> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/institutional-microeconomics/part-vii-market-design-and-welfare/ch24_market_design_mechanism_design.md). # Chapter 24 — Market Design and Mechanism Design *** ## The Economists Who Designed a Market Before 1994, the Federal Communications Commission gave away radio spectrum licenses — the invisible infrastructure of wireless communication — by two methods, both unhappy. The first was the "comparative hearing," a beauty contest in which applicants spent years and small fortunes arguing before administrative judges that they would best serve the public interest. In the 1980s Congress authorized a second method, random lotteries, hoping they would be faster and fairer. They were faster. Winners included speculators with no intention of building anything; in one celebrated case a group of dentists won the cellular license for Cape Cod and sold it to Southwestern Bell for $41 million (McMillan 1994). The scarcity value of the public's airwaves went to whoever held the winning ticket. In 1993 Congress authorized auctions, and the design problem was not trivial. Licenses are *complements* — Boston is worth more to a carrier that also holds New York, because a national network needs contiguous coverage — and *substitutes*: two licenses for the same city in similar bands are near-perfect alternatives. A local carrier values one city, a national carrier the bundle, and the government wanted both efficiency and revenue. Selling licenses one at a time would force a carrier to bid for Boston without knowing whether it would win New York: win Boston at a high price, lose New York, and it is stuck with a license worth less than it paid. This is the *exposure problem*. The answer came not from a single planner but from a contest of proposals. Economists advising bidders — Paul Milgrom and Robert Wilson for Pacific Bell, Preston McAfee for AirTouch — proposed designs in the FCC's rulemaking, and the Commission, advised by its staff economist Evan Kwerel and by John McMillan, adopted the *simultaneous multiple-round ascending auction* (SMRA): all licenses open at once, prices posted after every round, bidders free to shift among licenses as relative prices move (McMillan 1994; Milgrom 2004). The first auction, in July 1994, sold ten nationwide narrowband licenses for $617 million; FCC auctions have since raised more than $200 billion (Federal Communications Commission 2026). The most ambitious, the 2016–17 "incentive auction" designed by Milgrom and Ilya Segal, bought spectrum back from television broadcasters and sold it to wireless carriers in a single market (Milgrom and Segal 2020). The spectrum auctions are the signature achievement of *market design* — engineering institutions for allocation problems that existing markets handle poorly or not at all. This chapter develops its theory — auctions, matching mechanisms, and mechanisms that elicit private information — and §24.6 returns to the airwaves, where all three meet. If the preceding chapters argued that institutions matter, this one shows that some can be designed. *** ## The Spine **What problem exists?** Some goods cannot be allocated by spontaneous exchange: pricing them is ethically unacceptable (kidneys, school places), they are interdependent (spectrum licenses, landing slots), or thin markets with private information make bilateral bargaining inefficient. Chapter 4's silent trade showed exchange under spare institutional conditions; here those conditions fail and *design* is required. **What institution addresses it?** Auctions, matching algorithms, and mechanisms generally — formal procedures that structure strategic interaction to achieve efficiency, revenue, stability, or fairness despite private information. They are engineered rather than evolved, though designed markets are also tested, gamed, and revised (see §24.5). **What role do prices play?** An auction *creates* its price rather than taking it as given. Matching markets exclude prices deliberately, and the mechanism does the coordinating work prices normally do: the price system is one institutional solution among several. **Who captures the rents?** The allocation mechanism decides where an asset's scarcity rent goes. A comparative hearing dissipates it in the contest for the license (the Tullock rectangle of §31.3), a lottery hands it to the lucky, an auction turns it into public revenue. Inside an auction, bidders keep *information rents* — the gap between the winner's value and her payment — and the format, the reserve, and above all the number of bidders decide how much the seller recovers. Design also *creates* surplus, as in kidney exchange, and the choice of mechanism decides who captures it. *** ## 24.1 When Markets Must Be Designed *Mechanism design* is a theory of how to structure a game so that self-interested agents acting on private information produce a desired outcome. *Market design* applies it, with engineering judgment, to fix broken markets or build missing ones — the relation of physics to structural engineering, in the spirit of Alvin Roth's (2002) "economist as engineer." Three circumstances make design necessary. **Complementarities and thin markets.** When the value of one item depends on which others you hold — licenses, landing slots — bilateral negotiation is slow and exposed to holdup, and the indeterminacy of bilateral bargaining (see §4.3) becomes pathological. Prices across interdependent items must be coordinated by a common institution. **Ethical constraints on pricing.** Societies have decided that organs, school places, and residencies should not go to the highest bidder. Roth (2007) calls such constraints *repugnance* and treats them as facts to design around; in the United States, selling an organ for transplantation has been a crime since 1984. The allocation problem remains, so a matching mechanism must do the work of prices (on nonmarket allocation, see Chapter 5). **Private information and manipulation.** When agents have a motive to misreport values, preferences, or costs, unstructured interaction wastes gains from trade. Chapter 18 showed adverse selection unraveling markets and Chapter 19 a menu of contracts making people reveal their types; mechanism design generalizes the menu to many agents, continuous types, and multiple objectives. **What a working market needs.** Roth (2008) distilled the lessons of these cases into a checklist. A marketplace works when it is *thick*, bringing enough participants together at the same time that each has many options; when it overcomes the *congestion* that thickness brings, giving participants enough time, or a fast enough procedure, to consider the alternatives; and when participation is *safe and simple*, so that people do well without gaming the rules or hiding what they know. This chapter's cases fail one test or another. The medical market before 1952 lost thickness as offers were made ever earlier, spreading the market across time, and kidney exchange loses it as hospitals keep pairs to themselves (§24.3). New York's high school admissions were congested: a few rounds of offers could not clear a market of some 90,000 applicants. The Boston mechanism and the exposure problem of §24.6 fail on safety, because a participant who simply reports what she wants can be punished for it. Repugnance adds a further constraint, on which transactions may take place at all. The connecting thread is *incentive compatibility*: agents must find it in their own interest to behave as the designer intends. Mechanism design asks whether, where competitive markets are missing or broken, we can build institutions that channel self-interest as Smith's invisible hand does. *** ## 24.2 Auction Theory and Practice An auction is a mechanism for selling when the seller does not know what buyers will pay. It may be the oldest designed institution in economics — Herodotus (*Histories* 1.196) describes the auctioning of brides in Babylon — and it is among the most important: governments sell debt, timber, oil leases, and spectrum this way, and each time a search engine shows advertisements beside its results, an auction has just been run. ### The Four Standard Formats **English (ascending) auction.