Does maximin require scenario probabilities?

No. The basic scenario rule uses payoffs and a set of possibilities.

Can several decisions tie?

Yes. Equal minimum payoffs leave a tie that needs an explicitly stated secondary criterion.

Does a mixed maximin guarantee hold in every individual play?

No. It generally protects expected payoff against the opponent's strategy, not the realized payoff of each random action.

AI Glossary / Graphs and symbolic AI / Intermediate

Maximin

Definition

Maximin is a decision rule that chooses the option with the best worst-case payoff. For each available option, identify its smallest payoff across the situations or opposing choices being considered. Then select the option whose minimum is largest. The rule is appropriate when the objective is to protect a guaranteed payoff within a specified model.

What it means

An option can offer an attractive best case while exposing the decision-maker to a poor worst case. Maximin deliberately evaluates the lower end. It asks what the decision can secure even when the modeled circumstances are unfavorable.

The guarantee is only as broad as the modeled possibilities. If a scenario table leaves out a severe disruption, the selected minimum does not cover that disruption. Maximin also does not assert that the worst case is the most probable case. It applies a conservative comparison without needing to assign probabilities to the alternatives being minimized over.

How it works

Create a payoff function u(a,s), with decision a and scenario or opponent choice s. Calculate m(a)=min_s u(a,s), then choose an a that maximizes m(a). Payoffs must share an interpretable scale, and the feasible decisions and scenarios must be specified.

In game theory, a strategy can be randomized. A mixed maximin strategy maximizes the worst expected payoff over opposing strategies. The randomization is evaluated before its realized action is known to the opponent; revealing the actual action can change the problem.

Maximin and minimax are closely related but their order and sign conventions matter. Maximizing a minimum payoff corresponds to minimizing a maximum loss when loss is the negative of payoff. Interchanging the order of maximization and minimization is not valid without appropriate conditions.

Three practical examples

Service continuity: choose a backup arrangement with the highest minimum service score across several modeled equipment outages.
Competitive games: select a strategy that limits how badly an adversary can exploit it, including randomized strategies when allowed.
Robust planning: compare deployment configurations against a defined set of uncertain workload patterns when a guaranteed performance floor is the stated objective.

Think of it this way

Think of choosing among bridges by the weakest load rating along each complete route. You first find each route's weakest point, then choose the route whose weakest point is strongest.

The analogy captures the nested comparison, but real decisions may involve probabilities, several objectives, or recovery options. A single weakest-point score is only useful if it matches the actual decision criterion.

Worked example

Suppose three backup plans A, B, and C have service utility scores under scenarios X, Y, and Z. Plan A scores 4, 8, and 5; B scores 2, 12, and 7; C scores 3, 5, and 10. Higher is better, and these are hypothetical utility points, not money.

Their row minima are 4, 2, and 3. Maximin therefore chooses A and reports a modeled guarantee of 4. B has the largest possible score, 12, but its poor X outcome makes its minimum lower.

If credible scenario probabilities were instead 0.2, 0.5, and 0.3, expected scores would be 6.3 for A, 8.5 for B, and 6.1 for C. Maximizing expected utility would choose B. This does not make either calculation arithmetically wrong; they answer different questions.

Even allowing random mixtures of these plans cannot improve the maximin floor above 4 in this particular table: scenario X gives every plan at most 4, so any mixture also scores at most 4 there. Plan A already achieves that bound.

Illustrative case study

Fictional case: a regional event organizer compares three backup audio arrangements against power, network, and equipment disruptions. The organizer chooses a maximin plan because a minimum level of intelligible announcements is the stated priority.

During review, a technician identifies a missing combined disruption. The team adds it and recalculates the floors rather than presenting the old result as universal protection. It also checks whether some low-scoring situations can be mitigated after they occur. The fictional case shows how scenario coverage and the timing of corrective actions shape a robust decision; it does not report an operational success rate.

Common misconceptions and mistakes

Maximin does not maximize average performance, the best possible outcome, or the probability of success. It also differs from minimax regret, which compares each decision with what would have been best in each scenario.

Do not call every conservative decision “maximin” unless the minimum and the available options are defined. A strategy maximizing its own worst payoff is not automatically a Nash-equilibrium strategy in an arbitrary general-sum game. The standard equivalence has a specific two-player zero-sum setting with appropriate strategy spaces, including mixed strategies for finite games.

Business use

The rule provides an explicit way to discuss resilience requirements. A manager can see which scenario limits a plan's guaranteed score and whether improving that weak point changes the selected option.

Its conservatism also has a cost: it may give up considerable performance in common situations to protect against a rare possibility. Review the scenario set, utility scale, feasibility of randomization, and any hard minimum requirements. Where probabilities are dependable, compare the result with an expected-utility analysis rather than silently treating one criterion as a substitute for the other.

Check your understanding

a plan has scores 1, 20, and 20, while another has 3, 4, and 4. Which does maximin prefer?

Answer: the second plan, because its worst score is 3 rather than 1. The two high outcomes of the first plan do not raise its minimum.

Related terms

Learn next

apply maximin and expected utility to the same small scenario table and explain why their choices differ. Then study mixed strategies and the minimax theorem, paying attention to whether an opponent observes a probability distribution or the realized action before responding.

Sources

Roman Barták, “Decisions with multiple agents,” maximin and mixed-strategy lecture passages: https://ktiml.mff.cuni.cz/~bartak/ui2/lectures/lecture07eng.pdf. John Baez, “Game Theory, Part 16,” zero-sum security strategies: https://math.ucr.edu/home/baez/games/games_16.html. Primary teaching sources were read. The backup-plan calculations and fictional case are original.

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