Exclusive nightclubs and lonely hearts columns: Non-monotone participation in optional intermediation

► The paper studies markets where intermediation is possible. ► Intermediation is used by some traders only. ► The paper shows that sandwich equilibria exist. ► In a sandwich equilibrium, only middle type traders use intermediation. ► Examples are: dating agencies, job exchange, tourist operators and the housing market. In many decentralised markets, the traders who benefit most from an exchange do not employ intermediaries even though they could easily afford them. At the same time, employing intermediaries is not worthwhile for traders who benefit little from trade. Together, these decisions amount to non-monotone participation choices in intermediation: only traders of middle “type” employ intermediaries, while the rest, the high and the low types, prefer to search for a trading partner directly. We provide a theoretical foundation for this, hitherto unexplained, phenomenon. We build a dynamic matching model, where a trader's equilibrium bargaining share is a convex increasing function of her type. We also show that this is indeed a necessary condition for the existence of non-monotone equilibria.