** The price rises; bidders drop out as it passes their willingness to pay; the last bidder standing pays the price at which her last rival quit. Bidding is open, as at Sotheby's, so bidders see who drops out and when — information that matters when values have a common component. **Dutch (descending) auction.** The price falls until someone accepts. The flower auction at Aalsmeer, near Amsterdam, part of the growers' cooperative Royal FloraHolland, has long used descending clocks, because flowers wilt. Nothing is revealed along the way. **First-price sealed bid.** The highest sealed bid wins, and the winner pays her bid — the standard format for mineral rights and, with the lowest bid winning, for procurement. It is strategically equivalent to the Dutch auction: in both, a higher number raises the chance of winning but lowers the surplus from winning. **Second-price sealed bid (Vickrey 1961).** The highest bidder wins but pays the *second-highest* bid. William Vickrey's insight, recognized with the 1996 Nobel Prize, was that this format makes *truthful bidding a weakly dominant strategy*. Multi-unit versions are everywhere: the US Treasury sells its securities by sealed bid, with every winning bidder paying the same market-clearing price. ### The Logic of Truthful Bidding Why bid your value in a second-price auction? Your bid decides only *whether* you win, never *what you pay*. With a value of $100, bidding $100 wins whenever the highest rival bid is below $100. Shading to $80 changes the outcome only when the highest rival bid lies between $80 and $100 — and then it loses an auction you would have won at a profit. Bidding $120 changes it only when the rival bid lies between $100 and $120 — and then you pay more than the object is worth to you. Truthful bidding is never worse than any other bid, whatever the others do, and sometimes strictly better: it is *weakly* dominant. The argument requires *private* values; it fails when rivals' information bears on what the object is worth to you, as in the common-value settings below. ### Bid Shading in First-Price Auctions In a first-price auction you pay your own bid, so bidding your value guarantees zero surplus. Bidders *shade*, trading a lower chance of winning for a larger surplus when they win. With $$n$$ bidders whose values are drawn independently from the uniform distribution on $$\[0, 1]$$ — the canonical independent private values (IPV) model — the symmetric equilibrium bid is: $$ \beta(v) = \frac{n-1}{n} v $$ This is a *Bayesian Nash equilibrium*: each bidder's bid is a best response, for every value she might have, to the way the others' bids depend on their values (the formal definitions are collected in the Game Theory Toolkit, Appendix 0A–0C; §A0A.6 derives this bid and Appendix 24 the general case). With two bidders you bid half your value; with ten, nine-tenths. Because every bidder uses the same increasing bid function, the highest-value bidder always wins: the symmetric first-price auction is *efficient* for any $$n$$. What changes with $$n$$ is rent extraction: the winner's *information rent*, $$v - \beta(v) = v/n$$, shrinks as bidders multiply. ### Revenue Equivalence The Revenue Equivalence Theorem (Vickrey 1961; Riley and Samuelson 1981; Myerson 1981) is one of the most striking results in economic theory. Suppose bidders are risk-neutral and symmetric, with private values drawn independently from a common distribution — the IPV benchmark. Then *any* auction in which (i) the object goes to the bidder with the highest value and (ii) a bidder with the lowest possible value expects zero surplus raises the *same expected revenue*. All four standard formats qualify, and with uniform values on $$\[0, 1]$$ each raises the expected second-highest value, $$(n-1)/(n+1)$$. The second-price winner pays less than her bid; the first-price winner pays her bid but bids less than her value; in expectation the effects offset. Figure 24.1 follows one draw of five bidders through all four formats: the English and second-price auctions both end at the second-highest value, 0.65, the Dutch and first-price auctions at the winner's shaded bid, 0.60. Equivalence holds on average, not auction by auction. First-price revenue is also less variable: the winning bid is the expected second-highest value given the highest one, an average that smooths away some of the randomness of the second-highest value itself. ![Four auction formats, one draw](https://2171264621-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FgdyZSdo6Lnq7kmkrkZqF%2Fuploads%2Fgit-blob-35a3469f567eb7bb8b60a14e5f7629447a66f3d3%2Ffig24_auction_formats.png?alt=media) *Figure 24.1: Four Auction Formats, One Draw. The same five bidders, with values 0.20, 0.40, 0.50, 0.65 and 0.75, meet each format. The English and second-price auctions raise 0.65, the second-highest value; in the Dutch and first-price auctions the winner bids 80 percent of her value and pays 0.60. With five uniform bidders each format's expected revenue is (n − 1)/(n + 1) = 0.667: revenue equivalence concerns averages. Source: Author's illustration; model from Vickrey (1961) and Myerson (1981).* ### When Revenue Equivalence Fails Revenue equivalence fails when its assumptions do, and three failures are empirically common. **Risk aversion.** In a first-price auction shading is a gamble — more surplus if you win, a higher chance of losing — and a risk-averse bidder insures against losing by bidding closer to her value, so first-price revenue rises above second-price revenue. In a second-price auction there is nothing to insure: bidding your value stays dominant whatever your attitude to risk. **Affiliated (common) values.** When bidders estimate a common but uncertain value — the reserves under an oil lease — their information is *affiliated*: a high estimate by one makes high estimates by others more likely. Milgrom and Weber's (1982) *linkage principle* then ranks the formats: expected revenue is at least as high in the English auction as in the second-price auction, and at least as high in the second-price auction as in the first-price and Dutch auctions (with two bidders, English and second-price coincide). The more closely the winner's price is linked to others' information, the less she must shade against the winner's curse (below), and the English auction links most, because the survivors learn from the prices at which rivals quit. **Asymmetric bidders.** When values come from different distributions — an incumbent and an entrant — a weak bidder shades less than a strong one would at the same value, so a first-price auction can hand the object to the bidder who values it less. Condition (i) fails, and first- and second-price revenues generally differ, in a direction that depends on how the distributions differ (Maskin and Riley 2000). ### Revenue Equivalence Is Not Revenue Maximization Revenue equivalence says that the standard formats raise the same revenue, not the most revenue. Myerson (1981) found the revenue-maximizing auction, and it is not efficient. The key concept is the *virtual value*: $$ \phi(v) = v - \frac{1 - F(v)}{f(v)} $$ where $$F$$ is the distribution of values and $$f$$ its density: a bidder's value net of the information rent that must be left to every higher type. With symmetric bidders and a virtual value that rises with $$v$$, the optimal auction is a standard auction with a *reserve price* $$r^*$$ — a minimum bid below which the object stays unsold — set where $$\phi(r^*) = 0$$ (for a seller who values the object at nothing). The threshold depends only on the value distribution — strikingly, not on the number of bidders. With uniform values, $$\phi(v) = 2v - 1$$ and $$r^\* = 1/2$$, whether there are two bidders or twenty. A reserve does not overturn revenue equivalence. It changes who gets the object — sometimes no one — and so changes expected revenue, but first- and second-price auctions with the *same* reserve still raise the same amount: with two uniform bidders and a reserve of one-half, $$5/12 \approx 0.417$$ each, against $$1/3$$ with no reserve. Equivalence is not maximization, and the optimal reserve exposes the trade-off at the heart of design: to raise what high-value buyers pay, the seller commits to sometimes