Discovering Prices

Auction Design in Markets with Complex Constraints

Paul Milgrom

• Publisher: Columbia University Press
• Year: 2017
• ISBN:
  9780231544573
• Topic: Business
• Subtopic: Government & Business
• Index: Business

Traditional economic theory studies idealized markets in which prices alone can guide efficient allocation, with no need for central organization. Such models build from Adam Smith's famous concept of an invisible hand, which guides markets and renders regulation or interference largely unnecessary. Yet for many markets, prices alone are not enough to guide feasible and efficient outcomes, and regulation alone is not enough, either. Consider air traffic control at major airports. While prices could encourage airlines to take off and land at less congested times, prices alone do just part of the job; an air traffic control system is still indispensable to avoid disastrous consequences. With just an air traffic controller, however, limited resources can be wasted or poorly used. What's needed in this and many other real-world cases is an auction system that can effectively reveal prices while still maintaining enough direct control to ensure that complex constraints are satisfied.

In Discovering Prices, Paul Milgrom—the world's most frequently cited academic expert on auction design—describes how auctions can be used to discover prices and guide efficient resource allocations, even when resources are diverse, constraints are critical, and market-clearing prices may not even exist. Economists have long understood that externalities and market power both necessitate market organization. In this book, Milgrom introduces complex constraints as another reason for market design. Both lively and technical, Milgrom roots his new theories in real-world examples (including the ambitious U.S. incentive auction of radio frequencies, whose design he led) and provides economists with crucial new tools for dealing with the world's growing complex resource allocation problems.

1

INTRODUCTION

The mid-1990s was an auspicious time for the new academic discipline that was soon to become known as “market design.” This period saw the introduction of the first Internet Web browser, which provided consumers easy access to the World Wide Web and, soon after, to vast volumes of Web-based commerce. Online auctions like eBay, online stores and marketplaces like Amazon, and instantaneous advertising auctions like those run by Google emerged, and automation required that these markets operate with formal rules. These companies and others hired economists—who were thought to understand how markets actually work—to help engineers and programmers in designing the necessary rules.

Web-based companies were not the only ones looking for advice about how their markets should be organized. The same time period also saw the redesign of the National Resident Matching Program (NRMP). That program operates the market to match newly graduating doctors to hospital residency programs in the United States. The traditional matching algorithm, which had worked well for four decades, began by asking each hospital to rank the doctors who might enter its residency program and each doctor to rank the hospitals. In the usual mathematical model, those preferences are real things that doctors and hospitals know. According to that model, the NRMP system for determining matches between doctors and hospitals encouraged honest preference reporting and led to matches that were “stable,” meaning that there is no doctor and hospital who would both prefer to make a new deal with each other rather than to honor the one recommended by the match. But the model is not an exact match for reality, and one important omission became apparent only in the 1990s. What changed was the number of women in medical schools. Increasingly, graduating doctors were married to other doctors, and the couples insisted on compatible placements. The old system was not designed to accommodate that. Economists found themselves deeply engaged in new research about devising a new, replacement system that would have similar theoretical properties while still accommodating the needs of couples, in addition to the needs of single doctors and hospitals.

During the same period, the first U.S. auction of licenses to use radio spectrum to support services like pagers and mobile phones took place. These auctions, too, were designed with help and guidance from academic economists. There were thousands of licenses to be allocated, each described by the geographic area that it covered and the frequencies that it used. No two licenses were quite the same, but some buyers regarded certain licenses as economic substitutes—meaning roughly that the buyer would be less eager to acquire one license if it knew it could acquire the other more cheaply—and some buyers regarded certain licenses as economic complements—meaning that a buyer would be willing to pay a premium to acquire both licenses. In the absence of complements, the economic problem of efficiently assigning licenses to companies is similar to the problem of assigning single doctors to hospitals, but the possibility of complements makes the problem much more complex. Indeed, the doctors in a married couple would usually be willing to pay a premium (by accepting a lesser placement) if the two jobs were at the same or nearby hospitals. In both medical matching and license auctions, the presence of complements was what made the redesign of the market so challenging.

Despite all this practical activity, some economists schooled in traditional economic theory were skeptical of the field of market design altogether. Why, many asked, would markets need designing? Why can unregulated market participants not take care of themselves? According to a view that is still espoused by many economists, if resources are allocated in an inefficient way, and if parties can negotiate freely among themselves without artificially imposed constraints, then the parties will be sufficiently motivated to alleviate and eventually eliminate any important inefficiency without any outside assistance. According to that view, no organized market is needed to promote efficient trade.

