A geometric solution to the problem of existence and uniqueness of the value of the winning indicator at which the Wald-Savidge criterion has synthesis properties

In this paper, in games with nature, the concept of a payoff-indicator is considered, which quantitatively expresses the attitude of a decision-maker to payoffs. Based on the payoff indicator, the synthetic Wald-Savage criterion is determined as a linear convolution of the Wald and Savage criteria. The definition of a synthesized strategy is introduced as a strategy that is optimal according to the Wald-Savage criterion and not optimal according to any of the constituent criteria. A geometric condition is found, which is necessary and sufficient for the existence and uniqueness of the value of the payoff-indicator, under which the Wald-Savage criterion has the property of synthesis, which consists in the existence of a synthesized strategy. When this condition is met, a rule for finding synthesized strategies is given. A formula is obtained for the set of all strategies (synthesized and unsynthesized) that are optimal by the Wald-Savage criterion. The application of the results obtained is illustrated by analyzing the problem of financial and economic content.