Bidding with Securities: Auctions and Security Design

By PETER M. DEMARZO,ILAN KREMER, AND ANDRZEJ SKRZYPACZ*

We study security-bid auctions in which bidders compete for an asset by bidding with securities whose payments are contingent on the asset’s realized value. In formal security-bid auctions, the seller restricts the security design to an ordered set and uses a standard auction format (e.g., ﬁrst- or second-price). In informal settings, bidders offer arbitrary securities and the seller chooses the most attractive bid, based on his beliefs, ex post. We characterize equilibrium and show that steeper securities yield higher revenues, that auction formats can be ranked based on the security design, and that informal auctions lead to the lowest possible revenues. (JEL D44, G32, G34)

Auction theory and its applications have be- bid auction, and provide an extensive charac- come increasingly important as an area of eco- terization of such auctions. nomic research over the last 20 years. As a Formal auctions of this type are commonly result, we now have a better understanding of used in government sales of oil leases, wireless how the structure of an auction affects its out- spectrum, highway building contracts, and lead- come. Almost all the existing literature studies plaintiff auctions. Informal auctions of this type the case when bidders use cash payments, so (in the sense that formal auction rules are not set that the value of a bid is not contingent on future forth in advance) are common in the private events. sector. Examples include authors selling pub- In a few cases, such as art auctions, the real- lishing rights, entrepreneurs selling their firm to ized value is subjective and cannot be used as a an acquirer or soliciting venture capital, and basis for payment; however, this is the excep- sports associations selling broadcasting rights.1 tion. In many important applications, the real- The major difference between a formal and ization of the future cash flow generated by the an informal mechanism is the level of commit- auctioned asset or project can be used in deter- ment by the seller. In an informal mechanism, mining the actual payment. That is, the bids can bidders choose which securities to offer, and the be securities whose values are derived from the seller selects the most attractive offer ex post. In future cash flow. We call this setting a security- this case, the auction contains the elements of a signaling game because the seller may infer bidders’ private information from their security choices when evaluating their offers. In a for- * Stanford Graduate School of Business, 518 Memorial mal mechanism the seller restricts bidders to use Way, Stanford, CA 94305 (e-mails: DeMarzo: pdemarzo@ stanford.edu; Kremer: [email protected]; Skrzypacz: [email protected]). We thank two anonymous refer- 1 See Kenneth Hendricks and Robert H. Porter (1988) for ees, Kerry Back, Simon Board, Sandro Brusco, Nicolae a discussion of oil lease auctions, in which royalty rates are Garleanu, Burton Hollifield, John Morgan, and seminar commonly used. In wireless spectrum auctions, the bids are participants at Harvard University, Hebrew University, Uni- effectively debt securities (leading in some cases to default). versity of Iowa, Massachusetts Institute of Technology, Highway building contracts are often awarded through New York University, Stanford University, University of “build, operate, and transfer” agreements to the bidder that New York at Stony Brook, Technion, Tel Aviv University, offers to charge the lowest toll for a pre-specified period. UC Berkeley, UC Davis, UC San Diego, USC, Washington See Jill E. Fisch (2001) for the use of contingency-fee University, Yale University, WFA 2003, ASSA 2004, the auctions in the selection of the lead plaintiff in class action Stony Brook International Conference on Game Theory suits. In mergers, acquisitions, and venture capital agree- 2004, and the Utah Winter Finance Conference 2004 for ments, equity and other securities are commonly used (see useful comments. This material is based upon work sup- Kenneth J. Martin, 1996). John McMillan (1991) describes ported by the National Science Foundation under Grant the auction of the broadcast rights to the Olympic games, 0318476. We also acknowledge the financial support of the where bids contained revenue-sharing clauses. Similarly, Center for Electronic Business and Commerce. publishing contracts include advance and royalty payments. 936 VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 937 securities from a pre-specified ordered set, such reservation values. As a result, Alice as an equity share or royalty rate. The seller is would bid 4 Ϫ 1 ϭ $3M and win the committed to disqualify any offer outside this auction, paying Bob’s bid of 3 Ϫ 1 ϭ set. The seller also commits to an auction for- $2M. mat, such as a first- or second-price auction. One of our main results is that the revenues Now suppose that rather than bidding from an informal mechanism are the lowest with cash, the bidders compete by offering a fraction of the future revenues. As across a large set of possible mechanisms. In we discuss later, it is again a dominant other words, the seller benefits from any form of strategy for bidders to bid their reserva- commitment. Moreover, we show how to rank tion values. Alice offers 3⁄4 of future cash security designs and auction formats in terms of flows while Bob offers 2⁄3. As a result, their impact on the seller’s revenues, and that Alice wins the auction and pays according the design of the securities can be more impor- to Bob’s bid; that is, she gives up 2⁄3 of the tant than design of the auction itself. future revenues. This yields a higher payoff for the auctioneer; (2⁄3) ϫ $4M ϭ In our model, several agents compete for an Ͼ asset or project that requires an upfront invest- $2.67M $2M (in addition to any upfront payment). ment. This investment may correspond to an amount of cash that the seller needs to raise, or the resources required to run the project. Bid- This example is based on Robert G. Hansen ders observe private signals regarding the value (1985), who was the first to examine the use of they can expect if they acquire the asset. Our securities in an auction setting. Hansen showed initial structure is similar to an independent that a second-price equity auction yields higher private values model, so that different bidders expected revenues than a cash-based auction. In expect different payoffs upon winning, although a related paper, John G. Riley (1988) considers we also consider correlated and common val- first-price auctions where bids include royalty ues. The model differs from standard auction payments in addition to cash. He shows that models in that bids are securities. Bidders offer adding the royalty increases expected revenues. derivatives in which the underlying value is the The intuition in both cases is that adding an future payoff of the asset. Because the winner equity component to the bid lowers the differ- may make investments or take other actions that ence between the winner’s valuation and that of affect this payoff, we also discuss the possibility the second highest bidder. Because this differ- of moral hazard. ence is the rent captured by the winner, reduc- One might conjecture that the results from ing it benefits the seller. standard auction theory carry over to security- In this paper, we generalize this insight along bid auctions by simply replacing each security several dimensions. First, we consider a general with its cash value. Unlike cash bids, however, class of securities that includes debt, equity or the value of a security bid depends upon the royalty rates, options, and hybrids of these. Sec- bidder’s private information. This difference ond, we consider alternative auction formats can have important consequences as the follow- (e.g., first-price versus second-price). Third, we ing simple example demonstrates: consider informal auctions, in which the seller cannot commit to an auction mechanism in Consider an auction in which two bidders, advance. Alice and Bob, compete for a project. The The structure of the paper is as follows. The project requires an initial fixed investment basic model is described in Section I. We begin that is equivalent to $1M (we can inter- our analysis in Section II by examining formal pret some or all of this amount as a min- mechanisms, which consist of both an auction imum up-front cash payment required by format and a security design. There we establish the seller). Alice expects that if she un- dertakes the project, then on average it the following results: would yield cash flows of $4M; Bob expects future cash flows of only $3M. As- ● We characterize super-modularity conditions suming these estimates are private values, under which a monotone—and hence effi- in a standard second-price auction it is a cient—equilibrium is the unique outcome for dominant strategy for bidders to bid their the first- and second-price auctions. 938 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

