Breaking Exponential Complexity in Games of Ordered Preference: A Tractable Reformulation

2026-03-27

AI Summary

This work addresses the exponential computational complexity in generalized ordinal optimization problems (GOOPs) arising from an increasing number of preference levels. To overcome this challenge, the authors propose a compact reformulation of the Karush–Kuhn–Tucker (KKT) system that preserves the original hierarchical stationarity structure while reducing the number of variables to a polynomial scale in both the number of players and preference levels. Under quadratic objectives with linear constraints, the reformulated system is solution-equivalent to the original. For general nonlinear settings, the approach incorporates second-order sufficient conditions to identify genuine equilibria and employs a primal-dual interior-point algorithm with local quadratic convergence. This advancement substantially enhances the scalability of computing equilibria in multi-player GOOPs, effectively circumventing the exponential bottleneck inherent in conventional methods.

Abstract

Games of ordered preference (GOOPs) model multi-player equilibrium problems in which each player maintains a distinct hierarchy of strictly prioritized objectives. Existing approaches solve GOOPs by deriving and enforcing the necessary optimality conditions that characterize lexicographically constrained Nash equilibria through a single-level reformulation. However, the number of primal and dual variables in the resulting KKT system grows exponentially with the number of preference levels, leading to severe scalability challenges. We derive a compact reformulation of these necessary conditions that preserves the essential primal stationarity structure across hierarchy levels, yielding a "reduced" KKT system whose size grows polynomially with both the number of players and the number of preference levels. The reduced system constitutes a relaxation of the complete KKT system, yet it remains a valid necessary condition for local GOOP equilibria. For GOOPs with quadratic objectives and linear constraints, we prove that the primal solution sets of the reduced and complete KKT systems coincide. More generally, for GOOPs with arbitrary (but smooth) nonlinear objectives and constraints, the reduced KKT conditions recover all local GOOP equilibria but may admit spurious non-equilibrium solutions. We introduce a second-order sufficient condition to certify when a candidate point corresponds to a local GOOP equilibrium. We also develop a primal-dual interior-point method for computing a local GOOP equilibrium with local quadratic convergence. The resulting framework enables scalable and efficient computation of GOOP equilibria beyond the tractable range of existing exponentially complex formulations.

Problem

• Games of Ordered Preference
• exponential complexity
• scalability
• KKT system
• lexicographic equilibrium

Innovation

• ordered preference games
• polynomial reformulation
• reduced KKT system
• lexicographic equilibrium
• primal-dual interior-point method