Complex Dynamics of Mixed Triopoly Game with Quantity and Price Competition

Jing Wang,Yujing Song,Zhenhua Bao,Junqing Huang,Abdul Qadeer Khan

doi:10.1155/2021/9532340

Abstract

This article investigates the dynamics of a mixed triopoly game in which a state-owned public firm competes against two private firms. In this game, the public firm and private firms are considered to be boundedly rational and naive, respectively. Based on both quantity and price competition, the game’s equilibrium points are calculated, and then the local stability of boundary points and the Nash equilibrium points is analyzed. Numerical simulations are presented to display the dynamic behaviors including bifurcation diagrams, maximal Lyapunov exponent, and sensitive dependence on initial conditions. The chaotic behavior of the two models has been stabilized on the Nash equilibrium point by using the delay feedback control method. The thresholds under price and quantity competition are also compared.

Highlights

Cournot triopoly games considering heterogeneous players were investigated
Elabbasy et al [23] studied the dynamical behavior of the triopoly game with heterogeneous players with linear cost function. e three players were considered to be boundedly rational, adaptive, and naive. en, Elabbasy et al [24] further generalized the triopoly game with heterogeneous players with nonlinear cost function
We study the dynamics in a triopoly game with product differentiation, in which a state-owned public firm competes with two private firms

Summary

Model Specification

We adopt a standard differentiated oligopoly with a linear demand. e quasi-linear utility function of the representative consumer is. I 0 i 0 i 0 i≠j where q0 is the quantity produced by the public firm, qi(i 1, 2) is the quantity produced by the private firm, and y is the consumption of outside goods provided competitively (with a unitary price). For i 0, 1, 2, the inverse demand function is given by pi α − βqi − βδ 􏽘 qj,. Let c0 denote the state-owned public firm 0’s marginal cost. We assume that two private firms have the same marginal cost c1 c2 with α > c0 ≥ c1. We assume that the equilibrium quantities of public and private firms are strictly positive under both. Since firm 0 is a public firm, its payoff is the social surplus, given by SW 􏽘. Firm i(i 1, 2) is a private firm, and its payoff is its own profit: πi (pi − ci)qi

Cournot Competition

Bertrand Competition

Conclusion