Decision criteria

Decision criteria are, in Decision theory, formalized rules or methods used to select an optimal strategy under conditions of uncertainty or risk. They help systematize the selection process and determine the preferred solution based on available data or assumptions about the external environment.

Key Concepts

• Conditions of uncertainty — a situation where the exact outcome following a decision is unknown, and reliable probabilities of different outcomes are not available.
• Conditions of risk — a situation where the probabilities of different outcomes are known or can be reasonably estimated.

Criteria under Uncertainty

• Wald's criterion: Focuses on maximizing the minimum possible outcome. This represents a pessimistic strategy to guard against the worst-case scenarios.
• Maximax criterion: Focuses on the maximum possible outcome. This represents an optimistic approach, banking on the best-case scenario.
• Hurwicz's criterion: A compromise between optimism and pessimism. The choice is determined using an optimism parameter that balances between the worst and best possible outcomes.
• Savage's criterion: Minimizes the maximum regret of making the wrong decision. The evaluation is based on the difference between the optimal and actual outcomes.
• Laplace's criterion: Assumes that all outcomes are equally probable. The strategy with the highest average expected outcome is selected.

Criteria under Risk

• Bayes' criterion: Based on maximizing the expected outcome, taking into account known or estimated probabilities of outcomes. It is used when probabilities are reliably known.
• Hodge-Lehmann criterion: Combines the approaches of Wald and Bayes. It considers both the minimum possible outcome and the average value based on probabilities, with a specified weight.
• Bayes' minimax criterion: Applied when there is uncertainty about the probabilities themselves. The chosen strategy minimizes the worst-case expected risk across all plausible probability distributions.

Choosing a Criterion

The choice of a criterion depends on:

• the availability of information about probabilities,
• the attitude towards risk (inclination towards optimism or pessimism),
• the specifics of the particular problem.

Under conditions of complete uncertainty, Wald's, Maximax, Hurwicz, Savage's, and Laplace's criteria are more commonly used.

Under conditions of risk, Bayes', Hodge-Lehmann, or Bayes' minimax criteria are preferable.

Comparative Table

Criterion	Conditions of Application	Result Type	Strategy Type	Requires Probabilities	Optimization Type	Primary Calculation Method	Robustness to Errors
Wald's	Complete uncertainty	Minimum outcome	Pessimistic	No	Maximization of the minimum	Selecting the largest of the minimum outcomes	High
Maximax	Complete uncertainty	Maximum outcome	Optimistic	No	Maximization of the maximum	Selecting the largest of the maximum outcomes	Low
Hurwicz's	Complete uncertainty	Weighted between min and max	Compromise	No	Weighted optimization	Calculating a weighted average between the minimum and maximum outcomes	Medium
Savage's	Complete uncertainty	Maximum regret	Cautious	No	Minimization of regret	Constructing a regret matrix and selecting the minimum of the maximum regrets	High
Laplace's	Complete uncertainty	Average outcome	Neutral	No	Maximization of the average	Calculating the average value across all outcomes	Medium
Bayes'	Risk (known probabilities)	Expected outcome	Optimistic (rational)	Yes	Maximization of expected payoff	Calculating the mathematical expectation based on probabilities	Medium
Hodge-Lehmann	Partial uncertainty	Combined result	Compromise	Partially	Combined optimization	Combination of the minimum and the mathematical expectation with weights	Medium
Bayes' Minimax	Partial uncertainty (uncertainty about probabilities)	Maximum expected risk	Pessimistic	Partially	Minimization of maximum risk	Minimizing the worst-case Bayesian risk across all plausible probabilities	High