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

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.

Figures (4)

• Figure 1: (Left) For quadratic objectives and linear constraints, under the assumptions of \ref{['subsec:equivalence-quadratic-goop']}, the reduced and complete KKT systems share the same primal solution set, and every GOOP equilibrium is SOSC-qualified (hence the corresponding sets coincide). (Right) For general smooth GOOPs, the reduced KKT primal solution set is a relaxation (superset) of the complete KKT primal solution set, which contains the GOOP equilibria; \ref{['thm:sosc-goop']} certifies a subset of such equilibria. \newlabelfig:set-inclusions-rect-ell0
• Figure 1: Convergence of Algorithm \ref{['alg:goop-pdip']} for reduced and complete KKT systems under varying $\rho$. The curve and the shaded area denote the mean and the variance of $\log_{10}(\|\mathcal{K}_\rho(y)\|_2)$, respectively.
• Figure 2: Monte Carlo comparison over 100 randomly generated GOOP instances with nonquadratic objectives and nonlinear constraints, showing close agreement between primal solutions $z_{\mathrm{complete}}$ and $z_{\mathrm{reduced}}$, obtained from the complete and reduced KKT systems, respectively.
• Figure 3: Two-player intersection scenario solved using the reduced KKT system. We use the four-preference setup as in RAL-GOOP. Player 1 (blue) prioritizes reaching its goal and accelerates beyond the speed limit. Player 2 (red) prioritizes obeying the speed limit and proceeds toward its goal while remaining within the limit. The convergence plot shows local quadratic convergence of the KKT residual to $10^{-8}$. The homotopy parameter $\rho$ is reduced following a geometric schedule $\{1,2^{-1},\dots,2^{-10}\}$; the reported results correspond to $\rho = 2^{-10}$.

Theorems & Definitions (28)

• Definition 2.2: Local Generalized Nash Equilibrium for goop
• Theorem 2.3: Complete necessary conditions, cf. RAL-GOOP
• Proposition 2.4: Exponential growth of the complete KKT system
• Proof 1
• Theorem 3.1: Reduced KKT system is a relaxation of the complete KKT system
• Proof 2
• Proposition 3.2: Polynomial growth of reduced KKT system
• Proof 3
• Theorem 3.4: Primal solution equivalence in quadratic goop
• Proof 4