● First we compare security designs holding ● In the unique equilibrium satisfying standard fixed the auction format (first- or second- refinements of off-equilibrium beliefs, bid- price). We show that for either format, the ders use the “flattest” securities available, seller’s expected revenues are positively re- e.g., cash or debt. Moreover, the outcome is lated to the “steepness” (a notion that we equivalent to a first-price auction. As a result, define) of the securities. As a result, debt we conclude that this ex post maximization contracts minimize the seller’s expected pay- yields the worst possible outcome for the offs while call options maximize it. This re- seller. sult generalizes the observations of Hansen (1985) and Riley (1988). The intuition is that flattest security provides the ● Fixing the security design, we then consider cheapest way for a high type to signal his qual- the role of the auction format. We define two ity. Thus, bidders find it optimal to compete important classes of sets of securities: sub- using these securities. convex and super-convex sets. For sub- Section IV extends the model by considering convex sets—which include, for example, the the effects of relaxing liquidity constraints, set of debt securities—we show that a sec- moral hazard regarding the bidder’s investment, ond-price auction yields higher expected rev- reservation prices, and the introduction of affil- enues than a first-price auction. Alternatively, iated as well as common values. We demon- if the set is super-convex (e.g., call options), strate that the main insights of our analysis carry the reverse conclusion holds and first-price over to these settings. For example, we show that: auctions are superior. However, we find the effect of the auction format to be small rela- ● If the bidder’s investment in the project is tive to the security design. unverifiable and subject to moral hazard, ● We then ask whether the Revenue Equiva- then it is not optimal for the seller to offer lence principle for cash auctions, which states cash compensation to the winner for this that expected revenues are independent of the investment. auction format, can be extended to security- ● Combining cash payments with bids effec- bid auctions. We show it holds if the ordered tively “flattens” the bids and reduces the ex- set of securities is a convex set. This is true pected revenues of the seller. for important classes of securities, such as ● Our conclusions regarding the revenue con- equity. sequences of the security design carry over to ● Finally we combine these results to show that the case of affiliated values with both private the first-price auction with call options max- and common components. The impact of auc- imizes the seller’s revenue, while the first- tion formats is more complicated, especially price format with debt minimizes it, over a with common values. We identify a new way general set of auction mechanisms. security-bid auctions affect the winner’s curse and illustrate the differences in first- In Section III, we consider the case in and second-price auctions. Interestingly, rev- which the seller is unable to commit ex ante enue equivalence fails even with independent to a formal auction mechanism. Instead, he signals and equity bids. accepts all bids and chooses the security that is optimal ex post. Though often not labeled Section V concludes and the Appendix con- as “auctions” because they lack a formal tains proofs omitted in the text. Proofs of lem- mechanism, we believe that these informal mas are available at http://www.e-aer.org/data/ auctions represent the vast majority of sept05_app_demarzo.pdf. auction-like activity in practice, since in most transactions the seller is unable to commit to Related Literature.—As mentioned above, a decision rule ex ante. In this case, the task Hansen (1985) and Riley (1988) first demon- of selecting the winning bid is not trivial; it strated the potential advantages of equity versus involves a signaling game in which the seller cash auctions. In a more recent paper, Matthew uses his beliefs to rank the different securities Rhodes-Kropf and S. Viswanathan (2000) focus and choose the most attractive one. Our main on first-price auctions in a setting that is similar result is as follows: to the model we study in the first part of the VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 939 paper, and show that securities yield higher flows that he will use to rank securities. He revenues than a cash-based auction. However, examines a common value environment and none of these papers provides a general means obtains a partial characterization of the equilib- of comparing sets of nonlinear securities, as we rium in a binary model (two bidders, two types, do here. Nor do they compare auction formats two values). or consider informal auctions. Finally, the re- Other related literature includes R. Preston sults in Riley (1988) and Rhodes-Kropf and McAfee and John McMillan (1987), who solve Viswanathan (2000) are conditional upon the for the optimal mechanism in a model with a existence of a separating equilibrium in which a moral hazard problem. The optimal mechanism higher type bids a higher security. For example, is a combination of debt and equity, with the in Rhodes-Kropf and Viswanathan (2000), there mixture depending on the distribution of types. always exists a pooling equilibrium and in some Jean-Jacques Laffont and Jean Tirole (1987) cases it is the unique outcome. This is because examine a similar model. Board (2004) ana- they assume that the project does not require lyzes selling real options to competing buyers any costly inputs—thus the lowest type can with payments possibly conditional on exercis- offer 100 percent of the proceeds to the seller ing the option (where the exercise decision is and break even. Therefore, a low type is always subject to moral hazard). willing to imitate the bid of a high type. We use We can interpret our results in a setting in a framework that is closer to Hansen (1985), in which the seller needs to raise a fixed amount of which the project requires costly inputs. In this cash by selling a security backed by an asset. In case, we show conditions under which the first- this case, the security sold is the complement of price auction has a unique equilibrium, and it is the winning bid, and higher auction revenues separating. correspond to the seller raising the cash using One reason security-bid auctions may not “cheaper” securities. Thus, our result that call have received greater attention in the literature options are optimal is equivalent to the seller is perhaps due to Jacques Cre´mer (1987), who raising funds by issuing debt, with bidders com- argues that the seller can extract the entire sur- peting on the loan interest rate. This result plus if he can “buy” the winning bidder. Spe- therefore extends results in the security design cifically, the seller can offer cash to the bidder literature (e.g., David C. Nachman and Thomas to cover the costs of any required investment, H. Noe, 1994; DeMarzo and Darrell Duffie, ask all bidders to reveal their type, and award 1999; DeMarzo, 2002) which demonstrate debt the project to the highest type while keeping is optimal when the seller is privately informed 100 percent of the revenues. Initially, we rule to the case in which the bidders have superior out this solution by assuming the seller is cash information.2 Similarly, our result for informal constrained. Then we show in Section IV that, auctions suggests that when informed investors even if the seller is not cash constrained, are unconstrained, they will prefer to fund the Cre´mer’s approach does not survive if the bid- firm in exchange for equity or option-like der’s investment is not verifiable. In that case, securities. moral hazard forces the seller to offer only compensation that is contingent on the outcome I. The Model of the project. Yeon-Koo Che and Ian Gale (2000), Charles There are n risk neutral bidders who compete Z. Zheng (2001), Simon Board (2002), and for an asset, which we think of as the “rights to a Rhodes-Kropf and Viswanathan (2002) con- project.” The winner is required to make an in- sider auctions with financially constrained bid- vestment X Ͼ 0, which we can interpret as re- ders who use debt, or external financing, in their sources required by the project, or as a minimal bids. Hence, while bids may be expressed in amount of cash the seller must raise. In either case, terms of cash, they are in fact contingent claims X is known and equal across bidders. and are thus examples of the security bids that we examine here. Mark Garmaise (2001) studies a security-bid auction in the context of 2 Ulf Axelson (2004) derives a related result in a setting entrepreneurial financing. The entrepreneur in which the seller has a preference for cash, and can change commits to a probability distribution over cash the design of the security in response to the bids. 940 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

͉ ͉ ͉ ͉ Conditional on being undertaken by bidder i, In addition, the functions zh(z v), zhv(z v) and ͉ ͉ ͉ ʦ ϱ the project yields a stochastic future payoff Zi. zhvv(z v) are integrable on z (0, ). Bidders have private signals regarding Zi, which we denote by Vi. The seller is also risk These assumptions are weak and allow us to neutral and cannot undertake the project inde- take derivatives “through” expectation opera- pendently. The interest rate is normalized to tors. An example of a setting that fits our as- zero. We make the following standard eco- sumptions is the payoff structure: nomic assumptions on the signals and payoffs: (1) Zi ϭ ␪͑X ϩ Vi ͒ ASSUMPTION A: The private signals V ϭ ϭ ␪ (V1, ... , Vn) and payoffs Z (Z1, ... , Zn) satisfy where the project risk is independent of V and the following properties: log-normal with a mean of 1.5 The focus of this paper is on the case in (a) The private signals Vi are i.i.d. with density which bids are securities. Bidders compete for f(v) with support [vL, vH]. the project by offering the seller a share of the ϭ (b) Conditional on V v, the payoff Zi has final payoff. That is, the bids are in terms of ͉ ϱ density h(z vi) with full support [0, ). derivative securities, in which the underlying (c)(Zi, Vi) satisfy the strict Monotone Likeli- asset is the future payoff of the project Zi. Bids hood Ratio Property (SMLRP); that is, the can be described as function S(z), indicating the likelihood ratio h(z͉v)/h(z͉vЈ) is increasing3 payment to the seller when the project has final inzifv Ͼ vЈ.4 payoff z. We make the following assumptions regarding the set of feasible bids6: The important economic assumptions contained above are, first, that the private signals of DEFINITION: A feasible security bid is de- other bidders are not informative regarding the scribed by a function S(z), such that S is weakly signal or payoff of bidder i. (We extend our increasing, z Ϫ S(z) is weakly increasing, and results to allow for affiliated and/or common 0 Յ S(z) Յ z. values in Section IV C). Second, because Zi is not bounded away from zero, the project payoff The set of feasible securities encompasses cannot be used to provide a completely riskless standard designs used in practice. However, it is payment to the seller. Finally, the private signal not completely general. First, S(z) Յ z can be Vi is “good news” about the project payoff Zi viewed as a liquidity or limited liability con- using the standard strict version of the affiliation straint for the bidder; only the underlying asset assumption (see Paul R. Milgrom and Robert J. can be used to pay the seller. We assume, for Weber, 1982). now, that bidders do not have access to cash (or Given the above assumptions, we normalize other liquid assets) that they can pledge as pay- (without loss of generality) the private signals ment; they can transfer only property rights in so that the project.7 We make this assumption in order to focus first on pure security bids and simplify E͓Zi͉Vi ͔ Ϫ X ϭ Vi . the exposition. In Section IV we will generalize the setting to allow for cash payments. Thus, we can interpret the signal as the NPV of Similarly, S(z) Ն 0 corresponds to a liquidity the project, which we assume to be nonnegative. or limited liability constraint for the seller; the To simplify our analysis, we make several ad- seller cannot commit to pay the bidder except ditional technical assumptions: through a share of the project payoff. For ex-

