Multi-robot coordination often exhibits hierarchical structure, with some robots' decisions depending on the planned behaviors of others. While game theory provides a principled framework for such interactions, existing solvers struggle to handle mixed information structures that combine simultaneous (Nash) and hierarchical (Stackelberg) decision-making. We study N-robot forest-structured mixed-hierarchy games, in which each robot acts as a Stackelberg leader over its subtree while robots in different branches interact via Nash equilibria. We derive the Karush-Kuhn-Tucker (KKT) first-order optimality conditions for this class of games and show that they involve increasingly high-order derivatives of robots' best-response policies as the hierarchy depth grows, rendering a direct solution intractable. To overcome this challenge, we introduce a quasi-policy approximation that removes higher-order policy derivatives and develop an inexact Newton method for efficiently solving the resulting approximated KKT systems. We prove local exponential convergence of the proposed algorithm for games with non-quadratic objectives and nonlinear constraints. The approach is implemented in a highly optimized Julia library (MixedHierarchyGames.jl) and evaluated in simulated experiments, demonstrating real-time convergence for complex mixed-hierarchy information structures.

翻译：多机器人协调通常呈现层次化结构，其中部分机器人的决策依赖于其他机器人的规划行为。虽然博弈论为此类交互提供了理论框架，但现有求解器难以处理同时包含同步（纳什）与层次化（斯塔克尔伯格）决策的混合信息结构。本文研究具有森林结构的N机器人混合层次博弈，其中每个机器人作为其子树中的斯塔克尔伯格领导者，而不同分支中的机器人通过纳什均衡进行交互。我们推导了此类博弈的Karush-Kuhn-Tucker（KKT）一阶最优性条件，证明其涉及机器人最优响应策略随层次深度增加而不断升阶的导数，导致直接求解不可行。为克服这一挑战，我们提出一种消除高阶策略导数的拟策略逼近方法，并开发了用于高效求解近似KKT系统的非精确牛顿法。我们证明了该算法在非二次目标函数与非线性约束博弈中具有局部指数收敛性。该方法通过高度优化的Julia库（MixedHierarchyGames.jl）实现，并在仿真实验中验证了其对复杂混合层次信息结构可实现实时收敛。