withholding the object from a buyer who values it more than she does. Figure 24.2 puts the reserve in proportion. Revenue is flat near the optimum, so a seller who misjudges the distribution somewhat loses little, and the reserve's value fades quickly as bidders are added. Bulow and Klemperer (1996) proved the general version: with symmetric bidders and regular distributions, an ascending auction with no reserve and *one more bidder* raises more expected revenue than the optimal mechanism with the original number. For a seller, recruiting competition beats fine-tuning the rules. ![Reserve prices and competition](https://2171264621-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FgdyZSdo6Lnq7kmkrkZqF%2Fuploads%2Fgit-blob-9ddadb0792343279c2f3e6f1e22787c434604f55%2Ffig24_reserve_competition.png?alt=media) *Figure 24.2: The Optimal Reserve versus One More Bidder. Panel A: expected revenue against the reserve price for 1, 2, 3 and 5 bidders with values uniform on \[0, 1]; lines use the second-price formula, open dots the first-price auction computed from its equilibrium bids — the same numbers, so revenue equivalence survives a common reserve. Every curve peaks, flatly, at r = 0.5 whatever the number of bidders. Panel B: with n + 1 bidders and no reserve, revenue exceeds the optimal-reserve revenue with n bidders (Bulow and Klemperer 1996). Source: Author's calculations from the model of Myerson (1981).* ### The Winner's Curse In common-value settings, winning is *bad news*. If you outbid nine rivals for an oil lease, your estimate was the most optimistic of ten — and the most optimistic of ten estimates is, on average, too high. Bidders who ignore this overpay; sophisticated bidders ask what the lease is worth *given that I outbid everyone*. Capen, Clapp, and Campbell (1971), petroleum engineers, blamed poor returns on offshore leases on the curse; Hendricks and Porter (1988) found that on drainage leases, where owners of adjacent tracts know more, the better-informed firms earned higher returns, as theory predicts; and Roll's (1986) "hubris hypothesis" finds the curse in takeovers. *** ## 24.3 Matching Markets Some allocation problems are deliberately kept outside the price system: buying kidneys from the poor strikes most societies as repugnant, selling school places would entrench inequality, and medical residencies allocated by decentralized bargaining descended into chaos. The allocation problem remains, and a badly designed mechanism can be worse than none. ### Stability: The Design Criterion The organizing principle is *stability* (Gale and Shapley 1962): a matching is stable if no *blocking pair* exists — no student and school who both prefer each other to their assignments. Unstable matchings *unravel*: such pairs strike side deals and the formal mechanism loses credibility. The market for new American doctors is the cautionary tale (Roth 1984). In the 1940s hospitals competed by making offers ever earlier — by the mid-1940s, as much as two years before the internship began — while students held offers waiting for better ones. A clearinghouse begun in 1952, now the National Resident Matching Program, ended the unraveling, and Roth showed why: its algorithm produced stable matchings — in effect the hospital-proposing version of the deferred acceptance algorithm below, found by practitioners a decade before Gale and Shapley published it. In Britain, regional clearinghouses that produced stable matchings survived while most of the unstable ones were abandoned (Roth 1991). ### The Deferred Acceptance Algorithm Gale and Shapley (1962) proved that a stable matching always exists and gave an algorithm that finds one, *deferred acceptance* (DA). It is best understood through an example. **A worked example.** Three students — Alice, Bob, and Carol — apply to three schools with one seat each — Lincoln, Monroe, and Jefferson. Each student ranks the schools; each school ranks the students by its priority rules (siblings, neighborhood, lottery number). | Student | 1st choice | 2nd choice | 3rd choice | | ------- | ---------- | ---------- | ---------- | | Alice | Monroe | Lincoln | Jefferson | | Bob | Lincoln | Monroe | Jefferson | | Carol | Lincoln | Monroe | Jefferson | | School | 1st priority | 2nd priority | 3rd priority | | --------- | ------------ | ------------ | ------------ | | Lincoln | Alice | Carol | Bob | | Monroe | Bob | Alice | Carol | | Jefferson | Alice | Bob | Carol | **Round 1.** Each student applies to her first choice. Alice applies to Monroe; Bob and Carol apply to Lincoln. Lincoln tentatively holds Carol, whom it ranks higher, and rejects Bob. Monroe holds Alice. **Round 2.** Bob applies to Monroe, which ranks him above Alice, so it holds Bob and rejects Alice. **Round 3.** Alice applies to Lincoln, which ranks her above Carol, so it holds Alice and rejects Carol. **Round 4.** Carol applies to Monroe, which prefers Bob and rejects her. **Round 5.** Carol applies to Jefferson, which holds her. No one is rejected, and the algorithm stops. **Final matching:** Alice → Lincoln, Bob → Monroe, Carol → Jefferson. Figure 24.3A sets out the rounds. The matching is *stable*. Check every student who would rather be elsewhere: Alice prefers Monroe, but Monroe holds Bob, whom it ranks higher; Bob prefers Lincoln, but Lincoln holds Alice, whom it ranks higher; Carol prefers Lincoln and Monroe, but each holds a student it ranks above her. No blocking pair exists — indeed this is the only stable matching here. The "deferred" feature is crucial: a school holds an applicant only tentatively, so an early offer can be undone when a higher-priority applicant arrives, and a sequence of local adjustments produces a globally stable outcome. ### Properties and Trade-offs Student-proposing DA has three important properties. It always produces a stable matching. It is *student-optimal*: each student gets the best school she can get in *any* stable matching. And it is *strategy-proof for the proposing side*: truthful ranking is a dominant strategy for students — the matching counterpart of the second-price auction. Schools, by contrast, can sometimes gain by misrepresenting their rankings (Appendix 24 gives an example), one reason school systems put students on the proposing side and fix school priorities in public rules. Strategy-proofness matters. The "Boston mechanism," used in Boston until the mid-2000s, filled each school first with students who ranked it first, then with those who ranked it second, and so on, each assignment final (Abdulkadiroğlu and Sönmez 2003). Run it on the example with truthful rankings: Lincoln takes Carol and Monroe takes Alice in the first round, and Bob ends at Jefferson. But if Bob ranks Monroe first, Monroe, where he has priority over Alice, takes him at once — and Alice, who told the truth, falls to Jefferson, her last choice. Sophistication becomes a resource, and sincere families pay for it (Pathak and Sönmez 2008). Under DA every family's best strategy is the same, whatever its sophistication. The price of stability is efficiency. In the example Alice and Bob would *both* gain by swapping — Alice to Monroe, Bob to Lincoln — and Carol would be no worse off. But the swap is unstable: Carol prefers Lincoln to Jefferson, and Lincoln ranks her above Bob. Carol's priority blocks a trade that helps two students and hurts no one. An alternative algorithm, *top trading cycles* (TTC), makes the trade. Proposed by David Gale and reported in Shapley and Scarf (1974), it was adapted to school choice by Abdulkadiroğlu and Sönmez (2003). Each student points to her favorite remaining school and each school to its highest-priority remaining student; every cycle is carried out — each student in it gets the school she points to — and those students and seats leave the market. Here the first cycle is Alice → Monroe → Bob → Lincoln → Alice, so Alice gets Monroe and Bob gets Lincoln; Carol then gets Jefferson. But Carol has *justified envy*: she prefers Lincoln, which ranks her above the student it admitted. | Mechanism | Alice | Bob | Carol | Stable? | Pareto efficient for students? | Strategy-proof for students? | | ------------------- | ------------- | ------------- | --------------- | --------------------------- | ------------------------------ | ---------------------------- | | Deferred acceptance | Lincoln (2nd) | Monroe (2nd) | Jefferson (3rd) | Yes | No: Alice and Bob would swap | Yes | | Top trading cycles | Monroe (1st) | Lincoln (1st) | Jefferson (3rd) | No: Carol and Lincoln block | Yes | Yes | When the student-optimal stable matching is Pareto-inefficient, as here, no mechanism can be both stable and efficient, so the choice between DA and TTC is normative: stability honors the priorities a school system has promised; efficiency lets students trade them away. > **How We Know: School Choice in New York City** > > Until 2003, New York City's high school admissions were uncoordinated: schools made offers independently, some students received several and others none, and after three rounds about 30,000 students a year were assigned administratively to schools they had not listed. For students entering high school in 2004 the city adopted a single-offer, student-proposing DA designed with Atila Abdulkadiroğlu, Parag Pathak, and Alvin Roth, in which some 90,000 applicants rank up to twelve programs (Abdulkadiroğlu, Pathak, and Roth 2005). > > No one randomizes cities to mechanisms, and a before–after comparison mixes the reform with everything else that changed. Abdulkadiroğlu, Agarwal, and Pathak (2017) instead exploit the mechanism's design: because truthful ranking is, the cap aside, a dominant strategy under DA, submitted lists reveal preferences. They estimate a model of school demand from the lists and compute each student's welfare under the old process and the new. The identifying assumption is visible — rankings are taken at face value — and they find that coordination sharply reduced assignments to unranked schools and produced substantial welfare gains. > > Details matter. With a twelve-program cap, a student who likes more programs must choose which to list, and truth-telling is no longer dominant (Haeringer and Klijn 2009). And strategy-proofness has distributional consequences: under a manipulable mechanism the rents of sophistication accrue to the families that have it, at the expense of sincere ones (Pathak and Sönmez 2008), and a strategy-proof mechanism removes them — institutional design deciding who benefits. ### Kidney Exchange: When Design Saves Lives The most concrete vindication of the economist-as-engineer vision is kidney exchange (Roth, Sönmez, and Ünver 2004). Patient A's husband is willing to donate but is incompatible with her; Patient B's sister is in the same position with him. If A's husband is compatible with B and B's sister with A, a swap gives both patients a kidney. Because buying an organ is illegal, Congress amended the transplant law in 2007 to make clear that paired donation is not a purchase. Longer *cycles* — A's donor gives to B, B's to C, C's to A — raise the number of feasible transplants, but every operation in a cycle must be performed simultaneously, lest a donor renege after his own patient has received a kidney, and that caps cycles in practice at two or three pairs. The breakthrough came with *non-directed altruistic donors*, who give a kidney to a stranger and so start a *chain* rather than a cycle: the altruist gives to Patient A, A's donor to B, B's donor to C, and so on. Because each pair receives a kidney before its own donor gives one, a chain's links need not be simultaneous — they can be scheduled weeks apart — and a donor who reneges leaves no pair worse off than it started; the chain simply ends. The first such non-simultaneous chain, reported by Rees et al. (2009), produced ten transplants — matches that uncoordinated search would almost never have found. Figure 24.3B draws cycles and a chain. ![Deferred acceptance and kidney exchange](https://2171264621-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FgdyZSdo6Lnq7kmkrkZqF%2Fuploads%2Fgit-blob-68b3bf47f8e5881a5b891a74f04953685efd9999%2Ffig24_matching.png?alt=media) *Figure 24.3: Matching without Prices. Panel A: student-proposing deferred acceptance on the example of §24.3, round by round; each school holds its best applicant so far and rejects the rest, and no hold is final until no one is rejected. Panel B: kidney exchange, with each circle an incompatible patient–donor pair and each arrow a kidney from a pair's donor to another pair's patient; operations in a cycle must be simultaneous, while a chain started by a non-directed donor can be staggered. Source: Author's illustration.* > **How We Know: Agarwal, Ashlagi, Azevedo, Featherstone, and Karaduman (2019) — Counting the Cost of a Fragmented Market** > > Asking how well kidney exchange works requires a benchmark, and there is no counterfactual United States running a single national exchange. What exists is fragmentation: transplants are arranged inside hospitals or through several competing platforms, and hospitals choose which patient–donor pairs to submit. > > Agarwal and co-authors build the benchmark computationally. From data on the pairs submitted to major platforms they estimate the distribution of patient and donor types — blood type, tissue sensitization, and the compatibility structure that determines which exchanges are feasible — and simulate what a single unified platform would have achieved with the same population. The identification lies in the biology: whether two pairs can exchange is a medical fact, not a behavioral one, so the feasible set can be computed rather than assumed. > > The gap is large: unification could raise transplants by roughly 30 to 63 percent. Fragmentation persists because of an externality: a hospital that submits an easy-to-match pair helps patients elsewhere but gives up a transplant it could have arranged itself, so hospitals keep their easy cases. The algorithm is solved; the incentives of the institutions that feed it are not. *** ## 24.4 Mechanism Design as Institutional Engineering ### The General Problem A designer — government, platform, employer, auctioneer — wants to implement an allocation, but agents hold private information and may misreport it. Chapter 19 solved this problem for one principal and two types: an insurer's menu of deductibles induces each risk type to choose the contract meant for it (see §19.3). Mechanism design generalizes the logic. ### Social Choice Functions and Mechanisms Formally, each agent $$i$$ has a *type* $$\theta\_i \in \Theta\_i$$ — her private information: a valuation, a cost, a ranking of schools. A *social choice function* $$x(\theta\_1, \ldots, \theta\_n)$$ assigns an outcome in a set $$X$$ to every profile of types: who gets the object, how much each agent pays, which student attends which school. The designer wants to implement $$x(\cdot)$$ but cannot observe types. A *mechanism* $$\Gamma = (M, g)$$ gives each agent a message space $$M\_i$$ — what she can report or bid — and an outcome function $$g: M\_1 \times \cdots \times M\_n \to X$$ mapping messages to outcomes. A social choice function is *implementable* if some mechanism has an equilibrium in which agents' strategic messages produce exactly the outcome $$x(\cdot)$$ prescribes, profile by profile. ### The Revelation Principle The Revelation Principle (Gibbard 1973; Myerson 1979) is the foundation of the theory. If a social choice function can be implemented by *some* mechanism in *some* equilibrium of a given kind — dominant-strategy or Bayesian — then it can be implemented by a *direct* mechanism, in which each agent simply reports her type, with truth-telling an equilibrium of the same kind. The argument is constructive. Take any mechanism and any equilibrium of it. Build a direct mechanism that asks each agent her type and then plays, on her behalf, the strategy her type used in that equilibrium. Reporting truthfully gets an agent exactly what she got before. Reporting another type gets her what she would have got by imitating that type in the original game — an imitation she already rejected, since her equilibrium strategy was a best response. So no type gains by lying. To characterize what is achievable, then, we need only study truthful direct mechanisms. The principle is a tool of analysis, not a design recommendation. Practical designs are often indirect — an ascending auction asks for bids, not valuations — and even direct ones such as DA, in which each participant submits a ranking, are chosen for properties beyond incentive compatibility, for reasons §24.5 explains. ### Incentive Compatibility Two standards of incentive compatibility govern the theory. **Dominant-strategy incentive compatibility (DSIC):** truth-telling is optimal for each agent *whatever the others report*. The second-price auction is DSIC, and so, for students, is DA. DSIC is robust — agents need no beliefs about one another — but demanding, and it sharply limits what can be achieved. **Bayesian incentive compatibility (BIC):** truth-telling is optimal *in expectation*, given a common prior over the others' types. BIC is weaker — it leans on a commonly known distribution, and an agent who knew the others' types might want to lie — but it buys real possibilities. The mechanism of d'Aspremont and Gérard-Varet (1979) implements the efficient outcome with a balanced budget, paying each agent the expected benefit her report confers on everyone else and financing those payments from the other agents — something no dominant-strategy mechanism can do in general (Green and Laffont 1977). With a single agent the distinction disappears, which is why Chapter 19's screening constraints are simply incentive compatibility constraints; mechanism design extends them to many agents and continuous types. ### The VCG Mechanism The Vickrey–Clarke–Groves (VCG) mechanism (Vickrey 1961; Clarke 1971; Groves 1973) is the most powerful answer to the efficiency problem. With quasilinear preferences — utility is value minus payment — VCG asks each agent to report her valuation, chooses the allocation that maximizes total reported value, and charges each agent the *externality* she imposes on the others: the reduction in their total value caused by her presence. Consider three bidders for a single object: A values it at $100, B at $80, C at $50. | Bidder | Value | Wins? | Others' surplus with agent | Others' surplus without agent | VCG payment | | ------ | ----- | ----- | -------------------------- | ----------------------------- | ------------------ | | A | $100 | Yes | $0 (B and C get nothing) | $80 (B would win) | $80 − $0 = **$80** | | B | $80 | No | — | — | **$0** | | C | $50 | No | — | — | **$0** | A wins and pays $80, the surplus B loses because A is present. For a single object VCG *is* the second-price auction: the winner pays the second-highest value, exactly the externality she imposes on the runner-up. VCG is DSIC and efficient, but it has two significant limitations. First, it is generally *not budget-balanced*. In a single-object auction the payments simply flow to the seller; for a public good the budget can fail in either direction: the payments may collect more than the project costs, a surplus that cannot be handed back without destroying the incentive to report truthfully, or — if no one can be made to pay more than the project is worth to her — less than it costs. In bilateral trade, where the seller too must be paid enough to take part, VCG runs a deficit whenever trade occurs: the Myerson–Satterthwaite problem again (Appendix 24 works each case). Second, VCG is vulnerable to *collusion*, and it does badly when bidders want packages (see §24.6). ### The Limits of Design Two impossibility theorems mark the boundaries of design. The *Myerson–Satterthwaite theorem* (1983) concerns bilateral trade when the buyer's value and the seller's cost are private, independent, and drawn from overlapping ranges, so that it is uncertain whether trade is efficient. Then *no* mechanism is at once efficient, incentive compatible, individually rational (both parties take part voluntarily), and budget-balanced. Something must give: some efficient trades fail, trade is subsidized from outside, or participation is compelled. This is why bilateral bargaining is so hard to perfect (see §4.3). Private information has an irreducible cost; the second-best mechanism, which trades efficiency off against the other constraints, measures it (Appendix 24). The *Gibbard–Satterthwaite theorem* (Gibbard 1973; Satterthwaite 1975) states that with three or more alternatives and unrestricted preferences, the only strategy-proof social choice function that can select every alternative is a *dictatorship*: one agent's ranking decides. It is a cousin of Arrow's impossibility theorem for aggregating rankings (see §31.2), with a similar message: on an unrestricted domain, no nondictatorial voting rule is immune to manipulation. The escape routes are restrictions on the domain (single-peaked preferences over a line; quasilinear preferences with money transfers, where VCG lives) and weaker solution concepts such as BIC. ### Revenue Maximization versus Efficiency: A Normative Choice The designer faces a normative choice that Chapter 25 takes up directly and that returns in Part VIII, where the state becomes the designer. Myerson's optimal auction raises revenue with a reserve that sometimes leaves the object unsold; VCG achieves efficiency but not budget balance. Should the FCC maximize revenue, relieving taxpayers, or efficiency, putting licenses with the carriers that will use them best? Congress wrote a partial answer into the 1993 statute, which lists objectives from rapid deployment of new services to recovering for the public "a portion of the value of the public spectrum resource," and forbids the Commission, in designing its auctions, to base its findings of the public interest "solely or predominantly on the expectation of Federal revenues" (47 U.S.C. §309(j)). Mechanism design clarifies such trade-offs but cannot resolve them; the choice of objective is itself a political act (see §29.5 and Chapter 31). *** ## 24.5 Practical Market Design ### Why Simple Mechanisms Often Outperform "Optimal" Ones **Robustness matters more than optimality.** The "Wilson doctrine" (Wilson 1987) holds that good mechanisms should not depend on details the designer cannot know — the distribution of values, bidders' beliefs, the correlation of their information. It is the designer's answer to Knightian uncertainty (see §17.2): when no reliable probability distribution is available, prefer rules whose performance does not hinge on one. Ascending auctions and DA are "detail-free": they perform well across many environments, even if they are optimal only in special cases. **Participants must understand the mechanism.** Shengwu Li's (2017) *obviously strategy-proof* (OSP) mechanisms formalize what practitioners long knew: agents must not only *have* a dominant strategy but *see* that it is dominant. In an ascending auction the dominant strategy is transparent — stay in while the price is below your value. In a sealed-bid second-price auction it is the same strategy but less evident, and many laboratory subjects overbid. Only the ascending auction is OSP, and in Li's experiments subjects played the dominant strategy far more often there. Deferred acceptance is strategy-proof for students but not obviously so: on general preference domains no mechanism that always yields a stable matching is OSP (Ashlagi and Gonczarowski 2018), so a district must persuade families that truthful ranking is safe rather than rely on their seeing it. Mechanism design that ignores cognition is engineering that ignores the people who use the bridge. **Anticipate manipulation.