Discrete Choices under Social Influence:<br />Generic Properties

We consider a model of socially interacting individuals that make a<br />binary choice in a context of positive endogenous externalities. The model<br />encompasses, as particular cases, several models presented in the sociology<br />and economics literature. We mainly (but not only) decline the model<br />within a market context: the binary choice is to buy or not a given good<br />at a price posted by a monopolist, and the latter determines the price in<br />order to maximize his profit. Putting the price to zero in the customers<br />system allows to model adoption of norms and other collective issues. This<br />paper extends previous result to the case of a generic distribution of the<br />Individual Preferences or Willingnesses to Pay (IWP).<br />We show that the model properties depend on the modality class (the<br />number of maxima) of the IWP probability distribution function (pdf),<br />and on its smoothness properties. The main results are summarized on<br />phase diagrams whose axes are the parameters of the model, namely, the<br />difference between the average IWP over the population and the posted<br />price, and the strength of the social component (externality), both measured<br />in units of the variance of the IWP distribution. In such phase<br />diagram we represent the boundaries limiting regions where the system<br />presents different types of equilibria (e.g. unique or multiple).<br />We derive generic properties of the demand (or adoption) curve, which<br />are shown to depend only on qualitative features of the population heterogeneity:<br />the modality (number of maxima), the smoothness, the variance<br />(second moment) and the type of support (compact or infinite) of the underlying<br />IWP distribution. These alone determine the qualitative features<br />of the collective outcome.<br />The case of a monomodal pdf (a single maximum) is investigated in<br />details. When the social influence is small with respect to the width of the<br />IWP distribution, the aggregate demand exhibits features similar to the<br />one of “classical” demand curves in economics, with a continuously decreasing<br />adoption rate with increasing prices. However, beyond a critical<br />value of the ratio between social influence/variance of the IWP distribution<br />the inverse demand function exhibits a non classical behaviour: it<br />is a decreasing function at low and high adoption rates but has a non<br />monotonic behaviour in some intermediate range reaching a maximum at<br />some (possibly large) value of the adoption rate – and this even if the<br />distribution of preferences is monomodal –. As a result, depending on the<br />difference between the average IWP and the price, there are either one<br />or two stable equilibria. Thus, taking into account a positive (additive)<br />externality in the market context yields a more general family of demand<br />curves that the classical ones.<br />Finally, we study the generic properties of a monopolistic market. For<br />a large enough social influence strength (compared to the variance of the<br />IWP distribution), the optimal strategy for the monopolist exhibits a<br />drastic jump from one with a high price and a low number of customers,<br />to one of low price and a large number of customers. This discontinuity<br />arises even in the parameter region where the customers system exhibits<br />a single equilibrium