ASSUMPTION B: The conditional density function h(z͉v) is twice differentiable in z and v. 5 More generally, what is required for the SMLRP is that log(␪) have a log-concave density function. 6 These assumptions are typical of the security design 3 We use increasing/higher/lower in the strict sense and literature (e.g., Nachman and Noe, 1994; Oliver Hart and explicitly note weak rankings. John Moore, 1995; DeMarzo and Duffie, 1999). 4 Equivalently, h is log-supermodular, which can be written 7 Bidders can invest X in the project, but X might corre- as (Ѩ2/ѨzѨv)log h(z͉v) Ͼ 0 assuming differentiability. spond to an illiquid asset, such as human capital. VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 941 ample, the seller may not have the financial max(z Ϫ k, 0). Higher bids correspond to resources to do so, and is selling the project to lower strike prices. (This is equivalent to raise X. Because the seller cannot reimburse the the bidder retaining a debt claim.) bidder for the upfront investment, this assumption rules out a solution as in Cre´mer (1987). Given any security S, we let We take this constraint as given for now, but we show in Section IV that this constraint can fol- ES͑v͒ ϵ E͓S͑Zi ͉͒Vi ϭ v͔ low from an assumption that the bidder’s investment X is not verifiable. denote the excepted payoff of security S condi- ϭ Finally, we require both the seller’s and the tional on the bidder having value Vi v. Thus, bidder’s payment to be weakly increasing in the the expected payoff to the seller if the bid S is payoff of the project. Monotonicity is a standard accepted from bidder i is ES(Vi). On the other feature of almost all securities used in practice, hand, the bidder’s expected payoff is given by 8 Ϫ and so is a natural constraint to consider. Most Vi ES(Vi). Thus, we can interpret Vi as the importantly, without monotonicity for the bid- independent, private value for bidder i, and ders, equilibria would not be efficient, and with- ES(Vi) as the payment offered. The key differ- out monotonicity for the seller, the seller would ence from a standard auction, of course, is that have incentives to choose other than the highest the seller does not know the value of the bids, bid. but only the security, S. The seller must infer the Together, these requirements are equivalent value of this security. Since the security S is to S(0) ϭ 0, S is continuous, and SЈ(z) ʦ [0, 1] monotone, the value of the security is increasing almost everywhere. Thus, we admit standard with the signal Vi of the bidder, as we show sets of securities, including: below:

(a) Equity: The seller receives some fraction LEMMA 1: The value of the security ES(v) is ␣ ʦ [0, 1] of the payoff: S(z) ϭ ␣z. twice differentiable. For S 0, ESЈ(v) Ͼ 0, and (b) Debt: The seller is promised a face value for S Z,ESЈ(v) Ͻ 1. d Ն 0, secured by the project: S(z) ϭ min(z, d). II. Formal Auctions with Ordered Securities (c) Convertible debt: The seller is promised a face value d Ն 0, secured by the project, or In many auctions, bidders compete by offer- a fraction ␣ ʦ [0, 1] of the payoff: S(z) ϭ ing “more” of a certain security. For example, max(␣z, min(z, d)). (This is equivalent to a they compete by offering more debt or more debt plus royalty rate contract.) equity. We begin our analysis by examining (d) Levered equity: The seller receives a frac- formal auctions in which the seller restricts the tion ␣ ʦ [0, 1] of the payoff, after debt with bids to elements of a well-ordered set of secu- face value d Ն 0 is paid: S(z) ϭ ␣ max(z Ϫ rities. Bidders compete by offering a higher d, 0). (This is equivalent to a royalty agree- security. ment in which the bidder recoups some There are two main reasons why sellers re- costs upfront.) strict the set of securities that are admissible as (e) Call option: The seller receives a call option bids in the auction. First, it allows them to on the firm with strike price k: S(z) ϭ use standard auction formats—such as first- or second-price—to allocate the object and to determine the payments. Without an imposed 8 A standard motivation for dual monotonicity is that, if structure, ranking different securities is difficult it did not hold, parties would “sabotage” the project and and depends on the beliefs of the seller. There is destroy output. (Dual monotonicity is also implied if one party could both destroy and artificially inflate output, e.g., no clear notion of the “highest” bid. Ͼ Ͻ if S(z0) S(z1) for z0 z1, the bidder could inflate the cash The second reason a seller may want to re- Ϫ flows from z0 to z1 via a short-term loan to get payoff z0 strict the set of securities is that it can enhance S(z1).) Whether revenues can be distorted in this way de- revenues. We will demonstrate this result by pends on the context. We do not further defend this assumption here, but point out that it is a standard one, includes first (in this section) studying the revenues from typical securities used in practice, and guarantees a well- auctions with ordered sets of securities and then behaved equilibrium. (in Section III) comparing them to the revenues 942 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

TABLE 1—EXPECTED REVENUES FOR DIFFERENT SECURITY and show precisely when revenue equiva- DESIGNS AND AUCTION FORMATS IN EXAMPLE 1 lence will hold or fail. Overall, though, the impact of the auction format on revenues is Expected seller revenues minor compared to the security design. Security type First-price auction Second-price auction (c) Among the mechanisms examined, the first- Cash 50.00 50.00 price auction with debt yields the lowest Debt 50.05 50.14 expected revenues, while the first-price auc- Equity 58.65 58.65 tion with call options yields the highest Call option 74.53 74.49 expected revenues. In Section II D we shall see that these are the worst and best possible mechanisms in a broad class of security-bid from auctions in which the seller cannot commit auctions, and that all security-bid auctions to a restricted set and bidders can bid using any dominate cash auctions. feasible security. A. Securities, Auctions, and Mechanisms Example 1: Comparison of Revenues across Securities and Auction Formats.—Before pre- The first step in our analysis is to formalize senting the technical details of the analysis, we the notion of an ordered set of securities. An consider an example that illustrates our main ordered collection of securities can be defined ʦ results. Two bidders compete for a project that by a function S(s, z), where s [s0, s1]isthe requires an upfront investment of X ϭ 100. The index of the security, and S(s, ) is a feasible NPV of the project if run by bidder i is Vi, security. That is, S(s, z) is the payment of where Vi is uniform on the interval [20, 110]. security s when the output of the project has ϵ The project is risky with final value Zi, which is value z. As before we define ES(s, v) E[S(s, ϩ ͉ ϭ lognormal with mean X Vi and volatility of 50 Zi) Vi v]. percent. For the collection of securities to be ordered, Total surplus is maximized by allocating the we require that its value, for any type, is in- project to the highest type, in this case leading creasing in s. Then, a bid of s dominates a bid of ϭ Ј Ͼ Ј to an expected value of E[max(V1, V2)] 80. s if s s . We would also like to allow for a This is the maximum expected revenue achiev- sufficient range of bids so that for the lowest able by any mechanism. On the other hand, bid, every bidder earns a nonnegative profit, using a cash auction, the expected revenue is while for the highest bid, no bidder earns a ϭ given by E[min(V1, V2)] 50 (which is the positive profit. This leads to the following for- same for first- and second-price auctions by mal requirements for an ordered set of revenue equivalence). Table 1 shows the reve- securities: nues for different security designs and auction ʦ formats. DEFINITION: The function S(s, z) for s [s0, Several observations can be made, which co- s1] defines an ordered set of securities if: incide with our main results of this section: (a) S(s, ) is a feasible security. Ͼ (a) Fixing the auction format (first- or second- (b) For all v,ES1(s, v) 0. Յ Ն price), revenues increase moving from debt (c) ES(s0, vL) vL and ES(s1, vH) vH. to equity to call options. In Section II B we will define a notion of “steepness” for se- Examples of ordered sets include the sets of curities and show that steeper securities lead (levered) equity and (convertible) debt, indexed to higher revenues, and that all security by the equity share or debt amount, and call designs yield higher revenues than cash options, indexed by the strike price. Given an auctions. ordered set of securities, it is straightforward to (b) The auction format is irrelevant for a cash generalize the standard definitions of a first- and auction and for an equity auction. While the second-price auction to our setting: format does make a difference for debt and call options, the rankings are reversed. In Sec- First-Price Auction: Each agent submits a se- tion II C we will generalize these observations curity. The bidder who submitted the highest VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 943 security (highest s) wins and pays according to LEMMA 3: There exists a unique symmetric his security. equilibrium for the first-price auction. It is increasing, differentiable, and it is the unique Second-Price Auction: Each agent submits a solution to the following differential equation: security. The bidder who submitted the highest ͑n Ϫ 1͒f͑v͒ ͓v Ϫ ES͑s͑v͒, v͔͒ security (highest s) wins and pays the second- Ј͑ ͒ ϭ ϫ s 9 s v ͑ ͒ ͑ ͑ ͒ ͒ highest security (second-highest ). F v ES1 s v , v