** Porter and Zona (1993) documented bid rigging in Long Island highway contracts, where cartel members designated a winner and submitted "phantom" bids to simulate competition (on cartels, see §15.4). Misreporting under the Boston mechanism, shill bidding online, and signaling through bids in the spectrum auctions all show that incentive compatibility is necessary but not sufficient: the designer must also monitor, detect, and punish manipulation. ### Design versus Evolution The contrast between designed and evolved institutions is less sharp than it looks. The medical match was invented by practitioners and understood by theorists later, and designed markets, once running, evolve under pressure from their users. The Boston mechanism persisted for years after its flaws were known; spectrum bidders learned to signal to one another through the trailing digits of their bids, and the FCC responded by restricting bids to preset increments (Cramton and Schwartz 2000). Hayek's warning that no designer can gather the knowledge dispersed among participants (see §6.4) is not refuted by market design but absorbed into it. Prototype, test, watch for gaming, revise: the engineer's method makes a designed market one more round in competition as a discovery process (see §14.2). ### Empirical Market Design Guerre, Perrigne, and Vuong (2000) connected auction theory to data: bids can be inverted to recover the distribution of values, which then answers counterfactual questions — what another format or a higher reserve would raise, whether bidding looks competitive or collusive. The approach has been applied to timber sales, highway procurement, offshore oil leases, and online advertising. > **From Theory to Estimation: Nonparametric Auction Estimation** > > **The model**: In a symmetric IPV first-price sealed-bid auction, each of $$n$$ bidders draws a private value $$v\_i$$ from an unknown distribution $$F$$ and submits the bid $$b\_i$$ that maximizes $$(v\_i - b\_i) \cdot \Pr(\text{win} | b\_i)$$. The equilibrium bid is strictly increasing in value, and the first-order condition implies $$v\_i = b\_i + \frac{G(b\_i)}{(n-1) g(b\_i)}$$, where $$G$$ is the distribution and $$g$$ the density of observed bids — mapping each bid back to the value that generated it. > > **Identification**: Guerre, Perrigne, and Vuong (2000) showed that $$F$$ is point-identified from observed bids under IPV: estimate $$\hat{G}$$ and $$\hat{g}$$ by kernel methods and invert the first-order condition bid by bid to recover pseudo-values $$\hat{v}\_i$$, which then price counterfactual formats, optimal reserves, or mergers. For English auctions, Haile and Tamer (2003) derive bounds from two weak assumptions: bidders never bid above their values, and never let a rival win at a price they would have been willing to beat. > > **Data requirements**: All submitted bids (or at least the winning bid and the number of bidders) and auction characteristics, for many auctions of similar objects. > > **Key references**: Guerre, Perrigne, and Vuong (2000); Haile and Tamer (2003). ### Current Frontiers Search engines sell advertising positions by the *generalized second-price auction*, in which each advertiser pays the bid of the one ranked below it; truthful bidding is not an equilibrium, but the auction has one with the same positions and payments as VCG (Edelman, Ostrovsky, and Schwarz 2007). Platforms design the rules of their own marketplaces (see Chapter 16), and algorithmic mechanism design confronts computational limits, as the incentive auction had to (see §24.6). Speed is a newer frontier. On May 6, 2010, a cascade of automated trading drove the Dow Jones Industrial Average down roughly 1,000 points within minutes before prices recovered (Kirilenko et al. 2017). Continuous limit-order markets reward being a microsecond faster, and Budish, Cramton, and Shim (2015) argue that the resulting arms race is a design flaw; their remedy, *frequent batch auctions* — uniform-price sealed-bid auctions held at short, discrete intervals — removes the prize for tiny speed advantages. Whether mechanisms built for people work as intended when the participants are algorithms remains open, and here Knightian uncertainty about complex systems (see §17.6) outruns the tidy risk models of auction theory. *** ## 24.6 Selling the Airwaves: Auctions for Many Objects Single-object theory is where the FCC's problem begins, not where it ends. Sold one at a time, each license's price depends on guesses about licenses not yet sold; sold all at once, licenses raise the problem the vignette posed. ### The Exposure Problem A national carrier values the Boston–New York pair at 100 but either license alone at 20. Each city has a local carrier interested only in its own city, with a value of 40 or 90 — equally likely, independent across cities, and known only to itself. Efficiency requires the national carrier to win both licenses when both locals are weak (100 against 80) and the locals to win otherwise (130 or 180 against 100). Sell the licenses in a simultaneous ascending auction, prices rising together. Above 20 the national carrier is *exposed*: winning just one license loses money. If it stays in until 40 and both locals quit, it wins both for 80 and gains 20. If only one quits, it holds that license at 40 while the other's price climbs past 80, the most the second license is worth to it, and it ends up owning one license worth 20 that cost 40. Staying in gains 20 with probability one-quarter and loses 20 with probability one-half — an expected loss of 5. So the carrier quits at 20, and when both locals are weak the licenses go to bidders who value them at 80 rather than to the one who values them at 100. (With other values the carrier stays in, and the inefficiency takes the opposite form — it is stranded with one license; see Problem 7.) The SMRA mitigates exposure without solving it. Nothing closes until a round passes with no new bid on *any* license, so a bidder priced out of one license can switch to a substitute; with substitutes the design works as a real Walrasian tâtonnement (see §8.1; Milgrom 2004). An *activity rule* requires each bidder to bid on a minimum share of its eligibility every round or lose it, so no one can lie in wait while others reveal their values, and *withdrawal* rules let a bidder back out of a standing bid at a penalty, a partial insurance against exposure. But with strong complements, bidders must still assemble packages license by license. ### Package Bidding Accepting bids on *packages* removes the exposure problem. The theoretical ideal, VCG, fails badly in practice (Ausubel and Milgrom 2006). Let each local value its city at 100 and the national carrier bid 100 for the pair and nothing for either license alone. VCG gives each city to its local. Without the Boston local, the best use of the licenses for everyone else is worth 100 (the pair to the national carrier, or New York to its local); with her present, the others still get 100. She imposes no externality and pays nothing — and nor does the New York local. The seller collects zero although one bidder offered 100 for the pair — an outcome the seller and that bidder would both rather overturn, so it lies outside the *core*. Practical package auctions therefore charge core-selecting prices, high enough that no coalition of losers could have offered the seller more. The FCC experimented cautiously. For part of the 700 MHz band in 2008 it used *hierarchical package bidding*, which allows bids only on a preset nest of packages — individual regional licenses, larger groups, and a nationwide bundle (Goeree and Holt 2010). Regulators in Britain, Canada, and Australia adopted the *combinatorial clock auction*: an ascending clock phase, in which bidders discover prices, followed by sealed package bids priced by a core-selecting rule. The FCC itself has moved toward *clock auctions*: the auctioneer announces a price for each category of generic spectrum blocks, bidders state how many blocks they want, prices rise where demand exceeds supply, and a final phase assigns specific frequencies. ### The Incentive Auction The 2016–17 incentive auction had to buy spectrum from television broadcasters and sell it to wireless carriers in one market. Stations that stay on the air must be *repacked* into a narrower band without interfering with one another, and checking whether a given set of stations fits is a computationally hard problem, akin to coloring a map with few colors. Milgrom and Segal (2020) made the reverse auction a *descending clock*. Each station was offered a falling price to give up its license and could drop out at any time, keeping a channel in the repacked band; once a station could no longer be repacked alongside those that had dropped out, its price stopped falling and the FCC bought its license at that price. Dropping out when the offer falls below the value of staying on the air is *obviously* dominant, and because no offer is accepted until the end, the design is a *deferred-acceptance auction*, holding bids tentatively as a school in DA holds applicants. A forward clock auction sold