Expected Utility Calculator
Expected Utility (EU) helps evaluate decisions under uncertainty by combining the likelihood of different outcomes with your personal value (utility) for each outcome. It is calculated as the sum of the utility of each possible outcome multiplied by its probability.
Enter each possible outcome of a decision below. For each outcome, provide its **Probability** (a number between 0 and 1, where 1 represents 100%) and your subjective **Utility** value for that outcome (a number representing how much you value or dislike the outcome - it can be positive, negative, or zero).
Important: The probabilities of all outcomes you enter MUST sum up to 1 (or very close to it due to potential floating-point inaccuracies).
Enter Outcomes and Their Values
Understanding Expected Utility
What is Expected Utility (EU)?
Expected Utility Theory is a behavioral economics concept that describes how people make decisions when faced with uncertain outcomes. Unlike Expected Value (which uses objective monetary values), Expected Utility uses subjective 'utility' to represent a person's preference or satisfaction for an outcome. This acknowledges that people may not value money linearly (e.g., the satisfaction from gaining $100 might be different for someone with $100 compared to someone with $10,000).
Expected Utility Formula
The formula for Expected Utility is:
EU = Σ [ P(Outcomeᵢ) * U(Outcomeᵢ) ]
Σ(Sigma) means "sum of".P(Outcomeᵢ)is the Probability of a specific outcome (Outcomeᵢ).U(Outcomeᵢ)is the subjective Utility you assign to that outcome.
You calculate the Probability * Utility product for *each* possible outcome and then sum all these products together to get the total Expected Utility for the decision.
Expected Value vs. Expected Utility
While similar, Expected Value (EV) calculates the average outcome if the decision were repeated many times, using the actual monetary or objective value. EV = Σ [ P(Outcomeᵢ) * Value(Outcomeᵢ) ]. EU accounts for risk aversion or risk-seeking behavior by using subjective utility instead of objective value. For instance, a risky gamble with a high Expected Value might have a lower Expected Utility for a risk-averse person compared to a safer option with a lower EV.
Expected Utility Examples
Click on an example to see the scenario and calculation breakdown:
Example 1: Simple Gamble
Scenario: You are offered a gamble: Win $100 with 50% probability, or lose $50 with 50% probability.
1. Define Outcomes & Values:
- Outcome 1: Win $100
- Outcome 2: Lose $50
2. Assign Probabilities & Utilities: (Assuming simple utility = monetary value for this basic example)
- Outcome 1: P = 0.5, U = 100
- Outcome 2: P = 0.5, U = -50
3. Calculate Expected Utility:
EU = (P₁ * U₁) + (P₂ * U₂)
EU = (0.5 * 100) + (0.5 * -50)
EU = 50 + (-25)
4. Result: EU = 25
Conclusion: The Expected Utility of this gamble is 25.
(Using the calculator: Enter Outcome 1: P=0.5, U=100; Outcome 2: P=0.5, U=-50. Calculate.)
Example 2: Investment Choice
Scenario: You invest in a project. There's a 30% chance it yields high returns (Utility = 50), a 60% chance of moderate returns (Utility = 20), and a 10% chance it fails (Utility = -10).
1. Define Outcomes & Values: High Return, Moderate Return, Failure.
2. Assign Probabilities & Utilities:
- Outcome 1 (High): P = 0.3, U = 50
- Outcome 2 (Moderate): P = 0.6, U = 20
- Outcome 3 (Failure): P = 0.1, U = -10
3. Calculate Expected Utility:
EU = (P₁ * U₁) + (P₂ * U₂) + (P₃ * U₃)
EU = (0.3 * 50) + (0.6 * 20) + (0.1 * -10)
EU = 15 + 12 + (-1)
4. Result: EU = 26
Conclusion: The Expected Utility of this investment is 26.
(Using the calculator: Enter Outcome 1: P=0.3, U=50; Outcome 2: P=0.6, U=20; Outcome 3: P=0.1, U=-10. Calculate.)
Example 3: Insurance Decision (Simplified)
Scenario: Should you buy insurance? Insurance costs $10 (Utility = -10). There's a 1% chance of an incident costing $1000 (Utility = -1000). Without insurance, if an incident occurs, your utility is -1000. If no incident, utility is 0.
Option A: Buy Insurance
- Outcome 1: Incident occurs (P = 0.01). You pay $10 premium, insurance covers loss. Net effect on utility: -10. So U = -10.
