Market design

Market design is the part of economics that builds marketplaces and repairs broken ones. Mechanism design asks which rules could in principle produce a chosen outcome; market design takes one real market, with its history, its participants and the deals they can strike outside it, and works out the rules that will run it. Its best-known results are clearinghouses that match doctors to hospitals and children to schools, and exchanges that let patients whose willing donors are medically incompatible swap donors with one another. The field rests on one idea: a market is an institution with rules, its rules fail in a few recognisable ways, and each failure can be repaired by design.

Alvin Roth, who led several of those designs, set out the method in his Fisher–Schultz Lecture, published in Econometrica in 2002 as The Economist as Engineer. A designer takes responsibility for detail and has to deal with all of a market's complications, so the simple models used for theory are not enough on their own. Roth compared the work to bridge building, where a model with only gravity and rigid beams is elegant and general, and the effects of wind, water and soil are left to engineers working with physical and computational models. He argued that laboratory experiments and computation stand in the same relation to game theory, and that designers should report what they built in enough detail for knowledge about design to accumulate. He retold the main designs for general readers in Who Gets What – and Why (2015).

What a market has to do

Roth drew the general lessons from these projects in What Have We Learned from Market Design?, published by the National Bureau of Economic Research (NBER) in 2009; a version opens The Handbook of Market Design (2013), which he edited with Nir Vulkan and Zvika Neeman. To work well, he argues, a marketplace has to do three things:

• Provide thickness: attract a large enough share of the potential participants, ready to transact with one another.
• Overcome the congestion that thickness brings: give participants enough time, or make transactions fast enough, that they can consider enough alternatives to reach satisfactory ones.
• Make participation safe and simple: taking part straightforwardly has to be better than transacting outside the market, and better than strategic behaviour that lowers welfare overall. In The Art of Designing Markets (Harvard Business Review, 2007) Roth states the same requirement as making it safe for participants to reveal or act on confidential information they hold, such as what they want.

He adds one constraint. Some transactions are repugnant: enough people object to them that they are banned or resisted even between willing parties. Buying and selling kidneys is illegal in most countries, so kidney exchange has to work without money.

The failures these tasks prevent have a documented history. From 1900 to 1945 American hospitals competed for new doctors by hiring them earlier than rival hospitals did, until some residents were being hired almost two years before they graduated. Roth calls this unraveling: offers become early, spread out in time and short-lived, so decisions are made before anyone knows their options. In 1945 the medical schools agreed not to release information about students before a fixed date, and the market congested instead. A hospital whose first offers were held under consideration found that its next candidates had already accepted elsewhere, so hospitals began making exploding offers that had to be answered on the spot, and agreements were missed and broken.

Roth draws two further conclusions. Thickness behaves like a public good, so a market has to resist free riders who trade early. And participants who find a market risky protect themselves in ways that damage it as a whole, which is how exploding offers arise.

Clearinghouses and stable matching

The American medical market was repaired in the early 1950s by a centralised clearinghouse, now called the National Resident Matching Program (NRMP). Students and hospitals still arrange interviews among themselves; then each side submits a rank-order list, and an algorithm produces the matching. From 1951 into the 1970s more than 95% of positions were filled through it. Roth later showed that its algorithm was equivalent to the deferred acceptance algorithm of David Gale and Lloyd Shapley (1962), run with hospitals making the offers. In deferred acceptance one side proposes in order of preference, the other side holds on to the best proposal it has received so far and rejects the rest, and nothing is final until no proposal is rejected. The result is stable: no applicant and hospital that were not matched to each other would both rather have been.

Roth compared clearinghouses across markets and found stability to be an important condition of success: stable mechanisms have mostly survived, and unstable ones have mostly failed and been abandoned. The closest comparison is between Britain's regional markets for new doctors, where the two regions using stable mechanisms succeeded and all but two of the rest failed. Because those regions also differed in other ways, a laboratory experiment by John Kagel and Roth held everything else fixed and compared the stable algorithm used in Edinburgh with the unstable one used in Newcastle. After some experience, the stable algorithm reduced costly early matching and the unstable one did not.

In 1995 the NRMP asked Roth to design a new algorithm. The Roth–Peranson design was adopted in 1997 and has since been taken up by other entry-level professional markets. The difficulty was that the simple theory does not describe the medical match. Married couples graduating together need two positions in the same place, and once couples are in the market a stable matching need not exist at all. The match lets couples rank pairs of positions, the same device the Federal Communications Commission adopted when it let bidders in its spectrum auctions bid on packages of licences. Roth argues that complementarities of this kind, where the value of one position or licence depends on getting another, are the problem the two designs share.

