Decision Criteria Under Uncertainty and Risk

Understand Decision Criteria Under Uncertainty and Risk

Key rules to remember

Maximax

Choose the action with the highest of the row maximum payoffs

Optimist's rule. If the table shows costs, choose the lowest of the row minimum costs (minimin).

Maximin

Choose the action with the highest of the row minimum payoffs

Pessimist's rule. If the table shows costs, choose the lowest of the row maximum costs (minimax cost).

Regret (opportunity loss)

Regret = best payoff in that state (column) − payoff of the action in that state

Build a regret table state by state. For costs, regret = actual cost − lowest cost in that state.

Minimax regret

Choose the action with the smallest of the row maximum regrets

Savage's criterion. Regret is never negative.

Laplace (equal likelihood)

Average payoff = sum of payoffs in the row ÷ number of states

Choose the highest average (lowest for costs).

Hurwicz criterion

Weighted value = α × row maximum + (1 − α) × row minimum

α is the coefficient of optimism, between 0 and 1. Choose the action with the highest value.

Expected value

EV = Σ (p × X)

Use when probabilities are known (risk). Probabilities must add up to 1.

Variance and standard deviation

σ² = Σ p × (X − EV)² and σ = √σ²

Equivalent shortcut: σ² = Σ p × X² − EV². Weights are the probabilities, not 1 ÷ n.

Coefficient of variation

CV = σ ÷ EV (× 100 for a percentage)

Relative risk per rupee of expected return. Lower CV means lower relative risk.

How to solve Decision Criteria Under Uncertainty and Risk questions

1. 1Read the question and decide whether probabilities are given. If yes, it is risk: go to expected value and SD. If no, it is uncertainty: apply the named criteria.
2. 2Check whether the payoffs are profits or costs. This decides whether you maximise or minimise in each rule.
3. 3Redraw the matrix neatly with actions as rows and states as columns.
4. 4For maximax and maximin, write the row maximum and row minimum next to each action, then pick the best.
5. 5For minimax regret, find the best payoff in each column, build the regret table, write the row maximum regret, and pick the smallest.
6. 6For Laplace, add each row and divide by the number of states. For Hurwicz, apply α to the row maximum and 1 − α to the row minimum.
7. 7Under risk, compute EV, then variance using p × (X − EV)², then SD and CV. Compare the projects on both return and risk.
8. 8Write a one-line recommendation that names the action, the criterion and the reason, and note that different criteria may give different answers.

Quickest way: Row and column scan for payoff matrices

Common mistakes in Decision Criteria Under Uncertainty and Risk

• Building the regret table row by row instead of column by column.

  Maximax and maximin work on rows, so students carry the same habit over to regret.

  Fix: Regret is measured against the best payoff in each state. Find the best figure in each column first, then subtract each payoff from it.

• Applying maximax or maximin directly to a cost table.

  Students memorise the words 'maximum' and 'minimum' instead of the idea of the best and the worst outcomes.

  Fix: Check whether the payoffs are profits or costs. For costs, the best outcome is the lowest figure, so the optimist's rule picks the lowest cost and the pessimist's rule picks the lowest of the worst (highest) costs.

• Dividing by the number of outcomes when calculating variance under risk.

  Students confuse it with the standard deviation of raw data, where each observation has equal weight.

  Fix: When probabilities are given, weight each squared deviation by its probability. The probabilities already add up to 1, so no further division is needed.

• Choosing the project with the lowest SD when the EVs differ.

  SD is a measure of risk, so students stop there.

  Fix: SD is in rupees and is not comparable across projects with different EVs. Calculate CV for each project and compare that, then state the risk attitude behind your choice.

• Giving one 'best' answer and ignoring the decision maker's attitude.

  Students expect a single correct answer, as in most numerical questions.

  Fix: State the answer for each criterion. Then recommend one action and link it to the attitude: optimistic, pessimistic or regret-averse.

• Using Laplace or EV when the question says probabilities are unknown but then adding invented probabilities.

  Students want a familiar number to work with.

  Fix: Laplace assumes equal likelihood as a rule, not as a fact. Use it only when asked, and label it as an equal-weight criterion, not as an expected value from known probabilities.

Worked examples