On Matching Markets

Abstract: This thesis gives a contribution to matching theory. It examines three one-to-one matching models: the roommate problem, the partnership formation problem and the assignment game. In the former two, it is known that stable matchings may not exist. In the latter, stable outcomes do exist; we examine if they still do if we generalize the model. The thesis consists of three separate papers. In the first paper, "When Do Stable Roommate Matchings Exist? A Review", I compare different preference restrictions that ensure the existence of a stable roommate matching. Some of these restrictions are generalized to allow for indifferences as well as incomplete preference lists, in the sense that an agent may prefer remaining single to matching with some agents. I also introduce a new type of cycles and in greater detail investigate the domain of preferences that have no such cycles. In particular, I show how the absence of these cycles relates to the "symmetric utilities hypothesis" by Rodrigues-Neto (Journal of Economic Theory 135, 2007) when applied to roommate problems with weak preferences. The second paper, "A Competitive Partnership Formation Process" (co-authored with Tommy Andersson, Dolf Talman, and Zaifu Yang) considers a group of heterogeneous agents who may form partnerships in pairs. All single agents as well as all partnerships generate values. If two agents choose to cooperate, they need to specify how to split their joint value among one another. In equilibrium, which may or may not exist, no agents have incentives to break up or form new partnerships. The paper proposes a dynamic competitive adjustment process that always either finds an equilibrium or exclusively disproves the existence of any equilibrium in finitely many steps. When an equilibrium exists, partnership and revenue distribution will be automatically and endogenously determined by the process. Moreover, several fundamental properties of the equilibrium solution and the model are derived. In the third paper, "Assignment Games with Externalities" (co-authored with Helga Habis), we introduce externalities into a two-sided, one-to-one assignment game by letting the values generated by pairs depend on the behavior of the other agents. Extending the notion of blocking to this setup is not straightforward; a pair has to take into account the possible reaction of the residual agents to be able to assess the value it could achieve. We define blocking in a rather general way that allows for many behavioral considerations or beliefs. The main result of the paper is that a stable outcome in an assignment game with externalities always exists if and only if all pairs are pessimistic regarding the others' reaction following a deviation. The relationship of stability and optimality is also discussed, as is the structure of the set of stable outcomes.

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