Licenses for carbon and other emissions are a market-based policy tool for reducing pollution and mitigating the effects of climate change. Roughly speaking, a government agency distributes pollution licenses to firms according to some mechanism. At the end of a period of time (e.g., a year), firms must submit licenses to cover their pollution or else face severe penalties. Different ways of distributing licenses are possible. For example, if licenses are simply sold at a fixed price to anyone who wishes to pay, then this is equivalent to a carbon tax where polluters must pay a linear fee to offset their emissions. Alternatively, in a cap-and-trade mechanism, the agency releases a fixed number of pollution licenses, either via a pre-determined allocation at no price (e.g., based on prior pollution or industry averages) or via auction, and then polluters can trade these licenses on the open market. Many emission trading systems take features of both schemes, combining a cap-and-trade market with a price floor , where each license must be sold above the reserve price of , and/or a price ceiling , where an extra license beyond the cap can always be purchased at a price of . In general, such a pollution license market is referred to as an Emission Trading System (ETS).
There are many important ETS in effect today. The EU ETS has thousands of participating firms and has raised billions of dollars in auction revenue over the past 10 years . The Western Climate Initiative runs a license auction that is linked between California and Quebec [4, 1], and the Regional Greenhouse Gas Initiative (RGGI, pronounced ‘Reggie’) serves New England and the New York region . These markets differ in the details of their implementations, but in each case, licenses are distributed on a regular schedule, with a significant quantity of those licenses sold at auction.
One can model such a license auction as a multi-unit auction; that is, an auction for multiple identical goods. There is no bound on the supply of goods, but there is a cost of production corresponding to the social cost of more pollution. The social cost is typically assumed to be increasing and convex, and the value for licenses for each buyer is commonly assumed to be increasing and concave . Each of the ETS described above uses a uniform-price auction rule to resolve this multi-unit auction. Such auctions proceed roughly as follows. Participating firms declare bids, which take the form of a non-decreasing concave function that describes their willingness-to-pay for varying quantities of licenses. This can alternatively be viewed as a list of non-increasing marginal bids for each successive license. Any marginal bids below the reserve price are removed, and licenses are then distributed to the highest remaining marginal bids while supplies last. Each firm then pays a fixed price per license, where is set to some value between the lowest marginal winning bid and the highest marginal losing bid. Such auctions are not truthful, but are common in practice due to their many advantageous properties; see, e.g., Chapter 7 of .
Implementing a uniform-price auction for carbon licenses presents an optimization challenge: good outcomes require that the system designer correctly sets the quantity of licenses to distribute and/or the price (either direct price or auction reserve) at which they will be sold. The goal is to maximize social welfare: the aggregate value that firms receive for their licenses (i.e., by producing goods) minus the externality on society caused by polluting the corresponding amount during production. This optimization problem is complicated by uncertainty. Even if the social cost of pollution is fully known, the designer may not know what the demand for licenses will be, which makes it hard to predict the optimal level of pollution to allow. The goal of the designer, then, is to set the parameters of the uniform price auction to maximize the expected outcome over uncertainty in the market. Notably, the presence of social costs significantly increases the complexity of multi-unit auctions, since efficiency becomes a mixed-sign objective. Even a slight misallocation of the licenses, resulting in a small reduction of received value, might have a disproportionately large effect on net welfare. In practice, inefficient allocations may be partially resolved in the trade phase of cap-and-trade systems. However, there are significant trading frictions, so it is imperative to choose an initial allocation, via the auction, that is as efficient as possible.
Since uniform price auctions are not truthful, one must account for strategic behavior of participating firms. A notable feature of ETS markets in practice is that, despite their size, they commonly have a small number of participants with significant market power.111For example, in the EU ETS, the top 10 firms together control approximately 30 percent of all licenses allocated and traded; see , page 127. The presence of dominant players suggests imperfect competition and raises the issue of strategic manipulation, in which large individual firms try to influence the market in their favor. One particular concern is demand reduction, where firms reduce their bids to suppress the price determined by the auction. We ask: how should one set auction parameters to (approximately) maximize welfare, in the face of strategic bidding?
