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We analyse what happens when heterogeneous players can decide whether to enter and play a “Beauty Contest Game” (BCG) or stay out and get their common, fixed reservation utlity. We find that, if the BCG is “attractive” then at least some people enter, and at least one (stable) equilibrium always exist. Further, for low levels of “attractiveness” multiple equilibria exist: as many as extrema in the density function of types in the population. We find that the equilibria associated with maxima (minima) are stable (unstable). We characterise equilibra based on the size of the population of entrants and on the types that enter, and classify them into 4 categories that cover the full spectrum of possibilities. As an application, we consider a scenario where two distinct populations “merge”. The model could be helpful to understand, analyse and suggest policies for issues on political economy, immigration and education policy, among others.
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