In the traditional marriage model, each agent is matched to a single partner once and for all. Many-to-many models, on the other hand, allow each agent to be matched to several partners. Between these two models lie problems where each agent can be matched to several partners, but only to one at a given time or to a probability distribution over partners. These are fractional matching problems. For an instance of a fractional matching problem one can think of a law firm with many partners and associates. Each partner cares about which associate works for him and each associate cares about which partner he works for. Yet an associate can split his time between working for different partners. Or one can think of a surgical residency program with several attending physicians and residents. A resident cares about whom he assists while an attending physician cares about who assists her. Another example is a market where traders care about the identities of their trading partners. Among all of the rationales for considering fractional matching, two stand out. The first is fairness: the restriction to discrete matches precludes treating two, otherwise identical, agents equally. The second is efficiency: shoehorning a problem into a discrete model leads to inefficiency. To illustrate this, think of the following very unhappy Gale–Shapley world involving two boys Andy and Bob and two girls Yvonne and Zelda at a dance. Andy would like to dance with Yvonne. Unfortunately, Yvonne likes Bob and would prefer to dance with him. Bob, however, is hoping to dance with Zelda who, in turn, wants to dance with Andy. To make matters worse, suppose that each of them would rather not dance than dance only with the less preferred partner. Under the restriction that a pair can dance together the whole night or not at all, the only reasonable thing to do is for everyone to go home without dancing at all. Yet, it could be that each of these boys and girls would rather spend half the evening dancing with each partner than not dance at all. If they are willing to accept fractional matching, we could arrange it so that Andy dances with Yvonne and Bob dances with Zelda for the first half of the evening and for the second half Andy dances with Zelda and Bob dances with Yvonne. The focus of this paper is to define and show existence of competitive equilibria, thereby establishing that these problems can reasonably be dealt with as decentralized markets. I start with a version of the model where agents are not only matched, but also consume a tradable good that one might call “money.”۱ For such economies, I define and show existence of competitive equilibria (Theorem 1). A key insight to defining an equilibrium is that the goods that agents are trading (other than money) are not private goods. These equilibria involve price systems that are “double indexed” so that two different agents need not necessarily pay the same price to be matched with the same partner. Thus, I refer to these equilibria as double-indexed price (DIP) equilibria. The proof of existence involves some intricacies to handle the lack of free disposal: in our earlier example, Zelda cannot “discard” the time that Andy spends dancing with her. Next, I turn to a model without money.