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<p>This article explores a&#xa0;mathematical model of decision-making under uncontrollable factors, formalized as a&#xa0;hierarchical game with nature. The top-level manager is the decision maker. The strategies of lower-level players analyze the consequences of actions under strategic uncertainty. This player does not possess its own payoff function but influences the outcome of the conflict. A case of informed uncertainty is considered, i.e., the choice of the top-level player modifies the initial set of admissible strategies for the second player (nature). When choosing a&#xa0;strategy, the first player must consider the possibility of any uncertainties, taking into account uncertainty estimates. To constrain the mathematical model of the conflict, one possible approach to generating optimal solutions is described, based on the Savage principle. An algorithm for constructing the Savage regret function is proposed that takes into account the details of informed uncertainty. A special case of the problem is investigated where the upper-level player’s response speed and the constraint results are represented by numerical intervals, the objective function is quadratic, and the degree of possible responsive constraints on the player’s actions varies linearly. For this case, an explicit regret function is obtained and sufficient conditions for the power source for Savage’s solution are formulated.</p>

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The optimal Savage solution in a hierarchical game with nature

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Alexandr E. Bardin,
  • Juliya N. Zhiteneva,
  • Juliya A. Belskikh

摘要

This article explores a mathematical model of decision-making under uncontrollable factors, formalized as a hierarchical game with nature. The top-level manager is the decision maker. The strategies of lower-level players analyze the consequences of actions under strategic uncertainty. This player does not possess its own payoff function but influences the outcome of the conflict. A case of informed uncertainty is considered, i.e., the choice of the top-level player modifies the initial set of admissible strategies for the second player (nature). When choosing a strategy, the first player must consider the possibility of any uncertainties, taking into account uncertainty estimates. To constrain the mathematical model of the conflict, one possible approach to generating optimal solutions is described, based on the Savage principle. An algorithm for constructing the Savage regret function is proposed that takes into account the details of informed uncertainty. A special case of the problem is investigated where the upper-level player’s response speed and the constraint results are represented by numerical intervals, the objective function is quadratic, and the degree of possible responsive constraints on the player’s actions varies linearly. For this case, an explicit regret function is obtained and sufficient conditions for the power source for Savage’s solution are formulated.