The Parable of the Auctioneer: Complexity in Paul R. Milgrom’s Discovering Prices

Scott Duke Kominers∗ and Alexander Teytelboym∗∗

Designing marketplaces in complex settings requires both novel economic theory and real-world engineering, often drawing upon ideas from fields such as computer science and operations research. In Discovering Prices, Milgrom(2017) explains the theory and design of the United States “Incentive ” that reallocated wireless spectrum licenses from television broadcasters to telecoms. Milgrom’s account teaches us how economic designers can grapple with complexity both in theory and in practice. Along the way, we come to understand several different types of complexity that can arise in . Keywords: , pricing, market design, complexity JEL Classification: D47, C78, D44, D82, D02, D51, D71, D61, D62, D63

1. Introduction Meanwhile, the Ming and Qing Dynasties (1594– 1700) used randomized assignment systems to For hundreds of years, people have found in- eliminate the possibility of corruption in the se- genious ways to allocate resources and discover lection of candidates for civil service jobs (Will, prices. Merchants in medieval Europe, for ex- 2002).2 And even the great German poet Johann ample, set up multilateral clearing mechanisms Wolfgang von Goethe designed an auction: He for non-tradable debt, known as rescontre; these was interested in learning what value his publisher decentralized matching algorithms involved first put on a manuscript, and this led him to devise a clearing reciprocal debts and then clearing debt mechanism that turned out to be equivalent to a chains and cycles (e.g., in which i owes j who second-price, sealed-bid auction (Moldovanu and owes k who owes i). Rescontre appears to have Tietzel, 1998).3 been highly efficient (Börner and Hatfield, 2017).1 Although marketplace design has existed for ages, systematizing the field through the language

∗ Entrepreneurial Unit, Harvard Business School; Department of , Center of Mathematical Sciences and Applications (CMSA), and Center for Research on Computation and Society, Harvard University; and National Bureau of Economic Research. ∗∗ Department of Economics, Institute for New Economic Thinking at the Oxford Martin School, and St. Catherine’s College, University of Oxford. The authors appreciate helpful comments from the editor, Steven Durlauf, as well as from Zoë Hitzig, Paul Klemperer, Shengwu Li, Alex Nichifor, Yufei Zhao, and especially Paul R. Milgrom. Kominers is grateful for the support of the National Science Foundation (grants SciSIP-1535813 and SES-1459912), the Washington Center for Equitable Growth, the Ng Fund and the in Economics Fund of the CMSA, and the Human Capital and Economic Opportunity Working Group (HCEO) sponsored by the Institute for New Economic Thinking (INET). Teytelboym is grateful for the support of the Economic and Social Research Council (grant ES/R007470/1). Both authors are grateful for the support of an Alfred P. Sloan Foundation grant to the CMSA. 1In Venice, the rescontre mechanism was temporarily banned in 1593 (Börner and Hatfield, 2012, p. 15); this may have contributed to some moneylenders’ subsequent decision to demand corporal collateral, as in Shakespeare’s (1600) “pound of flesh.” 2Some have also speculated that Babylonians adopted lottery mechanisms for fairness reasons (Borges, 1959). 3Unfortunately, because Goethe’s legal counsel was unreliable, the publisher learned the reserve price in advance. 1 2 SCOTT DUKE KOMINERS AND ALEXANDER TEYTELBOYM and logic of economic theory has allowed us to ad- It seems intuitive that matching of couples dress problems in increasingly complex settings. to hospitals and optimal reserve pricing is some- The markets we consider today are more diverse how more “complex” than Goethe’s manuscript and interconnected than before—and our design allocation problem—and the problems that ride- goals have grown more ambitious. sharing companies face seems more complex still. In the 1950s, the American Medical Asso- But what does “complex” actually mean in this ciation (AMA) faced the problem of fairly and context? Economic theory does not on its own efficiently allocating residents to hospitals. The give us a language for understanding complex- AMA eventually arrived at an algorithm that ap- ity of market(place) design problems. Indeed, as peared to work rather well—the elegant Gale and Paul R. Milgrom argues in his new book, Dis- Shapley(1962) algorithm (Roth, 1984, 2008). But covering Prices: Auction Design in Markets with several decades later, the residency match faced a Complex Constraints: new problem: Doctors were marrying each other in greater numbers, necessitating a method to “most of economic theory incorporate couples’ joint preferences into the resi- treats simple models of the dency assignment process; solving this real-world economy in which prices alone “couples matching” problem relied on a subtle can guide efficient economic combination of economic theory and operations decisions, but that conclusion research (Roth and Peranson, 1999). Likewise, rarely applies exactly in com- economic theory has informed the adoption of plex systems” (Milgrom, 2017, Myerson(1981) optimal reserve prices in online p. 44). ad auctions (Ostrovsky and Schwarz, 2016), as Discovering Prices is based on the Arrow well as the eventual shift of ad auctions from Lecture Milgrom delivered at Columbia Univer- practitioner-designed “generalized