Statistical Dynamics in Piecewise Linear Cournot Game With Heterogeneous Duopolists

Abstract

We study a Cournot duopoly dynamic model in which reaction functions are piecewise linear. Such a model typically generates ergodic chaos when it involves strong nonlinearites. To investigate statistical properties, we construct explicit forms of density functions associated with chaotic trajectories. We demonstrate that the long-run average behavior possesses regular properties although each chaotic trajectory exhibits irregular motions. In particular, the ratios of the average outputs as well as the average profits are the same as those of Cournot outputs as well as Cournot profits.

Effects of Size, Composition, and Evolutionary Pressure in Heterogeneous Cournot Oligopolies with Best Response Decisional Mechanisms

“…Proof. The expression for ⋆ 2 can be immediately found by solving the fixed point equation ( ) = , with as in (23), noticing that no other solutions exist. The expression for ⋆ 1 directly follows by that for ⋆ 2 and by (16).…”

“…In particular, such works, in view of giving a foundation to the concept of Nash equilibrium, investigate under which conditions the dynamics converge towards the Nash equilibrium or towards a different attractor. To that research strand belong, for instance, the contributions by Agiza and Elsadany [8,9], by Agiza et al [10,11], by Ahmed and Agiza [12], by Angelini et al [13], by Bischi and Naimzada [14], by Canovas et al [15], by Cavalli and Naimzada [16,17], by Cavalli et al [18,19], by Elabbasy et al [20], by Ji [21], by Li and Ma [22], by Matsumoto [23], by Naimzada and Sbragia [24], by Pu and Ma [25], by Tramontana [26], and by Tuinstra [27].…”

Heterogeneity and the (de)stabilizing role of rationality

“…Conversely, the works concerning heterogeneous oligopolies, in which at least two firms adopt different decision mechanisms, mainly focused on studying different coupling of distinct decision mechanisms in oligopolies of fixed reduced sizes, usually duopolies or triopolies. We mention the papers by Agiza et al [8][9][10][11][12], Angelini et al [13], Bischi et al [14][15][16], Matsumoto [17], Cavalli et al [18][19][20] and Tramontana [21].…”

Density function of piecewise linear transformation

“…Matsumoto (2004) studies a Cournot duopoly dynamic model in which reactions functions are piecewise linear. As usual, the steady state becomes unstable if the slopes of reaction functions are strongly steep.…”