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Best-response Algorithms for Lattice Convex-Quadratic Simultaneous Games

Sriram Sankaranarayanan

TL;DR

This work analyzes best-response algorithms for lattice convex-quadratic simultaneous games, where players optimize convex-quadratic objectives over full-rank lattices. The key contribution is a sharp condition: if the interaction matrices $R_1=Q_1^{-1}C_1$ and $R_2=Q_2^{-1}C_2$ have all singular values below 1 (positively adequate), BR iterates are guaranteed to avoid divergence and terminate finitely, irrespective of initialization; the authors also develop a proximity-based framework linking lattice minimizers to continuous counterparts. When BR cycles within a finite trap, a corresponding finite-game mixed strategy Nash equilibrium over the trap yields a $(\Delta_x,\Delta_y)$-MNE for the original game, with explicit bounds that depend on lattice geometry and the proximity constant; under positively adequate objectives these bounds can be tightened, and a conjecture is posed that the bound may collapse to $\Delta=0$. The paper also proves tightness via counterexamples where some singular value exceeds 1 leads to divergence from certain starts, and provides a detailed proximity theorem grounded in the flatness theorem and covering radius of lattices. Computational experiments on pricing games and random instances show BR significantly outperforms SGM for more players, and consistently yields PNEs or MNEs with negligible deviations, supporting the practical viability of BR in complex lattice-structured 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.

Best-response Algorithms for Lattice Convex-Quadratic Simultaneous Games

TL;DR

This work analyzes best-response algorithms for lattice convex-quadratic simultaneous games, where players optimize convex-quadratic objectives over full-rank lattices. The key contribution is a sharp condition: if the interaction matrices and have all singular values below 1 (positively adequate), BR iterates are guaranteed to avoid divergence and terminate finitely, irrespective of initialization; the authors also develop a proximity-based framework linking lattice minimizers to continuous counterparts. When BR cycles within a finite trap, a corresponding finite-game mixed strategy Nash equilibrium over the trap yields a -MNE for the original game, with explicit bounds that depend on lattice geometry and the proximity constant; under positively adequate objectives these bounds can be tightened, and a conjecture is posed that the bound may collapse to . The paper also proves tightness via counterexamples where some singular value exceeds 1 leads to divergence from certain starts, and provides a detailed proximity theorem grounded in the flatness theorem and covering radius of lattices. Computational experiments on pricing games and random instances show BR significantly outperforms SGM for more players, and consistently yields PNEs or MNEs with negligible deviations, supporting the practical viability of BR in complex lattice-structured 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.
Paper Structure (34 sections, 12 theorems, 20 equations, 4 figures, 1 algorithm)

This paper contains 34 sections, 12 theorems, 20 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Literature
  4. Integer Programming Games.
  5. The BR algorithm.
  6. Definitions and Algorithm descriptions
  7. Lattices.
  8. The game and equilibria.
  9. The BR algorithm.
  10. Adequate objectives.
  11. Implication of positively and negatively adequate objectives.
  12. Main Results.
  13. Necessary and sufficient conditions for finite termination.
  14. Proximity in integer convex-quadratic programs.
  15. Flat direction of the ellipsoid.
  16. ...and 19 more sections

Key Result

Theorem 1

Given a lattice $\mathbf{\Lambda} \subset \mathbb{R}^n$ with full rank, and a convex set $C \subseteq \mathbb{R}^{n}$ such that $C \cap \mathbf{\Lambda} = \emptyset$, there exists a direction $\mathbf{z} \in \mathbf{\Lambda}^{*}$ such that $\max \left\{ \mathbf{z}^\top \mathbf{x} \mid \mathbf{x} \in

Figures (4)

  • Figure 1: Examples of lattices in $\mathbb{R}^2$
  • Figure 2: Implications among Theorems
  • Figure 3: Pricing game - performance profile. Shows the fraction of instances solved within given time
  • Figure 4: Random game - performance profile. Shows the fraction of instances solved within given time

Theorems & Definitions (33)

  • Definition 1
  • Example 1
  • Theorem 1: Flatness Theorem
  • Definition 2: $(\Delta_x, \Delta_y)$-Nash equilibrium
  • Definition 3
  • Definition 4
  • Theorem 2
  • Example 2
  • Theorem 3
  • Remark 1
  • ...and 23 more