Are the equilibria in these auction formats together with the boundary condition ES(s(vL), ϭ efficient? That is, does the highest value bidder vL) vL. win the auction? For second-price auctions the answer is straightforward; the standard charac- Assumption C is a joint restriction on the set terization of the second-price auction with pri- of securities and the conditional distribution of vate values generalizes to: Z.10 It can be shown to hold generally in the lognormal setting (1) in the case of debt, equity, LEMMA 2: The unique equilibrium in weakly and levered equity securities with d Յ X.Itcan undominated strategies in the second-price auc- be established numerically for other types of ϭ tion is for a bidder i who has value Vi v to securities, such as call options, under suitable submit security s(v) such that ES(s(v), v) ϭ v. parameter restrictions: for example, it holds in The equilibrium strategy s(v) is increasing in v. the numerical example computed earlier. Throughout our analysis, we assume that it In words, similarly to a standard second-price holds for all sets of securities under auction, each bidder submits bids according to consideration. his true value. We now turn our attention to the The first- and second-price auctions are two first-price auction. Incentive compatibility in standard auction mechanisms. They share the the first-price auction implies that no bidder features that the highest bid wins, and only the gains by mimicking another type, so that s(v) winner pays. The first property is necessary for satisfies efficiency, and the second is natural in our setting, since only the winner can use the assets of ͑ ͒ ϵ n Ϫ 1͑ ͒͑ Ϫ ͑ ͑ ͒ ͒͒ (2) U v maxvˆ F vˆ v ES s vˆ , v the project to collateralize the payment. One can construct many other auction mechanisms, ϭ Fn Ϫ 1͑v͒͑v Ϫ ES͑s͑v͒, v͒͒ however, that share these properties. For example, one can consider third-price auctions, or where U(v) is the expected payoff of type v. The auctions where the winner pays an average first-order condition of (2) then leads to a dif- of the bids, etc. Below we define a broad class ferential equation for s. However, an additional of mechanisms that will encompass these assumption is required to guarantee optimality: examples:

ASSUMPTION C: For all (s, v) such that the DEFINITION: A General Symmetric Mecha- bidder earns a positive expected proﬁt, i.e., nism (GSM) is a symmetric incentive compati- v Ϫ ES(s, v) Ͼ 0, the proﬁt function is log- ble mechanism in which the highest type wins supermodular: and pays a security chosen at random from a given set S. The randomization can depend on Ѩ2 the realization of types, but not on the identity of log͓v Ϫ ES͑s, v͔͒ Ͼ 0. ѨvѨs the bidders (so as to be symmetric).

Under this assumption, we have the following characterization: 10 Assumption C is the same condition imposed on util- ity functions in the auction literature, e.g., Eric S. Maskin and Riley (1984) use it to show existence and uniqueness of equilibria with risk-averse bidders. The requirements of symmetry and Assumption C underscore the fact that efﬁ- 9 Note that with private values, the second-price auction ciency is more fragile in ﬁrst-price than in second-price is equivalent to an English auction. auctions. 944 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

The ﬁrst-price auction ﬁts this description, with no randomization (the security is a function of the winner’s type). In the second-price auction, the security paid depends upon the realization of the second-highest type. GSMs also allow for more complicated payment schemes that depend on all of the bids. It will be useful in what follows to derive a basic characterization of the incentive compatibility condition for a GSM. We show that any GSM can be converted into an equivalent mechanism in which the winner pays a security that depends only on his reported type without fur- FIGURE 1. PAYOFF DIAGRAMS FOR CALL OPTIONS,EQUITY, ther randomization. AND DEBT LEMMA 4: Incentive compatibility in a GSM ˆ implies the existence of securities Sv in the 11 convex hull of S such that securities S1 is steeper than an ordered set S2 if ʦ ʦ for all S1 S1 and S2 S2, S1 strictly crosses ʦ n Ϫ 1͑ Ј͒͑ Ϫ ˆ ͑ ͒͒ v arg maxvЈ F v v ESvЈ v . S2 from below.

Thus, it is equivalent to a GSM in which the The following useful lemma shows that ˆ winner pays the non-random security Sv. steepness is naturally related to the shape of the underlying securities—if the payoffs of the se- This observation will allow us to compare rev- curities cross from below, then their expected enues across mechanisms by studying the rela- payoffs strictly cross: tionship between the set of securities S and its convex hull. LEMMA 5 (Single crossing): A sufﬁcient condition for S1 to strictly cross S2 from below is

B. Ranking Security Designs that S1 S2, and there exists z* such that Յ Ͻ Ն S1(z) S2(z) for z z* and S1(z) S2(z) for Recall from Table 1 that the seller’s revenues z Ͼ z*. varied greatly with the security design. As we will show, the revenues of different designs Comparing standard securities, note that a call depend upon the steepness of the securities. To option is steeper than equity, which in turn is do so, we need to formalize the notion of steep- steeper than debt. See Figure 1. ness of a set of securities. A simple comparison Why is steepness related to auction revenues? of the slopes of the securities is inadequate: Consider a second-price auction, where the win- comparing debt and equity, debt has higher ning bidder with type V1 pays the security bid slope for low cash flows and lower slope for by the second highest type V2. That is, the high cash flows. Rather, our notion of steepness winner pays ES(s(V2), V1). Since bidders bid is defined by how securities cross each other. their reservation value in a second-price auc- Intuitively, one security is steeper than another tion, ES(s(V2), V2) ϭ V2. Hence, the security if it crosses that security from below. Thus, we design impacts revenues only through the dif- introduce the following definition: ference

͑ ͑ 2͒ 1͒ Ϫ ͑ ͑ 2͒ 2͒ DEFINITION: Security S1 strictly crosses se- ES s V , V ES s V , V ϭ curity S2 from below if ES1(v*) ES2(v*) Ј Ͼ Ј implies ES1(v*) ES2(v*). An ordered set of which is just the sensitivity of the security to the true type. So to compare two securities we need to compare their slopes where the expected pay- 2 11 ϭ ¥ ␲ ments are the same and equal to V .Byour S is in the convex hull of S if S(z) k kSk(z) for all ␲ Ն ʦ ¥ ␲ ϭ z, with k 0, Sk S, and k k 1. definition, steeper securities are more sensitive VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 945 at the crossing point, and so lead to higher ble revenues, and call options yield the highest revenues. possible revenues, of any security-bid auction. More generally, steepness enhances competi- All security-bid auctions yield higher revenues tion between bidders since even with the same than a cash auction. bid, a higher type will pay more. This is the essence of the Linkage Principle, first used by Note that in all cases, these rankings are for any Milgrom and Weber (1982) to rank auction realization of types, and hence are stronger than formats for cash auctions when types are affil- the usual comparison based on an expectation iated.12 Applying the envelope theorem to the over types. incentive condition (2) for a first-price auction, we get C. Ranking Auction Formats

Ј͑ ͒ ϭ n Ϫ 1͑ ͒͑ Ϫ ͑ ͑ ͒ ͒͒ (3) U v F v 1 ES2 s v , v . In our setting of symmetric independent private values and risk neutrality, a well- To compare expected revenues in (efficient) known and important result for cash auctions auctions with different sets of securities, it is is the Revenue Equivalence Principle. It sufficient to rank expected payoffs of bidders. If implies that the choice of the auction for- ES2(s, v) are ranked for every s and v, then mat is irrelevant when the ultimate alloca- UЈ(v) are ranked and hence so are U(v) (using tion is efficient.14 We now turn to examining ϭ the boundary condition U(vL) 0). In general, the revenue consequences of the choice of we cannot rank ES2(s, v) everywhere, but it is auction format in a security-bid auction. As sufficient to rank ES2(s, v) at securities with the we have seen from the numerical analysis same expected payments. Our definition of of Example 1 in Table 1, revenue equiva- steepness again captures that ranking. This lence seems to hold for some security de- leads to the following main result: signs but not for others. To develop some further intuition, we begin with two simple PROPOSITION 1: Suppose the ordered set of examples. securities S1 is steeper than S2. Then for either a first or second-price auction, for any realiza- Example 2: Equity Auctions and Revenue tion of types, the seller’s revenues are higher Equivalence.—There are two bidders with in- using S1 than using S2. dependent types Vi distributed uniformly on [0, 1]. Upfront investment is X ϭ 1. The As a result, flat securities, like debt, lead to distribution of Zi has full support with mean ϩ low expected revenues, and steep securities, like X Vi. call options, lead to high expected revenues. In Consider a second-price equity auction. As fact, since debt and call options are the flattest13 we know, it is a dominant strategy to bid the and steepest possible securities, they represent reservation value: ␣SPA(v) ϭ [v/(v ϩ 1)], the worst and best designs for the seller. We can which is increasing in v. In a first-price auc- also extend the logic of Proposition 1 to cash tion it is an equilibrium strategy for agents to auctions, as a cash bid is flatter than any secu- bid ␣FPA(v) ϭ 1 Ϫ [ln(1 ϩ v)/v], which is rity. Thus, we have the following: also increasing.15 Now observe that both auctions yield equal COROLLARY: For a first- or second-price payoffs to the auctioneer, as in both auction auction, standard debt yields the lowest possi- formats the highest type wins and the average losing bid in a second-price auction equals the highest bid in the first-price auction: 12 See also Vijay Krishna (2002). The linkage principle is typically used to compare formats when signals are affiliated. In security-bid auctions, unlike cash auctions, even 14 William Vickrey (1961), Roger B. Myerson (1981), with independent types the winner’s expected payment de- Riley and William F. Samuelson (1981). pends on his true type, as pointed out by Riley (1988) in the 15 The payoff of a type v who pretends to be vЈ is context of royalty rates. vЈ{[ln(1 ϩ vЈ)/vЈ](1 ϩ v) Ϫ 1} ϭ (1 ϩ v)ln(1 ϩ vЈ) Ϫ vЈ, 13 By flattest, we mean that all other sets of securities are which is maximized by setting vЈϭv, verifying an steeper. equilibrium. 946 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

