Stable pairwise matching won Lloyd Shapley and Alvin Roth the Nobel prize for economics. It is an idea that is simple, slightly illuminating for economists, occasionally useful for everyone – and profoundly misleading.
The matches in question are between members of two groups, for example potential husbands and potential wives, or medical school graduates and hospitals that might employ them. The “stable” is defined narrowly: the pairing off is stable as long as no individual can find a way to improve his or her situation by trading partners. What counts as “improvement”? The game theory of Shapley and Roth does not really address that question.
The simple idea, demonstrated by Shapley a half century ago, is that under certain conditions a methodical process of elimination – many rounds of tentative pairings – leads to stability. Take a pool of equal numbers of would-be brides and grooms. The men keep on proposing to their favoured women. At first, only the irresistible men garner acceptances from the most appealing women. Gradually, though, each less attractive man will win the favour of some less attractive woman, who accepts the sad reality that she cannot do any better. At the end, while many people may wish they had a different spouse, no one will be able to arrange a trade. Any alternative pairing will be less desirable than the current one to one side or the other. That is exactly game theory stability.
The research is illuminating for economists because it teaches them that money is not needed to arrange an efficient allocation. Economists used to assume, and many still do, that cash markets are the best way to ensure that everyone is able to satisfy as many of his or her desires as possible. Shapley showed that in matching, under certain conditions and by some definitions, nothing more is needed than clear and consistent rankings of potential partners.
The illumination should be slight. Indeed, it probably takes a few years of economic training to be surprised that monetary values do not always lurk behind effective allocation decisions. After all, parents need neither game theory nor cash to divide a cake among the children. Money rarely plays a direct role in deciding who gets what in government programmes.
Roth showed that the Shapley technique is socially useful. He designed “clearinghouses” which have successfully matched doctors with hospitals, students with schools and kidney donors with recipients. The accomplishment is real, but limited. Roth’s clearinghouses require unusual conditions – not only consistent rankings of possible partners but also a single pairing decision, limited and basically equal information on both sides and the existence of alternatives which are fairly close substitutes.
Such clearinghouses are not suitable for most important allocation decisions. Marriages, for example, feature in Shapley’s original explanation of the technique, but few people would actually sign up for any sort of marriage clearinghouse. The choice of a life mate is too complicated to be trusted to an algorithm. In the West, the uncertainty of courtship is an integral part of the effort to create a bond that will be truly stable – able to last through thick and thin.
The unsuitability of a marriage clearinghouse hints at why the Nobel-winning idea is deeply misleading. It, like game theory as a whole, relies on four false assumptions about human nature.
First, satisfaction is defined in terms of preferences which may be arbitrary, unfair and temporary. That is dangerously simplistic. The hardest part of match-making is evaluating what the various candidates are really like and getting a good fit. Marriage match-makers, employment headhunters and university admissions officers earn their keep.
Second, game theory is individualistic, paying almost no attention to the social context of decisions. People, however, are profoundly social. Allocation shapes, and is shaped by, society. A serious study cannot simply ignore society’s needs and desires.
Third, the allocation games have no moral dimension. People are free to think about morality when they draw up their lists of favourites, but just as they are free to think about astrology. In reality, though, people should search for the truly good, and not merely for the pleasurable. In general, they do make an effort to be virtuous.
Finally, pairwise matching makes allocation into a non-monetary form of what economists call a market process: a collection of interactions between a large number of independent and fundamentally self-interested potential providers and users. In reality, such markets often make people uncomfortable. They prefer to decide on who gets what by co-operation, through committees or consensus, or by trusting some authority – whether a parent, a boss or a bureaucrat – to make just decisions.
There are times when it is helpful to treat life like a game. Shapley and Roth deserve thanks, and their prize, for showing how to design a good game. Life, however, should be taken more seriously.