Expected utility is a concept in economics and decision theory that combines the idea of utility, which represents an individual’s satisfaction or happiness, with the concept of probability. It is used to analyze decision-making under uncertainty and to model how individuals make choices when faced with uncertain outcomes.

The expected utility theory was developed to provide a rational framework for decision-making that considers both the preferences of individuals and the uncertainty associated with different outcomes. The theory assumes that individuals make decisions based on maximizing expected utility, which is the sum of the utilities of all possible outcomes, weighted by their probabilities.

The formula for calculating expected utility is as follows:

\[ \text{Expected Utility (EU)} = \sum_{i=1}^{n} P_i \times U(X_i) \]

Where:

– \( P_i \) represents the probability of outcome \( X_i \),

– \( U(X_i) \) represents the utility (satisfaction) associated with outcome \( X_i \),

– \( n \) is the total number of possible outcomes.

Key points about expected utility:

1. **Utility Function:**

– The function \( U(X_i) \) represents the utility or satisfaction derived from a specific outcome. Utility functions are typically assumed to be increasing, meaning that more of a good or service is preferred to less.

2. **Risk Aversion:**

– Expected utility theory helps explain risk aversion. If individuals are risk-averse, they will assign a higher value to a certain outcome with a lower utility than to a risky outcome with a higher expected utility. This reflects a preference for certainty.

3. **Risk-Seeking and Risk-Neutral Preferences:**

– Conversely, individuals with risk-seeking preferences may be willing to accept a lower expected utility for the chance of a higher utility outcome. Risk-neutral individuals are indifferent to risk and base decisions solely on expected utility.

4. **Decision-Making Under Uncertainty:**

– Expected utility theory is particularly relevant when individuals face uncertainty about future outcomes. It provides a way to model and analyze decision-making in situations where the probabilities and payoffs of different outcomes are not known with certainty.

5. **Critiques and Challenges:**

– While expected utility theory has been influential, it has faced critiques and challenges. Some criticisms include the assumption of consistent preferences, the difficulties in measuring and comparing utilities across individuals, and the recognition that people may not always behave according to the theory’s predictions.

6. **Application in Economics and Finance:**

– Expected utility theory has been widely used in economics and finance to analyze consumer choices, investment decisions, insurance, and other areas involving uncertainty. It is a foundational concept in the field of behavioral economics.

7. **Prospect Theory:**

– Prospect theory, developed by Daniel Kahneman and Amos Tversky, is an extension of expected utility theory that accounts for psychological factors and deviations from strict rationality in decision-making.

Expected utility theory provides a framework for understanding how individuals make decisions when faced with uncertainty. It has influenced economic and financial modeling and has contributed to the development of behavioral economics, which explores the psychological aspects of decision-making.