A dynamic marketing model with best reply and inertia

Abstract

a r t i c l e i n f o Article history: Available online xxx a b s t r a c tIn this paper we consider a nonlinear discrete-time dynamic model proposed by Farris et al. (2005) as a market share attraction model with two firms that decide marketing efforts over time according to best reply strategies with naïve expectations. The model also considers an adaptive adjustment toward best reply, a form of inertia or anchoring attitude, and we investigate the effects of heterogeneities among firms. A rich scenario of local and global bifurcations is obtained even with just two competing firms, and a comparison is proposed with apparently similar duopoly models based on repeated best reply dynamics with naïve expectations and adaptive adjustment.

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

“…In this section, the local stability around the equilibrium point will be analyzed through Jury condition [38]. The equilibrium points of system (17) can be solved by setting ( + 1) = ( ) = , that is,…”

“…Through solving the above nonlinear algebraic equations, four equilibrium points of system (17) can be obtained,…”

“…As all the boundary equilibrium points locate on the coordinate axes of ( 1 , 2 ), the stability of all the boundary equilibrium points of system (17) is equivalent to the corresponding points of the Logistic map. According to the stability theory, the local stability of an equilibrium point can be determined by analyzing the eigenvalues of Jacobian matrix of system (17) evaluated at the equilibrium point on the complex plane.…”

“…In system (17), the equilibrium 3 is stable when the parameters satisfy the stability condition, while, when one or more parameters violate the stability condition, the Nash equilibrium 3 will lose its stability through some types of bifurcation. In the next section, the stability of the equilibrium point is studied using numerical simulation.…”

“…Canovas et al [16] set up a piecewise smooth model for market competition, and the asymptotic dynamic behaviors of this model were discussed. Bischi et al [17] proposed an inertial model of market share based on the optimal response and naïve expectation, the influence of the heterogeneity of the enterprise was also analyzed, and it was found that there are very rich local and global bifurcation phenomena in the built system. Li and Ma [18] considered a bounded rational dual-channel game, and the complex dynamical behavior of their model was simulated in their works.…”