Influence of reflexion on the properties of equilibria in a nonlinear Stackelberg oligopoly model

We consider the game-theoretic problem of choosing optimal strategies for agents of the oligopoly market under a linear demand function and nonlinear agent cost functions. The influence of reflexive behavior on the number and properties of equilibria in the game is studied for agents with different types of cost functions: concave, corresponding to positive returns to scale, and convex, corresponding to a negative effect. It is proved that in the case of convex cost functions there is only one equilibrium, and in the case of concave cost functions there can be two equilibria, one less and the other greater than the equilibrium for linear costs. It has been established that for convex cost functions the equilibrium action increases with the growth of the agent's reflexion and decreases with the growth of the environment's reflexion, as in the model with linear cost functions. For concave cost functions, the influence of reflexion depends on the sign of the sum of conjectural variations $S$: with an increase in the agent's reflexion, the greater equilibrium increases, while the smaller one decreases for $S<0$, and vice versa for $S>0$.