We define the Shapley allocation function (SAF) based on the characteristic function on a set of strategy profiles composed of infinite strategies to establish an n-person biform game model. It is the extension of biform games with finite strategies and scalar strategies. We prove the existence of Nash equilibria for this biform game with SAF, provided that the characteristic function satisfies the linear and semicontinuous conditions. We investigate the essential stability of Nash equilibria for biform games when characteristic functions are perturbed. We identify a residual dense subclass of the biform games whose Nash equilibria are all essential and deduce the existence of essential components of the Nash equilibrium set by proving the connectivity of its minimal essential set.

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.