Re-matching, Experimentation, and Cross-subsidization

We study mediated many-to-many matching in dynamic two-sided markets in which agents private valuations for potential partners evolve stochastically over time, either as the result of exogenous shocks, or as the result of experimentation. In many environments, the platforms pro t-maximizing matching rule is either myopic or takes the form of a "virtual index rule" capturing the current and future pro\u85tability of each link between agents, accounting for the endogenous changes in the partnersmatching values. We show how the optimal matching rules can be implemented via a sequence of scoring auctions. We contrast matching dynamics under pro t maximization with their counterparts under welfare maximization. When all agents bene\u85t from interacting with all other agents from the opposite side, pro\u85t maximization involves fewer and shorter interactions. This conclusion, however, need not hold when certain agents dislike certain interactions