## 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.

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## 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 analyzingÂ situations where individuals must make a decision without knowing which outcomes may result from that decision, i.e., decision making underÂ uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability andÂ utilityover all possible outcomes. The decision made will also depend on the agentâ€™sÂ risk aversionÂ and the utility of other agents.

This theory also notes that the utility of a money does not necessarily equate to theÂ 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 ofÂ expected utilityÂ was first positedÂ byÂ Daniel Bernoulli,Â who used it as a tool to solve theÂ St. Petersburg Paradox.

The St. Petersburg ParadoxÂ can be illustrated as aÂ game of chance in whichÂ aÂ coin is tossedÂ at in each play of the game. For instance, if the stakes starts at \$2Â and doubleÂ every time heads appears, andÂ theÂ first time tails appears, the game ends and the player wins whatever is in the pot. Under such game rules, the player wins \$2Â if tails appears on the first toss, \$4Â if heads appears on the first toss and tails on the second, \$8Â if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2kÂ dollars, whereÂ kÂ equals number of tosses (k must be a whole number and greater than zero).Â Assuming the game can continue as long as the coin toss results in heads and in particular that the casino has unlimited resources, this sumÂ grows without boundÂ and so the expected win for repeated play is an infinite amount of money.

BernoulliÂ e solved the St. Petersburg ParadoxÂ by making the distinction between expected value andÂ expected utility, as the latter uses weightedÂ utilityÂ multiplied by probabilities, instead of using weighted outcomes.

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