second price” sity in November 2014. The introductory chap- auctions (Aggarwal et al., 2006; Edelman et al., ter is itself a mini-monograph accessible to non- 2007; Varian, 2009; Athey and Ellison, 2011) to specialists (including non-); there, Mil- -inspired Vickrey–Clarke–Groves grom illustrates the challenges that arise in de- mechanisms (Athey and Nekipelov, 2010; Varian signing markets when there are complex con- and Harris, 2014). It is hard to imagine any of straints on the allocative process. Milgrom dis- these innovations in marketplace design arising cusses three examples: farm consolidation after (or even being possible) without the groundwork land lotteries in Georgia; determining airspace laid by economic theory. rights for commercial space and atmospheric Nowadays, many markets are extraordinarily travel; and the Federal Communications Com- complex—both in theory and in practice. Con- mission’s (FCC’s) multibillion-dollar “Incentive sider the myriad difficulties a ride-sharing com- Auction,” which reallocated wireless spectrum to pany such as Lyft or Uber faces in organizing its repurpose it from broadcast television to telecom marketplace: The company must figure out where applications. In all three settings, there are sig- its customers want to go and how much they are nificant complementarities and interdependencies willing to pay, while tracking drivers and their between goods; as Milgrom shows, this makes it time-varying reserve prices for providing service. difficult to find price mechanisms that guide mar- The company then needs to efficiently calculate ket participants towards efficient allocations. Any journey prices and match drivers to riders. Over- economic theorist could read Discovering Prices all, the company must raise revenue to cover its from Chapter 2, which opens with a masterful costs (and reward its shareholders) while encour- exposition of the close analytical relationships aging drivers and customers to join and remain between existence and stability of competitive on the platform. And all of this has to happen equilibria, tâtonnement in general equilibrium “under the hood” in real time, and preferably be- theory, deferred acceptance algorithms in match- fore customers get tired of waiting and switch to ing markets, and auction design for indivisible another app. goods. As Milgrom points out, “substitutability” of goods links all four settings. Indeed, “gross THE PARABLE OF THE AUCTIONEER 3 substitutability” is crucial for uniqueness of com- (1) a market and petitive equilibrium and for tâtonnement in a (2) one or more design goals. classical general equilibrium setting (Arrow and Hurwicz, 1958), for existence of equilibrium in It seems plausible that Goethe’s primary design markets with indivisible goods (Kelso and Craw- goal was long-run profit—that is, he not only ford, 1982; Milgrom and Strulovici, 2009; Gul and wanted to maximize revenue from his manuscript, Stacchetti, 1999; Sun and Yang, 2006; Hatfield et but also to find what value the publisher placed al., 2013; Fleiner et al., forthcoming), for conver- on his work. Additional design goals common in gence of simple monotonic auctions (e.g., the de- practice include: ferred acceptance algorithm) to efficient outcomes • fairness (e.g., in school choice, Abdulka- (Gul and Stacchetti, 2000; Sun and Yang, 2009; diroğlu and Sönmez(2003), or food sup- Teytelboym, 2014), and for the numerous attrac- ply, Prendergast(2016)), tive properties of the Vickrey auction (Ausubel • revenue (e.g., in auctions, Myerson and Milgrom, 2006). (1981); Binmore and Klemperer(2002); Milgrom’s main theoretical innovations are Varian(2007); Ostrovsky and Schwarz the introduction and development of “near- (2016)), substitutability,” and associated approximate • simplicity/ease of use (e.g., of bidding equilibria (inspired by the pioneering work of languages, Lubin et al.(2008); Milgrom Dantzig(1957) and Korte and Hausmann(1978)), (2009, 2010); Klemperer(2010); Bich- along with the analysis of the performance of ler et al.(2014), or ordinal the deferred acceptance auction without substi- elicitation, Pathak and Sönmez(2008)), tutability (Milgrom and Segal, forthcoming). The • transparency (e.g., in financial markets, final two chapters connect the concepts of sub- Asquith et al.(2013)), stitutes and near-substitutes to powerful insights • privacy (e.g., in voting systems, McSh- from operations research, in order to explain the erry and Talwar(2007)), theoretical foundations behind the Incentive Auc- • speed (e.g., in algorithmic trading, Bud- tion, and to suggest future directions for the ish et al.(2015); Kyle and Lee(2017)), design of auctions under complex constraints. • entry or competition (e.g., in auctions In this essay, taking inspiration from Milgrom Levin and Smith(1994); Bajari and Hor- (2017), we offer a conceptual framework for think- taçsu(2003)), ing about complexity in markets. We show how • price discovery (e.g., in stock exchanges, Milgrom’s account of the design of the Incen- Biais et al.(1995, 1999); Cao et al. tive Auction serves as a sort of “parable” for the (2000)), design of complex markets. Along the way, we • elimination of adverse selection (e.g., highlight three different types of complexity and in ad auctions, Abraham et al.(2013); describe how they shape marketplace design. Arnosti et al.