An evolutionary algorithmic approach to determine the Nash equilibrium in a duopoly with nonlinearities and constraints

This paper presents an algorithmic approach to obtain the Nash equilibrium in a duopoly. Analytical solutions to duopolistic competition draw on principles of game theory and require simplifying assumptions such as symmetrical payoff functions, linear demand and linear cost. Such assumptions can reduce the practical use of duopolistic models. In contrast, we use an evolutionary algorithmic approach (EAA) to determine the Nash equilibrium values. This approach has the advantage that it can deal with and find optimum values for duopolistic competition modelled using non-linear functions. In the paper we gradually build up the competitive situation by considering non-linear demand functions, non-linear cost functions, production and environmental constraints, and production in discrete bands. We employ particle swarm optimization with composite particles (PSOCP), a variant of particle swarm optimization, as the evolutionary algorithm. Through the paper we explicitly demonstrate how EAA can solve games with constrained payoff functions that cannot be dealt with by traditional analytical methods. We solve several benchmark problems from the literature and compare the results obtained from EAA with those obtained analytically, demonstrating the resilience and rigor of our EAA solution approach.