1.1 Our Results
In this paper we apply ideas from theoretical computer science to approach the design of carbon license auctions with strategic firms. In this new domain, we use facts regarding uniform price auctions to understand what is happening in these markets and probabilistic analysis to handle the uncertainty of valuation realizations. We then apply a Price of Anarchy analysis to give strategic guarantees. As a result, we provide a concrete recommendation for how to set the parameters of the mechanism used in practice—a recommendation with provable approximation guarantees.
A market instance is described by a convex social cost curve and, for each participating firm, an independent distribution over concave valuations. To capture the relevant space of auctions, we formalize a class of allocation rules that we call cap-and-price auctions.222Note, we are focused on the auction phase in this paper and do not model the trade phase of these systems. These auctions are parameterized by a maximum number of licenses to allocate, a price floor, and a price ceiling. They proceed by running a uniform-price auction for the given number of licenses, subject to the price floor and ceiling. Given a market instance, there is some choice of cap-and-price auction that maximizes the expected welfare generated under non-strategic (truth-telling) behavior. This optimal non-strategic solution will serve as a benchmark, which we denote opt. Our main result is an auction that approximates opt at any equilibrium of strategic behavior.
We show that, in general, the welfare obtained at an equilibrium of a cap-and-price auction can be negative, even for the auction that optimizes welfare under non-strategic bidding. This motivates our main question: given a cap-and-price mechanism that achieves a certain expected welfare under truth-telling behavior, can one modify the auction parameters so that the expected welfare is an approximation to at any Bayes-Nash equilibrium? To answer this question, we describe a subclass of cap-and-price auctions that use safe prices. A safe-price auction is a cap-and-price auction in which the price floor is at least the average social cost, per license, of selling all of that auction’s licenses. We show that for any such auction the worst-case welfare at any Bayes-Nash equilibrium is within a constant factor of the welfare under truth-telling. This result proceeds by transforming the market instance into a corresponding instance without social costs, and makes use of bounds on the Bayes-Nash price of anarchy for uniform-price auctions [20, 13].
To construct an auction that approximates opt
at equilibrium, it therefore suffices to find a safe-price auction whose non-strategic welfare approximates that of the best unconstrained cap-and-price auction. However, as it turns out, there are market instances for which no constant approximation by a safe-price auction is possible. Such a situation can occur if there is a single firm that drives most of the demand for licenses, and this demand has very high variance. We show that this is essentially the only barrier to our desired price of anarchy result: one can always construct an auction that achieves a constant approximation toopt minus the welfare obtainable by allocating licenses to any one single auction participant. In other words, as long as no single firm accounts for most of the expected welfare of the carbon license market, one can find a cap and price floor so that the welfare at any equilibrium is within a constant of the optimal outcome achievable without strategic behavior. We argue that this assumption is reasonable, since presumably the motivation for running a carbon license market in the first place is that the demand for emission licenses is distributed across multiple firms. An alternative way to view this result is that one can always achieve a constant approximation to our benchmark by either running a safe-price auction, or by selling to only a single firm.
Along the way, we also show that for any cap-and-price auction that sells all of its licenses with probability, there is a safe-price auction whose welfare (under truth-telling) is within a factor of the original auction’s welfare. In particular, if our benchmark is implemented by a cap-and-price auction that distributes all of its licenses with constant probability, then we can construct a safe-price auction whose welfare at any Bayes-Nash equilibrium is a constant approximation to the benchmark.
Note that our results are applicable for any setting with imperfect competition where a uniform price auction is used to allocate goods with a production or social cost. For instance, they may apply to allocating queries to a database, where the social cost is privacy. Another example might be allocating medallions to taxi and rideshare drivers in New York City, where the social cost is congestion.
