Discrete Choices under Social Influence:Generic Properties

We consider a model of socially interacting individuals that make abinary choice in a context of positive endogenous externalities. The modelencompasses, as particular cases, several models presented in the sociologyand economics literature. We mainly (but not only) decline the modelwithin a market context: the binary choice is to buy or not a given goodat a price posted by a monopolist, and the latter determines the price inorder to maximize his profit. Putting the price to zero in the customerssystem allows to model adoption of norms and other collective issues. Thispaper extends previous result to the case of a generic distribution of theIndividual Preferences or Willingnesses to Pay (IWP).We show that the model properties depend on the modality class (thenumber of maxima) of the IWP probability distribution function (pdf),and on its smoothness properties. The main results are summarized onphase diagrams whose axes are the parameters of the model, namely, thedifference between the average IWP over the population and the postedprice, and the strength of the social component (externality), both measuredin units of the variance of the IWP distribution. In such phasediagram we represent the boundaries limiting regions where the systempresents different types of equilibria (e.g. unique or multiple).We derive generic properties of the demand (or adoption) curve, whichare shown to depend only on qualitative features of the population heterogeneity:the modality (number of maxima), the smoothness, the variance(second moment) and the type of support (compact or infinite) of the underlyingIWP distribution. These alone determine the qualitative featuresof the collective outcome.The case of a monomodal pdf (a single maximum) is investigated indetails. When the social influence is small with respect to the width of theIWP distribution, the aggregate demand exhibits features similar to theone of "classical" demand curves in economics, with a continuously decreasingadoption rate with increasing prices. However, beyond a criticalvalue of the ratio between social influence/variance of the IWP distributionthe inverse demand function exhibits a non classical behaviour: itis a decreasing function at low and high adoption rates but has a nonmonotonic behaviour in some intermediate range reaching a maximum atsome (possibly large) value of the adoption rate – and this even if thedistribution of preferences is monomodal –. As a result, depending on thedifference between the average IWP and the price, there are either oneor two stable equilibria. Thus, taking into account a positive (additive)externality in the market context yields a more general family of demandcurves that the classical ones.Finally, we study the generic properties of a monopolistic market. Fora large enough social influence strength (compared to the variance of theIWP distribution), the optimal strategy for the monopolist exhibits adrastic jump from one with a high price and a low number of customers,to one of low price and a large number of customers. This discontinuityarises even in the parameter region where the customers system exhibitsa single equilibrium.