Decision Theory: formula sheet

Key formulas

Payoff for an act under a state

Payoff = Revenue − Cost for that act and that state

Build each cell of the payoff matrix this way. Check whether the question wants profit or cost, since that changes whether you maximise or minimise.

Probability rule for states of nature

Σ P(state) = 1

States must be mutually exclusive and exhaustive. Check that the given probabilities add to 1 before using them.

Expected monetary value (EMV)

EMV of an act = Σ (Payoff × Probability of state)

Used under risk. Covered in detail in the risk topic.

Environment test

Probabilities known → Risk; probabilities unknown → Uncertainty; one sure state → Certainty; rational opponent → Conflict

Use this to name the environment before choosing any criterion.

Maximax

Choose the act with the maximum of [maximum payoff of each act]

Optimistic. For costs, use minimin (lowest of the lowest costs).

Maximin

Choose the act with the maximum of [minimum payoff of each act]

Pessimistic. For costs, use minimax (lowest of the highest costs).

Regret (opportunity loss)

Regret = Best payoff in that state − Payoff of the act in that state

Calculate column by column (state by state). The best act in each state has regret 0. For costs, regret = the act's cost − the lowest cost in that state.

Minimax regret

Choose the act with the minimum of [maximum regret of each act]

Uses the regret table, not the payoff table.

Hurwicz criterion

Hurwicz value = α × Maximum payoff + (1 − α) × Minimum payoff

α is the coefficient of optimism, 0 ≤ α ≤ 1. Choose the act with the highest value. α = 1 gives maximax, α = 0 gives maximin.

Laplace criterion

Laplace value = Sum of payoffs of the act ÷ Number of states

Equal probability for each state. Choose the highest value.

Expected Monetary Value of an act

EMV = Σ (Payoff of act in state i × P(state i))

Probabilities must sum to 1. Choose the highest EMV when payoffs are profits; choose the lowest when payoffs are costs.

Opportunity loss for an act in a state

Opportunity loss = Best payoff in that state − Payoff of the act in that state

For costs, use act cost − lowest cost in that state. The best act in each state has zero opportunity loss.

Expected Opportunity Loss

EOL = Σ (Opportunity loss of act in state i × P(state i))

Choose the act with the lowest EOL.

Link between EOL and EMV

Minimum EOL = EMV under perfect information − Maximum EMV

Use this as a cross-check. It equals EVPI.

Expected profit with perfect information

EPPI = Σ (best payoff in each state × probability of that state)

Take the highest payoff in each column (state), not the payoff of one action.

EVPI (main formula)

EVPI = EPPI − maximum EMV

Use the maximum EMV of the actions without information. For costs, use the lowest expected cost and EVPI = lowest EMV cost − expected cost with perfect information.

EVPI by opportunity loss

EVPI = minimum EOL

Opportunity loss = best payoff in the state − payoff of the action in that state. It holds when the payoffs are profits and the probabilities are known.

Expected monetary value of an action

EMV = Σ (payoff × probability)

Probabilities of all states must add up to 1.

EMV at a chance node

EMV = Σ (probability × payoff of each branch)

Probabilities of all branches from one chance node must total 1.

Value at a decision node

Value = highest EMV among the branches (for profit); lowest for cost

Reject the other branches; do not average them.

Net EMV of a path with a cost

Net EMV = EMV of outcomes − cost incurred on that path

Deduct costs such as test or investment cost on the branch where they arise. Do not deduct them twice.

Payoff of an outcome

Payoff = total inflow or profit on the path − all costs on the path

Use one consistent basis: either all payoffs net of costs, or gross payoffs with costs deducted at the end.

Expected utility

EU = Σ (probability × utility of outcome)

Use utilities, not money. Choose the action with the highest EU.

Utility of an intermediate outcome

U(x) = p × U(best) + (1 − p) × U(worst)

p is the probability at which the decision maker is indifferent between sure x and the gamble.

Certainty equivalent and risk attitude

Certainty equivalent < EMV: risk averse; = EMV: risk neutral; > EMV: risk seeking

Certainty equivalent is the sure amount the person accepts in place of the gamble.

Marginal analysis rule

Accept if incremental revenue ≥ incremental cost

Use only relevant (future, differing) costs and revenues.

Maximin and minimax (zero-sum game)

Saddle point exists if maximin = minimax

Row player: maximum of row minimums. Column player: minimum of column maximums.

Quick revision

• A payoff table lists actions against states of nature, with the payoff in each cell.
• Maximax picks the action with the best of the best payoffs; it is the optimist's rule.
• Maximin picks the action with the best of the worst payoffs; it is the pessimist's rule.
• Minimax regret: build the regret table, find each action's maximum regret, then choose the smallest.
• Hurwicz uses a coefficient of optimism α: α × best + (1 − α) × worst for each action.
• Laplace assumes equal probabilities for all states and picks the highest average payoff.
• EMV = Σ (probability × payoff) for each action; choose the highest for profits.
• EOL = Σ (probability × regret); choose the lowest. It selects the same action as the highest EMV.
• EVPI = EMV with perfect information − best EMV without it = minimum EOL.
• Decision trees: squares are decisions, circles are chance events; fold back from right to left.
• Subtract any cost of a stage from the branch value before comparing alternatives.
• Utility theory allows for risk attitude; EMV treats the decision maker as risk neutral.

Common mistakes

• Treating risk and uncertainty as the same thing. Fix: In this chapter, risk means probabilities are known and uncertainty means they are not. Check the question for probabilities.
• Listing a state of nature as an act, such as 'high demand' as an alternative. Fix: Acts are your choices. States are external events. Ask who controls it.
• Choosing the act with the highest minimum under minimax regret, or confusing maximin with minimax regret. Fix: Maximin works on payoffs: best of the worst payoffs. Minimax regret works on the regret table: smallest of the largest regrets.
• Calculating regret across a row instead of down a column. Fix: Regret belongs to a state. Find the best payoff in each column and subtract each entry from it.
• Choosing the act with the highest single payoff instead of the highest EMV. Fix: Always weight payoffs by probability and compare the totals, not the largest cell.
• Calculating opportunity loss row-wise instead of column-wise. Fix: Regret is computed within each state: the best value in that state minus the act's value in that state.
• Using the payoffs of one action to calculate EPPI. Fix: EPPI uses the best payoff in each column, so different actions can appear in different states.
• Subtracting the wrong EMV, such as the lowest or an average of EMVs. Fix: Subtract only the maximum EMV (for profits) from EPPI.
• Working left to right instead of folding back from the right. Fix: Always start at the final payoffs. Later decisions must be solved before earlier ones.
• Averaging the branches at a decision node. Fix: Use probabilities only at chance nodes. At a decision node, choose the best branch and ignore the others.

Exam tips

• In MCQs, the words 'probabilities are given' signal risk, and 'probabilities are not known' signal uncertainty. Read for them first.
• For a written answer on steps of decision making, give a short numbered list and one line on each step. This earns step marks.
• Always draw the payoff matrix with acts in rows and states in columns, and label units and profit or cost.
• In theory questions, define each element and give a one-line example, such as acts, states and payoff from a stocking decision.
• Learn the four environments with one example each. Short differences between risk and uncertainty are a common question.
• Read the question for the wording of each criterion. Also check whether the payoff is profit or cost before you start.
• Show the regret table in full. Step marks usually go for the table and the maximum-regret column.
• Always state the decision in words, for example 'Choose C under maximax', with its value.