(2016), or health insurance, 2. Markets, Marketplaces, and Finkelstein and Poterba(2004)), Complexity • trust (e.g., in financial benchmarks, Duffie and Stein(2015)), and/or A market arises whenever agents interact and • operation within ethical bounds (e.g., seek to transact, subject to economic incentives. in kidney exchange, Roth et al.(2004); Markets are everywhere—and many (but not all) Roth(2007); Leider and Roth(2010)). of them are at least partially organized through marketplaces consisting of rules guiding which Marketplace designers must overcome both transactions occur and how, and/or infrastruc- theoretical and practical challenges in order to ture that facilitates transaction execution (Komin- achieve their goals; complexity characterizes the ers et al., 2017; Eisenmann and Kominers, 2018; structure and difficulty of those challenges. Thus Kominers, 2018; Roth, 2018). complexity of a setting is somewhat like veloc- Marketplace design operates in settings con- ity: it has both a magnitude and a direction (or, sisting of rather, a type). Moreover, as much as we might 4 SCOTT DUKE KOMINERS AND ALEXANDER TEYTELBOYM want complexity to be an absolute property of de- local licenses, which needed to be aggregated into sign goals (a “topological property,” if you will), national networks for telecom use (Rosston, 2012; complexity is inextricably linked to the specific Leyton-Brown et al., 2017); this created a holdout market under consideration. For instance, bidder problem. Any television broadcaster who knew anonymity/privacy is easy to achieve in online that his or her license was crucial for assembling auctions, but is far more difficult to ensure in an a contiguous national network could in principle auction room at Sotheby’s. Transparency and have demanded compensation far above his or her trust, on the other hand, are a headache for eBay, true value (Kominers and Weyl, 2012; Rosston, but are less of a problem for brick-and-mortar 2012). Congress partially ameliorated holdout auction houses.4 risk by granting the FCC the authority to move broadcasters to different channels—i.e., to repack 3. The Auctioneer’s Parable them—in order to ensure that the spectrum freed up was on the same wavelengths nationwide. Milgrom’s Discovering Prices illustrates mar- But in order for the FCC to figure out which ketplace design in one of the most complex set- broadcasters it had to repack, it needed to know tings to date, showcasing how tools from eco- how much spectrum transfer the market would nomic theory, computer science, and operations bear, and which broadcasters would sell their research were combined in the design of Incen- licenses back. Working from the rough frame tive Auction. Along the way, Milgrom helps us of a double auction, the plan was first to run a understand how to identify and address different “reverse” auction to determine which broadcast sources of complexity. licenses could be bought back (with the remain- 3.1. The Setting. Radio spectrum is a lim- ing broadcasters being neatly repacked); then, ited resource: only a narrow band is useful for those licenses would be auctioned off to telecoms communication, and signals sent on overlapping in a “forward” (clock) auction. If the telecoms’ channels will interfere with each other. In the US, willingness to pay in the forward auction was not some of the most desirable spectrum—channels high enough to cover the spectrum acquisition on which signals can travel long distances and pen- cost determined in the reverse auction (plus a etrate walls—was allocated to television broad- revenue target), the FCC would reduce the spec- casters in the 1940s and 50s (Hazlett, 2008). To- trum clearing target, and return to the reverse day, however, fewer and fewer people in the US auction. watch broadcast television, while more and more access information through their mobile phones. The Incentive Auction sought to buy back 3.2. What made the Incentive Auction wireless spectrum licenses from hundreds of tele- complex? Underlying the setting for the In- vision broadcasters, while simultaneously selling centive Auction was a computational problem: those licenses to telecoms. The FCC hoped to Repacking the broacasters who did not sell back clear the market as efficiently as possible, with their licenses created the possibility of interfer- payments from the telecoms compensating broad- ence, whereby those broadcasters’ signals might casters who gave up their licenses. At the same partially or fully block each other; a given repack- time, the government sought to raise revenue to ing would be deemed appropriate only if it cre- cover the costs associated with the process. ated no more than “minimal” interference for If all spectrum licenses were homogeneous, every station.5 However, the general problem of then the FCC could have simply run a double auc- determining when a minimal-interference repack- tion to clear the market (Wilson, 1985; McAfee, ing exists is computationally intractable—it em- 1992). But licenses are by nature heterogeneous— beds a well-known, NP-complete graph-coloring they cover different areas and wavelengths (Cram- problem—making it effectively impossible to be ton, 2002). Moreover, most broadcasters owned certain which sets of broadcasters could be left

4But see recent work by Banksy(2018). 5According to the FCC criteria, a station j suffers minimal interference if no other station i interferes with more than 0.5% of station j’s pre-auction audience (Leyton-Brown et al., 2017, p. 7205). THE PARABLE OF THE AUCTIONEER 5

