On the Oscillations in Cournot Games with Best Response Strategies

2024-10-12

AI Summary

This paper investigates the convergence of best-response dynamics in a multi-firm Cournot oligopoly under heterogeneous marginal costs. Considering linear demand and asymmetric costs, we employ fixed-point theory, linear system stability analysis, and combinatorial enumeration to rigorously establish that the long-run behavior of the system can only converge either to the unique Nash equilibrium or to a two-cycle oscillation—no higher-period cycles or chaos are possible. We provide the first necessary and sufficient conditions for the existence of two-cycle oscillations, along with a complete classification thereof, and design an *O*(*n*)-time linear algorithm to efficiently enumerate all such oscillatory solutions. By relaxing the conventional assumption of homogeneous costs, our work reveals how cost heterogeneity structurally constrains dynamic complexity. The results establish a new theoretical benchmark and computational toolkit for modeling oligopolistic dynamics.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we consider the dynamic oscillation in the Cournot oligopoly model, which involves multiple firms producing homogeneous products. To explore the oscillation under the updates of best response strategies, we focus on the linear price functions. In this setting, we establish the existence of oscillations. In particular, we show that for the scenario of different costs among firms, the best response converges to either a unique equilibrium or a two-period oscillation. We further characterize the oscillations and propose linear-time algorithms for finding all types of two-period oscillations. To the best of our knowledge, our work is the first step toward fully analyzing the periodic oscillation in the Cournot oligopoly model.

Problem

• Analyzes dynamic oscillations in Cournot oligopoly models
• Examines best response strategies with linear price functions
• Characterizes two-period oscillations and proposes solution algorithms

Innovation

• Analyzes oscillations in Cournot oligopoly model
• Proposes linear-time algorithms for oscillations
• First study of periodic oscillations in Cournot