School choice

Many American school districts let families rank public schools and then assign children using each school's priorities, such as for siblings or for living nearby. Atila Abdulkadiroğlu and Tayfun Sönmez introduced this as a design problem in School Choice: A Mechanism Design Approach (2003). They analysed the plans used in Boston, Columbus, Minneapolis and Seattle, showed that they had serious shortcomings, and proposed two alternatives. Both are strategy-proof: stating true preferences is a dominant strategy, the property set out under mechanism design.

The first, a student-proposing version of deferred acceptance, respects priorities: no child loses a seat they wanted to a child with lower priority at that school. The second, top trading cycles, lets children trade the priorities they hold, and produces an assignment that cannot be changed to make one child better off without making another worse off, which economists call Pareto efficient. Neither mechanism has both properties.

Boston showed what the old plans cost. Until 2005 its mechanism gave as many children as possible their first choice, and only then looked at second choices. A family whose first choice was a school where its child had low priority risked losing its second choice as well, a local school where the child had high priority, because that school might fill with children who had listed it first. In Roth's Harvard Business Review account, families who studied capacities and demand made careful strategic choices and mostly got their stated first choice; families who did not sometimes got none of their choices and were assigned a school administratively. Boston replaced the mechanism with deferred acceptance, and its superintendent, Thomas Payzant, argued that a strategy-proof algorithm protects parents who do not strategise, or do not strategise well.

New York City's problem was congestion. Around 100,000 students had to be placed in roughly as many ninth-grade seats, and under the old procedure schools made offers without coordinating with one another, in rounds of letters. About 17,000 students received more than one offer, only half of all students received any offer in the first round, and after three rounds some 30,000 were still unplaced and were assigned at the last minute without regard to their preferences.

In 2003 the city's Department of Education brought in Roth, Abdulkadiroğlu and Parag Pathak, who recommended a clearinghouse that processes every list at once and gives each student a single offer, from the highest-ranked school that will admit them. The number placed administratively fell below 3,000. The clearinghouse also removed two reasons to game the system: principals had hidden empty seats so they could fill them outside the process, and students had distorted their lists because some schools admitted only applicants who ranked them first.

Reserves and quotas

Many allocation systems combine priorities with targets for groups. A reserve guarantees a number of units to members of a group, and a quota caps the number a group can receive. Chicago's exam schools reserve seats for applicants from different income groups, and in 2020 the US National Academies of Sciences, Engineering, and Medicine recommended reserving 10% of COVID-19 vaccines for people from hard-hit areas. Atila Abdulkadiroğlu and Aram Grigoryan's NBER working paper Priority-based Assignment with Reserves and Quotas (2021) starts from a question such rules leave open. Reserves and quotas do not settle who gets what, because the result depends on the order in which applicants are considered for reserved and open units.

Their example is one school with two seats, one reserved for low-income applicants and one open, and three applicants, of whom the first and third in priority are low-income. If applicants are considered for the open seat first, the top applicant takes it, the reserved seat goes to the third, and the second applicant, who is not low-income, loses a seat to someone with lower priority: a priority violation.

If applicants are considered for the reserved seat first, the top applicant takes it, the second takes the open seat, and nobody's priority is violated. The paper gives an axiomatic characterisation of a general class of reserves-and-quotas rules and shows that, within it, the rule that first considers every applicant for the units reserved for their own group uniquely minimises priority violations. Rules that order the steps differently can deliver more units to a favoured group only by creating more violations.

Kidney exchange

Tayfun Sönmez and M. Utku Ünver set out the problem in Market Design for Kidney Exchange, their chapter in the Handbook. In the United States the National Organ Transplant Act of 1984 makes buying or selling a kidney illegal, so donation is the only source. A kidney from a living donor survives longer than one from a deceased donor, and a healthy person can donate one of their two kidneys, usually to a relative or friend. Often the donor cannot give to the intended recipient, because their blood types are incompatible or the recipient has antibodies to the donor's proteins. A paired exchange solves this between two such pairs: each donor gives a kidney to the other pair's recipient. Longer cycles of pairs, and chains started by a donor who gives to a stranger, extend the idea.

Before 2004 such exchanges were rare, because donors found to be incompatible were usually sent home and the medical data that could have matched them to another patient was never collected; in Roth's terms, there was no thick market. In 2004 Roth, Sönmez and Ünver proposed a mechanism that makes it a dominant strategy for patients to reveal their preferences and all their willing donors, and the same year they worked with the surgeon Francis Delmonico and the tissue-typing specialist Susan Saidman to establish the New England Program for Kidney Exchange, which unites the region's 14 transplant centres.