“on the air” without interference.6 As a conse- the sense that bidders are incentivized to bid quence, there was some fundamental risk of error truthfully—and should be able to understand as in identifying the efficient set of broadcast licenses much. As a result, under the deferred acceptance to buy back. auction, bidders can figure out that truthful bid- If the efficient set of purchasable spectrum ding is optimal without subtle contingent reason- licences could not be identified with certainty, ing about other bidders. Additionally, the format then we could at best hope to approximate the ensures that bidders can trust the auctioneer not efficient allocation. But incentive-compatible to tamper with prices in an undetectable way mechanisms—like VCG—suggested by theory (Akbarpour and Li, 2018). (Vickrey, 1961; Clarke, 1971; Groves, 1973; Green But how can we ensure that a deferred and Laffont, 1977; Holmström, 1979) determine acceptance-based auction can actually clear the prices by comparing efficient and nearly-efficient market? In a classic paper, Kelso and Crawford allocations; in such mechanisms, small errors in (1982) showed that the deferred acceptance algo- assessing efficiency get magnified into large errors rithm can find market-clearing prices whenever in pricing (Milgrom, 2017, p. 168). The challenge goods are (gross) substitutes. If some broadcast- was therefore to design an auction that would ers’ licenses are instead complements from the be (approximately) efficient and robust to im- perspective of the FCC—in the sense that getting perfections in assessing repacking feasibility—yet access to one might make others more valuable— simple enough for broadcasters to participate in. then market-clearing prices might not even exist. Existing economic theory provided a useful guide, One of Milgrom’s key theoretical contributions but was only a starting point. in Discovering Prices (pp. 190–194) is that if The reverse auction that Milgrom and his goods are, in a formal sense “near-substitutes,” collaborators proposed would start by offering a then the deferred acceptance algorithm will reach high price for each broadcaster’s license. The auc- prices that are nearly efficient, and the resulting tioneer would then decrease the price offers in se- allocation will be close to efficient, as well. quence. As soon as a broadcaster rejected the auc- To be a bit more precise: Milgrom consid- tioneer’s offer, it exited the auction permanently— ers the structure of the FCC’s preferences over thus remaining on the air (Milgrom and Segal, broadcast licenses. The FCC wanted to buy back forthcoming). To deal with concerns about in- licenses in a way that would maximize the value terference, the auction would “freeze” the price of the broadcasters who kept their licenses (and of any broadcaster that could potentially create thus remained on the air) in the end. That is, the interference if it were to leave the auction. set of rejected broadcasters that the FCC does The reverse auction format just described not acquire licenses from should be those with corresponds exactly to a broadcaster-proposing highest value for operating as stations.7 From an deferred acceptance algorithm in which broad- auction design perspective, it would be ideal if casters propose prices at which they would be the FCC’s preferences were (one-for-one) substi- willing to sell their licenses, and any price re- tutable, meaning that if, at a given price profile, jected by the auctioneer cannot be offered again the FCC chooses to leave set of broadcasters S (Kelso and Crawford, 1982; Hatfield and Milgrom, on the air but some station s ∈ S indicates that 2005); such a mechanism is simple to implement, it will be willing to sell its license at a new, lower and makes it particularly easy for broadcasters price, then the FCC will still want to leave the to figure out how to bid. Indeed, in the deferred broadcasters in S \{s} on the air. acceptance auction, honest bidding is obviously Unfortunately, because of the practical con- optimal for unit-demand bidders (Li, 2017), in straints on repacking, the FCC’s preferences were

6A problem is in (complexity class) NP if it can be solved in nondeterministic polynomial time; a problem is NP-complete if it is in NP and all problems in NP can be reduced to it (Sipser, 2012). At least assuming a mathematical conjecture widely believed to be true (Cook, 2006), solutions to problems in NP cannot, in general, be found efficiently (although they can be verified quickly). 7For technical reasons, it is typically easier to work with the set of broadcasters whose offers to sell back their licenses are rejected, rather than the set whose are accepted. 6 SCOTT DUKE KOMINERS AND ALEXANDER TEYTELBOYM not substitutable. It was possible, for exam- auction was also quite intricate,10 as were the ple, that when a small-bandwidth broadcaster auction-closing rule and the “assignment round” reduced its buyout price, the FCC would want in which different frequencies were allocated to to reject a large-bandwidth broadcaster and in- telecoms. stead buy back licenses from a set of currently And with so many important details to get rejected small-bandwidth stations. To get around right, it would be unlikely for everything to go this problem, Milgrom showed how to approxi- entirely according to plan. Discovering Prices mate non-substitutable preferences, like those went to press as the assignment round was still of the FCC, with substitutable ones. The ap- running; that phase used a Vickrey auction to set proximation gives rise to an intuitive and robust final prices for particular bandwidths. It is well- “index of substitutability” ρ that is equal to 1 known that when bidders have complementary whenever preferences are exactly substitutable, preferences over frequencies, some Vickrey auc- and approaches 0 when stations become perfectly tion equilibria could result in low revenues (Mil- complementary. grom, 2004; Ausubel and Milgrom, 2006; Marsza- Milgrom shows that the worst-case efficiency lec, 2018). And indeed, this is exactly what ap- performance of the deferred acceptance auction pears to have happened: Ausubel et al.