It is known that Nash equilibria of potential game belong to the set of stationary points of the potential, moreover only the potential’s global maximum is an equilibrium in general case. In the paper, we consider Cournot oligopoly model with linear inverse demand function and S-shape players’ costs functions determined by cubical polynomials. S-shape form of a function means changing of function’s concavity by its convexity. Costs function of such form reflects changing of increasing return of the scale by decreasing return of the scale, what may be explained as a stage of introduction of a new capacities that is changed by a stage of its normal operation. Such a model is a potential game due to linearity of inverse demand function. The potential is constructed and it has a form of cubical polynomial. Nonconcavity of potential leads to non-uniqueness of equilibrium in general case. Author investigated in other papers the local search of stationary points with multi-start approach and with the following check of the point whether it is an equilibrium. That paper is concerned with an adaptation of branch and bound method for search of the potential’s global maximum which is always an equilibrium point. In the paper, the method is described and numerical experiment results are given.

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