Submitted to manuscript Fashion and Homophily

Fashion, as a “second nature ” of human beings, plays a quite non-trivial role not only in economy but also in many other areas, including even academia. We classify agents pursuing fashion into two types, conformists and rebels, who prefer to match and mismatch the majority action of their neighbors, respectively, and investigate their interactions through a social network. We are interested in how the structure of the social network affects the evolution of fashion, especially the emergence of fashion cycles. The conclusion is quite clean: homophily, in general, inhibits the emergence of fashion cycles. That is, to have the emergence of fashion cycles, conformists and rebels should be adequately mixed. This improves the insightful observation of Simmel (1904) that heterogeneity of consumers is essential to fashion. To establish the main result, a novel partial potential analysis is invented and recent approximation skills, which are developed by theoretical biologists and are in essence an elaborate mean-field method, are applied. Technically, this network game (referred to as the fashion game) can also be taken as a network extension of three elementary games: the coordination game, the anti-coordination game, and Matching Pennies. Through partial potential analysis, we show that the fashion game always possesses a pure Nash equilibrium if each conformist has no less conformist neighbors than rebel ones, improving a result of Jackson (2008). Through approximation skills, we transform the stochastic best response dynamic of the fashion game into a system of ordinary differential equations and give a fairly complete analysis of it. Simulations show that this approximation is quite reliable. Key words: network games; fashion cycle; homophily; Matching Pennies; partial potential analysis 1