We close with a few open questions and suggestions of possible directions for further research. In keeping with the emphasis of the book on both analysis and modeling, some of the questions and directions are of each type; that is, some call for the statement and proof of theorems, whereas progress on others will (first) involve the construction of new models.

1. Since many entry level labor markets and other two-sided matching situations don't employ centralized matching procedures, and yet aren't observed to experience the kinds of market failure that seem to be associated with unstable matching, we can conjecture that at least some of these markets reach stable outcomes by means of decentralized decision making. So one of the chief modeling problems that will arise in studying such markets will be to develop decentralized models of stable matching. (We noted that a consequence of Theorem 2.33 is that a random process that begins from an arbitrary matching and continues by satisfying a randomly selected blocking pair must eventually converge with probability one to a stable matching, provided each blocking pair has a probability of being selected that is bounded away from zero. Perhaps this kind of result will provide the building blocks for models of decentralized matching.)

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