DISCRETE CHOICES UNDER SOCIAL INFLUENCE: GENERIC PROPERTIES - Gordon, Mirta B. and Nadal, Jean-Pierre and Phan, Denis and Semeshenko, Viktoriya
MATHEMATICAL MODELS \& METHODS IN APPLIED SCIENCES 19, 1441-1481 (2009)
Abstract : We consider a model of socially interacting individuals that make a
binary choice in a context of positive additive endogenous
externalities. It encompasses as particular cases several models from
the sociology and economics literature. We extend previous results to
the case of a general distribution of idiosyncratic preferences, called
here Idiosyncratic Willingnesses to Pay (IWP). When j, the ratio of the
social influence strength to the standard deviation of the IWP
distribution, is small enough, the inverse demand is a classical
monotonic (decreasing) function of the adoption rate. However, even if
the IWP distribution is mono-modal, there is a critical value of j above
which the inverse demand is non-monotonic. Thus, depending on the price,
there are either one or several equilibria.
Beyond this first result, we exhibit the generic properties of the
boundaries limiting the regions where the system presents different
types of equilibria (unique or multiple). These properties are shown to
depend only on qualitative features of the IWP distribution: modality (
number of maxima), smoothness and type of support (compact or infinite).
The main results are summarized as phase diagrams in the space of the
model parameters, on which the regions of multiple equilibria are
precisely delimited. We also discuss the links between the model and the
random field Ising model studied in the physics literature.