Analysis of equilibria in a nonlinear oligopoly model

We consider the game-theoretic problem of choosing optimal strategies for agents of the oligopoly market for a linear demand function and nonlinear agent cost functions. A computational formula for the optimal action of an agent is derived in the form of a fractional irrational function; it is shown that the extrema of this function correspond to fixed points. To find a fixed point, an irrational equation is derived and approximate formulas for its solution are obtained. Necessary conditions for the existence and uniqueness or the multiplicity of equilibria depending on the agent type parameters are proved.