# Expected Utility: Definition, Calculation, and Examples

## 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, any particular event will occur.

### Key Takeaways

• Expected utility refers to the utility of an entity or aggregate economy over a future period of time, given unknowable circumstances.
• Expected utility theory is used as a tool for analyzing situations in which individuals must make a decision without knowing the outcomes that may result from that decision
• The expected utility theory was first posited by Daniel Bernoulli who used it to solve the St. Petersburg Paradox.
• Expected utility is also used to evaluate situations without immediate payback, such as purchasing insurance.

## Understanding 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 in which individuals must make a decision without knowing the outcomes that may result from that decision, i.e., decision making under uncertainty. These individuals will choose the action that will result in the highest expected utility, which is the sum of the products of probability and utility over 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 money does not necessarily equate to the total value of money. This theory helps explain why people may take out insurance policies to cover themselves for various risks. The expected value from paying for insurance would be to lose out monetarily. The possibility of large-scale losses could lead to a serious decline in utility because of the 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 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 each game's play. For instance, if the stakes start at $2 and double every time heads appear, once the first time tails appear the game ends, and the player wins whatever is in the pot. Under such game rules, the player wins$2 if tails appear on the first toss, $4 if heads appear on the first toss and tails on the second,$8 if heads appear on the first two tosses and tails on the third, and so on.

Mathematically, the player wins 2k dollars, where k equals the 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, in theory, the sum is limitless. Thus the expected win for repeated play is an infinite amount of money.

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

## Expected Utility vs. Marginal Utility

Expected utility is also related to the concept of marginal utility. The expected utility of a reward or wealth decreases when a person is rich or has sufficient wealth. In such cases, a person may choose the safer option as opposed to a riskier one.

For example, consider the case of a lottery ticket with expected winnings of $1 million. Suppose a person with comparatively fewer resources buys the ticket for$1. A wealthy person offers to buy the ticket off them for $500,000. Logically, the lottery holder has a 50-50 chance of profiting from the transaction. It is likely that they will opt for the safer option of selling the ticket and pocketing the$500,000. This is due to the diminishing marginal utility of amounts over $500,000 for the ticket holder. In other words, it is much more profitable for them to get from$0 - $500,000 than from$500,000 - \$1 million.

Now consider the same offer made to a very wealthy person, possibly a millionaire. Likely, the millionaire will not sell the ticket because they hope to make another million from it.

A 1999 paper by economist Matthew Rabin argued that the expected utility theory is implausible over modest stakes. This means that the expected utility theory fails when the incremental marginal utility amounts are insignificant.

## Example of Expected Utility

Decisions involving expected utility are decisions involving uncertain outcomes. An individual calculates the probability of expected outcomes in such events and weighs them against the expected utility before making a decision.

For example, purchasing a lottery ticket represents two possible outcomes for the buyer. They could end up losing the amount they invested in buying the ticket, or they could end up making a smart profit by winning either a portion of the entire lottery. Assigning probability values to the costs involved (in this case, the nominal purchase price of a lottery ticket), it is not difficult to see that the expected utility to be gained from purchasing a lottery ticket is greater than not buying it.

Expected utility is also used to evaluate situations without immediate payback, such as purchasing insurance. When one weighs the expected utility to be gained from making payments in an insurance product (possible tax breaks and guaranteed income at the end of a predetermined period) versus the expected utility of retaining the investment amount and spending it on other opportunities and products, insurance seems like a better option.

Article Sources
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1. UC Berkeley. "Risk Aversion and Expected-Utility Theory: A Calibration Theorem." Accessed May 7, 2021.

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