(2017) (across all valuations) is precisely ρ (Proposi- estimate that successful coordination by the bid- tion 4.6 of Milgrom(2017); see also de Vries ders cost the assignment round around $5 million and Vohra(2019)). 8 That is, the more substi- in revenue (admittedly just 3.5% of the round’s tutable the FCC’s preferences are according to revenue, and less than a tenth of a percent of Milgrom’s index, the closer the worst-case perfor- the overall auction revenue). Meanwhile, some mance of the deferred acceptance auction is to private equity funds appear to have coordinated the fully efficient . By making repack- ownership of “Class A” broadcast licenses in the ing rules quite flexible, the FCC sought to make reverse auction, reducing supply and substantially licenses as substitutable as possible (Kominers raising the FCC’s acquisition costs (Doraszelski and Weyl, 2012; Rosston, 2012). In simulations, et al., 2017). the deferred acceptance auction achieved around 95% efficiency (Leyton-Brown et al., 2017). 4. Three Types of Complexity Even with deferred acceptance auctions as- sured to produce nearly efficient outcomes, the The story of the Incentive Auction as told repacking problem would still need to be solved in Discovering Prices vividly illustrates at least at every round of the auction clock. Thus, the three types of complexity that also arise in other design team developed state-of-the-art infrastruc- design settings. First, how should marketplace de- ture for finding feasible repackings quickly, based signers specify and trade-off various design goals? on newly-developed deep optimization techniques Second, how can a marketplace designer overcome (Newman et al., 2017). The new computational in- difficulties of eliciting information from and/or frastructure ensured that the feasibility-checking providing information to market participants? software did not timeout too frequently—and it Third—perhaps least familiar to economists— would never make Type I (“false positive”) er- how should marketplace designer tackle funda- rors.9 mental computational constraints? As we already While the reverse auction was a “key and saw in Section3, computational complexity at innovative part of the FCC’s Incentive Auction” the heart of spectrum reallocation was a large (Cramton et al., 2015, p. 1), the forward clock part of what made the setting so challenging.

8The result uses insights from the theory of matroids (see also Korte and Hausmann(1978); Ostrovsky and Paes Leme (2015)). 9That is, the feasibility-checker would only deem a set of broadcasters non-interfering if a permissible repacking could be found. But by nature, there was still some possibility of Type II errors (“false negatives”): for some configurations of broadcasters, it might not be possible to find a feasible repacking sufficiently quickly; in that case, the system would declare the set of broadcasters potentially interfering. 10It featured intra-round bidding to avoid bidder exposure, as well as condition reserves to promote competition. THE PARABLE OF THE AUCTIONEER 7

4.1. Specification complexity. In many target was hit led the designers to run the for- market design settings, it can be difficult to trans- ward and reverse auctions sequentially, adjusting late abstract design goals into practical objectives. the clearing target as they went along. And even once we have done so, our design goals Specification complexity in other settings. may be hard—or even impossible—to achieve si- Even settings with few design goals often force multaneously. When goals conflict and/or are the market designer to make difficult decisions difficult to express, simply specifying what to pri- about ill-specified objectives. Take efficiency, for oritize becomes a design challenge all of its own; example. We often think of “efficient outcomes” specification complexity reflects these decisions as being those that maximize allocative surplus and tradeoffs. (the “Kaldor–Hicks criterion”). In Goethe’s case, Specification complexity in spectrum realloca- we might think of efficiency as purely allocative: tion. The FCC’s early remarks on its goals for re- the goal was simply to deliver the manuscript purposing spectrum were extremely abstract. For to the publisher who valued it the most. But example, the 2010 National Broadband Plan rec- efficiency really means maximizing welfare—and ommended that the policymakers “ensure greater thus our interpretation of efficiency as a design transparency concerning spectrum allocation and goal depends on the welfare function. If we have utilization” and “expand incentives and mecha- distributional preferences, then efficiency might nisms to reallocate or repurpose spectrum” (FCC, involve distorting the allocation—or even