Best-response Algorithms for Lattice Convex-Quadratic Simultaneous Games

TL;DR

This paper analyzes the convergence of best-response algorithms in lattice convex-quadratic games with nonlinear objectives, providing conditions for convergence, divergence, and equilibrium approximation.

Contribution

It introduces a sufficient condition based on singular values for the convergence of best-response algorithms in complex quadratic games and explores divergence scenarios.

Findings

Convergence guaranteed if all interaction matrix singular values are less than 1.

Divergence occurs when singular values exceed 1, with examples.

A relaxed Nash equilibrium can be computed within traps of strategies.

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.