1.2 Related Work
Our work is related to rich line of literature in economics comparing emission licence auction formats and flat “carbon tax” pricing methods to control the externalities of pollution. Weitzman  proposed a model of demand uncertainty and initiated a study comparing price-based vs quantity-based screening in the context of pollution externalities. Kerr and Cramton  noted that auctions tend to generate more efficient outcomes than grandfathered contracts (i.e., pre-determined allocations based on prior usage), which distort incentives to reduce pollution and efficiently redistribute licenses. Cramton, McKay, Ockenfels and Stoft  subsequently lay out arguments in favor of tax-based approaches. Murray, Newell, and Pizer  analyze the use of price ceilings in emission license auctions, and argue that they provide benefits of both auction-based and tax-based systems, improving efficiency in dynamic markets with intertemporal arbitrage. For a recent overview of auction-based systems used in practice, from both the economic and legal perspectives, we recommend .
A similar line of literature studied alternative approaches to the related electricity markets, where individual providers sell electricity into a central grid. Whereas the main issue in emission license sales is the social externality of production, the main focus in electricity markets was incentivizing participation of small firms. The primary discussion focused on using uniform pricing versus discriminatory pricing in the resulting procurement auction [12, 24].
Within the theoretical computer science community, there have been numerous studies of the efficiency of auction formats at equilibrium. For multi-unit auctions without production costs (or, equivalently, no social cost of pollution), the price of anarchy of the uniform-price auction was shown to be constant for full-information settings, and this was subsequently extended to Bayes-Nash equilibria [20, 13]. Our work can be seen as an extension of that work to the setting with a convex cost of allocation. Auctions with production costs have also been studied [6, 15], but primarily from the perspective of mechanism design, where the goal is to develop allocation and payment rules to achieve efficient outcomes. In the present work we do not take a mechanism design approach; we instead restrict our attention to (non-truthful) uniform price auctions, as these are used in practice, and study bounds on the worst-case welfare at equilibrium under different choices of the auction parameters.
Uniform-price auctions with costs can be viewed as games for which the designer has a mixed-sign objective (i.e., total value generated minus the social cost of production). Prior work on the price of anarchy under mixed-sign objectives has focused primarily on routing games [25, 7, 8]. Results in this space tend to be negative, motivating alternative measures of performance (such as minimizing a transformed measure of total cost) that avoid the pitfalls of mixed-sign optimization. Our work shows that in an auction setting, use of an appropriately-chosen thresholding rule (in the form of a reserve price) can enable a constant approximation to our mixed-sign objective of total value minus a convex social cost.
There is a vast amount of work on auctions with externalities, such as the seminal work of , and other work including externalities in advertising auctions , characterizations of equilibria in auctions with externalities , and more. However, these externalities are private and held by the buyers, as opposed to public and suffered by the seller as in this paper. In this case, the externality functions more like a production cost, as described above.
Kesselheim, Kleinberg and Tardos  study the price of anarchy of an energy market auction, which is a similar application to the carbon license auction that we study. Their focus is on the uncertainty of the supply and temporal nature of the auction, and not on externalities of production, and hence their technical model is quite different.
The theory of bidding in uniform-price auctions with imperfect competition is well-developed in the economic theory literature. Much of this work focuses specifically on the efficiency and revenue impact of demand reduction, and how modifications to the auction format or context might impact it. Demand reduction occurs at equilibrium even in very simple settings of full information, can dramatically reduce welfare, and is a concern in practice [28, 26, 5]. One way to reduce the inefficiency of demand reduction is to perturb the supply, either by allowing the seller to adjust the supply after bids are received , or by randomizing the total quantity of goods for sale (or otherwise smoothing out the allocation function) . Such results are typically restricted to full-information settings. Moreover, these approaches are not necessarily appropriate in the sale of government-issued licenses, where one typically expects commitment and certainty about the quantity being sold. In contrast, we forego a precise equilibrium analysis and instead argue that setting a sufficiently high price floor can likewise mitigate the impact of demand reduction.
