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Calculating Nash equilibrium on quantum annealers

Author

Listed:
  • Faisal Shah Khan

    (Dark Star Quantum Lab)

  • Olga Okrut

    (Dark Star Quantum Lab)

  • Keith Cannon

    (Dark Star Quantum Lab)

  • Kareem H. El-Safty

    (Dark Star Quantum Lab)

  • Nada Elsokkary

    (Dark Star Quantum Lab
    Khalifa University)

Abstract

Adiabatic quantum computing is implemented on specialized hardware using the heuristics of the quantum annealing algorithm. To solve a problem using quantum annealing, the problem requires formatting as a discrete quadratic function without constraints. 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 by seven to ten times is reported on quantum annealers developed by D-Wave System.

Suggested Citation

  • Faisal Shah Khan & Olga Okrut & Keith Cannon & Kareem H. El-Safty & Nada Elsokkary, 2025. "Calculating Nash equilibrium on quantum annealers," Annals of Operations Research, Springer, vol. 346(2), pages 1109-1126, March.
    • Handle: RePEc:spr:annopr:v:346:y:2025:i:2:d:10.1007_s10479-023-05700-z
    DOI: 10.1007/s10479-023-05700-z
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