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On the Oscillations in Cournot Games with Best Response Strategies

Zhengyang Liu, Haolin Lu, Liang Shan, Zihe Wang

TL;DR

This paper analyzes dynamic oscillations in a Cournot oligopoly with homogeneous products under simultaneous best-response updates and linear inverse demand. By examining the BR dynamics with nonnegativity constraints, it proves that the system converges to either a Nash equilibrium or a two-period cycle, and it shows that no oscillations of period greater than two can occur. The authors classify all possible two-period oscillations into three structural patterns and develop linear-time algorithms to detect and identify them, enabling efficient empirical analysis. The results hold for general cost structures and yield explicit forms in several cases, enhancing understanding of strategic cycling in oligopolies and laying groundwork for extensions to other competition models. The work provides a concrete methodological toolkit for diagnosing and studying cycling behavior in real markets.

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.

On the Oscillations in Cournot Games with Best Response Strategies

TL;DR

This paper analyzes dynamic oscillations in a Cournot oligopoly with homogeneous products under simultaneous best-response updates and linear inverse demand. By examining the BR dynamics with nonnegativity constraints, it proves that the system converges to either a Nash equilibrium or a two-period cycle, and it shows that no oscillations of period greater than two can occur. The authors classify all possible two-period oscillations into three structural patterns and develop linear-time algorithms to detect and identify them, enabling efficient empirical analysis. The results hold for general cost structures and yield explicit forms in several cases, enhancing understanding of strategic cycling in oligopolies and laying groundwork for extensions to other competition models. The work provides a concrete methodological toolkit for diagnosing and studying cycling behavior in real markets.

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.

Paper Structure

This paper contains 11 sections, 5 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Periodic Oscillation
  5. Case 1
  6. Case 2
  7. When $k_1=0$
  8. When $k_1>0$ and $k_2>0$
  9. When $0<k_1<3$ and $k_2>0$
  10. When $k_1> 0$ and $k_2=0$
  11. Conclusion

Key Result

Lemma 1

For any $i>j$ and any $t$, we let $Q^{t}=q_i^{t}-q_j^{t}$ and $\Delta c=c_i-c_j\geq 0$. There exists an integer $T>0$ such that $Q^{t}\leq0$ for any $t>T$. Moreover, if $\Delta c=0$, then $\lim_{t\rightarrow \infty}Q^t=0$.

Figures (1)

  • Figure 1: The shadowed area is the feasible domain for values of $(Q^t,Q^{t+1})$.

Theorems & Definitions (11)

  • Definition 1: Nash equilibrium
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 1 more