Expected utility is a formula that measures the desirability of a decision outcome, taking into account the probability and value of all possible outcomes. It considers the expected value of a decision alternative, which is determined by multiplying the probabilities of each possible outcome by their respective utilities. The formula for expected utility is:
EU = ∑(p * u)
where:
- EU is the expected utility
- p is the probability of an outcome
- u is the utility of an outcome
The Formula for Expected Utility: A Comprehensive Guide
The expected utility formula is a fundamental concept in decision theory that quantifies the value of uncertain outcomes. It helps decision-makers evaluate and select the option with the highest expected benefit.
Components of the Formula
The expected utility formula consists of several key components:
- Utilities: Utilities are numerical values assigned to different outcomes to represent their desirability. They are typically expressed as a number between 0 (worst outcome) and 1 (best outcome).
- Probabilities: Probabilities are the likelihood of occurrence for each outcome. They range from 0 (impossible outcome) to 1 (certain outcome).
Formula
The expected utility for a given decision is calculated as the weighted average of utilities of all possible outcomes, where the weights are the probabilities of those outcomes. The formula is:
Expected Utility = Σ(Utility of Outcome * Probability of Outcome)
Example
Suppose you are considering two investment options:
- Option A: Offers a 70% chance of gaining $50,000 with a utility of 0.8.
- Option B: Offers a 30% chance of gaining $80,000 with a utility of 0.9.
Using the expected utility formula:
| Outcome | Utility | Probability | Expected Utility |
|---|---|---|---|
| Gain $50,000 | 0.8 | 0.7 | 0.56 |
| Gain $80,000 | 0.9 | 0.3 | 0.27 |
| Total | – | – | 0.83 |
Based on this calculation, Option A has a higher expected utility (0.83) compared to Option B (0.56), making it the preferred choice under the expected utility criterion.
Factors to Consider
- Subjectivity: Utilities are subjective and can vary from person to person.
- Risk Tolerance: Risk-averse individuals will assign lower utilities to uncertain outcomes, while risk-seekers will assign higher utilities.
- Independence: The formula assumes that outcomes are independent, which may not always be true.
- Completeness: The formula requires that outcomes are complete and cover all possible scenarios.
Despite its limitations, the expected utility formula remains a valuable tool for decision-making under uncertainty. It provides a systematic and quantitative approach to evaluating and comparing different options based on their potential outcomes and risks.
Question 1:
What is the formula for expected utility?
Answer:
The formula for expected utility is EU = ∑(P(x) * U(x)), where:
- EU is the expected utility.
- P(x) is the probability of outcome x.
- U(x) is the utility of outcome x.
Question 2:
How does the expected utility formula incorporate both probabilities and utilities?
Answer:
The expected utility formula multiplies the probability of each possible outcome by its corresponding utility. This step acknowledges that an outcome with a higher probability should also have a higher weight in the overall calculation of expected utility.
Question 3:
What is the purpose of using the expected utility formula in decision-making?
Answer:
The expected utility formula provides a rational and objective method for evaluating decisions under uncertainty. By considering both the probabilities and utilities of different outcomes, it helps decision-makers select the option with the highest expected payoff.
Well, folks, that’s the lowdown on the formula for expected utility! I hope this article has been helpful in demystifying this important concept. Remember, making decisions under uncertainty can be tough, but having a framework to guide your choices can make all the difference. Thanks for taking the time to read! Be sure to check back later for more insightful content on all things finance and decision-making. Take care, and until then, keep on making those informed choices!