Hodges-Lehmann criterion
The Hodges-Lehmann criterion is a decision-making method under conditions of partial uncertainty that combines elements of pessimistic and probabilistic approaches. It is used in situations where probability estimates for outcomes are available, but the decision-maker also wishes to account for worst-case scenarios.
Essence of the Criterion
The Hodges-Lehmann criterion represents a compromise between:
- A focus on the minimum outcome for each strategy (as in Wald's criterion),
- And the calculation of the expected value of the outcome using estimated probabilities (as in Bayes' criterion).
For each strategy, a weighted value is calculated, combining the worst outcome and the expected value. The weight of each component is determined by a pre-selected coefficient that reflects the decision-maker's degree of caution or optimism.
Thus, the Hodges-Lehmann criterion allows for consideration of both the risk of worst-case losses and probabilistic expectations.
Applying the Criterion
The application process includes the following steps:
- For each strategy, its minimum outcome is determined.
- The expected value is calculated based on known or estimated outcome probabilities.
- The final score for each strategy is calculated as a weighted combination of the minimum outcome and the expected value.
- The strategy with the highest final score is selected.
A more cautious approach assigns a greater weight to the worst outcome, while a more optimistic approach assigns a greater weight to the expected value.
Advantages and Disadvantages
Advantages:
- Considers both probabilistic information and the risks of worst-case outcomes.
- Allows for flexible adjustment of the level of caution.
Disadvantages:
- Requires at least approximate estimates of event probabilities.
- The introduction of a weighting coefficient adds subjectivity to the decision-making process.