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Calculating Nash Equilibrium on Quantum Annealers

Olga Okrut, Keith Cannon, Kareem H. El-Safty, Nada Elsokkary, Faisal Shah Khan

TL;DR

This work addresses computing Nash equilibria for two-player, non-cooperative games using quantum annealing. It rewrites the standard doubly quadratic NE problem into a single quadratic program via Mangasarian–Stone, then converts the constrained problem into a QUBO by adding penalty terms and slack variables, enabling execution on quantum annealers. Three illustrative examples (Battle of the Sexes, the Bird Game, and an eight-strategy game) demonstrate NE-consistent results and report substantial hardware–software speedups (roughly seven to tenfold) over classical computation, with advantages observed on D-Wave systems. The study emphasizes the need to tune penalty weights, recognizes current limitations to pure strategies due to binary variables, and points to future work extending to mixed strategies and more complex game-theoretic models.

Abstract

Adiabatic quantum computing is implemented on specialized hardware using the heuristics of the quantum annealing algorithm. This setup requires the addressed problems to be formatted as discrete quadratic functions without constraints and the variables to take binary values only. The problem of finding Nash equilibrium in two-player, non-cooperative games is a two-fold quadratic optimization problem with constraints. This problem was formatted as a single, constrained quadratic optimization in 1964 by Mangasarian and Stone. Here, we show that adding penalty terms to the quadratic function formulation of Nash equilibrium gives a quadratic unconstrained binary optimization (QUBO) formulation of this problem that can be executed on quantum annealers. Three examples are discussed to highlight the success of the formulation, and an overall, time-to-solution (hardware + software processing) speed up of seven to ten times is reported on quantum annealers developed by D-Wave System.

Calculating Nash Equilibrium on Quantum Annealers

TL;DR

This work addresses computing Nash equilibria for two-player, non-cooperative games using quantum annealing. It rewrites the standard doubly quadratic NE problem into a single quadratic program via Mangasarian–Stone, then converts the constrained problem into a QUBO by adding penalty terms and slack variables, enabling execution on quantum annealers. Three illustrative examples (Battle of the Sexes, the Bird Game, and an eight-strategy game) demonstrate NE-consistent results and report substantial hardware–software speedups (roughly seven to tenfold) over classical computation, with advantages observed on D-Wave systems. The study emphasizes the need to tune penalty weights, recognizes current limitations to pure strategies due to binary variables, and points to future work extending to mixed strategies and more complex game-theoretic models.

Abstract

Adiabatic quantum computing is implemented on specialized hardware using the heuristics of the quantum annealing algorithm. This setup requires the addressed problems to be formatted as discrete quadratic functions without constraints and the variables to take binary values only. The problem of finding Nash equilibrium in two-player, non-cooperative games is a two-fold quadratic optimization problem with constraints. This problem was formatted as a single, constrained quadratic optimization in 1964 by Mangasarian and Stone. Here, we show that adding penalty terms to the quadratic function formulation of Nash equilibrium gives a quadratic unconstrained binary optimization (QUBO) formulation of this problem that can be executed on quantum annealers. Three examples are discussed to highlight the success of the formulation, and an overall, time-to-solution (hardware + software processing) speed up of seven to ten times is reported on quantum annealers developed by D-Wave System.
Paper Structure (10 sections, 31 equations, 5 figures)

This paper contains 10 sections, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Nash equilibrium
  3. Nash equilibrium as quadratic optimization
  4. Formatting for quantum annealers - QUBO
  5. Examples
  6. Battle of the Sexes
  7. A Bird Game: an example with three strategies
  8. An example with eight strategies
  9. Conclusion
  10. Acknowledgments

Figures (5)

  • Figure 1: QPU access time (in microseconds $\times 10e^6$) for two and three strategies game for D-Wave 2000 Q6 and D-Wave Advantage System 4.1. for the default annealing time of $20 \mu s$.
  • Figure 2: Frequency of pure Nash Equilibrium points based on 5000 samples for two-strategy game executed on D-Wave Advantage 4.1 QPU (bottom) and D-Wave 2000 Q6 QPU (top). The pure NE points occur at $p=(0,1), q=(0,1)$ and $p=(1,0), q=(1,0)$.
  • Figure 3: Frequency of pure Nash Equilibrium points based on 5000 samples for three-strategy game executed on D-Wave Advantage 4.1 QPU (bottom) and D-Wave 2000 Q6 (top). The pure NE points occur at $p=(0,0,1), q=(0,0,1)$ and $p=(1,0,0), q=(0,1,0)$.
  • Figure 4: Frequency of pure Nash Equilibrium points based on 5000 samples for the eight-strategy game executed on D-Wave Advantage 4.1.
  • Figure 5: QPU access time (in microseconds $\times 10e^6$) for eight-strategy game for D-Wave 2000 Q6 and D-Wave Advantage System 4.1 for an annealing time of $100 \mu s$.