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On the Interplay between Social Welfare and Tractability of Equilibria

Ioannis Anagnostides, Tuomas Sandholm

TL;DR

This work investigates when efficient Nash equilibria are computationally tractable by connecting Roughgarden's smoothness framework with no-regret learning dynamics in large $n$-player games. It shows that if full efficiency is guaranteed via $(\lambda,\mu)$-smoothness with $\rho_n=\lambda_n/(1+\mu_n)\to1$, decentralized no-regret dynamics (notably optimistic gradient descent) approximate Nash equilibria and converge rapidly in many natural game classes, with a deeper link to the Minty property. The paper also introduces clairvoyant gradient descent to achieve improved welfare while guaranteeing convergence to coarse correlated equilibria, and extends these results to Bayesian mechanisms, highlighting both tractable computation and welfare guarantees. Finally, it discusses the limits of smoothness-based tractability, including hardness results, and outlines directions for future work, such as refining convergence rates, exploring mean-field regimes, and extending to incomplete information settings.

Abstract

Computational tractability and social welfare (aka. efficiency) of equilibria are two fundamental but in general orthogonal considerations in algorithmic game theory. Nevertheless, we show that when (approximate) full efficiency can be guaranteed via a smoothness argument à la Roughgarden, Nash equilibria are approachable under a family of no-regret learning algorithms, thereby enabling fast and decentralized computation. We leverage this connection to obtain new convergence results in large games -- wherein the number of players $n \gg 1$ -- under the well-documented property of full efficiency via smoothness in the limit. Surprisingly, our framework unifies equilibrium computation in disparate classes of problems including games with vanishing strategic sensitivity and two-player zero-sum games, illuminating en route an immediate but overlooked equivalence between smoothness and a well-studied condition in the optimization literature known as the Minty property. Finally, we establish that a family of no-regret dynamics attains a welfare bound that improves over the smoothness framework while at the same time guaranteeing convergence to the set of coarse correlated equilibria. We show this by employing the clairvoyant mirror descent algortihm recently introduced by Piliouras et al.

On the Interplay between Social Welfare and Tractability of Equilibria

TL;DR

This work investigates when efficient Nash equilibria are computationally tractable by connecting Roughgarden's smoothness framework with no-regret learning dynamics in large -player games. It shows that if full efficiency is guaranteed via -smoothness with , decentralized no-regret dynamics (notably optimistic gradient descent) approximate Nash equilibria and converge rapidly in many natural game classes, with a deeper link to the Minty property. The paper also introduces clairvoyant gradient descent to achieve improved welfare while guaranteeing convergence to coarse correlated equilibria, and extends these results to Bayesian mechanisms, highlighting both tractable computation and welfare guarantees. Finally, it discusses the limits of smoothness-based tractability, including hardness results, and outlines directions for future work, such as refining convergence rates, exploring mean-field regimes, and extending to incomplete information settings.

Abstract

Computational tractability and social welfare (aka. efficiency) of equilibria are two fundamental but in general orthogonal considerations in algorithmic game theory. Nevertheless, we show that when (approximate) full efficiency can be guaranteed via a smoothness argument à la Roughgarden, Nash equilibria are approachable under a family of no-regret learning algorithms, thereby enabling fast and decentralized computation. We leverage this connection to obtain new convergence results in large games -- wherein the number of players -- under the well-documented property of full efficiency via smoothness in the limit. Surprisingly, our framework unifies equilibrium computation in disparate classes of problems including games with vanishing strategic sensitivity and two-player zero-sum games, illuminating en route an immediate but overlooked equivalence between smoothness and a well-studied condition in the optimization literature known as the Minty property. Finally, we establish that a family of no-regret dynamics attains a welfare bound that improves over the smoothness framework while at the same time guaranteeing convergence to the set of coarse correlated equilibria. We show this by employing the clairvoyant mirror descent algortihm recently introduced by Piliouras et al.
Paper Structure (43 sections, 31 theorems, 76 equations, 2 figures)

This paper contains 43 sections, 31 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Roadmap
  4. Background
  5. Notation
  6. Multilinear games
  7. Welfare and the price of anarchy
  8. Smooth games
  9. Nash equilibrium
  10. Regret
  11. Optimistic gradient descent
  12. Convergence to Nash equilibria via smoothness
  13. Proof sketch of \ref{['theorem:large_games-informal']}
  14. Connection with the Minty property
  15. Games with vanishing sensitivity
  16. ...and 28 more sections

Key Result

Theorem 1.1

Consider a sequence of $n$-player $(\lambda_n, \mu_n)$-smooth games $(\mathcal{G}_n)_{n \geq 1}$ such that $\rho_n \coloneqq \frac{\lambda_n}{1 + \mu_n} \to 1$ with a sufficiently fast rate. Then, there are decentralized and computationally efficient no-regret dynamics approaching an $(o_n(1), o_n(1

Figures (2)

  • Figure 1: $\mathsf{PoA}$ versus $\mathsf{rPoA}$ in random normal-form games.
  • Figure 2: The behavior of $\mathtt{OGD}$ in the bimatix game \ref{['eq:og']} with $\eta \coloneqq 0.01$. On the left, we plot the average social welfare of the dynamics. On the right, we plot the probability of playing a joint action in the diagonal.

Theorems & Definitions (63)

  • Theorem 1.1: Informal
  • Definition 2.1: Smooth game Roughgarden15:Intrinsic
  • Definition 2.2: Weak Nash equilibrium Babichenko16:Can
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.5
  • Corollary 3.5
  • Proposition 3.5
  • Theorem 3.6
  • ...and 53 more