# Expected Utility: Understanding, Calculating, and Examples

![Expected Utility: The utility that an entity or aggregate economy is expected to reach under any number of circumstances.](https://www.investopedia.com/thmb/C0PgAyBpweGNyuyMAyRJ3CA36lw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/expectedutility.asp-final-4ed3e56bbda54625894e1f2c666bba1a.png)

*— Investopedia / Julie Bang*

> **Definition**
>
> Expected utility is an economic concept that represents the anticipated satisfaction or benefit an individual or economy expects to achieve from various outcomes.

## What Is Expected Utility?

"Expected utility" is an economic term summarizing the [utility](https://www.investopedia.com/terms/u/utility.asp) 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](https://www.investopedia.com/terms/w/weightedaverage.asp) of all possible outcomes under certain circumstances. With the weights being assigned by the likelihood or probability, any particular event will occur.

Expected utility is used in decision-making in times of uncertainty, where individuals assess probabilities to choose actions with the highest expected utility. The theory originated with Daniel Bernoulli. It applies to real-life situations, such as insurance purchases and lottery participation. The St. Petersburg Paradox is a historical example of the concept.

### Key Takeaways

*   The expected utility is calculated using a weighted average of all possible outcomes, considering their probabilities.
*   Expected utility theory helps individuals make decisions under uncertainty by evaluating probable outcomes.
*   Daniel Bernoulli first introduced expected utility to solve the St. Petersburg Paradox.
*   The concept highlights that the utility of money does not equal its total value, explaining actions like buying insurance.
*   Expected utility relates to marginal utility, affecting decisions involving risks based on wealth levels.

## How Expected Utility Shapes Decision-Making

The expected utility of an entity is derived from the expected utility hypothesis. This hypothesis states that under uncertainty, the [weighted average](https://www.investopedia.com/terms/w/weightedaverage.asp) of all possible levels of utility will best represent the utility at any given point in time.

Expected utility theory helps analyze situations where individuals must decide without knowing the outcomes, often under uncertainty. These individuals choose the action with the highest expected utility, calculated by multiplying probability and utility for all outcomes. The decision made will also depend on the agent’s [risk aversion](https://www.investopedia.com/terms/r/riskaverse.asp) 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. Large-scale losses may seriously reduce utility due to the [diminishing marginal utility of wealth](https://www.investopedia.com/terms/l/lawofdiminishingutility.asp).

## The Origins and Evolution of Expected Utility

The concept of expected utility was first posited by [Daniel Bernoulli](https://www.investopedia.com/terms/b/bernoulli_hypothesis.asp), who used it to solve the [St. Petersburg Paradox](https://en.wikipedia.org/wiki/St._Petersburg_paradox).

The St. Petersburg Paradox is a game of chance where a coin is tossed each round. 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 2 dollars, where _k_ equals the number of tosses (_k_ must be a whole number and greater than zero). If the game continues with heads and the casino has unlimited resources, the sum is theoretically 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](https://www.investopedia.com/terms/e/expected-value.asp) and expected utility, as the latter uses weighted utility multiplied by probabilities instead of using weighted outcomes.

## Comparing Expected Utility and Marginal Utility

Expected utility relates to [marginal utility](https://www.investopedia.com/terms/m/marginalutility.asp). As a person's wealth increases, the expected utility of more money decreases. 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. They will likely choose to sell the ticket and take the $500,000 as the safer option. 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.

In 1999, economist Matthew Rabin argued that expected utility theory is unrealistic for modest stakes. This means that the expected utility theory fails when the incremental marginal utility amounts are insignificant.

## Examples of Expected Utility in Action

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. By assigning [probability values](https://www.investopedia.com/terms/p/p-value.asp) to the costs (like the lottery ticket price), the expected utility of buying the ticket often exceeds 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.

## The Bottom Line

Expected utility as the anticipated benefit or value that an individual or economy derives from uncertain circumstances. It's an important theory in decision-making under uncertainty, particularly in economic and financial contexts. Daniel Bernoulli's developed role the expected utility theory and it was significant in solving the St. Petersburg Paradox.  

Expected utility and other economic concepts like marginal utility and risk aversion relate to real-world decisions such as purchasing insurance or evaluating financial risk. Consider expected utility when you're faced with decisions involving uncertain outcomes to achieve maximum potential benefits, especially in financial planning and risk management.

> **Article Sources**
>
> 1.  UC Berkeley. "[Risk Aversion and Expected-Utility Theory: A Calibration Theorem](https://escholarship.org/content/qt731230f8/qt731230f8.pdf)." Accessed May 7, 2021.