Decision theory under risk and uncertainty

Payoff and regret tables, maximax, maximin, Hurwicz, Laplace and minimax-regret criteria under uncertainty, and EMV, EOL, EVPI and decision trees under risk.

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Why it matters

Choosing a plant capacity, deciding whether to launch a product or buy a costly machine are one-shot decisions whose payoff depends on future demand that nobody controls. Decision theory gives a disciplined way to choose: different rules when probabilities of the future are unknown (uncertainty) and when they can be estimated (risk), plus a way to put a rupee value on better information before paying for a market survey.

Key ideas

Elements. Alternatives (courses of action the decision maker controls), states of nature (future events not controlled), and a payoff table giving the outcome (profit or cost) for every alternative–state pair. Remove any dominated alternative (never better in any state) first.

Decision environments.

  • Certainty: the state is known — pick the best payoff (that is LP territory).
  • Risk: probabilities of the states are known or estimated — use expected values.
  • Uncertainty: probabilities unknown — use a criterion matching the decision maker's attitude.

Criteria under uncertainty (payoffs = profits).

  • Maximax (optimist): take each alternative's best payoff; choose the largest.
  • Maximin (Wald, pessimist): take each alternative's worst payoff; choose the largest. For costs the equivalent is minimax.
  • Hurwicz (realism): weighted value α·(best) + (1 − α)·(worst), with coefficient of optimism 0 ≤ α ≤ 1; α = 1 gives maximax, α = 0 gives maximin.
  • Laplace (equal likelihood): treat all states as equally likely and pick the best average.
  • Minimax regret (Savage): regret (opportunity loss) = best payoff in that state − payoff obtained. Take each alternative's largest regret; choose the smallest. Different criteria can recommend different alternatives — that is expected; each encodes a different attitude to risk.

Criteria under risk.

  • Expected monetary value (EMV): Σ p·payoff for each alternative; choose the largest (for profits).
  • Expected opportunity loss (EOL): Σ p·regret; choose the smallest. The EMV and EOL rules always pick the same alternative.
  • Expected value of perfect information (EVPI): the most it is worth paying for a perfect forecast. EVPI = expected value with perfect information − best EMV = minimum EOL.
  • Decision trees: for sequential decisions. Squares are decision nodes, circles chance nodes. Roll back from the right: at chance nodes take the expected value, at decision nodes take the best branch.

Limits of EMV. It is right for repeated decisions or a decision maker who is risk-neutral. For a single large bet, a risk-averse firm may rationally prefer a lower EMV with less downside; utility theory handles this by replacing money with utility.

Formulas

  • Hurwicz value: H = α · (max payoff) + (1 − α) · (min payoff)
  • Laplace value: L = (1/n) · Σ payoffs
  • Regret: Rᵢⱼ = max over i of (aᵢⱼ) − aᵢⱼ (for each state j)
  • Expected monetary value: EMVᵢ = Σⱼ pⱼ · aᵢⱼ
  • Expected opportunity loss: EOLᵢ = Σⱼ pⱼ · Rᵢⱼ
  • Expected value with perfect information: EVwPI = Σⱼ pⱼ · max over i of (aᵢⱼ)
  • EVPI = EVwPI − max EMV = min EOL

Symbols: aᵢⱼ = payoff of alternative i in state j (₹ lakh); pⱼ = probability of state j (Σ pⱼ = 1); n = number of states; α = coefficient of optimism; Rᵢⱼ = regret (₹ lakh).

Worked examples

Example 1 (decision under uncertainty). A firm can build a Large, Medium or Small plant. Profit (₹ lakh) for High, Moderate and Low demand: Large [200, 50, −120]; Medium [120, 80, −20]; Small [60, 40, 20]. Find the decision by each criterion (α = 0.6 for Hurwicz).

  1. Maximax: best payoffs 200, 120, 60 → Large.
  2. Maximin: worst payoffs −120, −20, 20 → Small (₹20 lakh guaranteed).
  3. Hurwicz: Large 0.6(200) + 0.4(−120) = 120 − 48 = 72; Medium 72 − 8 = 64; Small 36 + 8 = 44 → Large.
  4. Laplace: averages 130/3 = 43.3, 180/3 = 60, 120/3 = 40 → Medium.
  5. Regret: column best values are 200, 80, 20. Regrets: Large [0, 30, 140]; Medium [80, 0, 40]; Small [140, 40, 0]. Maximum regrets 140, 80, 140 → Medium (minimax regret 80).

Example 2 (GATE-type, decision under risk). Same table with P(High) = 0.3, P(Moderate) = 0.5, P(Low) = 0.2. Find the EMV decision and EVPI.

  1. EMV(Large) = 0.3(200) + 0.5(50) + 0.2(−120) = 60 + 25 − 24 = 61.
  2. EMV(Medium) = 0.3(120) + 0.5(80) + 0.2(−20) = 36 + 40 − 4 = 72.
  3. EMV(Small) = 0.3(60) + 0.5(40) + 0.2(20) = 18 + 20 + 4 = 42.
  4. Best: Medium, EMV = ₹72 lakh.
  5. With perfect information, choose the best plant for each state: EVwPI = 0.3(200) + 0.5(80) + 0.2(20) = 60 + 40 + 4 = 104.
  6. EVPI = 104 − 72 = ₹32 lakh. Check by EOL: Medium = 0.3(80) + 0.5(0) + 0.2(40) = 24 + 8 = 32 = min EOL ✓ (Large 43, Small 62).
  7. A market survey costing more than ₹32 lakh can never pay for itself.