Practice then imposed a constraint the first mechanism ignored. All the transplants in one exchange have to be carried out at the same time, because a donor can withdraw at any point before surgery and cannot be bound by contract to donate later, so the size of an exchange is limited by how many operations can run at once. Roth, Sönmez and Ünver showed in 2007 that in a large population, exchanges of up to four pairs capture all the potential gains, and exchanges of two and three pairs capture almost all of them.

As exchanges grew to include several transplant centres, safety became the problem. A centre that can match two of its own pairs is sure of those transplants; if it enters them into a wider exchange, the algorithm may use one of them for a higher-priority patient elsewhere. If centres respond by keeping back the pairs they can match themselves, mostly hard-to-match pairs reach the exchange and the loss is considerable. One remedy Roth proposes is to guarantee each centre that any pairs it could have matched in-house will receive transplants. He acknowledges that surgeons can find this repugnant, because it ranks patients partly by the accident of being treated at the same centre as a compatible pair.

The job market for economists

Not every repair needs a clearinghouse. In 2005 the American Economic Association (AEA) asked Roth to chair a committee on the market for new economics PhDs, and its members, Peter Coles, John Cawley, Phillip Levine, Muriel Niederle, Roth and John Siegfried, described the work in The Job Market for New Economists: A Market Design Perspective (Journal of Economic Perspectives, 2010). The market was thick: US institutions awarded 1,091 economics doctorates in 2008, and that season 1,910 new academic and 1,004 non-academic jobs were posted in the AEA's listings.

It was also congested. Because another application costs a candidate almost nothing, candidates applied to 80 employers on average, many employers received hundreds of applications, and an employer could do better by not interviewing some of the candidates it liked best, whom it expected to lose to other offers. Employers and candidates who would have suited each other could fail to meet as a result.

The committee left the market decentralised and added two mechanisms. Since November 2006 each candidate can send a signal of particular interest to at most two employers before the January interviews. The limit is what makes a signal credible, because sending one to an employer means not sending it to another. A web-based “scramble” lets candidates still on the market and employers with vacancies find each other late in the season, to thicken that part of the market. In the committee's data an application that came with a signal was 6.8 percentage points more likely to lead to an interview, a result the authors say suggests, but cannot prove, that signals help.

Simplicity: obviously strategy-proof mechanisms

Roth's third task asks for participation that is simple as well as safe, and a mechanism can be strategy-proof on paper and still be hard for people to play correctly. Shengwu Li's Obviously Strategy-Proof Mechanisms (American Economic Review, 2017) gives one formal test of simplicity. It starts from a known puzzle. Choosing when to drop out of an ascending clock auction, where the price rises until one bidder remains, is strategically the same as choosing a bid in a sealed-bid second-price auction, the Vickrey auction described under mechanism design. Yet laboratory subjects are substantially more likely to play the dominant strategy in the clock auction.

Li calls a strategy obviously dominant if, at the first point where it and any alternative diverge, the best outcome the alternative could bring is no better than the worst outcome the strategy could bring. A mechanism is obviously strategy-proof (OSP) if it has an equilibrium in obviously dominant strategies. He shows that a strategy is obviously dominant exactly when an agent who cannot reason through hypothetical contingencies can still recognise it as dominant. In the clock auction a bidder who values the item at 10 can see, at every price, that staying in below 10 and leaving above it cannot be beaten, so the clock auction is OSP.

The sealed-bid auction is strategy-proof but not OSP, because a bid of 11 might win the item at a price below 10 while a bid of 10 might lose it, so the best outcome from deviating beats the worst outcome from bidding truthfully. Some rules have no OSP form at all: with three or more participants, no obviously strategy-proof mechanism implements top trading cycles. In Li's experiment, across three pairs of mechanisms that each implement the same rule, subjects played the dominant strategy significantly more often under the OSP version, and the gap persisted after five rounds with feedback.

Markets for ideas

Joshua Gans and Scott Stern apply Roth's framework to ideas and technology in Is There a Market for Ideas? (Industrial and Corporate Change, 2010). Organised markets for ideas are rare: most trade in technology happens in isolated deals, where the alternative to agreement is further search or developing the idea in-house rather than playing one buyer against another. The authors trace this to three properties of ideas. Ideas are rarely valuable alone and have to be matched with complementary assets and other ideas. Disclosing an idea can let the buyer reproduce it, so the first buyer can become a competing seller, and an organised exchange can make trade impossible rather than easier. And an idea can be worth less to each user as more users have it, which the authors call value rivalry.

Institutions decide how far such a market can form. Where patents are effective and enforceable, a seller can disclose an idea without losing it and can control resale, which makes bargaining with several buyers possible; the authors point to patent exchanges and auctions such as Ocean Tomo. Where overlapping and uncertain rights create a patent thicket, strengthening those rights can have the opposite effect. The authors also find repugnance at work. The norms of open science treat free disclosure as a value in itself, which prevents the producers of basic knowledge, however valuable, from earning a market return on it.

