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

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.