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Convergence to stationary points in the Weisbuch-Kirman-Herreiner model for buyers' preferences in fish markets

Ali Ellouze, Bastien Fernandez

TL;DR

The paper rigorously analyzes the Weisbuch–Kirman–Herreiner buyer-preference dynamics in OTC fish markets under homogeneous buyers, proving almost-sure convergence to stationary points for arbitrary numbers of sellers. It characterizes stationary points and their stability for simple attractiveness distributions, revealing symmetry-breaking stationary states and non-monotone bifurcations as friction gamma varies. The authors develop a global well-posedness framework, identify an invariant attracting simplex, and, in key cases (notably N=2 and two-cluster configurations), provide explicit bifurcation scenarios. The work supplies a solid mathematical foundation for interpreting long-run buyer behavior in mean-field market models and guides future extensions to heterogeneous buyers and time-dependent attractiveness feedback.

Abstract

In a paper published in The Economic Journal in 2000, Weisbuch et al.\ introduce a model for buyers' preferences to the various sellers in over-the-counter (OTC) fish markets. While this model has become an archetype of economic conceptualization that combines bounded rationality and myopic reasoning, the literature on its asymptotic behaviours has remained scarce. In this paper, we proceed to a mathematical analysis of the dynamics and its full characterization in the simplest case of homogeneous buyer populations. By using elements of the theory of cooperative dynamical systems, we prove that, independently of the number of sellers and parameters, for almost every initial condition, the subsequent trajectory must asymptotically approach a stationary state. Moreover, for simple enough distributions of the sellers' attractiveness, we determine all stationary states and their parameter-dependent stability. This analysis shows that in most cases, the asymptotic preferences are ordered as the attractiveness are. However, depending on the parameters, there also exist robust functioning modes in which those sellers with highest preference are not the ones that provide highest profit.

Convergence to stationary points in the Weisbuch-Kirman-Herreiner model for buyers' preferences in fish markets

TL;DR

The paper rigorously analyzes the Weisbuch–Kirman–Herreiner buyer-preference dynamics in OTC fish markets under homogeneous buyers, proving almost-sure convergence to stationary points for arbitrary numbers of sellers. It characterizes stationary points and their stability for simple attractiveness distributions, revealing symmetry-breaking stationary states and non-monotone bifurcations as friction gamma varies. The authors develop a global well-posedness framework, identify an invariant attracting simplex, and, in key cases (notably N=2 and two-cluster configurations), provide explicit bifurcation scenarios. The work supplies a solid mathematical foundation for interpreting long-run buyer behavior in mean-field market models and guides future extensions to heterogeneous buyers and time-dependent attractiveness feedback.

Abstract

In a paper published in The Economic Journal in 2000, Weisbuch et al.\ introduce a model for buyers' preferences to the various sellers in over-the-counter (OTC) fish markets. While this model has become an archetype of economic conceptualization that combines bounded rationality and myopic reasoning, the literature on its asymptotic behaviours has remained scarce. In this paper, we proceed to a mathematical analysis of the dynamics and its full characterization in the simplest case of homogeneous buyer populations. By using elements of the theory of cooperative dynamical systems, we prove that, independently of the number of sellers and parameters, for almost every initial condition, the subsequent trajectory must asymptotically approach a stationary state. Moreover, for simple enough distributions of the sellers' attractiveness, we determine all stationary states and their parameter-dependent stability. This analysis shows that in most cases, the asymptotic preferences are ordered as the attractiveness are. However, depending on the parameters, there also exist robust functioning modes in which those sellers with highest preference are not the ones that provide highest profit.

Paper Structure

This paper contains 15 sections, 6 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction and definitions
  2. Definition of the dynamics.
  3. Considerations about the parameters
  4. Convergence to stationary points
  5. Analysis of the dynamics
  6. Well-posedness and a globally attracting set
  7. Proof of statement (ii) of Theorem \ref{['MAINRES1']}
  8. Proof of Proposition \ref{['CONTRACT']}
  9. Proof of statement (i).
  10. Proof of statement (ii).
  11. Analysis of the stationary points
  12. The homogeneous case of identical attractiveness.
  13. Markets composed by two sellers ($N=2$).
  14. Two-cluster dynamics in two-cluster markets.
  15. Conclusion

Key Result

Theorem 2.1

(i) For every initial condition $\mathbf{J}^0 \in \mathbb{R}^N$, the equation DYN_WKH admits a unique solution $t \mapsto \mathbf{J}(t)$ such that $\mathbf{J}(0)=\mathbf{J}^0$, which is defined for all $t \in \mathbb{R}^+_\ast$ and is bounded. (ii) For Lebesgue every initial condition $\mathbf{J}^0

Figures (1)

  • Figure 1: Illustration of the results of Proposition \ref{['FIX_HOMOG']}. Stream lines of the vector field $G$ of the system \ref{['DIFFDYN']} for $N=3$ and $a_1=a_2=a_3 = 1$. Left.$\gamma=\frac{2}{5}>\frac{1}{3}$, there is a unique stationary point of $G$ (and hence of $F$) with coordinates $\Delta_1=\Delta_2=0$. Right.$\gamma=\frac{2}{7}<\frac{1}{3}$, there are exactly 7 stationary points. Three points are stable. Following the Proposition (according to which, in the original variable $\mathbf J$, for each point, two coordinates are equal and the remaining one is larger), these points are respectively on the axis $\Delta_1=0$, on the axis $\Delta_2=0$ and on the diagonal $\Delta_1=\Delta_2$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Claim 3.2
  • proof
  • Claim 3.3
  • proof
  • Claim 3.4
  • proof
  • ...and 9 more