Finding the Nash equilibria of $n$-person noncooperative games via solving the system of equations

In this paper, we mainly study the equivalence and computing between Nash equilibria and the solutions to the system of equations. First, we establish a new equivalence theorem between Nash equilibria of $n$-person noncooperative games and solutions of algebraic equations with parameters, that is, finding a Nash equilibrium point of the game is equivalent to solving a solution of the system of equations, which broadens the methods of finding Nash equilibria and builds a connection between these two types of problems. Second, an adaptive differential evolution algorithm based on cultural algorithm (ADECA) is proposed to compute the system of equations. The ADECA algorithm applies differential evolution (DE) algorithm to the population space of cultural algorithm (CA), and increases the efficiency by adaptively improving the mutation factor and crossover operator of the DE algorithm and applying new mutation operation. Then, the convergence of the ADECA algorithm is proved by using the finite state Markov chain. Finally, the new equivalence of solving Nash equilibria and the practicability and effectiveness of the algorithm proposed in this paper are verified by computing three classic games.