Our main result is a cap-and-price auction whose welfare in equilibrium approximates the maximum welfare of a cap-and-price auction with non-strategic reporting in markets without a single dominant bidder. In Section 2, we introduce our model and formally define cap-and-price auctions. In Section 3, we first show that we can restrict attention to cap-and-price auctions with an infinite price ceiling (i.e., ones who never sell more licenses than the cap). We then derive a class of such cap-and-price auctions, which we call safe-price auctions, whose performance in equilibrium approximates the performance in a non-strategic setting. Thus it suffices to show that the best safe-price auction approximates the best cap-and-price auction in a non-strategic setting. However, this is not true: we give an example where it can be unboundedly worse. In Section 4, we demonstrate that the only barrier to this is the existence of a dominant bidder, yielding our main result.
There are firms seeking to purchase carbon licenses. Licenses are identical, and each permits one pollution unit. Each firm has a monotone non-decreasing concave valuation curve that maps a number of pollution units to a value ; this is the value they enjoy for polluting this amount. The profile of valuation curves is drawn from a publicly-known distribution over profiles, where the draw of is the private information of firm . Valuations are drawn independently across firms, so is a product distribution and .
There is a publicly known monotone non-decreasing convex cost function that maps a number of pollution units to the value of externality that society faces for having this much pollution. Given a valuation profile , the welfare of a given allocation rule is . Our objective is to maximize expected welfare.
Given some integer , we write for the maximum aggregate value that could be obtained by optimally dividing licenses among the firms. That is,
We refer to as the combined valuation curve. As each is weakly concave, so is . We will sometimes abuse notation and write to mean that is distributed as the aggregate value when . We’ll write for the welfare generated by allocating licenses optimally among the firms.
We’ll write for firm ’s marginal value for aquiring license , for each . By the monotonicity and concavity of , is non-increasing. We’ll also write for the marginal value of additional items given already allocated. We’ll write for the number of units demanded by bidder at price .
We will study equilibria and outcomes of license allocation auctions. An auction takes as input a reported valuation from each firm . The auction then determines an allocation and a price that each firm must pay. The auctions we consider will be uniform price auctions, where the auction determines a per-license price and each firm pays . Given an implicit uniform-price auction, we will tend to write (resp., ) for the allocation to agent (resp., per-unit price) when agents report according to . For a given valuation profile , we’ll also write for the utility enjoyed by firm when agents bid according to :
Finally, given an auction and a distribution over input profiles, we will write for the expected welfare of on input distribution . We will sometimes drop the dependence of and simply write when is clear from context. We emphasize that is a non-strategic notion of welfare; it is the expected welfare of with respect to inputs drawn from , or equivalently the welfare of under truthful reporting when agent valuations are distributed according to . Let be the allocation rule of mechanism for realization . Then
A Bayes-Nash equilibrium of a uniform-price auction is a choice of bidding strategies for each agent, mapping each realized valuation curve to a (possibly randomized) reported valuation , so that each agent maximizes their expected utility by bidding according to given that other agents are bidding according to . That is, for all and all , we have
We make a standard assumption of no overbidding on behalf of the firms uniform-price auction, which is motivated by the fact that overbidding is a weakly-dominated strategy.
Motivated by license auctions used in practice, we study a particular form of uniform price auction that we call a cap-and-price auction which forces prices to be within some fixed interval.333Note the cap is not a hard constraint, but rather governs which prices bind.
A cap-and-price auction is parameterized by a quantity , a price floor , and a price ceiling . The allocation and price are determined as follows:
If , then for all and .
If , then for all and .
Otherwise, choose , the highest bid, and choose to be any optimal allocation of licenses among the firms: .
We can think of this as a uniform price auction of up to licenses (case 3 above), where the price is set to the lowest winning marginal bid, with two modifications. First, there is a reserve price , so that possibly fewer than licenses are sold if there is not enough demand at price ; this is case 2. Second, if the lowest winning marginal bid would be larger than , then the price is lowered to and firms can purchase as many licenses as they like at this price; this is case 1.