wasting 2010, p. 75). One can imagine that there are resources—in order to improve the outcomes of many objective functions that would satisfy such poorer people in the market (Rawls(1971); Weitz- an abstract specification. The exercise of making man(1977); see also Dworczak R Kominers R the design goals more precise was the first prin- Akbarpour(2018)). Likewise, design goals such cipal complexity in the design of the Incentive as “fairness” and “transparency” depend crucially Auction. Following a wide-ranging consultation, on the market in question, and may be complex the policymakers decided that the goals of the to define in practice. auction would be: (1) to buy out hundreds of Even once we understand our design goals TV broadcasters with heterogeneous licenses, (2) individually, balancing among them can be com- to reallocate their licenses to telecoms as effi- plex. Sometimes, trade-offs arise from formal im- ciently as possible, and (3) to ensure a reasonable possibility results: Voting rules satisfying fairly amount of revenue for the government. But even mild desiderata end up being dictatorial (Arrow, these goals were not precise enough to serve as 1951; Gibbard, 1973; Satterthwaite, 1975). In guidance for how the auction should be run. If school choice and similar matching settings, it the market were one-sided (e.g., if the FCC only is typically impossible to simultaneously ensure had to sell available spectrum), then the problem (Pareto) efficiency, -proofness, and elimi- would have been easier—the FCC could have de- nation of justified envy (Abdulkadiroğlu and Sön- signed an efficient auction and simply increased mez, 2003; Abdulkadiroğlu et al., 2009; Kesten, the amount of spectrum on sale until the revenue 2010). Other times, some of our design goals target was hit. But designing efficient centralized pin down a unique rule or a class of rules—but two-sided markets with privately informed par- all the mechanisms in that class might fail to ticipants typically requires a subsidy (Myerson satisfy some other design goal. For example, in and Satterthwaite, 1983). Thus, the revenue vs. an object allocation setting with independent efficiency trade-off is much starker in broadcast private values and transfers, if we always want spectrum reallocation settings than in typical to choose efficient outcomes in a strategy-proof spectrum auctions (Loertscher et al., 2015). way, we must use a Groves mechanism (Groves, The auctioneers resolved the objective specifi- 1973; Green and Laffont, 1977; Holmström, 1979; cation complexity in two ways. First, in order to Williams, 1999). But Groves mechanisms are not increase competition between broadcasters, spec- budget-balanced and might be susceptible to col- trum was homogenized by allowing repacking. lusion (Klemperer, 2004; Ausubel and Milgrom, Second, the need to figure out how much spec- 2006). Another common example arises when trum to clear while ensuring that the revenue 8 SCOTT DUKE KOMINERS AND ALEXANDER TEYTELBOYM governments run spectrum auctions and set effi- then broadcaster holdout on its own would have ciency and revenue as their two key goals—it is made reassembly impossible: there was likely well-known that these objectives conflict even in so much complementarity between TV stations single-object auctions (Myerson, 1981). Yet more that the FCC would have had no chance of de- complex trade-offs arise from practical considera- termining the appropriate compensation to take tions: in ecosystem services markets in develop- a nationwide band of them off-air (Mailath and ing countries, for example, market designers must Postlewaite, 1990). Fortunately, the ability to balance development targets with environmental repack TV stations created enough competition concerns (Jayachandran et al., 2017; Salzman et to be able to think of the market as approximately al., 2018; Wunder et al., 2018; Teytelboym, 2019). competitive (Cramton et al., 2015).11 While the Specification complexity can also arise in the forward auction rules could, in principle, build on seemingly simple process of choosing which trans- previous package auctions for spectrum, running actions should be allowed. When goods are com- the reverse auction to clear the TV stations off- modities, every transaction is like every other air was no trivial matter. The natural strategy- (for a given transaction volume). But in mar- proof way to find the efficient outcome—running kets with substantial heterogeneity, it may be a VCG auction—would have been confusing for hard to determine which types of transactions the broadcasters and, in the presence of possible to “conflate” (i.e., treat as identical), and which complementarities, would have had undesirable to keep distinct. These sorts of questions arise properties (Rothkopf, 2007). Informational com- in ride-sharing (how local should pricing be?), plexity therefore remained in two forms: how to online advertising (how fine-grained should audi- elicit broadcasters’ valuations as accurately as ence targeting be?), and most spectrum auctions possible and how to give broadcasters clear and (which bands should be sold together?). By con- credible signals about how to bid. The deferred flating sets of slightly different transactions, the acceptance auction elegantly tackled this informa- market designer might be able to increase mar- tional complexity; meanwhile, Milgrom’s results ket thickness, reduce cherry-picking, and make on the