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Decision Theory & Expected Utility Cheat Sheet
A printable reference covering expected value, expected utility, risk attitudes, decision trees, and maximin and minimax regret criteria for grades 11-12.
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Decision theory helps students compare choices when outcomes are uncertain. This cheat sheet covers how to organize options, probabilities, payoffs, and preferences in a clear mathematical way. Students need these tools for finance, statistics, economics, engineering, and everyday risk decisions. It is especially useful when the highest average payoff is not the same as the best personal choice. The core idea is to calculate expected value by multiplying each outcome by its probability and adding the results. Expected utility goes further by replacing money or payoff with a utility value that represents preference or satisfaction. Decision trees show choices, chance events, and outcomes in order. Other rules, such as maximin and minimax regret, help compare decisions when probabilities are unknown or when caution matters.
Key Facts
- Expected value is calculated by EV = p1x1 + p2x2 + ... + pnxn, where each p is a probability and each x is an outcome value.
- Expected utility is calculated by EU = p1u(x1) + p2u(x2) + ... + pnu(xn), where u(x) is the utility of outcome x.
- A rational expected utility choice selects the option with the greatest expected utility, not always the greatest expected monetary value.
- Probabilities for all outcomes in one chance event must add to 1, so p1 + p2 + ... + pn = 1.
- For a risk-neutral decision maker, utility is often modeled as u(x) = x, so maximizing expected utility matches maximizing expected value.
- A risk-averse decision maker has diminishing marginal utility, meaning each additional dollar adds less utility than the previous dollar.
- The maximin rule chooses the option with the best worst-case payoff, so it focuses on protection against the lowest outcome.
- Minimax regret chooses the option with the smallest possible maximum regret, where regret = best payoff in that state minus chosen payoff.
Vocabulary
- Expected Value
- The probability-weighted average outcome of a decision or random process.
- Utility
- A numerical measure of how much a person values or prefers an outcome.
- Expected Utility
- The probability-weighted average utility of the possible outcomes of a decision.
- Decision Tree
- A diagram that shows decision points, chance events, probabilities, and final outcomes in sequence.
- Risk Aversion
- A preference for a safer option over a risky option with the same expected monetary value.
- Regret
- The amount lost by choosing one option instead of the best option for the state that actually occurs.
Common Mistakes to Avoid
- Adding payoffs without multiplying by probabilities is wrong because expected value must weight each outcome by how likely it is.
- Choosing the highest expected value automatically is wrong because a person with risk aversion may prefer a lower expected value with greater certainty.
- Using probabilities that do not add to 1 is wrong because one complete chance event must include all possible outcomes.
- Confusing utility with dollars is wrong because utility measures preference, and the same dollar amount can have different value to different people.
- Applying minimax regret to the original payoff table is wrong because minimax regret must first convert payoffs into regret values for each state.
Practice Questions
- 1 A game pays 10 with probability 0.7. What is the expected value of the game?
- 2 Option A gives 250 with probability 0.5 and $0 with probability 0.5. Find the expected value of each option.
- 3 A decision has utilities u(100) = 8, and u(200 and a 0.75 chance of $100?
- 4 Explain why a risk-averse person might choose a guaranteed 125.