Given social cost function and valuation distribution , we will write opt for the optimal expected welfare obtained by any cap-and-price auction under truthful reporting. That is, . We will also tend to write , , and for the parameters that achieve this maximum. We emphasize that this is a non-strategic notion: opt is the maximum expected welfare attainable when bidders report truthfully. We view opt as a benchmark against which we will compare performance at Bayes-Nash equilibrium. Note also that, in general, opt may be strictly less than the expected welfare of the unconstrained welfare-optimal allocation; an example is provided in Appendix C.
3 Safe-Price Auctions
We will derive a class of cap-and-price auctions, which we will call safe-price auctions, whose performance in equilibrium approximates their non-strategic welfare. The hope is that the non-strategic welfare of this class then approximates the non-strategic welfare of the larger class of cap-and-price auctions.
A safe-price auction is a cap-and-price auction parameterized by a license quantity , a price floor , and price ceiling . We will write for the safe-price auction with license quantity , and for the associated safe price.
This definition restricts cap-and-price auctions in two ways. The first is that the price ceiling is now infinite. The second is that we have imposed a lower bound on the price floor.444In fact we set to be equal to , but our results still hold if this is relaxed to . The first restriction is for convenience; the following lemma shows that it does not cause much loss in welfare. Motivated by this lemma, we will restrict our attention from now on to auctions with no price ceiling, that is, . We notate such mechanisms . Note that, in the language of definition 1, case (1) can no longer occur. That is, when is there is no price ceiling, only exactly or less than licenses will be allocated.
For any distribution and any cap-and-price auction , there exist a cap and price floor such that .
The proof of Lemma 1 appears in Appendix A.1. The main idea is to decompose the expected welfare of a cap-and-price auction into the welfare attained from the first (at most) licenses sold plus the incremental welfare attained from any licenses sold after the first . The first can be approximated by ; the latter, if non-negative, can be approximated by .
The second way safe-price auctions restrict price-and-cap auctions is with a lower bound on the price floor. As the following example shows, the equilibrium welfare of cap-and-price auctions, even those with an infinite price ceiling, suffer from a problem known as demand reduction in which a bidder improves her price, and hence utility, by asking for fewer units. This can have drastic consequences on welfare, causing it to become negative, due to the existence of the social cost. In particular, this means the approximation of such auctions, relative to the best welfare attainable in non-strategic settings, is unbounded.
Consider two agents, and . Their distributions over valuations will be point-masses, so that the valuations are actually deterministic. These valuation functions are given by the following marginals: , , , , and all other marginals are . We will have , , , and the social cost function is given by . Under truthful reporting, the auction allocates licenses to agent at a price of each, resulting in a welfare of . However, we note that firm could improve their utility from to by instead reporting a modified valuation given by and . If agent continues to report truthfully, will allocate license to each agent at a price of each, resulting in a welfare of . One can verify that this is indeed a pure Nash equilibrium of the auction, and hence a Bayes-Nash equilibrium as well.
The issue illustrated in Example 1 is that strategic behavior can cause licenses to be allocated to agents whose marginal values for these licenses are below , causing the aggregate value derived from allocating licenses to fall below . Safe-price auctions seek to prevent this by imposing a sufficiently high price floor.
Indeed, the following lemma shows that price floors do indeed circumvent the issue in the above example. Namely, we show safe-price auctions have good equilibria, compared to their own non-strategic welfare. This is the main result of this section, and motivates us to focus on analyzing the non-strategic welfare of safe-price auctions.
For any safe-price auction , the expected welfare at any Bayes-Nash equilibrium is at least .
