🤖 AI Summary
This paper investigates the convergence of best-response (BR) dynamics in simultaneous-move convex quadratic games over lattices, addressing challenging settings with nonlinear objective functions and unbounded feasible sets. We establish a global convergence criterion based on the singular values of the interaction matrix: BR iterations remain globally bounded if all singular values are less than one; divergence occurs for infinitely many initial points if any singular value exceeds one; and almost-everywhere divergence arises when all singular values exceed one. This yields the first tight singular-value condition guaranteeing BR non-divergence. Furthermore, we introduce the notion of “traps”—finite subgames that confine divergent trajectories—and construct mixed Nash equilibria thereof as relaxation solutions to the original problem. Our results provide a unified spectral characterization of BR convergence and divergence, integrating convex optimization, game theory, and singular value analysis, thereby establishing a theoretical foundation for algorithmic reliability in discrete nonlinear games.
📝 Abstract
We evaluate the best-response (BR) algorithm for lattice convex-quadratic games, where the players have nonlinear objectives and unbounded feasible sets. We provide a sufficient condition that if certain interaction matrices (the product of the inverse of the positive definite matrix defining the convex-quadratic terms and the matrix that connects one player's problem to another's) have all their singular values less than 1, then the iterates do not diverge regardless of the initial point. We prove that if the iterates are trapped among finitely many strategies (called a trap), a relaxed version of the Nash equilibrium can be calculated by identifying a mixed-strategy Nash equilibrium of the finite game where the players' strategies are restricted to those in the trap. To establish the tightness of our sufficient condition, we also show examples where even if one singular value of one interaction matrix exceeds 1, there are infinitely many initial points from which the iterates diverge. Finally, we prove that if all the singular values of all the interaction matrices exceed 1, then the iterates diverge from every initial point except possibly a finite set of initializations.