### What is Expected Utility

Expected utility is an economic term summarizing the utility that an entity or aggregate economy is expected to reach under any number of circumstances. The expected utility is calculated by taking the weighted average of all possible outcomes under certain circumstances, with the weights being assigned by the likelihood, or probability, that any particular event will occur.

### BREAKING DOWN Expected Utility

The expected utility of an entity is derived from the expected utility hypothesis. This hypothesis states that under uncertainty, the weighted average of all possible levels of utility will best represent the utility at any given point in time.

Expected utility theory is used as a tool for analyzing situations where individuals must make a decision without knowing which outcomes may result from that decision, i.e., decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utilityover all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

This theory also notes that the utility of a money does not necessarily equate to the total value of money. This theory helps explains why people may take out insurance policies to cover themselves for a variety of risks. The expected value from paying for insurance would be to lose out monetarily. But, the possibility of large-scale losses could lead to a serious decline in utility because of diminishing marginal utility of wealth.

### History of the Expected Utility Concept

The concept of expected utility was first posited by Daniel Bernoulli, who used it as a tool to solve the St. Petersburg Paradox.

The St. Petersburg Paradox can be illustrated as a game of chance in which a coin is tossed at in each play of the game. For instance, if the stakes starts at $2 and double every time heads appears, and the first time tails appears, the game ends and the player wins whatever is in the pot. Under such game rules, the player wins $2 if tails appears on the first toss, $4 if heads appears on the first toss and tails on the second, $8 if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2^{k} dollars, where *k* equals number of tosses (k must be a whole number and greater than zero). Assuming the game can continue as long as the coin toss results in heads and in particular that the casino has unlimited resources, this sum grows without bound and so the expected win for repeated play is an infinite amount of money.

Bernoulli e solved the St. Petersburg Paradox by making the distinction between expected value and expected utility, as the latter uses weighted utility multiplied by probabilities, instead of using weighted outcomes.