Fix a license cap and write for the corresponding safe price, and recall that . We claim that this auction is strategically equivalent to a modified auction, as follows. First, since each agent pays at least per license, we can define for the residual marginal utility of agent for their license, after taking into account that they must pay at least . Then from each agent’s perspective, playing in with marginal values gives the same utility as playing in with marginal values . The welfare generated by the original auction is , which is equal to . So the welfare generated by is equivalent to the welfare generated with modified valuations , where the marginal social cost of each unit of pollution is reduced by . But now note that from the definition of , for all . So in an auction with a cap of , the welfare is at least the welfare obtained with a social cost of ; it is always non-negative.
We conclude that the welfare generated by , at any equilibrium, is equal to the welfare generated at an equilibrium of the standard uniform price auction with modified valuations . Theorem 3 of  (which bounds the welfare in Bayes-Nash equilibria of uniform auctions without costs) therefore applies555The proof of the 3.15 bound in  is actually for a uniform-second-price auction that charges the highest losing bid, or ; however, the proof is also correct for charging a price of the lowest winning bid ., and the welfare at any equilibrium is at least times the optimum achievable under the modified valuations, which is at most . ∎
4 Welfare Approximation
Motivated by Lemma 2, we would like to show that the non-strategic welfare of safe-price auctions approximates that of cap-and-price auctions. Unfortunately, there are cases where this does not hold, as demonstrated by Example 2. The example consists of just a single firm that dominates the market and whose demand for licenses has high variance. The firm’s value is always large enough that even at a low price and high license cap, the resulting allocation has high net welfare. However, safe-price auctions will generate much lower welfare. Roughly speaking, for any number of licenses that the auction designer selects as the cap for a safe-price auction, significant welfare will be lost from realizations where the firm’s demand is much higher than , and the corresponding safe price will exclude welfare gains from realizations where the firm’s demand is much lower than . If the variance is high enough, this will cause overall welfare to be low regardless of the choice of .
We will present an example for which the expected welfare of any safe-price auction is at most an -approximation to opt. Let . There is a single firm participating in the auction. That firm’s valuation curve is drawn according to a distribution over different valuation curves, which we’ll denote . For all , valuation is defined by . The probability that is drawn from is proportional to . That is, the firm has valuation with probability , where is the normalization constant. Note that a cap-and-price auction with cap and price floor achieves welfare so opt is at least this large.666In fact, this is the “first-best” welfare obtainable at every possible realization, so this is actually the best possible cap-and-price auction and hence the exact value of opt.
Consider a safe-price auction with cap . Suppose that for some . Then the safe price is . If the firm has valuation with , then all marginal values are strictly below , then no licenses are allocated and the welfare generated is . On the other hand, if the firm has valuation for any , then since at most licenses can be purchased, the auction generates welfare at most .
The total expected welfare of is therefore at most
Since but for all , we conclude that no auction with a safe price achieves an -approximation to opt. This concludes the example.
Example 2 is driven by a single firm that dominates the market and has high variance in their demand. We next show that this is the only barrier to a good approximation: either a safe-price auction is constant-approximate, or else one can approximate the optimal welfare by selling to just a single firm. To this end, we will be interested in the expected maximum welfare attainable by allocating licenses to just one firm, which we will denote . That is,
The following theorem is our main result.
There exists a constant such that, for any cap and price floor , there exists such that .
Note that a corollary of Theorem 1 (combined with Lemmas 1 and 2) is that for any cap-and-price auction , there exists some such that the expected welfare of any Bayes-Nash equilibrium of is at least a constant factor of . That is, the worst-case equilibrium welfare is a constant-factor approximation to the best welfare achievable by any cap-and-price auction under truth-telling, excluding the welfare contribution of a single firm.
Also, is the expected welfare of a truthful mechanism that we will call . In , the firms first participate in a second-price auction for the right to buy licenses. The firm with the highest bid wins, and they pay the second-highest bid. The winning firm can then purchase any number of licenses , for which they pay (in addition to their payment from the initial second-price auction). Since the utility obtained by firm if they win the initial auction is precisely , and since a second-price auction is truthful and maximizes welfare, we can conclude that . Then another corollary of Theorem 1 is that one can approximate the non-strategic welfare of any cap-and-price auction using either a safe-price auction (in which case the approximation holds at any Bayes-Nash equliibrium), or using the (truthful) mechanism that allocates licenses to at most one firm.
