Albrecht, J., Cai, X., Gautier, P. and Vroman, S. (2023). On the foundations of competitive search equilibrium with and without market makers Journal of Economic Theory, 208:1--33.

The literature offers two interpretations of competitive search equilibrium, one based on a Nash approach and the other on a market-maker approach. When each buyer visits only one seller, the two approaches are equivalent. However, when each buyer visits multiple sellers, this equivalence can break down. We present a model in which every buyer visits 2 sellers. A buyer who trades with one seller receives a value of s, while a buyer who trades with 2 sellers receives value 1. Letting s vary from 0 (perfect complements) to 1 (perfect substitutes) we characterize the competitive search equilibrium under the two interpretations. We show that for low values of s, the Nash and market-maker competitive search equilibria coincide, but the common equilibrium is inefficient. For intermediate values of s, the two equilibria again coincide and are efficient. Finally, for high values of s, the Nash and market-maker equilibria differ, and only the latter is efficient.