Continuity of the first price auction Nash equilibrium correspondence. (Q1852669)

In the textbook first a price auction, where an item is to be sold to the highest bidder, players have comonly known valuations \((v_1,\dots,v_n)\) of the object, and strategies are bid vectors \((b_1,\dots,b_n)\) is considered. In the general, in more realistic versions studied in current literature valuations are drawn from a commonly known vector of independent probability measures \((F_1,\,F_n)\), but are private information. Strategies are basically distributions over pairs of valuations and bids; but to avoid using (privately known) valuations to break ties, players also send messages as part of their strategies. The present paper studies the asymmetric case where the \(F_i\)'s are not necessarily identically distributed, and shows that the equilibrium correspondence of such games has a nonempty closed graph. Given compactness of the domain, this is equivalent to upper hemicontinuity (in the weak topologies of domain and range spaces). Under conditions, derived in companion papers of the same author, ensuring uniqueness of equilibrium, this gives continuity of the equilibrium map. And continuity gives robustness to comparative statics results. Examples of utilization of continuity in the latter way are provided.