Common mistakes

  • Computing regret as payoff minus the row maximum instead of the column (state) maximum minus payoff.
  • Using maximin on a cost table instead of minimax (for costs, choose the smallest worst cost).
  • In Hurwicz, applying α to the worst payoff instead of the best.
  • Forgetting that EVPI is measured against the best EMV, not against an arbitrary alternative.
  • Rolling a decision tree forward from the left instead of back from the right.
  • Treating EMV as the profit that will actually occur; it is a long-run average.

For GATE PI

  • Payoff or cost tables with three alternatives and three states: pick the decision by a named criterion (MCQ) or compute the Hurwicz value, EMV or EVPI (NAT).
  • Regret tables and minimax regret.
  • Small decision trees with one chance node and one decision.
  • Practise building the regret table column by column and double-check EVPI with min EOL.

Quick check

  1. Payoffs of an alternative are 50, 10 and −30. What is its Hurwicz value with α = 0.4?
  2. If EVwPI = 90 and the best EMV = 75, what is EVPI?
  3. Which criterion uses the column maxima of the payoff table?
  4. Under which criterion do all states get equal weight?

Answers: 1. 0.4(50) + 0.6(−30) = 2. 2. 15. 3. Minimax regret (and EVwPI). 4. Laplace.

Interview questions

All Operations Research interview questions

Try answering each one aloud before you open it.

  1. 1.Explain the difference between risk and uncertainty in decision-making.Concept

    In decision-making, risk refers to situations where the probabilities of different outcomes are known or can be estimated. Uncertainty, on the other hand, refers to situations where these probabilities are unknown or cannot be reliably estimated. This distinction affects how decisions are made, as different tools and approaches are used to handle risk versus uncertainty.

  2. 2.What is the expected value criterion in decision theory?Concept

    The expected value criterion is a decision-making tool used under risk. It involves calculating the weighted average of all possible outcomes, where each outcome is weighted by its probability of occurrence. The decision with the highest expected value is typically chosen, as it represents the most favorable average outcome over time.

  3. 3.Why is the maximin criterion used in decision-making under uncertainty?Application

    The maximin criterion is used in decision-making under uncertainty to ensure a conservative approach. It involves selecting the decision that maximizes the minimum payoff. This approach is useful when decision-makers want to avoid the worst-case scenario, as it focuses on minimizing potential losses in the absence of known probabilities.

  4. 4.How does the minimax regret criterion work in decision theory?Concept

    For each state of nature, the regret (opportunity loss) of an alternative is the best payoff available in that state minus the payoff the alternative gives, so regrets are never negative. Build the regret table column by column, note each alternative's largest regret, and choose the alternative whose largest regret is smallest. It suits a decision maker who wants to limit how badly a choice can look in hindsight.

  5. 5.What happens if probabilities are incorrectly estimated in decision-making under risk?Application

    If probabilities are incorrectly estimated in decision-making under risk, it can lead to suboptimal decisions. The expected value calculations will be inaccurate, potentially leading to choices that do not maximize the actual expected outcome. This highlights the importance of accurate probability estimation in risk-based decision-making.

  6. 6.Explain how sensitivity analysis is used in decision-making under uncertainty.Application

    Sensitivity analysis is used to assess how the outcomes of a decision change with variations in input parameters. In decision-making under uncertainty, it helps identify which variables have the most impact on the decision outcome. This analysis aids in understanding the robustness of a decision and in identifying critical factors that require more accurate estimation.

  7. 7.Calculate the expected value of a decision with outcome A: payoff ₹100 thousand with probability 0.4, and outcome B: payoff ₹200 thousand with probability 0.6.Numerical

    EV = Σ p·payoff = 0.4 × 100 + 0.6 × 200 = 40 + 120 = ₹160 thousand. It is a long-run average: in any single trial the actual payoff will be 100 or 200, never 160.

  8. 8.Project X has a 50% chance of ₹300 thousand and a 50% chance of ₹100 thousand. Project Y has a 70% chance of ₹200 thousand and a 30% chance of ₹50 thousand. Which should be chosen on expected value?Numerical

    EMV(X) = 0.5 × 300 + 0.5 × 100 = ₹200 thousand; EMV(Y) = 0.7 × 200 + 0.3 × 50 = 140 + 15 = ₹155 thousand. On EMV, X is chosen. X also has the higher worst case (100 against 50), so even a risk-averse decision maker would prefer it here.

  9. 9.What is the role of utility theory in decision-making under risk?Concept

    Utility theory plays a role in decision-making under risk by considering the decision-maker's preferences and risk tolerance. Instead of focusing solely on monetary outcomes, utility theory evaluates the satisfaction or utility derived from different outcomes. This approach helps in making decisions that align with the decision-maker's subjective preferences and risk attitudes.

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