V2 v1 ͑ Ϫ ͒2 ͓␣SPA͑ ͉͒ Յ ͔ ϭ ͫ ͯ Յ ͬ 1 v1 v2 1 E V2 V2 v1 E ϩ V2 v1 ϭ ͵ d ϭ . V2 1 v2 v1 v1 v1 3 0 ͑ ϩ ͒ ln 1 v1 ϭ 1 Ϫ . v1 We conclude that bidders’ welfare is higher in the first-price auction and, since in both auction While in this example revenue equivalence formats the highest type wins, revenues are holds, as the next example shows this is not the higher in the second-price auction. Thus, reve- case for all securities. nue equivalence fails. To gain some insight into why revenue Example 3: Second-Price Auctions Yield equivalence fails, note that in the second-price Higher Revenues for Debt.—Consider a debt auction, the winner pays a random security (de- auction. There are two bidders, types Vi are termined by the second highest bid). The win- independent and uniform on [0, 1], X ϭ 0, and ner’s payment is equivalent to paying the the distribution of Zi given Vi is uniform on [0, “expected security” 16 2Vi]. If a bidder wins and pays according to a debt bid with face value b, the payoff to the z2 S ͑z͒ ϭ E͓min͑z,2V ͉͒V Յ v ͔ ϭ z Ϫ seller is min(b, z), which for a type v yields on v1 2 2 1 4v average: 1 in a first-price auction.18 This security is not a (4) E͓Z Ϫ min͑b, Z ͉͒V ϭ ͔ i i i v debt security, and therefore is steeper than debt.19 As a result of this steepness, the seller’s 1 2v ͑2v Ϫ b͒2 ϭ ͵ ͑z Ϫ b͒ dz ϭ . revenues are enhanced. On the other hand, in 2v 4v b the case of equity, a convex combination of securities is also an equity security. Thus, there In a second-price auction it is an equilibrium is no change in steepness, and so no change in strategy for agents to bid their reservation val- revenues. ues: bSPA(v) ϭ 2v. In a first-price auction it is an FPA 17 equilibrium strategy to bid b (v) ϭ 2⁄3 v. Sub- and Super-Convex Sets of Securities.— Suppose bidder 1 wins the auction. In a first- The previous examples suggest that the revenue price auction, his payoff is differences across auction formats will stem from the differences in steepness between the ͑2v Ϫ bFPA͑v ͒͒2 ͑2v Ϫ 2v /3͒2 4 set of securities and its convex hull. This moti- 1 1 ϭ 1 1 ϭ v1 , vates the following formal classification: 4v1 4v1 9 while in a second-price auction, his payoff is DEFINITION: An ordered set of securities S ϭ ʦ {S(s, ):s [s0, s1]} is super-convex if it is ͑ Ϫ SPA͑ ͒͒2 2v1 b V2 steeper than any nontrivial convex combination Eͫ ͯV Յ v ͬ 4v 2 1 of the securities in S. It is sub-convex if any 1 nontrivial convex combination of the securities 20 ͑2v Ϫ 2V ͒2 in S is steeper than S. ϭ ͫ 1 2 ͯ Յ ͬ E V2 v1 4v1

18 In this example the support of Z1 is bounded by 2v1 , so the security is monotone. Note that the bidder’s expected E Z Ϫ S Z ͉V ϭ ϭ payoff with this security is [ 1 v1( 1) 1 v1] 16 2 ͉ ϭ ϭ While this example violates some of our technical E[Z1/4v1 V1 v1] v1/3, as before. assumptions (X Ͼ 0 and Z has full support), it provides a 19 In general, a convex combination of debt securities is simple closed-form solution (and suggests that our results steeper than debt. For example, a 50-50 mix of debt with are more general). We thank a referee for this example. face value 50 and face value 100 has slope 1⁄2 for z ʦ (50, 17 To verify that this is an equilibrium, note from (4) that 100), and so crosses debt from below. the payoff of a type v who pretends to be vЈ is vЈ[(2v Ϫ 20 A nontrivial convex combination puts positive weight 2vЈ/3)2/4v], which is maximized for vЈϭv. on more than one security. VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 947

Not every set falls into one of the categories Here we ask whether it can be recovered for above. Still, there are some important examples some classes of securities—that is, what is spe- of sub- and super-convex sets: cial about cash? From Proposition 2, revenue equivalence LEMMA 6: The set of standard debt contracts fails in one direction for a super-convex set, and is sub-convex. The set of convertible debt con- in the opposite direction for a sub-convex set. tracts indexed by the equity share ␣, the set of Hence, a natural candidate is a set in the middle, levered equity contracts indexed by leverage, i.e., a convex set: and call options are super-convex sets. DEFINITION: An ordered set of securities S is Based on the characterization above, we can convex if it is equal to its convex hull. again use the Linkage Principle to rank the expected revenues of first- and second-price In fact, convex sets of securities have a sim- auctions. Here the proof relies on Lemma 4, ple characterization—each security is a convex which allows us to interpret the second-price combination of the lowest security s0 and the format as a first-price mechanism in the convex highest security s1. Thus, each security can be hull of the set of securities: thought of as s0 plus some “equity shares” of the Ϫ security (s1 s0), and so it can be thought of as PROPOSITION 2: If the ordered set of secu- a generalization of a standard equity auction. rities is sub-convex, then the first-price auction Our main result in this section is that under yields lower expected revenues than the second- convexity, the Revenue Equivalence holds. price auction. If the ordered set of securities is super-convex, the first-price auction yields PROPOSITION 3 (Revenue equivalence): Ev- higher expected revenues than the second-price ery efficient equilibrium of a GSM with securi- auction. (This revenue comparison also holds ties from an ordered convex set yields the same conditional on the winner’s type, for all but the expected revenues. (This equivalence also holds lowest type.) conditional on the winner’s type.)

One subtlety in the proof of Proposition 2 is Note that this is a stronger statement than that the security paid by the lowest type is the equivalence between a first- and second-price same for both auction formats (and is defined by auction, as it holds for any symmetric mecha- the zero-profit condition). Thus, neither format nism. Also note that the standard envelope ar- employs a “steeper” security for that type. We gument behind Revenue Equivalence does not get around this problem by slightly perturbing extend directly to security auctions. For cash, the support of the types for one of the auction there is no linkage between the true type and the formats, comparing revenues, and taking the bidder’s expected payment when types are in- limit. dependent, so revenues depend only upon the Proposition 2 reveals that the auction format allocation.21 That is not the case with security can have an impact on revenues. As we have bids, as we have seen. However, when the se- seen, however, this revenue impact stems from curity set is convex, because paying a random the difference in steepness between the set of security is equivalent to paying the expected securities and their convex hull. This difference security drawn from the same set, the expected is always less extreme than the difference in linkage across all mechanisms is identical. steepness that can be obtained by changing the security design directly. In that sense, the design D. Best and Worst Mechanisms of the securities is much more important than the design of the auction format in determining We can combine the results of the previous revenues. two sections to determine the best and worst security design and format combinations. Note Revenue Equivalence for Convex Sets of Se- curities.—While revenue equivalence does not hold for general security auctions, it does hold 21 That is, in the case of cash auctions, (3) reduces to for cash, and holds for equity in our examples. UЈ(v) ϭ F nϪ1(v). 948 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 that, since debt is a sub-convex set, from Prop- design” is in the hands of the bidders, who can osition 2 the first-price auction is inferior to the choose to bid using any feasible security. second-price auction, and conversely for call Without the structure of a well-ordered set, options, which are super-convex. The following once the bids are submitted there is no obvious proposition establishes that a first-price auction notion of a “highest” bid. In this case, the seller with debt and with call options provides lower faces the task of choosing one of the submitted and upper bounds for the seller’s revenue across bids. Since there is no ex ante commitment by a broad class of mechanisms. the seller to a decision rule, the seller will choose the winning bid that offers the highest PROPOSITION 4: A first-price auction with expected payoff. Since the payoff of the secu- call options yields the highest expected reve- rity depends on the bidder’s type, the seller’s nues among all general symmetric mechanisms. choice may depend upon his beliefs regarding A first-price auction with standard debt yields the bid each type submits in equilibrium. Thus, the lowest expected revenues among all general this setting has the features of a signaling game symmetric mechanisms. that takes the form:

Proposition 4 establishes that the design of the (a) Bidders submit simultaneous bids that are security is more important than the specific auc- feasible securities. tion format: the revenue consequences of shift- (b) The seller chooses the winning bid. ing from debt to call options in a first-price (c) The winner pays his bid and runs the auction exceeds the consequences of any project. change in the auction mechanism. We remark that Proposition 4 is stated with We consider a sequential equilibrium of this respect to the particular set of feasible securities game. We argue that in the informal auctions, we have allowed thus far. It can be extended in bidders will choose the flattest securities possi- the obvious way: for any feasible set, if there is ble. That is, they will bid with cash, if it is a steepest set of securities which is (super-) feasible; otherwise, they will bid with debt. convex, then a (first-price) auction using this set Thus, Proposition 4 implies that the seller’s yields the highest possible revenues. Similarly, expected revenues are the lowest possible from if there is a flattest set which is (sub-)convex, any general auction mechanism. then a (first-price) auction using this set yields To gain some insight, consider first a case in the lowest possible revenues. which bidders can use cash. We argue that in For example, if bidders can pay cash, a cash equilibrium bidders use only cash. The intuition auction is the worst possible auction for the for this result is as follows. Let Sv be the secu- seller. This is because cash, which is insensitive rity bid by type v. When a bidder of type v to type, is even flatter than standard debt secu- decides on his bid, he has the option to mimic rities. Alternatively, the seller may be able to other types. In particular, he can mimic a type increase revenues by using securities that are vЈϭv Ϫ dv just below him. Such a deviation even more leveraged than call options. For ex- has two effects. First, it will reduce his proba- ample, the seller might pay the bidder cash for bility of winning to that of type vЈ. Second, it additional equity. (See Section IV A for a fur- will lower his expected payment if he wins from ther discussion of this case.) ESv(v)toESvЈ(v). On the margin, these two effects must balance out (otherwise there is a III. Informal Auctions: The Signaling Game profitable deviation). But now suppose types just below v use se- In the previous section we considered formal curity bids rather than cash. Consider the devi- auctions in which bidders are restricted to ation by type v to a cash bid of amount b(vЈ) ϭ Ј choose securities from a specific well-ordered ESvЈ(v ). Since the seller values this cash bid Ј set. In reality, there is often no such restriction. the same as the bid SvЈ by type v , the marginal That is, the seller is unable to commit to ignore effect on the probability of winning is the same offers that are outside the set. As a result, the as if he deviates to SvЈ. However, because the seller will consider all bids, choosing the most expected value of the security is increasing in attractive bid ex post. In this case, the “security the true type, the expected payment is b(vЈ) ϭ VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 949

Ј Ͻ ESvЈ(v ) ESvЈ(v). Thus, if type v is indifferent the bid came from the type “most eager” to to a deviation using securities, he will profit make the deviation. from a deviation using cash. As a result, an In order to define the D1 criterion in our equilibrium will involve only cash bids. context, we introduce the following notation. The second step of the logic above can be First, let Si be the random variable representing applied even when cash bids are not available. the security bid by bidder i, which will depend i When mimicking a lower type, it is cheaper for on Vi. For any feasible security S, let R (S)be a higher type to use a security that is less sen- the scoring rule assigned by the seller, repre- sitive to the true type—i.e., a flatter security— senting the expected revenues the seller antici- than that used in the proposed equilibrium. This pates from that security, given his beliefs. reasoning suggests an equilibrium will involve Along the equilibrium path, the seller’s beliefs the flattest securities available. are correct, so that the scoring rule satisfies However, there is a difficulty with extending i i this result when cash is not available. The gain (5) R ͑S͒ ϭ E͓ES͑Vi ͉͒S ϭ S͔. from a deviation depends crucially on the seller’s evaluation of the bid. The argument we Given the seller’s scoring rule, Ri, it must gave above is simplified by the fact that the also be the case in equilibrium that bidders are ϭ value of a cash bid is unambiguous; the seller bidding optimally. That is, conditional on Vi does not need to rely on his beliefs. But when v, Si solves bidders do not use cash, the value of any off- i i i equilibrium bid depends on the seller’s off- (6) U ͑v͒ ϭ maxS P ͑R ͑S͒͒͑v Ϫ ES͑v͒͒ equilibrium beliefs. As with general signaling games, there are many equilibria of this game if where Pi(r) is the probability that r is the high- we do not impose any restrictions on the beliefs est score.25 Thus, Ui(v) is the equilibrium ex- of the seller when an “unexpected” bid is ob- pected payoff for bidder i with type v. served.22 We turn to such restrictions next. Suppose the seller observes an out-of- equilibrium bid, so that the score is not deter- A. Refining Beliefs: The D1 Criterion mined by (5). Which types would be most likely to gain from such a bid? For each type v,wecan To rule out equilibria supported with arbi- determine the minimum probability of winning, trary off-equilibrium beliefs, the standard re- Bi(S, v), that would make bidding S attractive: finement in the signaling literature is the notion of strategic stability, introduced by Elon Kohl- Bi͑S, v͒ ϭ min͕p : p͑v Ϫ ES͑v͒͒ Ն Ui͑v͖͒. berg and Jean-Francois Mertens (1986). For our purposes, a weaker refinement, known as D1, is Then the D1 criterion requires that the seller sufficient to identify a unique equilibrium.23 believe that a deviation to security S came from The D1 refinement (see In-Koo Cho and David the types which would find S attractive for the M. Kreps, 1987; Cho and Joel Sobel, 1990) is a lowest probability of winning26: refinement commonly used in the security de- 24 i͑ ͒ ʦ ͑ i͑ ͒͒ sign literature. Intuitively, the D1 refinement (7) R S ES arg minv B S, v . criterion requires that if the seller observes an out-of-equilibrium bid, the seller should believe Thus, a sequential equilibrium satisfying the D1 criterion for the auction game can be described by scoring rules Ri and bidding strategy Si for all i satisfying (5)–(7). 22 If the seller believes all off-equilibrium bids are made by the lowest type, the gain from deviating is minimized. These beliefs seem unreasonable, however, since many securities would be unprofitable for the lowest type. 23 Strictly speaking, D1 is defined for discrete type 25 If there are ties, we require that Pi be consistent with spaces. However, it can be naturally extended to continuous some tie-breaking rule. types (see, e.g., Garey Ramey, 1996). 26 We have economized on notation here. If the set of 24 See, e.g., Nachman and Noe (1994) and DeMarzo and minimizers is not unique, the score is in the convex hull of Duffie (1999). ES(v) for v in the set of minimizers. 950 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

B. Equilibrium Characterization other than debt contracts, then expected revenues are higher than without such commitment. Using the D1 refinement, we can now extend the argument we made for cash deviations to PROOF: other securities. Suppose type v mimics type Follows immediately from Proposition 4 and vЈϭv Ϫ dv. The cost of doing so for type v is Proposition 5. ESvЈ(v). Now suppose v instead deviates to a security S that is flatter than SvЈ and such that IV. Extensions Ј ϭ Ј ES(v ) ESvЈ(v ). Because the security is flatter, it has a lower cost for type v, ES(v) Ͻ A. Relaxing the Liquidity Constraints ESvЈ(v). How would the seller respond to the deviation S? We have assumed that both the seller and the Because S is flatter than SvЈ, it is a more bidders are liquidity constrained. We now ex- expensive security for types below vЈ, and plore implications of either the seller or bidders Ј cheaper for types above v , than SvЈ. Therefore, having access to cash. the types that are “most eager” to deviate to S must be above vЈ. By D1, this implies that the Moral Hazard: Non-Contractible Invest- seller will evaluate S as at least as valuable as ment.—The main setting for our model is the SvЈ. Therefore, if type v is indifferent to a de- case in which the seller is liquidity constrained; viation to SvЈ, he will profit from a deviation to indeed, we can interpret X as an amount of cash a flatter security S. As a result, an equilibrium the seller must raise. But if X corresponds to will involve only the flattest possible securities. resources required for the project, and if the We now proceed with a formal statement of seller has surplus cash, securities in which the our results. As is standard in the auction setting, seller reimburses the winner for a portion of the we will focus on symmetric equilibria.27 We initial investment (and thus have S(0) Ͻ 0) are maintain Assumption C for the flattest securi- feasible. Importantly, these securities can be ties, so that existence of an efficient equilibrium steeper than call options and so increase reve- of the first-price auction is assured. Then we nues. For example, the seller could auction off have: the rights to a fraction ␧ of the cash flows and reimburse the winner directly for the investment PROPOSITION 5: Given symmetric strategies, (1 Ϫ ␧)X. By making ␧ arbitrarily small, the there is a unique equilibrium of the informal seller can extract the entire surplus. While this auction satisfying D1. This equilibrium is equiv- theoretical mechanism was proposed by Cre´mer alent, in both payoffs and strategies, to the (1987), it is not observed in practice. A likely equilibrium of a first-price auction in which reason is moral hazard: if the winner’s invest- players bid with the flattest securities feasible. ment is not fully contractible, and if the winner In particular, if they can bid with cash, they will receives only a small fraction of future reve- use only cash; if cash is not feasible, they will nues, then he may underinvest.28 bid with standard debt contracts. For example, suppose that after the auction the winning bidder i can choose whether to invest X. Again, we can now combine this result with If X is invested, the payoff of the project is Zi as the result of the previous section to formalize before, and his payment to the seller is S(Zi). If X the value of the seller’s ability to commit to a is not invested, the payoff is 0, and his payment to restricted set of securities. the seller is S(0). If S(0) Ն 0, the bidder’s payoff is nonpositive without investment, and so the op- COROLLARY: If the seller can commit to a formal auction with an ordered set of securities 28 For example, in several oil lease auctions run by the U.S. Department of the Interior in which the bidders bid high royalty rates, the oil fields were left undeveloped (and the government received almost no revenues) because bid- 27 That is, we restrict attention to equilibria in which the ders did not capture enough of the revenues to warrant their bidders use symmetric strategies and the seller uses the private investment (see Ken Binmore and Paul Klemperer, same scoring rule for all players. 2002). VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 951 tion not to invest is irrelevant. But suppose a bid the project (for amounts above B). Because with S(0) Ͻ 0 is accepted with positive probabil- these securities become flatter as B increases, ity. Then every bidder, including the lowest type, the seller’s expected revenues decrease with B. can earn positive profits by making such a bid and A pure cash auction, yielding the lowest possi- not investing. Yet if bidders do not invest and ble revenues, is possible for a second-price auc- Ͻ S(0) 0, the seller loses money. As a result, the tion if B exceeds vH, and for a first-price auction seller would choose not to accept such securities, if B exceeds the expected maximum type for Ϫ as shown below: n 1 bidders (which is less than vH). Thus, a first-price auction yields lower revenues for the Ͻ PROPOSITION 6: Suppose that the seller is seller as long as B vH. These results are not liquidity constrained and the investment X is consistent with Board (2002), who considers not contractible. Then: debt auctions and shows that they yield higher revenues than cash auctions, with the smallest (a) In a first- and second-price formal auction effect for first-price auctions. (with an ordered set of securities): (i) If a security without reimbursement is B. Reservation Prices allowed, then with probability 1 the winning bid satisfies S(0) Ն 0. That is, In this section we discuss briefly how reserva- competition between bidders rules out tion prices can be incorporated into our analysis. reimbursement. Commitment to a reservation price can improve (ii) If all securities involve reimbursement, the seller’s revenues, and even absent commit- then all bidders bid the highest allowed ment, a reservation price may be relevant if selling security and do not invest, leading to the project entails an opportunity cost. negative revenues for the seller. In the case of formal auctions, we assumed Ͻ (b) Any mechanism in which bids with S(0) 0 earlier that the lowest security, s0, was such that win with positive probability cannot be all types earn nonnegative profits; that is, ES(s0, Յ efficient. vL) vL. A reservation price is equivalent to (c) In an informal auction, securities with assuming that s0 restricts that set of types that S(0) Ͻ 0 are used with probability 0. can profitably participate. In particular, if we choose s0 so that Thus, when X is not contractible, even if the ͑ ͒ ϭ seller has cash we can rule out reimbursement ES s0 , vr vr from the seller: it would either not occur in ʦ equilibrium or not be in the seller’s best interest. for some type vr [vL, vH], then vr is the reservation price, and types below vr will not be Partial Cash Bids.—Suppose bidders have allocated the project. All of our results general- cash equal to B, where B is known and common ize to this case. In case of informal auctions, 29 to all bidders. Cash relaxes the limited liabil- since there is no commitment, we interpret vr as ity restriction for bidders so that the flattest the outside opportunity for the seller. securities are now debt claims on the total assets of the bidder (cash ϩ project), defined by C. Affiliated Private and Common Values

