非完整梯度博弈：叶式纳什均衡、稳定性与欺骗
Non-Holonomic Gradient Play: Leafwise Nash Equilibria, Stability, and Deception
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中文总结 AI 辅助
本研究提出非完整梯度博弈动力学框架，通过内在线性化分析平衡稳定性，并发展欺骗几何理论，证明欺骗可倾斜黎曼几何以操纵博弈平衡。
中文摘要 AI 辅助
我们研究多智能体系统中的广义学习动力学，其中联合状态在流形上演化，且智能体通过依赖于状态、可能非完整的向量场行动。在束分裂条件下，我们证明这些动力学具有作为投影黎曼梯度的内在表示，从而产生一类非完整梯度博弈动力学。我们通过内在线性化刻画其平衡点的局部稳定性，该线性化明确捕捉了黎曼联络和执行框架非完整性的影响。该框架将欧几里得空间和流形上的经典梯度博弈作为特例恢复，同时适应非完整执行。接着，我们发展了不对称信息下欺骗的几何理论，其中智能体利用对其他智能体学习规则的优先知识来操纵涌现的平衡。我们证明欺骗有效地倾斜了不知情智能体所感知的黎曼几何，扭曲了它们的投影梯度方向。我们建立了在欺骗参数开集上指数稳定平衡点的持久性，并推导了由此产生的平衡位移的显式一阶特征。解析和数值例子说明了该框架及其在博弈中平衡操纵的含义。
英文摘要
We study generalized learning dynamics in multi-agent systems whose joint state evolves on a manifold and whose agents act through state-dependent, potentially nonholonomic vector fields. Under a bundle-splitting condition, we show that these dynamics admit an intrinsic representation as projected Riemannian gradients, giving rise to a class of \emph{nonholonomic gradient play} dynamics. We characterize the local stability of its equilibria through an intrinsic linearization that explicitly captures the effects of the Riemannian connection and the nonholonomy of the actuation frame. The framework recovers classical gradient play on Euclidean spaces and manifolds as special cases while accommodating nonholonomic actuation. We then develop a geometric theory of deception under asymmetric information, whereby an agent exploits privileged knowledge of other agents' learning rules to manipulate the emerging equilibrium. We show that deception effectively \emph{tilts the Riemannian geometry} perceived by the oblivious agents, distorting their projected gradient directions. We establish persistence of exponentially stable equilibria over an open set of deception parameters and derive an explicit first-order characterization of the resulting equilibrium displacement. Analytical and numerical examples illustrate the framework and its implications for equilibrium manipulation in games.
发表机构
- University at Buffalo, State University of New York（纽约州立大学布法罗分校）
- University of California San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。
