Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players

A heterogeneous Bertrand duopoly game with bounded rational and adaptive players manufacturing differentiated products is subject of investigation. The main goal is to demonstrate that participation of one bounded rational player in the game suffices to destabilize the duopoly. The game is modelled with a system of two difference equations. Evolution of prices over time is obtained by iteration of a two dimensional nonlinear map. Equilibria are found and local stability properties thereof are analyzed. Complex behavior of the system is examined by means of numerical simulations. Region of stability of the Nash equilibrium is demonstrated in the plane of the speeds of adjustment. Period doubling route to chaos is presented on the bifurcation diagrams and on the largest Lyapunov characteristic exponent graph. Lyapunov time is calculated. Chaotic attractors are depicted and their fractal dimensions are computed. Sensitive dependence on initial conditions is evidenced.Bertrand duopoly, heterogeneous expectations, nonlinear dynamics, chaos

A Differential game approach in the case of a polluting oligopoly

In this paper we propose an oligopolistic market model of pollution, where demand is not linear and firms are revenue maximizers. Additionally we assume that the rate of purification is very small tending to zero and that each firm accumulates a pollution share depending for example on firm’s size. The game ends up with Markov strategies employed by all firms. Our findings show that under conditions it is possible a marginal decrease on the total pollution stock to increase firms’ discounted revenues. A reallocation caused by a uniform decrease in all firms pollution, reorders the marginal change of the pollution stocks in reverse of the original order of the allowed pollution.Non linear strategies; Markov equilibrium; allowed pollution stock.

Controlling Chaos Through Local Knowledge

We propose an duopoly game where quantity-setting firms have incomplete information about the demand function. In each time step, they solve a profit maximization problem assuming a linear local approximation of the demand function. In particular, we construct an example using the well known duopoly Puu's model with isoelastic demand function and constant marginal costs. An explicit form of the dynamical system that describes the time evolution of the duopoly game with boundedly rational players is given. The main result is the global stability of the system.Cournot duopoly, incomplete information, isoelastic demand function, time evolution, boundedly rational players.

Double route to chaos in an heterogeneous triopoly game

We move from a triopoly game with heterogeneous players (E.M.Elabassy et al., 2009. Analysis of nonlinear triopoly game with heterogeneous players. Computers and Mathematics with Applications 57, 488-499). We remove the nonlinearity from the cost function and introduce it in the demand function. We also introduce a different decisional mechanism for one of the three competitors. A double route to complex dynamics is shown to exist, together with the possibility of multistability of different attractors, requiring a global analysis of the dynamical system.Triopoly game; Heterogeneous players; Global analysis

Mathematical Properties of a Combined Cournot-Stackelberg model.

The object of this work is to perform the global analysis of a new duopoly model which couples the two points of view of Cournot and Stackelberg. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigates as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which di¤er significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.Cournot-Stackelberg duopoly, Isoelastic demand function, Discontinuous reaction functions, Multistability, Border-collision bifurcations.

Collusion and Reciprocity in Infinitely Repeated Games

This paper extends the standard industrial organization models of repeated interaction between firms by incorporating preferences for reciprocity. A reciprocal firm responds to unkind behavior of rivals with unkind actions (destructive reciprocity), while at the same time, it responds to kind behavior of rivals with kind actions (constructive reciprocity). The main finding of the paper is that, for plausible perceptions of fairness, preferences for reciprocity facilitate collusion in infinitely repeated market games, that is, the critical discount rate at wish collusion can be sustained tends to be lower when firms have preferences for reciprocity than when firms are selfish. The paper also finds that the best collusive outcome that can be sustained in the infinitely repeated Cournot game with reciprocal firms is worse for consumers than the best collusive outcome that can be sustained in the infinitely repeated Cournot game with selfish firms.

New properties of the Cournot duopoly with isoelastic demand and constant unit costs.

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu (1991). The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark-Shacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties di¤er significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.Cournot duopoly, isoelastic demand function, multistability, border-collision bifurcations.