- Outcome 2: No incident (P = 0.99). You pay $10 premium. Net effect on utility: -10. So U = -10.
EU(Buy) = (0.01 * -10) + (0.99 * -10) = -0.1 + (-9.9) = -10.
Option B: Don't Buy Insurance
- Outcome 1: Incident occurs (P = 0.01). No premium, but suffer the full loss. Utility = -1000.
- Outcome 2: No incident (P = 0.99). No premium, no loss. Utility = 0.
EU(Don't Buy) = (0.01 * -1000) + (0.99 * 0) = -10 + 0 = -10.
Conclusion: In this *simplified* example where utility equals monetary value, both options have the same EU of -10. Real decisions involve subjective risk aversion, which the utility values would capture (e.g., the *feeling* of losing $1000 might be disproportionately worse than 100 times the feeling of losing $10).
(Using the calculator for Option A: Outcome 1: P=0.01, U=-10; Outcome 2: P=0.99, U=-10. Calc EU. For Option B: Outcome 1: P=0.01, U=-1000; Outcome 2: P=0.99, U=0. Calc EU.)
Example 4: Career Choice (Simplified)
Scenario: You are choosing between two career paths. Path A has a 60% chance of leading to a highly satisfying job (Utility=80) and a 40% chance of a less satisfying job (Utility=30). Path B has an 80% chance of the less satisfying job (Utility=30) and a 20% chance of a moderately satisfying job (Utility=50).
Evaluate Path A:
- Outcome 1 (Highly Satisfying): P = 0.6, U = 80
- Outcome 2 (Less Satisfying): P = 0.4, U = 30
EU(Path A) = (0.6 * 80) + (0.4 * 30) = 48 + 12 = 60.
Evaluate Path B:
- Outcome 1 (Less Satisfying): P = 0.8, U = 30
- Outcome 2 (Moderately Satisfying): P = 0.2, U = 50
EU(Path B) = (0.8 * 30) + (0.2 * 50) = 24 + 10 = 34.
Conclusion: Based on these utilities and probabilities, Path A has a higher Expected Utility (60 vs 34).
(Using the calculator for Path A: Outcome 1: P=0.6, U=80; Outcome 2: P=0.4, U=30. Calc EU. For Path B: Outcome 1: P=0.8, U=30; Outcome 2: P=0.2, U=50. Calc EU.)
Example 5: Product Launch Decision
Scenario: Your company considers launching a new product. Outcome possibilities: High Success (Utility=200, P=0.4), Moderate Success (Utility=80, P=0.5), Failure (Utility=-50, P=0.1).
1. Assign Probabilities & Utilities:
- Outcome 1 (High Success): P = 0.4, U = 200
- Outcome 2 (Moderate Success): P = 0.5, U = 80
- Outcome 3 (Failure): P = 0.1, U = -50
2. Calculate Expected Utility:
EU = (0.4 * 200) + (0.5 * 80) + (0.1 * -50)
EU = 80 + 40 + (-5)
3. Result: EU = 115
Conclusion: The Expected Utility of launching the product is 115.
(Using the calculator: Outcome 1: P=0.4, U=200; Outcome 2: P=0.5, U=80; Outcome 3: P=0.1, U=-50. Calculate.)
Example 6: Litigation Gamble
Scenario: You are a plaintiff considering a lawsuit. Your lawyer says you have a 70% chance of winning (Utility=10000) and a 30% chance of losing, costing you legal fees (Utility=-3000).
1. Assign Probabilities & Utilities:
- Outcome 1 (Win): P = 0.7, U = 10000
- Outcome 2 (Lose): P = 0.3, U = -3000
2. Calculate Expected Utility:
EU = (0.7 * 10000) + (0.3 * -3000)
EU = 7000 + (-900)
3. Result: EU = 6100
Conclusion: The Expected Utility of pursuing the lawsuit is 6100.
(Using the calculator: Outcome 1: P=0.7, U=10000; Outcome 2: P=0.3, U=-3000. Calculate.)
Example 7: Simple Lottery Ticket
Scenario: A lottery ticket costs $2 (Utility = -2 if you don't win anything). There's a 1 in 100 chance of winning $500 (Utility = 498, net of ticket cost). There's a 99 in 100 chance of winning nothing (Utility = -2, the ticket cost).
1. Assign Probabilities & Utilities:
- Outcome 1 (Win): P = 1/100 = 0.01, U = 498
- Outcome 2 (Lose): P = 99/100 = 0.99, U = -2
2. Calculate Expected Utility:
EU = (0.01 * 498) + (0.99 * -2)
EU = 4.98 + (-1.98)
3. Result: EU = 3
Conclusion: The Expected Utility of buying the ticket is 3.