SD͑d, z͒ ϭ min͑d, B ϩ z͒ Our model thus far is based on the classic independent private values framework. We dis- ϭ min͑d, B͒ ϩ min͑͑d Ϫ B͒ϩ, z͒. cuss here how our results generalize when values are afﬁliated (see Milgrom and Weber, As the decomposition above reveals, we can 1982) and may have common, as well as pri- think of this security as an immediate cash vate, components.30 Formally we assume that: payment (up to B), plus a standard debt claim on

29 Che and Gale (2000), Zheng (2001), and Rhodes- 30 Afﬁliation (essentially, the log-supermodularity of the Kropf and Viswanathan (2000) consider models where the joint density function) implies that “good news” about one cash amount is heterogeneous and privately known. of the variables (learning it lies in a higher interval), raises 952 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

ASSUMPTION D: The private signals V ϭ This result again follows from the linkage princi- ϭ (V1, ... , Vn) and payoffs Z (Z1, ... , Zn) satisfy ple, since the second-highest bid is affiliated with, the following properties: or “linked” to, the winner’s value. This linkage creates an advantage for the second-price auction. (a) The private signals Vi are affiliated and On the other hand, if there is a common value distributed symmetrically with full support component to the asset, this can have an oppos- [vL, vH]. ing effect regarding the optimal auction format, ϭ (b) Conditional on V v, the payoff Zi has and create an advantage for a first-price auction. ͉ ϱ density h(z vi, vϪi) with full support [0, ). We show this below for a case of independent The distribution is symmetric in the last n Ϫ signals and common value: 1 arguments. ● ͉ ϭ (c)(Z, V) are strictly affiliated. Suppose Vi are independent, and E[Zi V] ¥ 34 j Vj. Then for an equity auction, the First we consider formal auctions with a fixed first-price auction generates higher expected auction format. Under appropriate conditions revenues than a second-price auction. there exists a unique symmetric increasing equilibrium for both the first- and the second-price The intuition for this result is that, since the auction.31 Given an efficient equilibrium, we equity-share is increasing with the bidder’s can generalize our result regarding the impact of type, and therefore correlated with the asset’s the security design on the seller’s revenues32: value, to generate the same expected revenues the expected equity-share in the second-price ● Given a symmetric increasing equilibrium, auction is lower than in the first-price auction. then fixing the mechanism (first- or second- But the lower average equity share reduces the price), steeper sets of securities yield higher linkage to the winner’s own type, reducing rev- revenues for the seller.33 enues in the second-price auction. Finally, consider the setting of an informal The intuition for this result is the same as be- auction. With affiliated private values, our con- fore—steeper securities increase the effective clusions regarding the informal auction hold— competition between bidders since they are high types prefer to use flat securities to more costly for higher types. separate from lower types: What about the comparison of auction formats? Here it is useful to consider first the case ● In the unique D1 equilibrium, bidders use the ͉ of affiliated private values (i.e., h(z vi, vϪi) does flattest possible securities leading to the low- 35 not depend on vϪi). In this setting, revenue est revenues for the seller. equivalence fails even for cash auctions, as shown by Milgrom and Weber (1982). In our With common values the signaling game setting: becomes much more complex. Now, after observing the bids, the seller is potentially more ● With affiliated private values, for both convex informed than the bidder. Thus the bidder and sub-convex sets of securities, second- faces an adverse selection problem. This ad- price auctions generate higher expected rev- verse selection is mitigated by making the enues than first-price auctions. seller’s payoff sensitive to revenues, potentially leading bidders to bid using steeper securities. We leave the analysis of this case for future research. the expectation of any monotone function of the variables. With two random variables, it is equivalent to the MLRP. 31 For a second-price auction we do not need extra conditions. For a first-price auction, we again need log-super- 34 This is the so-called “Wallet Game”; see Jeremy Bu- modularity of the winner’s profit, which becomes more low and Klemperer (2002). complicated in this case. 35 The proof is the same as that of Proposition V with a 32 See Appendix for proofs. minor modification to step 3. Intuitively, affiliation provides 33 In this case, the notion of steepness is that given in the seller with information about the bidder’s type after Lemma 5, which, combined with affiliation, will imply that observing other bids, but this plays no role in a separating securities strictly cross in terms of their expected costs. equilibrium. VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 953

