Wald-Savage synthetic criterion for playing with nature and its economic applications

In the game with nature, it`s proposed to use the Wald-Savage synthetic criterion as an optimality principle, which allows to evaluate the optimality of the strategy planned in the game from the synthetic (joint) point of view of wins and risks, which distinguishes it from Wald’s criterion, which allows to evaluate the strategies optimality from the wins position, abstracting from risks, and from the Savage criterion characterizing the optimality in terms of game risks, abstracting from wins. The authors give a definition of a synthesized strategy that is optimal according to the «joint» Wald-Savage criterion and is not optimal, either according to the Wald criterion or Savage criterion. The property of synthesizing the Wald-Savage criterion is considered, the absence of which means that the use of this criterion for the search for synthesized strategies does not make sense. However, usage of the Wald-Savage criterion in this case allows us to reveal the dependence of the Wald - Savage criteria on the determined win indicator. The problem of synthesizing is formulated in the work and its solution is given at the level of substantiating the necessary and sufficient conditions for the Wald-Savage criterion to have no synthesizing property.