“Competing Pre-match Investments Revisited: A Characterization of Monotone Bayes-Nash Equilibria in Large Markets”
Abstract: We solve an open problem pertaining to the relationship between competitive and non-cooperative models of pre-match investment. We study an incomplete information version of Peters and Siow’s (2002) model of competing pre-marital investments and NTU matching, with finitely many agents and i.i.d. types. Our main result is a precise characterization of side-symmetric, strictly monotone Bayes-Nash equilibrium (SSMBNE) behavior when the market grows large and the empirical type distributions converge to those of an unbalanced continuum economy à la Peters and Siow (2002). We find that the limits of SSMBNE strategies always differ from competitive (hedonic) equilibrium strategies, and that there is always inefficient overinvestment on both sides of the market. Our analysis relies on a novel way of using results from the theory of approximate distributions of order statistics that allows us to characterize equilibrium behavior even though the limit strategies must be discontinuous and the size of the discontinuities is determined by a complex, two-sided interaction. These techniques should be useful for the analysis of other large Bayesian games with discontinuous payoffs and interacting distributions of outcomes.
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