the chance to participate in only one of these lotteries.)

A particular type of utility function is used to make such comparisons, and it’s called the von Neumann − Morgenstern utility function, or the expected utility function. This function is actually a combination of a utility function under certainty and the probabilities of the different prizes. First we have to specify a utility function when we receive the prizes with certainty, which we write u(Y), and then we multiply these functions with the respective probability. Hence, in our case the expected utilities of the three lotteries, called L1, L2 and L3, respectively,

U (L1) = (1/2) u (100) + (1/2) u (0),
U (L2) = (1/4) u (200) + (3/4) u (0),
U (L3) = (1/10) u (500) + (9/10) u (0).

Obviously to know which lottery a particular consumer/player will choose we have to know the form of the “certainty” utility function u(Y).

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