Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations

Abstract

In a differentiated triopoly model with heterogeneous firms, the local stability of the Nash equilibrium under both quantity and price competition is analyzed. We find that the presence of a firm following a gradient rule based on marginal profits, and a player with adaptive expectations, determines the local stability of the Nash equilibrium, regardless the competition type, while the effects of the degree of product differentiation on the stability depend on the nature of products. Moreover, the Nash equilibrium is more stable under quantity competition than under price competition.

Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

“…Let ω * ∈ {ω 1 , ω 2 , ω 3 } and τ j * (j = 1, 2, 3) its corresponding value of τ be defined as in Equations (11) and (12).…”

“…Li and Ma [10] considered a small rational dual-channel game and simulate their model's complex dynamic behaviour in their research. Many researchers have explored the complex dynamical behaviours of this type of models from different aspects, such as differentiated goods [11][12][13][14][15], bounded rationality [16], heterogeneous firms [7,[17][18][19], delayed decisions [20][21][22][23][24][25] and other factors [26][27][28].…”

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

“…And they found that the Nash equilibrium point may lose its stability through a flip bifurcation or a Neimark-Sacker bifurcation. In addition, a lot of researchers have discussed the complex dynamical behaviors of nonlinear oligopolies from different aspects, such as differentiated goods [21][22][23][24][25], heterogeneous firms [26][27][28][29][30][31], and delayed decisions [32,33].…”

An evolutionary Cournot oligopoly model with imitators and perfect foresight best responders

“…Moreover, the circumstance of heterogeneous degrees of rationality and computational abilities among players lead to consider the simultaneous presence of different decision mechanisms. Various pairings of heterogeneous behaviors, including the best response, the gradient rule and the LMA rule, are considered in Leonard and Nishimura (), Den Haan (), Agiza and Elsadany (), Angelini, Dieci, and Nardini (), Tramontana (), Cavalli and Naimzada (), Andaluz and Jarne (), Cavalli, Naimzada, and Tramontana (), Cavalli and Naimzada (), Pireddu (), Tramontana, Elsadany, Xin, and Agiza (), Naimzada and Tramontana () and in Andaluz, Elsadany, and Jarne ().…”