Expected Utility
What is Expected Utility?
Expected utility is the probability-weighted average of the utility of each possible outcome, used to rank risky choices.
Expected utility ranks risky options by first converting each outcome into utility and then averaging those utilities using the probabilities. The order matters: you take the utility of each outcome and weight it, rather than taking the utility of the average outcome. For a risk-averse person with a concave utility curve, the expected utility of a gamble is below the utility of its expected value, which is exactly what makes the sure thing more attractive. Expected value is the simpler cousin that averages the dollar payoffs and ignores attitudes toward risk, so it treats a coin flip for $1,000 and a sure $500 as identical. A decision maker who maximizes expected utility picks the option with the highest weighted utility, not the highest average payout.
Expected Utility: a worked example
Let utility equal the square root of wealth. Option A is a sure $900, and since the square root of 900 is 30, its utility is 30. Option B is a coin flip between $400 and $1,600; the square roots are 20 and 40, so its expected utility is 0.5 × 20 + 0.5 × 40 = 30. The two tie on expected utility even though B has the higher expected value, 0.5 × $400 + 0.5 × $1,600 = $1,000 against A's $900. Giving up $100 of expected value to escape the risk is what risk aversion looks like in numbers.
The mistake students make with expected utility
The classic error is computing the utility of the expected value instead of the expected value of the utility. Those are different numbers whenever the utility curve is not a straight line, and for a concave curve the utility of the average is larger. Take each outcome, convert it to utility first, then weight by probability. Reversing the order erases risk aversion from the answer.
Expected Utility questions
What is the difference between expected value and expected utility?
Expected value averages the dollar outcomes, while expected utility averages the satisfaction those outcomes deliver. Expected value ignores how a person feels about risk, so it rates a fair gamble the same as the equivalent sure amount. Expected utility separates them, because a concave utility curve makes the gamble worth less.
How do you calculate expected utility?
Multiply the utility of each outcome by the probability of that outcome and add the products together. Write the utility function first, plug in each possible wealth level, then weight by the probabilities, which must sum to 1. The result is a utility number, not a dollar amount, so convert it back with the certainty equivalent if you need dollars.
Does maximizing expected utility always mean avoiding risk?
No, maximizing expected utility means taking whichever option has the highest weighted utility, and that is often the risky one. A gamble with a big enough payoff beats a sure thing even for a cautious person. Only risk-averse preferences and a fair gamble together make the certain option win.
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