the freed spectrum; a *clearing target*, lowered in stages until carriers' bids covered the payments to broadcasters and the cost of relocating stations, linked the two. The auction closed in 2017 having cleared 84 megahertz; carriers bid about $19.8 billion, of which about $10 billion went to broadcasters (Federal Communications Commission 2026). > **How We Know: Testing Auction Rules before Betting Billions on Them** > > Field data cannot say which multi-object format works best. Values are unobserved, so efficiency cannot be measured; each auction sells a different band at a different moment, so revenue comparisons confound format with spectrum, timing, and demand; and a new format has no field record. Structural work infers values from the final allocation, but under assumptions about bidding that carry the conclusions (Bajari and Fox 2013). > > The laboratory solves the measurement problem by construction. With *induced values* (Smith 1976), the experimenter pays each subject according to a value schedule she assigns — 20 for either license alone and 100 for the pair, say — so the efficient allocation is known in advance, and the *same* environment can be run under several formats with fresh subjects. Differences in efficiency and revenue are then caused by the rules, because nothing else differs: random assignment of institutions, impossible in the field. Such *testbeds* probe the failures theory warns of — exposure losses, confusion, collusive signaling — and the hierarchical package bidding the FCC adopted in 2008 was developed and tested in experiments of this kind (Goeree and Holt 2010). > > The limits mirror the strengths. Students bidding modest stakes for an afternoon are not carriers bidding billions over weeks, so testbeds are best at detecting flaws that bite at any stakes and weakest at predicting magnitudes. *** ## 24.7\* Formal Appendix: Outline Appendix 24 collects the formal derivations: the Bayesian Nash equilibrium of the first-price auction; revenue equivalence via the envelope theorem; Myerson's optimal auction, reserve prices, and ironing; the Revelation Principle; VCG's dominant-strategy property and its budget in auctions, public projects, and bilateral trade; the existence, student-optimality, and strategy-proofness of deferred acceptance (via the Blocking Lemma), with an example of manipulation by a school; the Myerson–Satterthwaite theorem and the second-best mechanism; and the Gibbard–Satterthwaite theorem. *** > **Common Confusions** > > *"Auctions always lead to efficient outcomes."* Only under specific conditions. With symmetric bidders and private values, all four standard formats award the object to the highest-value bidder. Efficiency fails when bidders are asymmetric (a first-price auction can hand the object to a weaker bidder who shades less), when goods are complements (the exposure problem deters efficient package bids), when bidders collude, or when the seller imposes a revenue-maximizing reserve that sometimes leaves the object unsold. Efficiency is a design *choice*, not an automatic property of auctions. > > *"A reserve price breaks revenue equivalence."* A common reserve changes who gets the object — sometimes no one — and so changes expected revenue, but first- and second-price auctions with the same reserve still raise the same amount. Revenue equivalence compares formats; it says nothing about whether any of them maximizes revenue. > > *"Mechanism design can solve any allocation problem."* When it is uncertain whether trade is efficient, Myerson–Satterthwaite shows that bilateral trade with private values cannot be made at once efficient, incentive compatible, individually rational, and budget-balanced; Gibbard–Satterthwaite shows that on an unrestricted domain the only strategy-proof voting rule is dictatorship. The impossibility theorems matter as much as the constructive results. > > *"Matching markets are inferior because they don't use prices."* They exclude prices for reasons — ethical, distributional, practical — and face their own trade-offs: deferred acceptance is stable and strategy-proof but not efficient, top trading cycles is efficient and strategy-proof but not stable, and in some markets, as §24.3's example shows, no mechanism is both stable and efficient. The choice reflects social values, not technical inferiority. > > *"The Revelation Principle means the designer has full information."* It says agents can be induced to *report* truthfully, not that the designer knows their types in advance. Truthful revelation must be *paid for* with information rents — the surplus left to high types in screening, the winner's surplus in an auction. *** ## Summary Auctions are designed institutions too. The four standard formats raise the same expected revenue when symmetric, risk-neutral bidders have independent private values; equivalence fails with risk aversion, affiliated values, or asymmetric bidders. Myerson's optimal reserve shows that equivalence is not maximization — the seller gains by sometimes withholding the object — but one more bidder beats the best reserve (Bulow and Klemperer 1996). The winner's curse shows the information environment shaping bids. Matching markets replace prices with algorithms that must balance stability, strategy-proofness, and efficiency: deferred acceptance delivers the first two, top trading cycles the last two, and sometimes no mechanism is both stable and efficient. Kidney exchange, with non-simultaneous chains started by altruistic donors, shows design creating value no spontaneous process would find. The Revelation Principle reduces the search for mechanisms to truthful direct ones; VCG achieves efficiency in dominant strategies but not budget balance, and fails when bidders want packages; Myerson–Satterthwaite and Gibbard–Satterthwaite mark costs of private information that no design removes. Practice adds robustness, cognition, and manipulation, and designed markets keep evolving. The spectrum auctions assemble every tool — the exposure problem, the SMRA, package and clock auctions, an incentive auction built on deferred acceptance — and show that the designer's objective is political. Design is strongest where rules need not depend on fragile details, and humbler where Knightian uncertainty rules. Chapter 25 turns to the welfare foundations that justify choosing one mechanism over another. *** ## Key Terms * **Market design** — The applied field that uses mechanism design and engineering judgment to create or improve markets and allocation systems * **Mechanism design** — The theory of designing games so that self-interested agents with private information produce desired outcomes * **Revelation Principle** — Whatever a mechanism implements in some equilibrium, a direct mechanism implements with truthful reporting as an equilibrium of the same kind * **Incentive compatibility (DSIC)** — Truth-telling is optimal whatever the others report (strategy-proofness) * **Incentive compatibility (BIC)** — Truth-telling is optimal in expectation over others' types, given a common prior * **VCG mechanism** — Efficient allocation with dominant-strategy truth-telling, achieved by charging each agent the externality she imposes on others * **Revenue Equivalence Theorem** — With symmetric, risk-neutral bidders and independent private values, all auctions that award the object to the highest-value bidder and leave the lowest type zero surplus raise the same expected revenue * **Linkage principle** — With affiliated values, tying the price to other bidders' information raises expected revenue, so English auctions raise at least as much as sealed-bid formats * **Virtual value** — A bidder's value net of the information rent left to higher types, $$\phi(v) = v - (1 - F(v))/f(v)$$ * **Reserve price** — A minimum price below which the seller keeps the object; the revenue-maximizing (Myerson) reserve sets the virtual value to zero * **Winner's curse** — The winner of a common-value auction tends to hold the most optimistic estimate, and overpays unless she shades her bid * **Stable matching** — A matching with no blocking pair: no two agents who both prefer each other to their assigned partners * **Deferred acceptance** — The Gale–Shapley