Entry in Beauty-Contest Games

We study how voluntary participation in Beauty Contest Games (BCGs) affects the actions and payoffs of type-heterogenous players. In a BCG, players have two goals --one personal, the other social-- and so BCGs appropriately model relevant economic situations like participating in a social network, partaking in the coding of an open-source software, or the choice of research topics by academics, among others. Key in these and other cases is the concept of "social norm" that will emerge in the associated "community", and so people's entry choices will depend crucially on their expectations regarding not only how many but who (which types) will join in. We find that in equilibrium there is entry as long as the BCG is "attractive" and that there might be multiple equilibria, each indexed by its associated social norm. We also find that, when finite, there is an odd number of equilibria and that --if ordered based on the value of the associated social norm-- odd/even equilibria are stable/unstable. Further, for low attractiveness, equilibrium social norms are univocally associated with the extrema of the distribution of types in the economy, so that stable/unstable equilibria are linked to maxima/interior minima. We find that "universal" communities in which everybody joins the BCG (as implicitly assumed by the literature) only occur when the BCG is sufficiently attractive and the economy's average type is not extreme. In non-universal communities, social norms are affected by the attractiveness of the BCG and the functional form of the distribution of types in the economy (especifically, its skewness around extrema). Attractiveness affects both the size and the composition of the community. Thus, an increase in attractiveness could lead both to the entry of new members and to the exit of some others.