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Learning Optimal Tax Design in Nonatomic Congestion Games

Qiwen Cui, Maryam Fazel, Simon S. Du

TL;DR

This work tackles learning an optimal tax to maximize social welfare in nonatomic congestion games under equilibrium feedback, where only Nash equilibria resulting from applied taxes are observable. It introduces a novel online framework that uses piecewise-linear approximations of the marginal-cost tax, enforces a strongly convex potential via an auxiliary linear term, and employs an exploratory tax to reveal information about unknown facilities. The authors prove that their method can achieve an $ε$-optimal tax with $O(F^2β/ε)$ samples, and they provide computationally efficient implementations for network congestion settings. The approach bridges learning under partial information with the structural properties of congestion games, offering a practical route to welfare-improving toll design in large-scale traffic and networked systems.

Abstract

In multiplayer games, self-interested behavior among the players can harm the social welfare. Tax mechanisms are a common method to alleviate this issue and induce socially optimal behavior. In this work, we take the initial step of learning the optimal tax that can maximize social welfare with limited feedback in congestion games. We propose a new type of feedback named \emph{equilibrium feedback}, where the tax designer can only observe the Nash equilibrium after deploying a tax plan. Existing algorithms are not applicable due to the exponentially large tax function space, nonexistence of the gradient, and nonconvexity of the objective. To tackle these challenges, we design a computationally efficient algorithm that leverages several novel components: (1) a piece-wise linear tax to approximate the optimal tax; (2) extra linear terms to guarantee a strongly convex potential function; (3) an efficient subroutine to find the exploratory tax that can provide critical information about the game. The algorithm can find an $ε$-optimal tax with $O(βF^2/ε)$ sample complexity, where $β$ is the smoothness of the cost function and $F$ is the number of facilities.

Learning Optimal Tax Design in Nonatomic Congestion Games

TL;DR

This work tackles learning an optimal tax to maximize social welfare in nonatomic congestion games under equilibrium feedback, where only Nash equilibria resulting from applied taxes are observable. It introduces a novel online framework that uses piecewise-linear approximations of the marginal-cost tax, enforces a strongly convex potential via an auxiliary linear term, and employs an exploratory tax to reveal information about unknown facilities. The authors prove that their method can achieve an -optimal tax with samples, and they provide computationally efficient implementations for network congestion settings. The approach bridges learning under partial information with the structural properties of congestion games, offering a practical route to welfare-improving toll design in large-scale traffic and networked systems.

Abstract

In multiplayer games, self-interested behavior among the players can harm the social welfare. Tax mechanisms are a common method to alleviate this issue and induce socially optimal behavior. In this work, we take the initial step of learning the optimal tax that can maximize social welfare with limited feedback in congestion games. We propose a new type of feedback named \emph{equilibrium feedback}, where the tax designer can only observe the Nash equilibrium after deploying a tax plan. Existing algorithms are not applicable due to the exponentially large tax function space, nonexistence of the gradient, and nonconvexity of the objective. To tackle these challenges, we design a computationally efficient algorithm that leverages several novel components: (1) a piece-wise linear tax to approximate the optimal tax; (2) extra linear terms to guarantee a strongly convex potential function; (3) an efficient subroutine to find the exploratory tax that can provide critical information about the game. The algorithm can find an -optimal tax with sample complexity, where is the smoothness of the cost function and is the number of facilities.
Paper Structure (29 sections, 20 theorems, 55 equations, 2 figures, 4 algorithms)

This paper contains 29 sections, 20 theorems, 55 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions and Technical Novelties
  3. Related Work
  4. Learning in congestion games.
  5. Optimal tax design in congestion games.
  6. Stackelberg games.
  7. Mathematical programming under equilibrium constraint.
  8. Preliminaries
  9. Nonatomic congestion games.
  10. Cost function.
  11. Nash equilibrium.
  12. Potential Function.
  13. Network congestion games.
  14. Tax Design for Congestion Games
  15. Polytope Description for Congestion Games
  16. ...and 14 more sections

Key Result

Lemma 1

$\Phi^{\mathrm{repa}}$ is convex under Assumption asp:cost. If $y^*=\mathop{\mathrm{argmin}}\limits_y \Phi^{\mathrm{repa}}(y)$, then for any $x\in\phi^{-1}(y^*)$, $x$ is a Nash equilibrium.

Figures (2)

  • Figure 1: Social Welfare Curves of the Algorithm for various values of $c$ and $p$. We can observe that the social welfare converges to the optimal one quickly.
  • Figure 2: Estimated Tax Functions at the Last Iteration for various values of $c$ and $p$. The estimation is not uniformly accurate but they are accurate at the induced Nash equilibrium.

Theorems & Definitions (43)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • ...and 33 more