The auction shows the design deciding the outcome of a market that still clears with money. But some of the most important markets have no prices at all — not because no one has thought to set them, but because money is forbidden, or useless, or repugnant. A child is assigned to a public school, but the seat is not sold. A new doctor is matched to a hospital for training, but the post does not go to the highest bidder. A patient receives a donated kidney, but the organ may not be bought. In these markets the thing to be allocated is scarce and the parties have preferences, as in any market; but the price, the great clearing device of every market the price families studied, is unavailable. The allocation must be accomplished by rules alone — and here, where there is no price to do the work, the design must do all of it.
The problem in such markets is one of matching: pairing the members of two sides — students and schools, doctors and hospitals, in the simplest case the partners contemplating a marriage — each of whom has preferences over the other side. What makes a matching good? The key idea, which Lloyd Shapley and David Gale introduced, is stability. A matching is stable if there is no pair who would both prefer each other to their assigned partners — no student and school, say, who are not matched to each other but would each rather be. Such a pair is a blocking pair, and if one exists, the matching is fragile: the two will find each other and defect, undoing the assignment. A stable matching has no blocking pair; it is self-enforcing, in the sense that no two participants can both do better by abandoning it. Stability is to a matching market what market-clearing is to a price market: the condition under which the allocation holds together, with no one able to improve by going around it.1
Gale and Shapley did not merely define a good matching; they gave a rule — an algorithm — that always finds one. In the deferred-acceptance procedure, one side proposes and the other tentatively accepts. Each student applies to his favorite school; each school tentatively holds its favorite applicants up to its capacity and rejects the rest; each rejected student applies to his next favorite; each school again holds its favorites from the new pool together with those it was holding, perhaps releasing one it now likes less; and so on, the rejected always reapplying and the holders always keeping their best so far, until no one is rejected and the tentative holds become final. Gale and Shapley proved that this procedure always terminates in a stable matching — that the simple discipline of proposing, tentatively holding, and deferring final acceptance until the end is guaranteed to leave no blocking pair. It is a remarkable thing: a rule that, run mechanically, solves a problem of mutual preference that no central planner could solve by inspection, and solves it stably every time.2
The deferred-acceptance rule has subtle properties that matter to its design. Which side proposes turns out to matter: the proposing side gets its best possible stable matching, and the other side its worst. When students propose, every student gets the best school he could have in any stable matching, and every school the worst student it could have — and conversely if schools propose. So the very same idea of stability admits more than one stable matching, and the rule's design, in choosing who proposes, chooses among them, favoring the proposers. The rule has a second valuable property: for the proposing side, it is strategy-proof — no participant can do better by misrepresenting his preferences; honesty is the best policy, as Roth's criterion for a good market demanded. This is precious in a market without prices, where the temptation to game the assignment is strong: a well-designed matching rule lets participants simply tell the truth about what they want, and assigns them as well as any rule could, stably and without reward for strategizing.3
These ideas are not academic. The medical residency match — the system that places thousands of new doctors into hospital training posts each year — was redesigned by Alvin Roth and colleagues around the deferred-acceptance algorithm, replacing a process that had unraveled into chaos with one that produces a stable, strategy-proof match. School-choice systems in great cities were rebuilt on the same foundation, so that families could rank schools honestly and be assigned fairly, without the strategic agonizing the old systems forced. And most striking of all, the algorithm's logic was extended to kidney exchange: patients with willing but incompatible donors are matched in chains and cycles — my donor gives to you, yours to another, another's to me — so that incompatible pairs become compatible transplants, by a designed matching that money is forbidden to arrange. Here the design does not merely allocate efficiently; it saves lives that no market with prices was permitted to save. The matching market is design at its most consequential, doing with rules alone what no price was allowed to do.4
The match completes the demonstration the auction began. In the auction, the design decides the outcome of a market that money still clears; in the match, money cannot clear at all, and the design does everything — turning a tangle of mutual preferences into a stable, fair, honest assignment by the discipline of a rule. Together they establish the author's claim past doubt: that the rules of a market shape its outcome as much as, and sometimes entirely more than, the competition within it; that markets are made, and made well or badly, by the rules their designers choose. But all of this has been the bright face of market design — the engineer improving the world, clearing the medical match, saving the dying. There is a darker face, which the volume must now turn to, and which the author names at the center of the subject. For the rules are chosen by someone; and the someone has interests. Who designs the market, and in whose interest? The bright engineer is one answer; the captured regulator is another. The next plate turns from the design to the designer. The next plate is the interest.5
The matching markets are the purest case of the volume's thesis: markets without prices, where the design must do all the work, and does it. Gale and Shapley defined what a good matching is — a stable one, with no pair who would defect — and gave a rule, the deferred-acceptance algorithm, that finds one every time, favoring whichever side proposes and rewarding honesty over strategy. And the rule, carried into the world, has placed doctors, assigned schoolchildren, and matched the sick with the kidneys that save them, accomplishing by design what money was forbidden to accomplish. There is no clearer proof that the market is its rules, and the rules a choice. But every design has a designer, and every designer an interest; and the volume must now ask the question its bright examples have postponed. Who chooses the rules, and for whom? The next plate is the interest.