robustness of the deferred acceptance auc- the marketplace safer for participants (Levin and tion in the presence of complementarities gave Milgrom, 2010; Milgrom, 2010). But at the same some reassurance about how closely the outcome time, conflation might harm efficiency or other might approximate efficiency. allocative goals by limiting the marketplace’s abil- Informational complexity in other settings. In ity to respond to heterogeneity in preferences. many settings, of course, market-clearing prices are enough to both aggregate preferences and 4.2. Informational complexity. Even af- guide market-clearing. But communication fric- ter the designer’s objectives are clear, information tions, preference heterogeneity, and/or comple- frictions often stand in the way. The market needs mentarities among goods may make it hard to to somehow “discover” participants’ preferences. determine market prices (Hurwicz, 1960; Mount And before that, participants need to acquire and Reiter, 1974; Nisan and Segal, 2006; Segal, and process information in order to access the 2007). Pricing is also difficult whenever utility is market and form their preferences. Informational not fully transferable, either because of societal complexity reflects the difficulties that arise in ag- preferences or incomplete markets (Fleiner et al., gregating information from market participants forthcoming). Additionally, efficient pricing may and/or in providing information to them. conflict with other design goals.12 Informational complexity in spectrum reallo- When prices are unavailable to aid decen- cation. Figuring out how much to compensate tralized market-clearing, we sometimes develop broadcasters for selling their licenses was a daunt- ing challenge. If repacking had not been possible,

11However, as we noted briefly in Section 3.2, Doraszelski et al.(2017) estimate that strategic supply reduction did impact the market, and may have increased FCC’s acquisition costs by 22%. 12For example, in auctions to sell complements, Vickrey prices sit outside the , result in low revenue, and encourage . In response, many different core-pricing rules have been proposed (Day and Milgrom, 2008; Erdil and Klemperer, 2010; Day and Cramton, 2012; Bünz et al., 2018). THE PARABLE OF THE AUCTIONEER 9 centralized mechanisms in which we elicit par- identify high-quality trading partners (Pallais, ticipants’ preferences directly. But here again 2014; Li et al., 2016).13 complexity can arise: Many systems for assigning children to schools, for example, ask parents to rank-order the schools their children may attend. 4.3. Computational complexity. Ques- If the assignment mechanism is strategy-proof, tions of how to specify objectives and structure then it is straightforward for parents to submit information have been central in market(place) preferences—they simply list the schools they design—and economics, more broadly—for a long find acceptable in true preference order. But sub- time. But many market design settings have a mitting even a simple list may be difficult for further layer: computational complexity, under parents if the assignment system is manipulable; which fundamental computational constraints im- moreover, in that case, the designer struggles to pede the designer’s ability to achieve his or her determine how the submitted rank-order lists re- goals.14 Computational problems affect the ex- late to parents’ true preferences. Moreover, even tent of specification and informational complex- if a marketplace can elicit true preferences from ity, and can also introduce independent practical individual agents, it may be unable to prevent problems. more complicated manipulations. For example, Computational complexity in spectrum reallo- Vickrey auctions can not only be manipulated cation. As we have already discussed, the prob- by coordinated misreporting (i.e., they are not lem of determining which sets of broadcasters “collusion-proof”), but also by using shill bidders could be left on the air with minimal interference (i.e., they are not “false-name-proof”; see Yokoo was computationally intractable. This computa- et al.(2004)). tional complexity constrained the design of the Complexity might also arise when the de- Incentive Auction substantially—in particular, it signer needs to elicit combinatorial preferences made computing efficient allocations impractical, over a large set of objects. In contexts like dis- which ruled out VCG and similar mechanisms. In play ad auctions (Lahaie et al., 2008), electricity VCG, errors in solving the repacking problem are markets (Schnizler and Neumann, 2007), and amplified by interdependence of prices and by the business school course allocation (Sönmez and need to compute prices in one step. By adjusting Ünver, 2010; Budish, 2011; Budish and Cantillon, prices gradually, the deferred acceptance auction 2012), some goods might be complements, while was less sensitive to imperfections in feasibility others are substitutes—meaning that rank-order checking and more likely to find market-clearing lists are insufficient to characterize participants’ prices (and thus an efficient allocation). Even so, preferences. Yet submitting high-dimensional advances in deep optimization were required in combinatorial preferences is infeasible, necessi- order to ensure that the repacking problems could tating the design of simpler bidding languages be solved quickly and accurately enough that the (Milgrom, 2009; Bichler