This strongly held belief in the power of unregulated markets was baked into the formal models economists traditionally used to understand the world. Formal claims in economics are often presented in mathematical terms as theorems that are based on the assumptions of a particular mathematical model. Formalization is important to economics, because it allows readers and others to identify the precise assumptions that underpin any purported conclusion, to verify that the assumptions really do imply this conclusion, and to check how deviations from the assumptions might alter the conclusion. In the case of the traditional view described earlier, the relevant claim is known as the Coase theorem, named for its originator, British economist Ronald Coase. The theorem relies on four assumptions about the parties involved in any transaction: that they have secure, transferable property rights; can bargain freely and effectively; can transact without costs or regulatory constraints; and will transact whenever it is mutually beneficial to do so. Most important from Coase’s perspective was that the efficiency of the outcome does not depend on who initially owns any property rights, because ownership can be changed, if necessary, as part of the bargain.

Coase understood that this model would not apply exactly to any real situation, so the legal default situation could be important in practice. Many barriers to securing property rights and making them transferable, bargaining effectively, making and enforcing contracts, and conducting trade often stand in the way. In a straightforward bargain between two people, the conclusion described by the Coase theorem might be reasonably realistic. But bargaining is especially difficult when an agreement among multiple parties is needed to achieve much benefit, and the conclusion of the theorem is therefore least likely to describe real outcomes in such cases. Despite these qualifications, reasoning along “Coasian” lines bolstered a deeply held belief among many economists that regulations on markets should be minimal and that market participants are usually best left to take care of their own affairs, without being subjected to “designs” imposed by regulators or, certainly, by academic economists.

Long before Coase, an even older strand known as classical economic theory emphasized how markets could run by themselves, seemingly without the need for explicit design. The eighteenth-century Scottish philosopher and economist Adam Smith famously described how the “invisible hand” of the market refuted his contemporaries’ concerns that, with the decline of feudalism, the absence of anyone to control production would lead to economic chaos. The reason he gave was that if any goods were in short supply, prices for those goods would rise to promote increased production and to encourage reduced usage, and similarly, surpluses would lead producers to cut back—all as if guided by an invisible hand.

A more modern account highlights the assumptions that would be required for various of Smith’s conclusions to be justified. Kenneth Arrow and Gerard Debreu famously formulated a model that addresses the conclusion that prices can guide the economy to an efficient outcome and includes the assumption known as perfect competition. A market is competitive to the extent that whenever one party to a transaction demands significantly more favorable terms than the prevailing ones, there are other suppliers or customers who are willing to replace that party and participate in the same transaction according to the prevailing terms. In a perfectly competitive economy, each individual participant, acting alone, has zero influence on the terms of trade. The economic system with all its participants, balancing supply and demand, determines those terms. Adding other assumptions, including that each household cares only about its own consumption and is never satiated, always wanting more of at least some goods, leads to the first welfare theorem: in a perfectly competitive economy, if the prevailing prices are such that the supply is equal to the demand for every type of good, then there is no other feasible allocation that makes one agent better off without making another worse off. An allocation with the italicized property is said to be Pareto efficient, in honor of the famous economist Vilfredo Pareto, who introduced this criterion.

Like the Coase theorem, the first welfare theorem relies on assumptions that in some real situations fail to hold even approximately. For example, the mathematical model used to prove the theorem assumes that each market participant affects others only by trading with them. When one person’s or company’s consumption or production decision directly affects another person’s welfare or another company’s ability to produce, that is called an “externality.” Externalities are common and can be negative or positive. For example, a homeowner may use her noisy lawnmower too early in the morning, disturbing her neighbors’ ability to sleep. This is a negative externality, because the neighbors’ welfare is harmed by the homeowner’s choice. An example of a positive externality is the consequence of Apple’s development and marketing of its iPhone. That decision spawned valuable new opportunities for app developers, whose products were complementary to Apple’s. Like many new products, by making consumers aware and by proving that consumers would demand this product, the iPhone also created new market opportunities for competing products like Google’s Android operating system and the smartphones produced by Samsung, Lenovo, and HTC. According to the neoclassical theory, markets do not sufficiently deter activities with negative externalities nor do they sufficiently reward activities with positive externalities. Many of the rules of social interaction in markets and other settings are designed to mitigate or eliminate negative externalities. For example, rules that prevent drivers of cars from blocking an intersection can enable other drivers to reach their destinations more quickly and safely.

Externalities are not the only real-world complication that upsets the conclusion of the first welfare theorem. Its foundational assumption is that markets are perfectly competitive, but some are far from it, because some participants have substantial power to set or affect prices. Apple, for example, had considerable flexibility in pricing the iPhone: its high price relative to the products of its competitors cost it some sales, but the sales it did make were much more profitable.

