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Solving Monotone Variational Inequalities with Best Response Dynamics

Yu-Wen Chen, Can Kizilkale, Murat Arcak

Abstract

We leverage best response dynamics to solve monotone variational inequalities on compact and convex sets. Specialization of the method to variational inequalities in game theory recovers convergence results to Nash equilibria when agents select the best response to the current distribution of strategies. We apply the method to generalize population games with additional convex constraints. Furthermore, we explore the robustness of the method by introducing various types of time-varying disturbances.

Solving Monotone Variational Inequalities with Best Response Dynamics

Abstract

We leverage best response dynamics to solve monotone variational inequalities on compact and convex sets. Specialization of the method to variational inequalities in game theory recovers convergence results to Nash equilibria when agents select the best response to the current distribution of strategies. We apply the method to generalize population games with additional convex constraints. Furthermore, we explore the robustness of the method by introducing various types of time-varying disturbances.
Paper Structure (14 sections, 3 theorems, 37 equations, 8 figures)

This paper contains 14 sections, 3 theorems, 37 equations, 8 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Variational Inequalities
  4. Best response dynamics
  5. Best response dynamics in atomic games
  6. Best response dynamics in population games
  7. BEST RESPONSE DYNAMICS FOR VARIATIONAL INEQUALITIES
  8. ROBUSTNESS ANALYSIS
  9. APPLICATIONS
  10. Traffic network
  11. Cost disturbances and dynamics disturbances
  12. Cost disturbances
  13. Dynamics disturbances
  14. CONCLUSIONS AND FUTURE WORK

Key Result

Theorem 1

Consider the variational inequality (eq:VI) where $K \subseteq\mathbb{R}^n$ is compact and convex and $F:K \rightarrow \mathbb{R}^n$ is $C^1$ monotone. Then SOL($K$,$F$) is globally asymptotically stable under the best response dynamics (eq:BRD) with $\pi(t)=F(x(t))$.

Figures (8)

  • Figure 1: The network is composed of three Y-intersections. Three possible routes are given. The delay functions are provided in the contexts.
  • Figure 2: The figure describes the Y-intersection where the traffic lights induce delay. The traffic flows are in red, while the delay functions are in black. The branch line, Link 2, and the mainline, Link 1, affect each other through the traffic light making their delay functions depend on $x_1$ and $x_2$.
  • Figure 3: Since the trajectory evolves on the 3-dimensional probability simplex, we project it and draw it on a plane. Compared with Fig. \ref{['fig:traffic_wo']}, the trajectory in Fig. \ref{['fig:traffic_w']} changes drastically and remains in the feasible set.
  • Figure 4: delay functions for links
  • Figure 5: without disturbance
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1: Variational Inequality
  • Definition 2: Strong Monotonicity / Monotonicity
  • Definition 3: Best response
  • Definition 4: Best response dynamics
  • Theorem 1
  • proof
  • Definition 5: Cost disturbance and Dynamics disturbance
  • Definition 6: Admissible Dynamics Disturbance
  • Theorem 2
  • proof
  • ...and 3 more