(Using the calculator: Outcome 1: P=0.01, U=498; Outcome 2: P=0.99, U=-2. Calculate.)
Example 8: Grading Decision
Scenario: A student studied for an exam. They estimate a 0.8 probability of getting a B (Utility=80) and a 0.2 probability of getting a C (Utility=70).
1. Assign Probabilities & Utilities:
- Outcome 1 (Grade B): P = 0.8, U = 80
- Outcome 2 (Grade C): P = 0.2, U = 70
2. Calculate Expected Utility:
EU = (0.8 * 80) + (0.2 * 70)
EU = 64 + 14
3. Result: EU = 78
Conclusion: The Expected Utility associated with the exam outcome is 78.
(Using the calculator: Outcome 1: P=0.8, U=80; Outcome 2: P=0.2, U=70. Calculate.)
Example 9: Simple Choice with Certainty
Scenario: What is the EU of accepting a guaranteed $50? This is a decision with only one outcome.
1. Assign Probabilities & Utilities:
- Outcome 1 (Receive $50): P = 1.0, U = 50
2. Calculate Expected Utility:
EU = (1.0 * 50)
3. Result: EU = 50
Conclusion: The Expected Utility of a certain outcome is simply its utility value.
(Using the calculator: Enter Outcome 1: P=1.0, U=50. Delete or leave other rows blank/zero. Calculate.)
Example 10: Decision with Many Outcomes
Scenario: A complex scenario with four possible outcomes.
1. Assign Probabilities & Utilities:
- Outcome 1: P = 0.2, U = 150
- Outcome 2: P = 0.3, U = 70
- Outcome 3: P = 0.4, U = -20
- Outcome 4: P = 0.1, U = -100
2. Calculate Expected Utility:
EU = (0.2 * 150) + (0.3 * 70) + (0.4 * -20) + (0.1 * -100)
EU = 30 + 21 + (-8) + (-10)
3. Result: EU = 33
Conclusion: The Expected Utility for this complex decision is 33.
(Using the calculator: Enter all four outcomes with their P and U values. Calculate.)
Frequently Asked Questions about Expected Utility
1. What is Expected Utility (EU)?
EU is a concept used to evaluate decisions with uncertain outcomes. It weighs each potential outcome's subjective value (utility) by its probability and sums them up to give a single measure of the desirability of the decision.
2. How is Expected Utility calculated?
The formula is EU = Σ [ P(Outcomeᵢ) * U(Outcomeᵢ) ]. You multiply the probability of each outcome by its utility and then add up all these products for every possible outcome.
3. What is the difference between Probability and Utility?
Probability is the objective likelihood of an outcome occurring (a number between 0 and 1). Utility is your subjective value or satisfaction level for that outcome (a numerical representation of how good or bad it feels to you, can be positive, negative, or zero).
4. Do the probabilities need to sum to 1?
Yes, the probabilities of all possible, mutually exclusive outcomes for a single decision must sum exactly to 1 (or 100%) to cover all possibilities.
5. Can Utility values be negative?
Absolutely. Negative utility values represent outcomes you dislike or that cause you dissatisfaction, loss, or pain.
6. How many outcomes can I enter?
The calculator allows you to add multiple outcomes. You should enter all mutually exclusive outcomes that could result from the decision you are evaluating.
7. What are valid inputs for Probability and Utility?
- Probability: Must be a number between 0 and 1 (inclusive).
- Utility: Can be any finite number (positive, negative, or zero).
8. What does the resulting Expected Utility value mean?
The EU value represents the average utility you can "expect" to receive if you make this decision. A higher EU suggests a more desirable decision compared to one with a lower EU (or a negative EU). When comparing options, you would typically choose the one with the highest Expected Utility.
9. How is Expected Utility different from Expected Value?
Expected Value uses the objective, numerical value of an outcome (like money). Expected Utility uses the subjective value (utility) of an outcome. EU is often used in economics and decision theory because it better accounts for human behavior like risk aversion.
10. Why might someone choose an option with lower Expected Value but higher Expected Utility?
This happens due to risk aversion. A person might assign disproportionately low utility to large losses, making a risky gamble with a high potential payoff (and high EV) less attractive in terms of *felt satisfaction* (lower EU) than a safer option with a smaller guaranteed gain (lower EV but higher EU).