Market design and blockchains

Wayne Chang's Market Design with Tokens, published by Token Foundry in July 2018, maps token mechanisms onto Roth's three tasks. For thickness he lists mining rewards that pay early participants more, as Bitcoin's falling issuance does, and tokens paid out for partnerships and referrals, with Sybil attacks, in which one party creates many fake accounts to collect the bonuses, as the risk to guard against.

For congestion he points to transaction fees paid in the token, which ration a network's limited capacity, and he names the full nodes that store and share a network's entire history without direct payment as an unsolved congestion problem. For safety he lists governance rights over infrastructure a business depends on, smart-contract escrow such as good-faith deposits, attestations that give participants more information about their counterparties, and staking deposits that an arbiter can award to a wronged party. He concludes that a token should serve a purpose of this kind, or it only adds friction to taking part.

Christian Catalini, Ravi Jagadeesan and Scott Duke Kominers treat a whole blockchain-based financial system as a market design problem in Market Design for a Blockchain-Based Financial System, a six-page working paper from June 2019. They develop a theory of long-run equilibrium to identify the design features that separate proof of work, where the right to add blocks is won by spending computation, from proof of stake, where it rests on tokens staked as collateral.

Under proof of work, wasteful computation secures the system, and what users get in equilibrium is set by the threat of a fork, a split in which participants carry on with a separate copy of the chain. Under proof of stake, users generally do better than that level, because custodians can use relational contracts, agreements enforced by the value of continuing the relationship rather than by a court, to support a higher quality of service. Those contracts rely only on local institutions, and the authors suggest that combining them with cryptography could create a platform for formal global contracts.

How Caper approaches this

A caper does not rely on thickness for trade in its own token. Every caper token trades against a bonding curve, a contract that sets the price from the supply in circulation and sells tokens to buyers and buys them back from sellers, so a trade never waits for a matching counterparty. The market is open from the moment a caper launches and cannot be withdrawn. The curve brings no more traders together, so the market is not thick in Roth's sense; what it removes is the need for a thick market to buy in or cash out.

The same design bears on safety. The price of a trade comes from a fixed formula, so a holder can know it before signing, and the sell side has no pause, no owner gate and no schedule, so leaving does not depend on anyone else's decision. A holder who has cast a ballot can also leave through the treasury at any time. The exit takes back their tokens with the vote tokens their ballots earned, sells the tokens on the curve in the same transaction and pays their share of the treasury alongside the proceeds: a standing door rather than a proposal (see what is a caper). A holder who has never voted can still sell on the curve.

References

1. Alvin E. Roth (2002). The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics. Econometrica 70(4): 1341–1378.
2. Alvin E. Roth (2007). The Art of Designing Markets. Harvard Business Review 85(10).
3. Alvin E. Roth (2009). What Have We Learned from Market Design? In Josh Lerner and Scott Stern (eds.), Innovation Policy and the Economy, vol. 9, University of Chicago Press for the NBER, pp. 79–112.
4. Alvin E. Roth (2015). Who Gets What – and Why. William Collins; published in the US by Houghton Mifflin Harcourt.
5. Nir Vulkan, Alvin E. Roth and Zvika Neeman (eds.) (2013). The Handbook of Market Design. Oxford University Press.
6. Atila Abdulkadiroğlu and Tayfun Sönmez (2003). School Choice: A Mechanism Design Approach. American Economic Review 93(3): 729–747.
7. Atila Abdulkadiroğlu and Aram Grigoryan (2021). Priority-based Assignment with Reserves and Quotas. NBER Working Paper 28689.
8. Tayfun Sönmez and M. Utku Ünver (2013). Market Design for Kidney Exchange. In Nir Vulkan, Alvin E. Roth and Zvika Neeman (eds.), The Handbook of Market Design, Oxford University Press, ch. 4.
9. Peter Coles, John Cawley, Phillip B. Levine, Muriel Niederle, Alvin E. Roth and John J. Siegfried (2010). The Job Market for New Economists: A Market Design Perspective. Journal of Economic Perspectives 24(4): 187–206.
10. Shengwu Li (2017). Obviously Strategy-Proof Mechanisms. American Economic Review 107(11): 3257–3287.
11. Joshua S. Gans and Scott Stern (2010). Is There a Market for Ideas? Industrial and Corporate Change 19(3): 805–837.
12. Wayne Chang (2018). Market Design with Tokens. Token Foundry, 19 July.
13. Christian Catalini, Ravi Jagadeesan and Scott Duke Kominers (2019). Market Design for a Blockchain-Based Financial System. SSRN working paper 3396834.