Efficiently Solving Mixed-Hierarchy Games with Quasi-Policy Approximations

2026-02-02

AI Summary

This work addresses the computational challenges in solving forest-structured mixed-hierarchical games in multi-robot systems, where high-order derivatives of strategies hinder tractability. To overcome this, the authors propose a quasi-strategy approximation that effectively eliminates high-order derivatives and yields a simplified Karush–Kuhn–Tucker (KKT) system. Building upon this, they design an inexact Newton method to solve the reduced system efficiently and, for the first time, establish its local exponential convergence under non-quadratic objectives and nonlinear constraints. The accompanying Julia library, MixedHierarchyGames.jl, demonstrates real-time convergence in simulations involving complex mixed information structures, significantly surpassing existing solvers in both scalability and computational efficiency.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

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.

Problem

• mixed-hierarchy games
• multi-robot coordination
• Stackelberg equilibrium
• Nash equilibrium
• hierarchical decision-making

Innovation

• quasi-policy approximation
• mixed-hierarchy games
• inexact Newton method
• KKT conditions
• multi-robot coordination