Maximin Criterion

Concept

Theoretically, the Maximin Criterion is a decision rule under uncertainty that selects the alternative whose worst possible outcome is better than the worst possible outcome of every competing alternative. It reflects a strongly risk-averse or precautionary position and is associated with Rawlsian distributive reasoning when applied to welfare, where priority is given to improving the position of the least advantaged. In health economics, it may be used when decision-makers place overriding importance on avoiding the most adverse health, financial or distributional outcome.

Mathematically, the criterion identifies the minimum payoff associated with each decision and then selects the decision with the largest of these minimum values. Where outcomes are expressed as losses or costs, the equivalent minimax formulation selects the decision with the smallest maximum loss. The framework does not require probabilities and therefore differs from expected-value and expected-utility approaches.

In practice, the criterion is implemented by constructing a payoff matrix for alternative interventions across plausible states of the world, identifying the lowest payoff for each intervention, and selecting the intervention with the highest minimum payoff. It may be applied in priority setting, robust policy appraisal and decisions involving severe uncertainty where probability distributions are unavailable, disputed or considered unreliable.

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Purpose

Used to select a healthcare intervention or policy that provides the most favourable worst-case outcome when decision-makers adopt a highly precautionary approach to uncertainty.

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Mathematical Formulae

Primary Formula

a = arg max?a ? A? [min?s ? S? u(a, s)]*

where:

• a* = selected alternative
• A = set of available alternatives
• S = set of possible states
• u(a, s) = payoff or utility from alternative a under state s

Supporting Formulae

For losses or costs:

a = arg min?a ? A? [max?s ? S? L(a, s)]*

where:

• L(a, s) = loss associated with alternative a under state s

Related Mathematical Methods

• Decision matrix analysis
• Minimax criterion
• Robust decision analysis
• Multi-criteria decision analysis
• Scenario analysis

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Example

A health authority compares three vaccination strategies under three uncertain epidemic scenarios. Outcomes are expressed as QALYs gained.

Strategy	Mild Epidemic	Moderate Epidemic	Severe Epidemic	Minimum QALYs
Strategy A	1,200	1,000	700	700
Strategy B	1,050	950	850	850
Strategy C	900	880	820	820

The minimum outcomes are 700, 850 and 820 QALYs respectively. Applying the criterion:

max(700, 850, 820) = 850

Strategy B is selected because it provides the largest minimum health gain.

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Excel Implementation

Function	Example Formula	Health Economics Application
MIN	=MIN(B2:D2)	Identifies the worst outcome for each healthcare strategy.
MAX	=MAX(E2:E4)	Identifies the largest minimum outcome across strategies.
XLOOKUP	=XLOOKUP(MAX(E2:E4),E2:E4,A2:A4)	Returns the strategy selected by the criterion.
INDEX and MATCH	=INDEX(A2:A4,MATCH(MAX(E2:E4),E2:E4,0))	Provides an alternative method for identifying the preferred strategy.

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VBA (Optional)

Automate the evaluation of large payoff matrices and identify the maximin strategy across multiple healthcare scenarios.

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Sources

• Luce RD, Raiffa H. Games and Decisions: Introduction and Critical Survey. Wiley.
• Rawls J. A Theory of Justice. Harvard University Press.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.