Decision Making under Uncertainty: Maximax, Maximin, Regret, Hurwicz, Laplace

Understand Decision Making under Uncertainty

Key rules to remember

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.

How to solve Decision Making under Uncertainty questions

1. 1Draw the payoff table with acts as rows and states as columns. Note whether payoffs are profit or cost.
2. 2If payoffs are not given, compute them first from the data, for example contribution less fixed costs, or revenue less cost.
3. 3Write the row maximum and row minimum next to each act.
4. 4Apply maximax and maximin using these row figures and name the chosen act for each.
5. 5Build the regret table: find the best payoff in each column, then subtract each entry from it (reverse for costs). Take the row maximum regret and choose the smallest.
6. 6For Hurwicz, calculate α × max + (1 − α) × min for each act. For Laplace, calculate the row total divided by the number of states.
7. 7State the final decision under each criterion in a sentence, with its value.
8. 8 Add one line of comment on the attitude to risk each criterion reflects.

Quickest way: Row-and-column scan

Common mistakes in Decision Making under Uncertainty

• Choosing the act with the highest minimum under minimax regret, or confusing maximin with minimax regret.

  Both names contain 'min' and 'max' and both are cautious.

  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.

  Students subtract from the best payoff of the act rather than of the state.

  Fix: Regret belongs to a state. Find the best payoff in each column and subtract each entry from it.

• Reporting the regret value as the answer to maximin or maximax, or reporting the wrong unit.

  The tables sit side by side and the numbers get mixed.

  Fix: Label each table. Name the act and quote its payoff or regret under each criterion.

• Using α the wrong way round in Hurwicz.

  Students attach α to the minimum.

  Fix: α always weighs the maximum (optimism) and (1 − α) weighs the minimum.

• Applying profit rules to a cost table.

  The rules are memorised as 'highest' without checking the data.

  Fix: With costs, the best is the lowest. Maximin becomes minimax cost and regret is the cost minus the column's lowest cost.

• Using probabilities or expected values in Laplace and calling it a risk problem.

  Laplace gives an average, which looks like EMV.

  Fix: Laplace is an equal-probability assumption used only when no probabilities are given. Use EMV when probabilities are stated.

Worked examples