It is well-known that in the absence of transfers, stable matchings are positive assortative when... more It is well-known that in the absence of transfers, stable matchings are positive assortative when agents’ preferences are strictly monotonic in each other’s attributes and attributes are commonly known. Instead, monotonicity is consistent with the existence of stable matchings that exhibit negative sorting when the attributes on one side of the market are private information. This paper sheds light on the scope of this consistency, showing that in most monotonic markets not every agent can know that a stable matching is negative, but not positive, assortative.
We explore the relationship between evidence and knowledge, when knowledge is described by a part... more We explore the relationship between evidence and knowledge, when knowledge is described by a partition of a finite state space and evidence is represented by a collection of sets of messages that is measurable with respect to the partition. We show that there is evidence for an event only if the event is self-evident—i.e., known at every state. Thus, an event is public—commonly known at all of its states—whenever there is mutual evidence for it. It follows that the notion of mutual evidence captures the infinite regress embedded in the notion of common knowledge. We show that these results offer a novel foundation for understanding knowledge and common knowledge, implying that total provability is possible only if the partition is either the coarsest or the finest one. We also show that the existence of partial provability outside these two extremes hinges on the non-monotonic nature of provability.
Many matching markets run clearinghouses that impose quotas to achieve some desirable allocation.... more Many matching markets run clearinghouses that impose quotas to achieve some desirable allocation. Achieving quotas that are not satisfied by any stable allocation requires enforcement ex post, once individuals have been allocated. When participation in the clearinghouse is voluntary and non-participating agents are not bounded by the desired quotas, however, achieving the desired quotas requires, in addition, full participation ex ante. Thus, deterministic clearinghouses can achieve no unstable distributional goal. This paper explores the scope of random clearinghouses as an effective redesign. I show that a random mechanism inducing full participation exists if and only if agents’ preferences are sufficiently heterogeneous. The main finding is a polynomial-time algorithm that identifies the class of quotas that cannot be achieved by any random mechanism.
It is well-known that in the absence of transfers, stable matchings are positive assortative when... more It is well-known that in the absence of transfers, stable matchings are positive assortative when agents’ preferences are strictly monotonic in each other’s attributes and attributes are commonly known. Instead, monotonicity is consistent with the existence of stable matchings that exhibit negative sorting when the attributes on one side of the market are private information. This paper sheds light on the scope of this consistency, showing that in most monotonic markets not every agent can know that a stable matching is negative, but not positive, assortative.
We explore the relationship between evidence and knowledge, when knowledge is described by a part... more We explore the relationship between evidence and knowledge, when knowledge is described by a partition of a finite state space and evidence is represented by a collection of sets of messages that is measurable with respect to the partition. We show that there is evidence for an event only if the event is self-evident—i.e., known at every state. Thus, an event is public—commonly known at all of its states—whenever there is mutual evidence for it. It follows that the notion of mutual evidence captures the infinite regress embedded in the notion of common knowledge. We show that these results offer a novel foundation for understanding knowledge and common knowledge, implying that total provability is possible only if the partition is either the coarsest or the finest one. We also show that the existence of partial provability outside these two extremes hinges on the non-monotonic nature of provability.
Many matching markets run clearinghouses that impose quotas to achieve some desirable allocation.... more Many matching markets run clearinghouses that impose quotas to achieve some desirable allocation. Achieving quotas that are not satisfied by any stable allocation requires enforcement ex post, once individuals have been allocated. When participation in the clearinghouse is voluntary and non-participating agents are not bounded by the desired quotas, however, achieving the desired quotas requires, in addition, full participation ex ante. Thus, deterministic clearinghouses can achieve no unstable distributional goal. This paper explores the scope of random clearinghouses as an effective redesign. I show that a random mechanism inducing full participation exists if and only if agents’ preferences are sufficiently heterogeneous. The main finding is a polynomial-time algorithm that identifies the class of quotas that cannot be achieved by any random mechanism.
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