Expected Utility Calculator

Enter outcomes (comma-separated):


Enter probabilities (comma-separated):


Utility function (optional, use x for outcome):


Calculate

Decision-making under uncertainty is one of the most critical challenges in economics, finance, game theory, and psychology. The Expected Utility Calculator helps simplify that process by evaluating choices based on the expected value of utility rather than just raw outcomes.

The concept of expected utility is central to rational choice theory. Instead of maximizing expected monetary value, individuals are assumed to maximize their expected utility, a subjective measure that reflects their personal preferences, risk tolerance, and satisfaction.

This calculator allows users to enter different outcomes and their associated probabilities, and optionally apply a utility function (such as √x or log(x)) to capture individual risk attitudes.

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📐 The Formula

The expected utility is calculated as:

Expected Utility (EU) = Σ [U(xᵢ) × pᵢ]

Where:

• xᵢ is the i-th outcome,
• pᵢ is the probability of xᵢ occurring,
• U(xᵢ) is the utility derived from xᵢ,
• Σ denotes summation over all outcomes.

If no utility function is applied, the formula becomes:

EU = Σ [xᵢ × pᵢ]

This simpler form calculates the expected value, not utility.

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🛠️ How to Use the Expected Utility Calculator

Follow these steps:

1. Enter Outcomes: Input all possible outcomes separated by commas.
   • Example: 100, 200, 300
2. Enter Probabilities: Input the corresponding probabilities for each outcome.
   • Example: 0.2, 0.5, 0.3
3. (Optional) Enter Utility Function: Customize utility by entering a JavaScript-compatible function like:
   • Math.sqrt(x) for risk-averse utility
   • Math.log(x) for logarithmic utility
   • If left blank, outcomes will be used as-is.
4. Click Calculate: Get the expected utility value based on your input.

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🔍 Example

Let’s say a person has three potential outcomes for a business venture:

• $100 with 20% probability
• $200 with 50% probability
• $300 with 30% probability

They are risk-averse and define their utility as the square root of the monetary outcome.

Input:

• Outcomes: 100, 200, 300
• Probabilities: 0.2, 0.5, 0.3
• Utility Function: Math.sqrt(x)

Step-by-step:

• √100 × 0.2 = 10 × 0.2 = 2.0
• √200 × 0.5 ≈ 14.14 × 0.5 = 7.07
• √300 × 0.3 ≈ 17.32 × 0.3 = 5.20

Expected Utility = 2.0 + 7.07 + 5.20 = 14.27

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❓ Frequently Asked Questions

1. What is expected utility?

Expected utility represents the average satisfaction or value an individual expects from different uncertain outcomes, weighted by their probabilities.

2. Why not just use expected value?

Expected value ignores personal preferences and risk tolerance. Expected utility incorporates how outcomes are felt, not just their numeric value.

3. What is a utility function?

A utility function translates an objective outcome (like money) into subjective satisfaction. Common examples include √x, log(x), or x².

4. Can I leave the utility function blank?

Yes! The calculator will treat outcomes as utilities directly and compute expected value.

5. How are probabilities entered?

Enter them as decimal numbers (e.g., 0.5 for 50%). They should sum to 1 for accuracy, though the calculator won’t enforce it.

6. Can I use negative outcomes?

Yes, but be careful with utility functions like Math.sqrt(x) or Math.log(x) which are undefined for negative numbers.

7. What does it mean to be risk-averse?

A risk-averse person prefers a guaranteed outcome over a risky one with the same expected value. Their utility functions grow slowly, e.g., √x or log(x).

8. What’s the opposite of risk-averse?

Risk-seeking. These individuals prefer risky outcomes and often use functions like x² or exponential functions.

9. What is neutral utility?

Risk-neutral individuals value outcomes exactly as they are. Their utility function is linear (e.g., U(x) = x).

10. Can this be used in game theory?

Yes, it’s widely used to analyze strategies under uncertainty where players weigh outcomes and probabilities.

11. What fields use expected utility?

Economics, finance, decision theory, insurance, psychology, and AI use expected utility to model rational decision-making.

12. How do I interpret the result?

The number is a weighted average utility. Compare expected utilities of different choices — the higher, the better.

13. Does the calculator normalize probabilities?

No, it doesn’t. You should ensure your probabilities sum to 1 for accurate computation.

14. Can I use scientific functions like log10 or exp?

Yes, use JavaScript-compatible expressions like Math.log10(x), Math.exp(x), etc.

15. What if I input invalid math in the utility function?

You’ll get an error message. Double-check your syntax (e.g., Math.sqrt(x), not just sqrt(x)).

16. Can this be used for real-life investment analysis?

Absolutely. It’s ideal for modeling investor preferences and comparing risky assets.

17. What happens if probabilities exceed 1?

The result may be mathematically incorrect. Probabilities should always be ≤ 1 and sum to 1.

18. Can I use percentages?

Yes, but convert them to decimals. For example, 50% = 0.5.

19. Is this calculator mobile-friendly?

Yes, the form and calculations are simple and responsive.

20. What if I have more than 3 outcomes?

No problem. You can enter as many as needed, as long as the number of outcomes matches the number of probabilities.

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🧾 Conclusion

The Expected Utility Calculator is a powerful tool for analyzing decisions where uncertainty and personal preferences come into play. Whether you’re an economist, investor, student, or game theorist, this tool helps quantify what really matters: not just what you might get, but how much value it brings you.