## Nobel 2012, Roth and Shapley

There will be many posts summarizing the modern market design aspect of Roth and Shapley, today’s winners of the Econ Nobel. So here let me briefly discuss certain theoretical aspects of their work, and particularly my read of the history here as it relates to game theory more generally. I also want to point out that the importance of the matching literature goes way beyond the handful of applied problems (school choice, etc.) of which most people are familiar.

Pure noncooperative game theory is insufficient for many real-world problems, because we think that single-person deviations are not the only deviations worth examining. Consider marriage, as in Gale and Shapley’s famous 1962 paper. Let men and women be matched arbitrarily. Do we find such a set of marriages reasonable, meaning an “equilibrium” in some sense? Assuming that every agent prefers being married (to anyone) to being unmarried, then any set of marriages is a Nash equilibrium. But we find it reasonable to think that two agents, a man and a woman, can commit to jointly deviate, breaking their marriage and forming a new one. Gale and Shapley prove that there always exists a match that is “pairwise stable” meaning that no pair of men and women wish to deviate in this way.

Now, if you know your game theory, you may be thinking that such deviations sound like a subset of cooperative games. After all, cooperative (or coalitional) games involve checking for deviations by groups of agents, who may or may not be able to arbitrarily distribute their joint utility among their coalition. Aspects of such cooperation are left unmodeled in their noncooperative sense. It turns out (and I believe this is a result due to Mr. Roth, though I’m not sure) that pairwise stable matches are equivalent to the (weak) core of the same cooperative game in one-to-one or many-to-one matching problems. That means both that checking deviations by one potential set of marrying partners is equivalent to checking deviations by any sized group of marrying partners. But more importantly, this link between the core and pairwise stability allows us to utilize many results in cooperative game theory, known since the 1950s and 60s, to answer questions about matching markets.

Indeed, the link between cooperative games and matching, and between cooperative and noncooperative games, allows for a very nice mathematical extension of many well-known general problems: the tools of matching are not restricted solely to school choice and medical residents, but indeed can answer important questions about search in labor markets, about financial intermediation, etc. But to do so requires reframing matching as simply mechanism design problems with heterogeneous agents and indivisibility. Ricky Vohra, of Kellogg and the Leisure of the Theory Class blog, has made a start at giving tools for such a program in his recent textbook; perhaps this post can serve as a siren call across the internet for Vohra and his colleagues to discuss some examples on this point on their blog. The basic point is that mechanism design problems can often be reformulated as linear programs with a particular set of constraints (say, integer solutions, or “fairness” requirements, etc.). The most important set of constraints, surely, are incomplete information which allows for strategic lying, as Roth discovered when he began working on “strategic” matching theory in the 1980s.

My reading of much of the recent matching literature, and there are obviously exceptions of which Roth and Shapley are both obviously included as well as younger researchers like Kojima, is that many applied practitioners do not understand how tightly linked matching is to classic results in mechanism design and cooperative games. I have seen multiple examples, published in top journals, of awkward proofs related to matching which seem to completely ignore this historical link. In general, economists are very well trained in noncooperative game theory, but less so in the other two “branches”, cooperative and evolutionary games. Fixing that imbalance is worthwhile.

As for extensions, I offer you a free paper idea, which I would be glad to discuss at further length. “Repeated” matching has been less often studied. Consider school choice. Students arrive every period to match, but schools remain in the game every period. In theory, I can promise the schools better matches in the future in exchange for not deviating today. The use of such dynamic but consistent promises is vastly underexplored.

Finally, who is left for future Nobels in the areas of particular interest to this blog, micro theory and innovation? In innovation, the obvious names are Rosenberg, Nelson and Winter; Nelson and Winter’s evolutionary econ book is one of the most cited texts in the history of our field, and that group will hopefully win soon as they are all rather old. Shapley’s UCLA colleagues Alchian and Demsetz are pioneers of agency theory. I can’t imagine that Milgrom and Holmstrom will be left off the next micro theory prize given their many seminal papers (along with Myerson, they made the “new” game theory of the 70s and 80s possible!), and a joint prize with either Bob Wilson or Roy Radner would be well deserved. An econ history prize related to the Industrial Revolution would have to include Joel Mokyr. There are of course many more that could win, but these five or six prizes seem the most realistically next in line.