# Expected utility theory

Expected utility theory is a mathematical model for decision making under uncertainty. It is used to assess the decision-making process of an individual in a given context, and determine which option is the most rational based on an evaluation of the expected value of each possible outcome. This model is based on the assumption that people are rational and make choices that maximize their expected utility, or the sum of the utilities of all the possible outcomes weighted by their respective probabilities. This theory can be used by managers to identify the most logical choice in a given situation, and to assess the riskiness of a particular course of action.

## Example of expected utility theory

• A manager is faced with the decision of whether to invest in a new product line. Using expected utility theory, they can evaluate the expected value of each potential outcome, weighing the expected profits and losses. Based on this analysis, they can determine which option is the most rational and will maximize their expected utility.
• An investor is considering two different stocks. Using expected utility theory, they can analyze the expected returns of each stock and weigh the expected profits and losses in order to decide which one is the more logical choice.
• A consumer is deciding whether to purchase a new car or a used car. Using expected utility theory, they can compare the expected costs and benefits of each option and determine which one is the more rational choice.

## Formula of expected utility theory

The expected utility theory can be mathematically expressed as:

$$EU = \sum_{i=1}^{n} p_i U(x_i)$$,

where EU is the expected utility, $$p_i$$ is the probability associated with each outcome $$x_i$$, and $$U(x_i)$$ is the utility associated with each outcome. This formula is used to calculate the expected utility of a decision by summing the product of the probability and the utility of each outcome.

The expected utility of a decision is a measure of the average benefit or satisfaction that an individual is likely to experience if they choose a particular course of action. It is calculated by summing the product of the probability and the utility of each outcome and dividing the total by the total number of possible outcomes.

The utility of an outcome is a measure of the satisfaction or benefit that an individual would gain from the outcome. It is calculated by assigning a numerical value to each outcome that reflects the individual's subjective evaluation of its value or benefit. The higher the utility of an outcome, the more desirable it is to the individual.

The probability of an outcome is a measure of the likelihood that it will occur if the decision is made. It is calculated by assigning a numerical value to each outcome that reflects the individual's subjective evaluation of its likelihood. The higher the probability of an outcome, the more likely it is to occur.

By combining the probability and the utility of each possible outcome, the expected utility theory allows managers to determine which option is the most rational based on an evaluation of the expected value of each. This can help managers make more informed decisions and reduce the risk associated with their decisions.

## When to use expected utility theory

Expected utility theory can be used in a variety of contexts, such as economics, finance, and operations management. Specifically, it is most applicable when an individual or company needs to make a decision in the face of uncertainty, and when the decision involves multiple possible outcomes with differing levels of utility. Some applications of expected utility theory include:

• Risk management: Expected utility theory can be used to assess the relative risk of different decisions and identify the most suitable option. This can be especially useful when making financial decisions, such as investments.
• Strategic management: The model can be used to analyze strategic decisions and identify the most suitable course of action.
• Forecasting: Expected utility theory can be used to predict the future behavior of individuals and organizations based on their current actions.
• Behavioral economics: The model helps to explain how individuals make decisions, and how they weigh the expected utility of different outcomes.

## Types of expected utility theory

Expected utility theory is a mathematical model for decision making under uncertainty that is used to assess the decision-making process of an individual. There are several different types of expected utility theory, including:

• The Von Neumann–Morgenstern expected utility theory which is based on the assumption that an individual's preferences can be represented by a set of utility functions which represent their preferences for each possible outcome of a decision.
• The Prospect Theory which is based on the idea that people are risk-averse in certain situations, and instead of maximizing expected utility, they seek to minimize losses.
• The Cumulative Prospect Theory which is an extension of the Prospect Theory, and takes into account the fact that people are more risk-averse when the outcome is uncertain.
• The Dispositional Theory which is based on the idea that people make decisions based on their underlying preferences and beliefs, rather than purely on the expected utility of the outcomes.
• The Heuristics and Biases Theory which suggests that people often make decisions based on heuristics, or mental shortcuts, rather than on a rational evaluation of the expected utility of each possible outcome.

## Steps of expected utility theory

The decision-maker is the person or entity making the decision, and the available options are the different courses of action that are available for consideration.

• Step 2: Assign utilities to each of the options.

Utilities are the numerical values assigned to each option, which represent the degree to which the decision-maker values each possible outcome.

• Step 3: Assign probabilities to each of the options.

Probabilities are the chances that each option will occur, and should be based on past data or expert knowledge.

• Step 4: Calculate the expected utility of each option.

The expected utility of each option is the sum of the utility of the outcome multiplied by the probability of that outcome occurring.

• Step 5: Choose the option with the highest expected utility.

The option with the highest expected utility is the most rational choice, as it has the highest chance of yielding the desired outcome.

## Advantages of expected utility theory

Expected utility theory provides a structured and systematic approach to decision making under uncertainty, and offers several advantages. These include:

• The ability to compare different options in a rational manner, by comparing their expected values and determining which option is most likely to provide the highest utility.
• The ability to assess risk and uncertainty when making decisions, by considering the probability of each outcome and its potential utility.
• The potential to identify a course of action that is likely to lead to a higher expected value than other alternatives.
• The capacity to incorporate uncertain information and subjective preferences into the decision-making process.
• The potential to take into account changes in the environment that may affect the outcome of the decision.

## Limitations of expected utility theory

Expected utility theory has several limitations that should be taken into consideration when making decisions. These limitations include:

• Human behavior is not always rational, as people often make decisions based on emotions or other biases that cannot be accounted for in the expected utility model.
• The expected utility theory assumes that all outcomes are equally likely, when in reality this is rarely the case.
• It is difficult to accurately estimate the utility of each possible outcome, as the value of a given outcome will vary from person to person.
• The expected utility theory does not take into account non-monetary costs and benefits, such as the impact of a decision on the environment or on a person's mental health.
• The expected utility theory does not account for uncertainty, as it assumes that the probabilities of each outcome are known.
• The expected utility theory does not consider other decision-making criteria such as time preferences or risk aversion.

## Other approaches related to expected utility theory

Expected utility theory is not the only approach used to assess decision making under uncertainty. Other approaches related to expected utility theory include:

• Subjective expected utility theory (SEU), which is based on the assumption that an individual’s decision-making process is influenced by their own personal beliefs and values.
• Prospect theory, which focuses on how people make decisions in situations where the outcomes are uncertain. It takes into account the fact that people tend to be risk-averse, and will often make decisions that minimize losses.
• The Bayesian approach, which uses probability theory to evaluate the expected outcomes of different decisions.