These first two reasons for failures of unorganized markets—externalities and imperfect competition—are discussed at length in all elementary microeconomics textbooks. But there are two more assumptions that receive much less discussion in textbooks but are of critical importance for market design. The first is the assumption that consumers and firms do not care which units they may receive or supply of each product and consequently that the only relevant constraints on market transactions are that the quantity demanded must be equal to the quantity supplied. The second is that prices exist at which supply is equal to demand. The Arrow-Debreu model incorporates the first of these assumptions. A theorem about that model identifies mathematical conditions, related to the convexity of certain sets, that are sufficient to imply the second—that market-clearing prices exist.

Why does the Arrow-Debreu model, like most other economic models, assume that products in a category are homogeneous? The traditional answer is that if two items are different in any important way, whether that be physical characteristics or their time and place of availability, then they can just be treated as different products with different prices. A traveler who needs a hotel room in New York City on Tuesday is not likely to be satisfied by a room in a different city or in the same city on a different day, so rooms in different cities or on different days are different products and can have different prices. The trouble with this answer is that it can be carried only so far. There can be only one physical product at any exact location and time. If product descriptions must respect every distinction, then every supplier of any physical product is a monopolist and every unit of every item has its own price! These are troubling conclusions for a model of an economy based on competition or in which individual choices are supposed to be guided by the knowledge of every price of every product.

In reality, some details of time and place and even physical characteristics are always overlooked in defining a product category. As a consequence, some heterogeneity always remains. For example, although nonexperts may think of bushels of wheat as homogeneous, the physical characteristics defining “number 2 red wheat” include limits on the minimum weight per bushel of wheat, the maximum fraction of damaged kernels, the percentage of white wheat kernels mixed in with the red wheat kernels, the amount of foreign material, and so on.¹ The commodity “number 2 red winter wheat in Chicago” is defined by a range of times and places that the wheat will be available, in addition to the range of physical characteristics just described.

In some applications, very fine differences among products within a category are critically important for serving demand. The electricity market may be organized to pay generators the same price for electric power delivered at 5:00 p.m. and 5:04 p.m., but a user who flips a switch at 5:00 p.m. cannot use the power available at 5:04 p.m. The electricity system needs to be managed to deliver what users demand when they demand it, and not just a certain total number of megawatts at a particular location in the time interval from 5:00 to 5:05 p.m.

Resource constraints—for example, the statement that the supply of electricity is sufficient to meet demand—differ among markets in qualitatively important ways. I will call a resource constraint “simple” if, when people attempt to violate it, the only consequence is that some demand is left unserved. For example, if three people show up to drive two cars at a car rental company, one of them will be disappointed. Traditional economic analyses, which extoll the ability of price adjustments to correct imbalances of demand and supply, implicitly assume that all constraints are simple. However, for some resource constraints, attempted violations can have less benign consequences. For example, if two trains try to use the same segment of track at the same time, the consequence is not merely that one finds the track unavailable: they risk a disastrous collision. Adam Smith’s account of prices eventually adjusting so that resources are not continually over-demanded is poor consolation for the riders on those trains! When even temporary imbalances of supply and demand are intolerable, a system of prices alone is just not good enough: some other means of coordination is needed to ensure no imbalance. Another example is drawn from electricity markets. When the demand for electrical power exceeds the capacity of the electricity transmission network, there can be brownouts or blackouts that affect all consumers.

In this monograph, I recognize two kinds of complexity that often interact to make centralization of markets desirable and good market design valuable. First, when the plans of many individuals need to satisfy constraints to avoid incurring the very high costs associated with such events as train crashes and brownouts, the constraint is not simple. Second, when heterogeneity within a product category implies that acceptable market performance may require producing and assigning the right units of products to individual users, I refer to those acceptability constraints as complex as well. The two sources of complexity are often found together. When they are present, the classical economic conception—in which individual adjustments in firms’ and consumers’ decisions, guided by prices, resolves temporary conditions of excess supply or demand—is not a suitable basis for a theory of market design.²

The second underappreciated assumption that I have highlighted is that market-clearing prices must exist. Arrow and Debreu have proven in their model that if certain sets are convex, then market-clearing prices exist. But their convex-sets assumptions are not innocuous. Among the implications of those assumptions are that all goods can be made and used, not only in whole units but in fractional units as well, and that production processes can be scaled up or down without losing efficiency. In reality, there are some goods like sugar and wheat and paint that can be consumed in fractional units, but goods like houses are...