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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 the products. Their decision of which product to choose 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<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1080/15427951.2011.579849">doi:10.1080/15427951.2011.579849</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e5dlpppybnf7dakn6dlepxotkq">fatcat:e5dlpppybnf7dakn6dlepxotkq</a> </span>
more »... of social influence and the inherent fitness of the products. In a special case, when the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and with strong enough social influence, monopoly occurs. These results support the observations made by Salganik et. al.  in their experimental study of cultural markets.
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