et al., 2014; Budish and auctioneer would know which prices could be Kessler, 2016; Budish et al., 2016). reduced at each stage. In still other markets, complexity can arise Computational complexity in other settings. from the need to provide information to help par- More generally, we know from computer science ticipants learn their own preferences. It can be that some types of computation (say, the “set nontrivial, for example, to enable parents to un- cover” problem or finding (Bayesian) Nash equi- derstand the trade-offs between different school- libria) are difficult to execute quickly. On the ing options for their children (Hastings et al., other hand, it is straightforward to run English 2007; Corcoran et al., 2018). And even struc- auctions to find efficient allocations of heteroge- tured marketplace exchanges may lack ratings or neous, substitutable items (Kelso and Crawford, certification systems that would help participants 1982; Demange et al., 1986; Gul and Stacchetti,

13For further discussion, see Sections 4 and 5 of Kominers(2018). 14Our usage of the term “computational complexity” is not too distant from its use in theoretical computer science. Complexity theory characterizes the inherent difficulty of different computational problems; our broader usage here also includes practical computational challenges. 10 SCOTT DUKE KOMINERS AND ALEXANDER TEYTELBOYM

2000). But when markets are dynamic or fea- and Prestwich(2014); Masoud and Jayakrishnan ture substantial heterogeneity, computing market- (2017); Ma et al.(2018); Rheingans-Yoo et al. clearing prices is often very difficult (Nisan and (2019)). And surely even more complex settings Segal, 2006; Gonczarowski et al., 2014). Likewise, will yield to our design efforts in years to come. in two-sided matching markets we can find stable Marketplace design in increasingly complex outcomes via the deferred acceptance algorithm settings may bring political and societal chal- (Gale and Shapley, 1962). But in markets with lenges. As Hitzig(2018) argues, the Incentive combinatorial preferences, it can be computation- Auction is a signal that a future market designer ally intractable to compute a stable outcome or advising the government might no longer be able even check whether one exists (McDermid and to act simply as competent technocrat or a hum- Manlove, 2010; Manlove, 2013). ble “economic engineer” (Roth, 2002), but would Sometimes, we can “relax” computationally have to become “a craftsman [and] a technol- complex problems to solve them faster and/or ap- ogist who implements as well as designs, and proximately (see, e.g., Nguyen and Vohra(2018, who creates as well as conceives”. As market forthcoming); Nguyen et al.(2019))—yet practi- design moves into complex settings, it becomes cal concerns may rule such relaxations out. For less clear how designs can be audited and how example, computing fully efficient allocations in designers can be scrutinized. Market designers electricity markets is computationally difficult, must not lose not sight of the need to maintain yet if we seek approximate solutions we might public trust. But checks on the remit of market end up with mismatches in supply and demand designers should not necessarily place limits on that cause blackouts (Cramton, 2017). the expanding role of market design. At a time when politicians say that “the people . . . have had enough of experts,”15 Milgrom(2017) gives us 5. Outlook hope that market designers will still be called So what have we discovered from Prices? upon to showcase their craft. Modern marketplace design increasingly wres- tles with complexity; as it does so, we need novel, tailor-made theory as well as supporting infras- tructure. Complexity has real economic meaning— References and can take multiple forms. Discovering Prices teaches us how to map and Abdulkadiroğlu, Atila and Tayfun Sönmez, isolate different forms of complexity—to pin down “School choice: A approach,” exactly where the standard economic toolkit can , 2003, 93 (3), 729– be applied, where existing methods can be ex- 747. tended, and where entirely new tools are required. Abdulkadiroğlu, Atila and Tayfun Sön- In doing so, Milgrom(2017) proves that market- mez, “School Choice: A Mechanism Design place design can work in increasingly complex Approach,” American Economic Review, 2003, settings. Briefly returning to the ride-sharing ex- 93, 729–747. ample from the introduction. Ride-share market- , Parag A. Pathak, and Alvin E. places face specification complexity in determin- Roth, “ versus Efficiency in ing how to trade-off efficiency and revenue goals. Matching with Indifferences: Redesigning the They must then elicit information about riders’ NYC High School Match,” American Economic and drivers’ multidimensional preferences over Review, 2009, 99, 1954–1978. cost, direction, and speed. Computational com- Abraham, Ittai, , Moshe plexity plays a role, too, as optimal allocations Babaioff, and Michael Grubb, “Peaches, and travel schedules must be computed quickly, lemons, and cookies: Designing auction mar- in real time. Once isolated, these different forms kets with dispersed information,” in “Proceed- of complexity become approachable both in the- ings of the Fourteenth ACM Conference on ory in and practice (see, for example, Manna Electronic Commerce” 2013, pp. 7–8.

15See, for example, Mance(2016). THE PARABLE OF THE AUCTIONEER 11

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