Calculating Nash equilibrium and Nash bargaining solution on quantum annealers
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Adiabatic quantum computing is implemented on specialized hardware using heuristics of the quantum annealing algorithm. This set up requires that the problems to be solved are formatted as discrete quadratic functions without constraints, and the variables taking 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 quadratic function optimization in 1964 by Mangasarian and Stone. Similarly, the Nash bargaining problem is a constrained polynomial optimization problem of degree higher than 2. Here, I show that adding penalty terms to the quadratic function formulation of Nash equilibrium, and reducing the higher powers of the polynomial expression for the Nash bargaining problem, together with addition of penalty terms, gives QUBO formulation of these problems for execution on quantum annealers.
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