algorithm, in which receivers hold offers tentatively; proposer-optimal, stable, and strategy-proof for the proposing side * **Boston mechanism** — Immediate acceptance: seats go first to students who rank a school first, each assignment final; manipulable * **Top trading cycles** — An algorithm that carries out cycles of students pointing to schools and schools to priority students; Pareto efficient and strategy-proof but not stable * **Justified envy** — A student prefers a school that ranks her above a student it admitted * **Obviously strategy-proof (OSP)** — A mechanism whose dominant strategy is recognizably dominant, as in an ascending clock auction (Li 2017) * **Kidney chain** — A sequence of transplants started by an altruistic donor; unlike a cycle, its operations need not be simultaneous * **Myerson–Satterthwaite theorem** — With independent private values drawn from overlapping ranges, no bilateral trading mechanism is efficient, incentive compatible, individually rational, and budget-balanced * **Gibbard–Satterthwaite theorem** — With three or more alternatives and unrestricted preferences, every strategy-proof social choice function that can select every alternative is dictatorial * **Exposure problem** — The risk that a bidder who values licenses as a package wins only part of it at prices above its stand-alone value * **Simultaneous multiple-round ascending auction (SMRA)** — All licenses open at once and close together, with prices posted each round and an activity rule that forces bidders to bid actively to keep their eligibility * **Package (combinatorial) bidding** — Bids on bundles of items, which removes the exposure problem but complicates pricing *** ## Problems ### \[Core] 1. **Second-price intuition.** You are bidding in a second-price sealed-bid auction for a painting you value at $500. Explain why bidding $500 is optimal. Then explain why bidding $600 could make you worse off but bidding $400 could never make you *better* off (only worse off or the same). 2. **Gale-Shapley practice.** Three students (X, Y, Z) and three schools (A, B, C) have the following preferences: | Student | 1st | 2nd | 3rd | | ------- | --- | --- | --- | | X | A | B | C | | Y | B | A | C | | Z | A | C | B | | School | 1st | 2nd | 3rd | | ------ | --- | --- | --- | | A | Y | X | Z | | B | X | Y | Z | | C | Z | X | Y | (a) Run the student-proposing DA algorithm. Show each round. (b) Verify that the final matching is stable by checking all potential blocking pairs. (c) Would any student benefit from misreporting their preferences? 3. **VCG payment.** Two bidders compete for a single object. Bidder 1 values it at $60; Bidder 2 values it at $45. (a) Who wins under VCG? (b) What does the winner pay? (c) Verify that this equals the second-price auction payment. ### \[Extension] 4. **Bid shading and competition.** In a first-price sealed-bid auction with $$n$$ bidders and values uniformly distributed on $$\[0, 100]$$: (a) Derive the equilibrium bidding function. (b) Calculate the expected revenue for $$n = 2, 5, 10, 100$$. (c) Show that as $$n \to \infty$$ expected revenue approaches 100, as does the expected highest value, so the winner's information rent vanishes. (d) Explain intuitively why more competition benefits the seller. (e) With $$n = 2$$, the seller can either set the optimal reserve price or attract a third bidder and set no reserve. Find the optimal reserve and compare the expected revenue of the two options. 5. **Revenue equivalence breakdown.** Suppose bidders are risk-averse with utility $$u(x) = \sqrt{x}$$, and values are uniform on $$\[0, 1]$$ with $$n = 2$$. (a) Argue qualitatively why a risk-averse bidder shades less in a first-price auction than a risk-neutral bidder. (b) Does this increase or decrease expected revenue relative to the risk-neutral benchmark? (c) Is the second-price auction's expected revenue affected by risk aversion? Why or why not? 6. **Myerson–Satterthwaite.** A buyer values an object at $$v\_B \sim U\[0, 1]$$ and a seller values it at $$v\_S \sim U\[0, 1]$$, independently; each knows only her own value. Efficiency requires trade whenever $$v\_B > v\_S$$. (a) What is the probability that trade is efficient, and what are the maximum expected gains from trade? (b) Suppose instead that trade takes place at a posted price $$p = 0.5$$ whenever both parties agree to it. Show that this mechanism is incentive compatible, individually rational, and budget-balanced; compute the probability of trade and the share of the maximum expected gains it realizes; and explain why it is not efficient. (c) Explain intuitively why no mechanism can achieve efficiency, incentive compatibility, individual rationality, and budget balance at once. 7. **Exposure and package bidding.** Two licenses, A and B, are for sale. A national carrier values the pair at 100 and either license alone at 20. Local carrier $$L\_A$$ wants only A and local carrier $$L\_B$$ only B; each local's value is 40 or 90 with equal probability, independently, and known only to that local. The licenses are sold in a simultaneous ascending clock auction: prices on both rise together in small steps, a bidder that leaves a license cannot return to it, and a license is sold at the current price once a single bidder remains on it. Each local stays in until the price reaches its value. (a) For each of the four value profiles, find the efficient allocation. (b) Suppose the national carrier stays in on both licenses until prices reach 40 and then acts optimally. Find its payoff in each profile and its expected payoff. Should it bid above 20? (c) What is the expected loss of efficiency when the national carrier bids optimally? (d) Repeat (b) and (c) when each local's value is 30 or 90, so that a national carrier that stays in learns the locals' values at 30. (e) Now each local values its license at 60, and the national carrier bids 100 for the pair, and nothing for either license alone, in a VCG auction that accepts package bids. Find the allocation, each bidder's payment, and the seller's revenue. Why might the seller and the national carrier both object? ### \[Empirical] 8. **FCC spectrum auctions.** The FCC's auction pages (Federal Communications Commission 2026) give each auction's dates, format, and winning bids, the population of each license area, and the bandwidth of its licenses. (a) For at least one auction of each format described in §24.6 — an SMRA (for example Auction 97, AWS-3, 2014–15), the SMRA with hierarchical package bidding (Auction 73, 700 MHz, 2008), the incentive auction's forward auction (600 MHz, 2016–17), and a clock auction (for example Auction 107, C-band, 2020–21) — compute the price per MHz-pop: net winning bids divided by the product of bandwidth in megahertz and population covered. (b) Are there systematic differences across formats? (c) What confounding factors make it difficult to attribute the differences to format alone, and how does the evidence described in the spectrum How We Know box avoid them? 9. **School choice reform.** Abdulkadiroğlu, Agarwal, and Pathak (2017) evaluate New York City's move from uncoordinated offers to a single-offer deferred acceptance mechanism for students entering high school in 2004. (a) Why would a simple before–after comparison of student outcomes be potentially misleading? (b) The authors estimate students' preferences from the rank-order lists submitted under the new mechanism. Why does the strategy-proofness of DA make this possible, and how does the twelve-choice cap complicate it? (c) Pathak and Sönmez (2008) argue that manipulable mechanisms such as the Boston mechanism transfer surplus from sincere to sophisticated families. What pattern in rank-order lists submitted under a Boston-style mechanism would suggest strategic reporting, and how could a switch to DA be used to test for it? 10. **Detecting collusion.** Porter and Zona (1993) study bidding in Long Island highway construction contracts. (a) Under competitive bidding, what should the relationship be between a firm's cost estimate and its bid? (b) Under a bid-rigging cartel with bid rotation, how would this relationship differ for cartel members versus competitive fringe firms? 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