We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of a number of available products. Their choice of product is a stochastic function of the inherent quality of the product and its market share. Using techniques from stochastic approximation theory, we show that market shares converge to an equilibrium. We also derive the market shares at equilibrium in terms of the level of social influence and the inherent quality of the products. In a special case, in which the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and that with strong enough social influence, monopoly occurs. These results support the observations made in the experimental study of cultural markets in Salganik et al., Experimental Study of Inequality and Unpredictability in an Artificial Cultural Market..
"Social Influence and Evolution of Market Share." Internet Math. 7 (2) 107 - 134, 2011.