Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Multi-Agent Decision-Making in Finite-Population Games

Shinkyu Park, Lucas C. D. Bezerra

TL;DR

This work advances robust decision-making in finite-population games by analyzing the Kullback-Leibler Divergence Regularized Learning (KLD-RL) protocol, which couples stochastic population dynamics with a softmax-like update controlled by $\eta$ and a reference distribution $\theta$. By leveraging $\delta$-passivity and $\delta$-antipassivity, the authors derive stability results for the closed-loop EDM under delayed, noisy payoff observations and quantify how the revision rate $\lambda$ and regularization $\eta$ trade off convergence speed, noise sensitivity, and long-run variance. A central bound shows the deviation between the interpolated population state $\widehat{X}^N$ and the regularized best-response $C^{\eta,\theta}(p)$ scales with $(\lambda^2\eta)^{-1}$, while a Lipschitz property of $C^{\eta,\theta}$ in $p$ allows mitigating reduced $\lambda$ via larger $\eta$. Simulations on a finite population with task-allocation dynamics confirm that properly tuned $\lambda$ and $\eta$ improve noise robustness and convergence to the equilibrium $x^*$, and the work highlights the potential to learn $x^*$ and $\theta$ in future data-driven extensions.

Abstract

We study the robustness of an agent decision-making model in finite-population games, with a particular focus on the Kullback-Leibler Divergence Regularized Learning (KLD-RL) model. Specifically, we examine how the model's parameters influence the impact of various sources of noise and modeling inaccuracies -- factors commonly encountered in engineering applications of population games -- on agents' decision-making. Our analysis provides insights into how these parameters can be effectively tuned to mitigate such effects. Theoretical results are supported by numerical examples and simulation studies that validate the analysis and illustrate practical strategies for parameter selection.

Robust Multi-Agent Decision-Making in Finite-Population Games

TL;DR

This work advances robust decision-making in finite-population games by analyzing the Kullback-Leibler Divergence Regularized Learning (KLD-RL) protocol, which couples stochastic population dynamics with a softmax-like update controlled by and a reference distribution . By leveraging -passivity and -antipassivity, the authors derive stability results for the closed-loop EDM under delayed, noisy payoff observations and quantify how the revision rate and regularization trade off convergence speed, noise sensitivity, and long-run variance. A central bound shows the deviation between the interpolated population state and the regularized best-response scales with , while a Lipschitz property of in allows mitigating reduced via larger . Simulations on a finite population with task-allocation dynamics confirm that properly tuned and improve noise robustness and convergence to the equilibrium , and the work highlights the potential to learn and in future data-driven extensions.

Abstract

We study the robustness of an agent decision-making model in finite-population games, with a particular focus on the Kullback-Leibler Divergence Regularized Learning (KLD-RL) model. Specifically, we examine how the model's parameters influence the impact of various sources of noise and modeling inaccuracies -- factors commonly encountered in engineering applications of population games -- on agents' decision-making. Our analysis provides insights into how these parameters can be effectively tuned to mitigate such effects. Theoretical results are supported by numerical examples and simulation studies that validate the analysis and illustrate practical strategies for parameter selection.
Paper Structure (13 sections, 2 theorems, 40 equations, 2 figures)

This paper contains 13 sections, 2 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Description
  3. Evolutionary Dynamics and Task Allocation Games
  4. Passivity Techniques for Stability Analysis
  5. Problem Formulation
  6. Illustrative Example
  7. Analysis of the KLD-RL Protocol
  8. Conclusions
  9. Linear Interpolation $\widehat{X}^N(t)$ of Population State $X^N(t)$
  10. Proof of Theorem \ref{['theorem:effect_of_noise']}
  11. Derivation of \ref{['eq:effect_of_noise']}
  12. Derivation of \ref{['eq:choice_function_lipschitz_continuity']}
  13. Proof of Proposition \ref{['proposition:mean_variance_of_stationary_distribution']}

Key Result

Theorem 1

Consider the closed-loop model eq:closed_loop_model, and recall the $\delta$-storage function $\mathcal{S}$ and $\delta$-antistorage function $\mathcal{L}$, defined for EDM eq:edm and the game model eq:task_allocation_games, respectively. For any time horizon $T>0$, the following inequality holds: where $\alpha_\lambda = \lambda \mathcal{S}(p(0), \widehat{X}^N (0)) + \mathcal{L} (q(0), \widehat{X

Figures (2)

  • Figure 1: A feedback diagram consisting of the task allocation game model \ref{['eq:task_allocation_games']} and the KLD-RL model \ref{['eq:kld_rl']}. Each agent engaged in the game receives opportunities to revise its strategy at the arrival times $\mathbb T_\lambda$ of its associated Poisson process with rate parameter $\lambda$. The parameter $\eta$ in the KLD-RL model regulates the trade-off in decision-making between the payoff vector $p(t)$ and the reference distribution $\theta$.
  • Figure 2: (a) Comparison of $\| q(t) \|_\infty$ using the KLD-RL ($\eta=0.04, \lambda = 0.1$) and Smith protocols ($\lambda=1.0$) for $N=10$. (b) Trajectories of $\| q(t) \|_\infty$ under the KLD-RL protocol for $\eta=0.001, 0.04, 10.0$ with fixed $\lambda = 0.1$ and $N=10$. (c) Trajectories of $\| q(t) \|_\infty$ under the KLD-RL protocol for $\lambda=0.01, 0.1, 1.0$ with fixed $\eta = 0.04$ and $N=10$. (d) Trajectories of $\| q(t) \|_\infty$ under the KLD-RL protocol for $N = 10, 20, 40$ with fixed $\eta=0.04$ and $\lambda=0.1$.

Theorems & Definitions (6)

  • Example 1
  • Definition 1
  • Definition 2
  • Remark 1: On Assumption \ref{['assumption:bounded_game_state']}
  • Theorem 1
  • Proposition 1