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Near-Optimal Quantum Algorithms for Computing (Coarse) Correlated Equilibria of General-Sum Games

Tongyang Li, Xinzhao Wang, Yexin Zhang

TL;DR

This paper investigates quantum algorithms for computing $\varepsilon$-approximate CE and $\varepsilon$-approximate CCE in $m$-player normal-form games. It introduces a quantum CE approach based on a quantum-quantized multi-scale MWU and a quantum Gibbs sampler on amplitude-encoded loss vectors, and a quantum CCE approach extending Grigoriadis–Khachiyan with a ghost-iteration technique, leveraging QRAM to manage history samples. The results yield near-optimal quantum query complexities in $m$ and $n$ with $\tilde{O}(m\sqrt{n})$ for CE (fixed $\varepsilon$) and $\tilde{O}(m\sqrt{n}/\varepsilon^{2.5})$ for CCE, plus corresponding time bounds and matching quantum lower bounds $\Omega(m\sqrt{n})$ up to polylog factors. The work highlights the role of amplitude encoding, QRAM-based sampling, and quantum Gibbs sampling in accelerating equilibrium computations for general-sum games and outlines open questions on improving $\varepsilon$-dependence and extending to broader game classes.

Abstract

Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing $\varepsilon$-approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of $\tilde{O}(m\sqrt{n})$ for fixed $\varepsilon$. For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity $\tilde{O}(m\sqrt{n}/\varepsilon^{2.5})$. Both algorithms demonstrate a near-optimal scaling in the number of players $m$ and actions $n$, as confirmed by our quantum query lower bounds.

Near-Optimal Quantum Algorithms for Computing (Coarse) Correlated Equilibria of General-Sum Games

TL;DR

This paper investigates quantum algorithms for computing -approximate CE and -approximate CCE in -player normal-form games. It introduces a quantum CE approach based on a quantum-quantized multi-scale MWU and a quantum Gibbs sampler on amplitude-encoded loss vectors, and a quantum CCE approach extending Grigoriadis–Khachiyan with a ghost-iteration technique, leveraging QRAM to manage history samples. The results yield near-optimal quantum query complexities in and with for CE (fixed ) and for CCE, plus corresponding time bounds and matching quantum lower bounds up to polylog factors. The work highlights the role of amplitude encoding, QRAM-based sampling, and quantum Gibbs sampling in accelerating equilibrium computations for general-sum games and outlines open questions on improving -dependence and extending to broader game classes.

Abstract

Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing -approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of for fixed . For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity . Both algorithms demonstrate a near-optimal scaling in the number of players and actions , as confirmed by our quantum query lower bounds.
Paper Structure (25 sections, 15 theorems, 64 equations, 3 tables, 4 algorithms)

This paper contains 25 sections, 15 theorems, 64 equations, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Motivations.
  3. Contributions.
  4. Techniques.
  5. Open questions.
  6. Preliminaries
  7. Game theory and no-regret learning
  8. Game theory
  9. Online learning
  10. No-regret learning in normal-form games
  11. Quantum computing
  12. Quantum algorithms for games
  13. Quantum algorithm for computing correlated equilibria
  14. Quantum algorithm for computing coarse correlated equilibria
  15. Quantum lower bounds
  16. ...and 10 more sections

Key Result

Theorem 1

algo:quantum-ce computes an $\varepsilon$-correlated equilibrium of an $m$-player normal-form game with $n$ actions for each player using $m\sqrt{n}(\log(mn))^{O(1/\varepsilon)}$ queries to $\mathcal{O}_{\mathcal{L}}$ and $m^2\sqrt{n}(\log(mn))^{O(1/\varepsilon)}$ time.

Theorems & Definitions (27)

  • Definition 1
  • Theorem 1: Informal version of \ref{['thm:quantum-ce']}
  • Theorem 2: Informal version of \ref{['thm:quantum-cce']}
  • Theorem 3: Restatement of \ref{['thm:lower']}
  • Definition 2: Correlated equilibrium
  • Definition 3: Coarse correlated equilibrium
  • Theorem 4: Theorem 1.5 in hazan2016introduction
  • Theorem 5: Theorem 1.1 in peng2023fast
  • Definition 4: Amplitude encoding
  • Theorem 6: Quantum Gibbs sampler gao2024logarithmic
  • ...and 17 more