Generalization of minimax and maximin criteria in a game against nature for the case of a partial a priori uncertainty

Abstract

This study proposes a new criterion for choosing the optimal decision in a game against nature under a partial a priori uncertainty. The paper's main novelty consists in examining the situation when a part of the a priori probabilities of states of nature is known, and the other part is unknown. We prove the theorems for choosing the optimal decision as for the payoff and risk matrix, as well as for the profit matrix in the situation of a partial a priori uncertainty. The proposed approach also generalizes the Bayes, Wald, Savage, Hurwicz, and Laplace criteria since the minimum average payoff (or risk) for each of these criteria we can quickly obtain from the article's derived formulas. A practical example of a game against nature under a partial a priori uncertainty illustrates the proposed approach and shows its effectiveness compared to well-known criteria. We show that the introduced criterion provides the choice of a decision that is also optimal in conditions of risk, which indicates the effective use of the vector of known a priori probabilities.