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.
Understanding Decision Theory & Expected Utility
Expected utility becomes important because money does not mean the same thing in every situation. Losing one hundred dollars may be annoying for a person with large savings but serious for a student with little money. A utility scale turns outcomes into personal value.
It does not need to measure happiness perfectly. It only needs to rank outcomes consistently. One common way to build a scale is to set the utility of a worst outcome to zero and a best outcome to one hundred.
Intermediate outcomes can then be judged by comparing them with simple lotteries. A person who is indifferent between a sure amount and a gamble reveals how much risk that amount is worth to them.
Risk attitudes change with context. Risk aversion often makes sense when a loss could affect rent, food, safety, or an important goal. Diminishing marginal utility explains part of this pattern.
The first amount of money may solve an urgent problem. Later amounts improve life less. A risk seeking choice can occur too, especially when someone faces a loss and sees a small chance of recovery as preferable to a certain loss.
This does not prove that one attitude is always correct. Students should state whose preferences are being modeled and what the payoff includes. Time, stress, reputation, and safety can matter alongside money.
A decision tree is evaluated from right to left. Start at each final outcome and attach its payoff or utility. At a chance point, combine the possible branch values using their probabilities.
At a choice point, keep the branch that gives the decision maker the preferred value. This backward process prevents a common mistake of choosing an attractive early branch while ignoring what happens later. Trees are useful for choices such as buying insurance, choosing a phone plan, testing a product, or deciding whether to gather more information.
Information has value when it can change a later choice enough to improve the expected result. Information is not automatically worth paying for, since a test can be inaccurate or too expensive.
When reliable probabilities are unavailable, different rules express different priorities. Maximin treats the lowest payoff as the key concern. It can be sensible for emergency planning, safety design, or a choice where failure would be hard to recover from.
Minimax regret compares each choice with the best choice that would have been available after the true state became known. It focuses on avoiding a large feeling of missed opportunity. Neither rule discovers the single correct answer.
They show how caution is being defined. Check that outcomes use the same units, probabilities are realistic, and every relevant outcome is included. Small changes in assumptions can reverse a result, so sensitivity checks are part of responsible decision making.