Consider the example in which Joe must choose between two lotteries with four possible prizes…

Consider the example in which Joe must choose between two lotteries with four possible prizes (Example 5.8). Does either lottery stochastically dominate the other?

Example 5.8

This example is adapted from one in Luce and Raiffa (1957). Joe may choose one of the two lotteries, each over four possible prizes: (A) a new automobile, (B) a trip to the beach, (C) a computer, and (D) dinner for four at the best restaurant in town. Joe prefers A to B, B to C, and C to D. In the first lottery, every prize is equally likely (the probabilities of winning A, B, C, and D are all 25%). In the second lottery, the probabilities of winning A, B, C, and D are 15%, 50%, 15%, and 20%, respectively, so the probability of the trip has increased. Which lottery should Joe choose?

Luce and Raiffa (1957) provided a set of relevant axioms from which one can justify using expected utility to compare alternatives similar to those in this example. These will be briefly described here.

Luce and Raiffa (1957) provided a set of relevant axioms from which one can justify using expected utility to compare alternatives similar to those in this example. These will be briefly described here. this idea and states that a lottery in which B is one possible outcome is equivalent to a lottery in which B is replaced by the equivalent lottery over outcomes A and C. The fifth axiom extends transitivity to lotteries. The sixth axiom states that, if the decision-maker prefers A to C, then, for two lotteries over outcomes A and C, the decision-maker prefers the lottery in which A is more likely.

From these axioms, it can be shown that any lottery is equivalent to a simpler lottery involving only the best possible outcome and the worst possible outcome. Moreover, when comparing two lotteries, the decision-maker should prefer the lottery that is equivalent to the simpler lottery with the larger probability of the best possible outcome. (That is, given two lotteries L1 and L2, L1 is equivalent to a simpler lottery SL1, and L2 is equivalent to a simpler lottery SL2. If the probability of the best possible outcome in SL1 is greater than the probability of the best possible outcome in SL2, then the decision-maker should prefer the original lottery L1.) Finally, there exists a utility function that describes the preferences of a decision-maker who accepts these axioms.

 

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