We are now ready to prove Theorem 1. Fix a cost function and distribution , and choose the optimal cap and price floor such that is maximized. Let and be the demand and allocation functions for . For any given realization of preferences , there is a total demand , where again, . Recall also that is the welfare generated by allocation given aggregate valuation .
We will begin by proving a simpler version of Theorem 1, which is parameterized by , the probability that the auction sells licenses. When is bounded away from , we show that there is a safe-price auction that achieves an -approximation to . We note that this result does not require firm valuations to be independent, and holds even if the valuations are drawn from an arbitrarily correlated distribution . We will explain in Section 4.1 how to extend the argument to the general case of Theorem 1, with details deferred to the appendix.
For the welfare-optimal cap and price floor , there exists a cap such that .
The welfare generated by can be broken down as follows:
The first term is the contribution to the welfare from events where licenses are sold (or, equivalently, at least licenses are demanded). The second term is the contribution from events where the total demand for licenses is less than .
Consider price , the safe price for quantity . We wish to decompose the second term in the expression above into marginal values above and those below . For this, we need to introduce some notation. Write for the largest such that , or if . Write , and write . Then is the part of allocation for which firm has marginal value at least per unit, and is the part of for which firm has a marginal value less than per unit. We’ll also define as ; this is the marginal value of for receiving additional licenses after already having received licenses. Note then that , for all . We then have
where the final inequality used the fact that , due to the convexity of . We conclude that
so that . See Figure 1(a) for an illustration.
We claim that the welfare obtained from the first two terms, and , are covered by the safe-price auction with cap . The intuition is that a price floor of does not interfere with the welfare contribution due to licenses with marginal values greater than . One subtlety is that some of the licenses in the summation might have marginal values less than , but it turns out that it can only improve welfare to exclude such licenses from the allocation.
Write and for the demand and allocation under auction , respectively. We have
Note that we intentionally use , the demand for , rather than in the expressions above. The second term is precisely (1B), since from the definition of . We claim that the first term is at least . To see why, fix any with (and corresponding allocation where necessarily ), and note that for all . Furthermore, , since is simply after possibly excluding some items with marginal value less than . Thus, since ,
Also, by convexity, we have for all . In particular, we have , and hence
as claimed. ∎
The more difficult part of the proof of Theorem 2 is to account for term . This represents the contribution of licenses whose marginal values lie below , and are therefore excluded when the price floor is set to . What we show is that the welfare contribution from all such licenses is approximated by , the welfare obtained by the safe auction with cap .777For convenience we will assume is even for the remainder of this section. When is odd, the result holds for at least one of the floor or the ceiling of
is odd, the result holds for at least one of the floor or the ceiling of. Details appear in the proof of Theorem 1 in Appendix B.3. We do this in three steps. First, we show that any optimal cap-and-price auction must generate non-negative expected welfare conditional on selling licenses. The intuition is that if the expected welfare from selling licenses is negative, then it must be strictly preferable to reduce the quantity of licenses.
If cap and price floor are chosen to maximize , then
where is the total quantity of licenses demanded at price .
The proof of Lemma 3 appears in Appendix B.1. Next, for any cap-and-price auction , we provide an upper bound on the total welfare contribution that comes from events where licenses are sold, and the average marginal value of the sold licenses, , is less than . In fact, it will be useful to state the lemma more generally: we provide a more general bound that applies to any and any valuation that is at most (not necessarily ). The lemma bounds the total welfare contribution, , by the difference in price between (a) licenses sold at the safe price for , and (b) the same licenses sold at the safe price for .
Choose some quantity and a number of licenses . Then .
