# What is an Expected Utility?

Michael Ryan

Expected utility is a theory commonly used in game theory and economics. It is a statistical measurement of the probability of a good outcome to a risky decision. "Utility" is the relative measurement of satisfaction to the outcome. For example, consider a person who is offered two jobs. Job A offers higher pay, but there's a 50 percent chance he'll have to move to an undesirable location; Job B offers much lower pay, but he won't have to move. Weighing the options to make the decision is an example of expected utility. Decisions to participate in lotteries and other gambling situations also are good examples.

Expected utility is a weighted average; to calculate it, multiply the utility of each possible outcome by the probability of that outcome actually taking place. So, if there is a 50% chance of making 10 US Dollars (USD) dollars and a 50% chance of making no money, the expected utility is \$5 USD. If a second course of action had an expected utility of 3 USD, then the first course of action would be the more logical choice because it has a higher expected utility.

Economics normally assumes that people act rationally, making the choices that have the highest expected utility, but this assumption is problematic. Many problems posed by game theory, such as the "Prisoner's Dilemma," put people in a situation where if each individually acts to increase expected utility, none of the participants will get a good outcome. If the participants were allowed to communicate and participate, they would be able to plan a better overall outcome for all participants; this sort of communication is normally prohibited in the statement of the problem, however.

The situation changes if player's interact over a long period of time. A longer period of time requiring a series of decisions allows participants to experiment with cooperation and to continue cooperating if the other participant reciprocates. Communication is not necessary for this strategy to succeed. In this situation, the expected utility of cooperating and giving other players the chance to cooperate is high because the long-term gains of cooperation outweigh the short-term losses caused by a brief deviation from the strategy of non-cooperation. Determining the best overall strategy, the best mix of cooperation and non-cooperation, is an open problem. Some economics departments even hold "Prisoner Dilemma" contests that let people test out their strategies against each other.