Does maximin assume all states are equally likely?

No. It uses the worst payoff without assigning state probabilities.

Is the maximin action always the safest real-world choice?

No. That conclusion depends on how safety, outcomes, and possible states were modeled.

Can the worst state differ between actions?

Yes. The minimum is calculated separately for each action; different actions can have different least favorable states.

AI Glossary / Mathematics and optimization / Intermediate

Wald's maximin model

Definition

Wald's maximin model is a decision rule that selects the action with the best worst-case payoff among the states considered possible. For each action, identify its least favorable payoff; then choose an action whose minimum payoff is largest. It does not require assigning probabilities to those states.

What it means

The rule addresses a decision made before the uncertain state is known. A payoff table describes what each action would produce in each state, with larger values preferred. The decision maker focuses on protecting the worst outcome represented in that table.

This is a criterion for choosing, not a prediction that the worst state will occur. Nor is it a guarantee against events omitted from the model. The uncertainty set and payoff estimates are therefore central: changing which states are considered possible can change the selected action even when the rule stays the same.

How it works

List feasible actions and the relevant possible states. Express their consequences on a consistent payoff or utility scale. For each action a, calculate m(a)=min over states of its payoff. Select an action maximizing m(a).

If consequences are costs to minimize, use the equivalent minimax-cost formulation: minimize each action's maximum cost. Do not apply the payoff version directly to a cost table without reversing the direction.

Several actions may tie on their worst outcomes, so a separate, stated tie-break rule may be needed. This introductory formulation compares fixed actions. Allowing randomized decisions or sequential adaptation changes the feasible decision policies and requires a corresponding model of how uncertainty interacts with those policies.

Three practical examples

1. Capacity planning: compare configurations by their minimum supported throughput across a specified set of workloads, when that minimum is the chosen planning objective.

2. Service continuity: assess response plans by the lowest service score achieved across explicitly defined disruption scenarios.

3. Algorithm selection: choose among implementations using their worst observed scenario score as a deliberately conservative criterion, while distinguishing observed scenarios from every possible input.

Think of it this way

Imagine selecting a route by comparing its slowest plausible journey with the slowest plausible journey on each alternative route. You favor the route with the least troublesome worst case.

For travel times, that is a minimax-cost version. The analogy also reveals the limitation: an implausible scenario can dominate the choice if it is included without qualification. The rule itself does not determine which scenarios deserve inclusion.

Worked example

Suppose three plans have benefit scores under quiet, normal, and demanding workloads. Plan A scores (5,5,5), B scores (2,8,12), and C scores (4,7,9). Larger scores are better.

Their minimum scores are A:5, B:2, and C:4. Wald's maximin rule selects A because 5 is the largest minimum. It does not select B despite B's best-case score of 12.

Now compare a different rule, minimax regret. The best score available in each state is (5,8,12). Subtract each plan's scores from those statewise best values. A's regrets are (0,3,7), B's are (3,0,0), and C's are (1,1,3). Their maximum regrets are 7,3,3, so B and C tie under minimax regret.

The two rules answer different questions: protecting absolute worst performance versus limiting the largest missed opportunity. No probabilities or expected-value calculations are used in either part of this original example.

Illustrative case study

Fictional case: an operations team must choose a processing configuration before it knows which of several workload patterns will arrive. It has scenario-based estimates of the service score each configuration can deliver.

The team initially chooses the configuration with the highest minimum score. During review, it discovers that one severe workload was generated using an assumption that no longer applies. It revises the scenario set and recalculates the table before making its recommendation.

The report identifies the decision criterion and its scenario limits. It describes the selected plan as best by the modeled minimum score, rather than universally best or guaranteed to handle every future workload.

Common misconceptions and mistakes

Maximin is not maximax, which chooses the action with the best best-case payoff. It is also not expected-utility maximization, which uses probabilities and utilities.

Do not confuse minimax payoff or cost reasoning with minimax regret. Regret compares each action with the best action for that particular state.

Worst-case protection can be sensitive to an extreme but poorly supported scenario. That is a modeling issue worth examining, not a reason to silently remove inconvenient outcomes.

The rule can leave ties between actions with very different non-worst outcomes. It supplies no automatic preference between those tied actions unless an additional rule is specified.

Business use

Wald's rule provides a transparent way to express a deliberate preference for a stronger modeled performance floor. It can support planning where reliable probabilities are unavailable and stakeholders agree that worst-case performance is the relevant criterion.

Document the uncertainty set, outcome scale, feasibility assumptions, and tie-break policy. Compare the decision with alternative criteria to expose what is sacrificed for the stronger floor. If credible probabilistic evidence becomes available, consider whether the original non-probabilistic formulation still represents the intended decision.

Check your understanding

A plan scores (3,20) and another scores (6,7), with higher scores preferred. Which does maximin choose?

Answer: The second plan. Its minimum is 6, compared with 3 for the first. The first plan's larger best-case score does not change that comparison.

Related terms

Robust optimization
Minimax
Expected utility
Decision under uncertainty
minimax regret
uncertainty set
payoff matrix
scenario analysis
pessimistic criterion

Learn next

Build a small action-by-state table and calculate maximin, maximax, and minimax-regret decisions. Then remove or add one explicitly justified scenario and compare the results. Study robust optimization next to see how the same worst-case idea extends to continuous decision variables and uncertainty sets.

Sources

Primary university teaching material read: Craig Boutilier, CSC2534 decision-theory lecture, sections on strict uncertainty, Wald maximin, and Savage minimax regret: https://www.cs.toronto.edu/~cebly/2534/Notes/CSC2534_Lecture2.pdf . The payoff tables and fictional operations scenario are original teaching material. This entry concerns the basic non-probabilistic decision rule, not all of Wald's statistical decision theory.

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