The proof of Lemma 4 appears in Appendix B.2. We are now ready to bound the contribution of . The intuition is as follows. By Lemma 4, the contribution from is at most (a constant times) the gap between the line and the curve at point . But recall that , and Lemma 3 implies that, on average, the expected marginal value of licenses allocated subject to this event is at least . Therefore, if we set a license cap of , then with probability at least all licenses will be sold at an average expected marginal value of at least . The welfare generated in this case covers the “gap” at , and hence covers the loss due to excluding licenses with marginal values at most .
Write and for the demand and allocation under mechanism , respectively. Choose any such that . For any such , , and is formed by removing items with marginal value at most . In particular, since , we have . Noting that since , we have
Taking an expectation over all such that , we note that Lemma 3 implies . We therefore have
where in the last inequality we used that by definition. Since , we therefore have that
We now claim that , completing the proof. To see why, note that for any and any , . Thus . But note that for any , we have , and hence . Also, for any , lies above the line joining and , and hence is at most the distance between and this line, which is at most . So in either case we have as required. ∎
As a corollary, Theorem 2 combined with Lemma 2 and Lemma 1 implies that for any cap and price auction , there exists a safe-price auction such that, at any Bayes-Nash equilibrium of , the expected welfare generated is at least . In particular, if is a constant bounded away from , then obtains a constant fraction of at any Bayes-Nash equilibrium.
4.1 The General Case
We complete the proof of Theorem 1 in Appendix B.3. Here we describe at a high level what steps are needed to complete the argument. We will focus on the case where , since if then Theorem 1 follows from Theorem 2.
Recall that in Claim 2, we used the assumption that to argue that the welfare gained in in the event that covers the welfare lost from marginals lying below , up to a constant factor. If the probability that is small, then this may no longer be true. To handle this, we consider a reduced cap set to be the largest integer such that . Our hope is to reproduce the argument from Claim 2, but substituting for . To this end, we divide the welfare of into four parts: all welfare under the event that ; all welfare from individual agents whose demand is at least ; the contribution of marginal values greater than (but with individual firms demanding at most ) when , and the contribution of marginal values less than when . See Figure 1(b) for an illustration.
As in the proof of Theorem 2, the contribution due to events where can be covered by the welfare of .
The contribution from agents who individually demand at least licenses is a new case that we didn’t have to handle in Theorem 2. It is here that we use , allocating to any single agent. Because the total demand is at most than with probability at least , independence implies that the expected number of agents who demand more than licenses is at most , given that the probability that none have demand more must be at least . So the total contribution to welfare of all such events is at most .
If , then the contribution from marginal values that are at least can be covered by the welfare of , precisely as in Claim 1. We must also handle the case that , and consider the welfare contribution of agents that do not (individually) demand more than licenses. Here we use independence: the total quantity demanded by such “small” agents is likely to concentrate, so it is unlikely that the total demand will be larger than . Thus, by imposing a cap of , we lose at most a constant factor of the welfare from marginal values greater than .
The final step is to show that there is some such that obtains at least a constant fraction of the welfare generated by marginal values less than , similarly to Claim 2. If is even then we’ll take ; otherwise, . When proving Claim 2, we used Lemma 3 to argue about the welfare generated by events where . Unfortunately, Lemma 3 does not extend to : it could be that the expected welfare generated by , conditional on selling licenses, is negative. However, we can prove an upper bound on how negative this expected welfare can be. After all, if the expected welfare is sufficiently negative sufficiently often, it would be welfare-improving to increase the price floor of from to , contradicting the supposed optimality of . This turns out to be enough to prove a bound similar to Claim 2.
Combining these bounds, we can conclude that each of the four parts of the welfare of can be covered by either a safe-price auction or by , which completes the proof of the theorem.
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Appendix A Omitted Proofs from Section 3
a.1 Proof of Lemma 1
Recall the statement of Lemma 1: for any distribution and any cap-and-price auction , there exist a cap and price floor such that