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<p>This study addresses the uncertainty in information provided by players during a game through the application of rough sets, a framework widely used for tackling such challenges. It further examines the compatibility of continuous differential games with rough programming, leading to the exploration of Stackelberg equilibrium solutions for differential games characterized by rough controls and rough state trajectories. Given that the Stackelberg approach offers a hierarchical solution framework with a leader-follower structure, the analysis is divided into two parts: Part 1 considers dependent followers, while Part 2 addresses independent followers. For each scenario, necessary and sufficient conditions for optimal control were derived. To address the inherent uncertainty, the rough problem was transformed into a crisp equivalent using two methods: the expected value operator and the trust measure approach, enabling the derivation of expected equilibrium and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> </InlineEquation>-trust strategies for the Stackelberg rough differential game. Numerical examples were provided for both cases to illustrate the theoretical findings, demonstrating the determination of rough intervals for players and their corresponding state trajectories. The study successfully identifies optimal solutions by transforming rough problems into crisp ones, highlighting the practical utility of the proposed methods in handling uncertainty within hierarchical differential games.</p>

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Stackelberg open-loop equilibrium solutions for rough differential games

  • 期刊论文

  • 发表日期：

    2026年2月7日

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  • Abd El-Monem A. Megahed,
  • Mohamed R. Zeen El Deen,
  • Asmaa A. Ahmed

摘要

This study addresses the uncertainty in information provided by players during a game through the application of rough sets, a framework widely used for tackling such challenges. It further examines the compatibility of continuous differential games with rough programming, leading to the exploration of Stackelberg equilibrium solutions for differential games characterized by rough controls and rough state trajectories. Given that the Stackelberg approach offers a hierarchical solution framework with a leader-follower structure, the analysis is divided into two parts: Part 1 considers dependent followers, while Part 2 addresses independent followers. For each scenario, necessary and sufficient conditions for optimal control were derived. To address the inherent uncertainty, the rough problem was transformed into a crisp equivalent using two methods: the expected value operator and the trust measure approach, enabling the derivation of expected equilibrium and \(\alpha \) -trust strategies for the Stackelberg rough differential game. Numerical examples were provided for both cases to illustrate the theoretical findings, demonstrating the determination of rough intervals for players and their corresponding state trajectories. The study successfully identifies optimal solutions by transforming rough problems into crisp ones, highlighting the practical utility of the proposed methods in handling uncertainty within hierarchical differential games.