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Advertising competition among firms may give rise to very complex dynamical behaviors. In this article, under the assumption of spillover effects, a two-stage dynamical Cournot game of advertising competition between two firms, which produce homogeneous products, is developed. Then, the local stability of the equilibrium points is discussed, and stability conditions of the equilibrium points are obtained. In order to reveal the complex dynamical behaviors of the model, both analytical and numerical methods are employed. The research results show that the two coordinate axes and the diagonal of the system are invariant manifolds, and one can obtain the dynamical behaviors on the invariant manifolds by analyzing the Logistic map. In addition, it is found that the advertising efforts of these two firms may be synchronized, even if the system is in a chaotic state. Multistability is another topic of focus. It is found that not only two attractors but also three attractors can coexist in the phase space, and contact bifurcation can also occur during the evolution of the attracting basins. Finally, the impact of corporate advertising efforts on corporate profits is analyzed. It is found that too much advertising effort is harmful to the firms, and a firm with low adjustment speed can even earn more when the system is chaotic.

We analyze the global stability of the equilibria in Bertrand and Cournot duopolies. Assuming a set of sufficient conditions for the global stability of the Bertrand duopoly equilibrium, we derive additional conditions which are sufficient for the global stability of the Cournot duopoly equilibrium. We use the relationships among the first and second order partial derivatives of the ordinary and inverse demand functions in deriving our results.

In potential games, the best-reply dynamics results in the existence of a cost function such that each player's best-reply set equals the set of minimizers of the potential given by the opponents' strategies. The study of sequential best-reply dynamics dates back to Cournot and, an equilibrium point which is stable under the game's best-reply dynamics is commonly said to be Cournot stable. However, it is exactly the best-reply behavior that we obtain using the Lyapunov notion of stability in game theory. In addition, Lyapunov theory presents several advantages. In this paper, we show that the stability conditions and the equilibrium point properties of Cournot and Lyapunov meet in potential games.

The properties of Cournot mixed oligopoly consisting of one public firm and one or more than one private firms have mostly been analyzed for simple cases on the basis of numerical calculations of the equilibrium values for a linear market demand function and linear or quadratic cost functions. In this paper, after proving the existence of a unique equilibrium in Cournot mixed oligopoly under general conditions on the market demand and each firm’s cost function, we derive conditions ensuring the existence of a unique Nash equilibrium for the mixed oligopoly where one public firm and at least one of the private firms are active in a general model of Cournot mixed oligopoly with one public firm and several private firms.

The so-called “Win-Continue, Lose-Reverse” (WCLR) rule is a simple iterative procedure that can be used to choose a value for any numeric variable (e.g., setting a price or a production level). The rule dictates that one should evaluate the effect on profits of the last adjustment made to the value (e.g., a price variation), and keep on changing the value in the same direction if the adjustment led to greater profits, or reverse the direction of change otherwise. Somewhat surprisingly, this simple rule has been shown to lead to collusive outcomes in Cournot oligopolies, even though its application requires no information about the other firms’ profits or choices. In this paper, we show that the convergence of the WCLR rule toward collusive outcomes can be very sensitive to small independent perturbations in the cost functions or in the income functions of the firms. These perturbations typically push the process toward the Nash equilibrium of the one-shot game. We also explore the behavior of WCLR against other strategies and demonstrate that WCLR is easily exploitable. We then conduct a similar analysis of the WCLR rule in a differentiated Bertrand model, where firms compete in prices.

The properties of Cournot mixed oligopoly consisting of one public firm and one or more than one private firms have mostly been analyzed for simple cases on the basis of numerical calculations of the equilibrium values for a linear market demand function and linear or quadratic cost functions. In this chapter, after proving the existence of a unique equilibrium in Cournot mixed oligopoly under general conditions on the market demand and each firm’s cost function, we derive conditions ensuring the existence of a unique Nash equilibrium for the mixed oligopoly where one public firm and at least one of the private firms are active in a general model of Cournot mixed oligopoly with one public firm and several private firms.