V. Conclusion revelation principle argument, we can restrict attention to securities that do not induce diver- We have examined an aspect of bidding rel- sion, i.e., such that 1 Ϫ SЈ(z) Ն ␦. Because this atively ignored in auction theory—the fact that limits the steepness of the security, it lowers the bidders’ payments often depend on the realiza- revenues the seller can achieve using the opti- tion of future cash flows. This embeds a security mal formal auction. On the other hand, it also design problem within the auction setting. First rules out debt. So if bidders are cash con- we analyzed formal auctions in which the seller strained, the flattest possible securities have the chooses the security design and restricts bidders form S(d, z) ϭ min(d,(1Ϫ ␦)z). Because these to bid only using securities in an ordered set. securities are not as flat as standard debt, the This enables a simple ranking of the securities revenues of the seller are in fact enhanced by and the use of standard auction formats. We this restriction.36 showed conditions for which Revenue Equiva- More generally, our analysis provides clear lence holds, and determined the optimal and intuition for the way in which moral hazard worst format and security design combinations. concerns will interact with competition and rev- In particular, we showed that revenues are in- enues in the auction. For example, if the reve- creasing in the steepness of the securities, and nues of the project are costly to verify, we know demonstrated that the first-price debt auction from Robert Townsend (1979) that it is optimal yields the lowest revenues, whereas a first-price for the party who observes the cash flows to be auction with call options yields the highest the residual claimant. Thus, if the seller ob- revenues, across a broad class of possible serves the cash flows, the optimal agency con- mechanisms. tract is a call option, which also maximizes the Next, we considered informal auctions in auction revenues. If the winner observes the which the seller does not restrict the set of cash flows, there is a tradeoff between verifica- securities or the mechanism ex ante, but tion costs, which are minimized with debt se- chooses the most attractive bid ex post. In this curities, and auction revenues. case, security design is in the hands of the For mergers and acquisitions, tax implica- bidders. We show that this yields the lowest tions and accounting treatment are likely to be possible expected revenues for the seller, and is important. For example, the deferral of taxes equivalent to a first-price auction using the flat- possible with an equity-based transaction may test feasible securities, such as debt or cash. give rise to the use of equity bids even in an Thus, there are strong incentives for the seller to unrestricted setting. Our results imply that this be actively involved in the auction design and tax preference can also lead to enhanced reve- select the securities that can be used. nues for the seller. Finally, we generalized our results to include It would be useful to allow for more compli- relaxed liquidity constraints (incorporating as- cated information structures. For example, bid- pects of moral hazard), partial cash bids, reser- ders may have private information not only vation prices, and affiliated and common about V but also about X, or the seller may have values. All of our main insights and results are private information. Another extension of our robust to these features. model that would be useful in applications There are a number of natural extensions to would be to allow for asymmetries in bidders’ our framework. For example, the role of the valuations and costs. One difficulty is that in security design often extends beyond the auc- such a setting, it may be impossible to ensure an tion to determine the winner’s and seller’s in- efficient outcome (see, for example, recent work centives ex post. While we discuss a simple moral hazard setting in Section IV A, more general settings can be considered. Some of 36 Similarly, if the seller can divert cash flows, the con- these can be modeled as further restrictions on straint becomes SЈ(z) Ն ␦, which rules out cash or debt and the set of feasible securities. For example, con- enhances revenues in an informal auction. Other moral sider the case in which the winner has the op- hazard settings can also be considered. If the winner can add arbitrary risk (see, e.g., S. Abraham Ravid and Matthew I. portunity to divert cash flows from the project, Spiegel, 1997), then bidders are restricted to using convex with each dollar diverted generating a private securities. See also Samuelson (1987) for a discussion of payoff of ␦ Ͻ 1. In this case, by the usual other potential constraints on the security design. 954 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 by Sandro Brusco et al., 2004, in the context of PROOF OF PROPOSITION 2: mergers). Consider the direct revelation game corre- 1 sponding to the two auctions. Let Sv be the 2 APPENDIX security bid in the first-price auction, and let Sv be the expected security payment in the second- Proofs of all technical lemmas are available price auction for a winner with type v, defined from the AER Web Appendix. We provide here as in Lemma 4. Then, if the set of securities is 1 2 the proofs of all propositions in the text. super-convex, Sv crosses Sv from below, and a nearly identical argument to that used in the PROOF OF PROPOSITION 1: proof of Proposition 1 for first-price auctions The reasoning for the second-price auction can be applied to prove that the seller’s ex- was provided in the text. For the first-price pected revenues are higher in the first-price auction, let Uj(v) be the equilibrium payoff for auction. j type v, and Sv the security bid, with the set Sj. The only complication is that the securities 1 ϭ 2 ϭ Efficiency implies U (vL) U (vL) 0. By issued by the lowest types are identical in the condition (2), if U1(v) ϭ U2(v), we have first- and second-price auctions, so that the 1 ϭ 2 1Ј ϭ ESv(v) ESv(v). If S1 is steeper than S2, then securities do not strictly cross and U (vL) 1Ј Ͼ 2Ј 2Ј ESv (v) ESv (v). Thus, from the envelope U (vL). To resolve this, we can change the ϩ ␧ condition, support in the first-price auction to [vL , ϩ ␧ vH], with an atom at vL with mass equal 1Ј͑ ͒ ϭ n Ϫ 1͑ ͒͑ Ϫ 1Ј͑ ͒͒ ϩ ␧ U v F v 1 ESv v to that originally on the set [vL, vL ]. 1 ϩ ␧ ϭ Ͻ 2 ϩ ␧ Now, U (vL ) 0 U (vL ), and by Ͻ n Ϫ 1͑ ͒͑ Ϫ 2Ј͑ ͒͒ ϭ 2Ј͑ ͒ F v 1 ESv v U v . the same argument as in the proof of Propo- 1 Ͻ 2 Ն ϩ ␧ sition 1, U (v) U (v) for all v vL . 1 Ͻ 2 Ͼ Hence, U (v) U (v) for v vL. Since bid- Then by the continuity of the strategies and ϩ ders’ payoffs are lower, the seller’s expected payoffs in the boundary of the support vL revenue is higher for each realization of the ␧, the first-price auction has weakly higher winning type under S1. revenues. However, since the securities strictly cross for higher types, the revenues PROOF OF COROLLARY TO cannot be equal. PROPOSITION 1: The proof for sub-convex sets is identical, Since debt has slope 1 and then 0, it strictly with the inequalities reversed (and taking the crosses any other nondebt security from above. limit of the support for the second-price Call options have slope 0 and then 1, and so auction). strictly cross any other nonoption security from below. The result then follows directly from PROOF OF PROPOSITION 3: Proposition 1. In a GSM, the winner pays according to a What about the comparison of debt securi- random security. From Lemma 4, the expected ties to other sets that may include debt? The payment by type v reporting vЈ can be written as ˆ ˆ proof for second-price auctions is unchanged. ESvЈ(v), where SvЈ is in the convex hull of the But for first-price auctions, we have the minor ordered set of securities S. Since S is convex, we difficulty that a debt security does not strictly can define s*(vЈ) such that cross another debt security. In that case we ␧ ͑ ͑ Ј͒ ϭ ˆ modify the set of debt securities by adding S s* v , ) Sv Ј( ). in cash, i.e., the security payoff is S( z) ϭ min( z, d) ϩ ␧. This set strictly crosses any Because S is ordered, incentive compatibility other set from above, and so the revenues can implies s*(v) must be increasing; otherwise a be ranked as in Proposition 1. The result then bidder could raise the probability of winning follows from the continuity of the equilibrium without increasing the expected payment. Thus, strategies and payoffs in the first-price auc- s*(v) defines an efficient equilibrium for the tion as we take the limit as ␧ 3 0. We can do first-price auction. The result then follows from the same for call options by subtracting cash the uniqueness of equilibrium in the first-price ␧ from each security. auction. VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 955

PROOF OF PROPOSITION 4: Note that The proof follows that of Proposition 2, except that instead of the second-price auc- v Ϫ ESd͑v͒ arg min B͑Sd, v͒ ϭ arg max . tion we consider a general symmetric mech- v v U͑v͒ anism over some subset of the feasible securities. The result follows, as a call op- Because the objective function is strictly log- tion contract is steeper and a standard debt supermodular by Assumption C, from Donald contract is flatter than any convex combina- M. Topkis (1978) we know that v is weakly tion of feasible securities (where we use the increasing with d. Thus, (7) implies R(Sd)is same trick as in the proof of the Corollary increasing in d. to Proposition 1 if the sets of securities intersect). Step 1b: R supports an equilibrium in the unrestricted auction. PROOF OF PROPOSITION 5: We focus on the no-cash case, as the argu- Consider any deviation to a debt contract. From ment with cash is similar (and even simpler as Step 1a, the probability of winning the auction off-equilibrium beliefs do not play any major is the same as in a first-price auction. Since we role). In step 1, we show existence of a debt- have a first-price equilibrium, there is no gain to based equilibrium that survives the D1 refine- the deviation. Consider a deviation to a nondebt ment. In steps 2 and 3, we show that no other contract S. To show it is not profitable for any Յ equilibrium exists. In step 2, we prove that type, we must show that P(R(S)) minvB(S, v). any equilibrium is equivalent to the equilib- Let v be the highest type in the set that mini- rium of a first-price auction in debt contracts. mizes B(S, v). Then ES(v) Ն R(S). It is suffi- In step 3, we use the envelope condition to cient to show that type v does not find the argue that the securities used in the original deviation to S profitable, i.e., to show that equilibrium must have been debt as well. P(R(S)) Յ B(S, v).

The proof differs somewhat from the intuition Find d such that ESd(v) ϭ ES(v). Then B(Sd, in the text, which considers deviations by type v v) ϭ B(S, v). From Lemma 5, types vЈϽv ﬁnd to mimic type vЈϭv Ϫ dv. This “local devia- Sd more expensive than S, so that B(Sd, vЈ) Ͼ Ј Ն d tion” is more intuitive but is not precise since B(S, v ) B(S, v). Therefore, arg minvЈB(S , types are not discrete and therefore there is no vЈ) Ն v, and so from (7), nearby type vЈ that type v is truly indifferent toward mimicking. R͑Sd͒ Ն ESd͑v͒ Ն R͑S͒.

Step 1: The equilibrium from a first-price Thus, if a deviation to S is profitable, so is a debt auction is a D1 equilibrium in the unre- deviation to Sd. But this contradicts the fact that stricted auction (with appropriate off-equilibrium no deviation to a debt contract is profitable. beliefs). Step 2: A symmetric D1 equilibrium in the We need to demonstrate that given the strate- unrestricted auction has the same payoffs as the gies from the debt auction, there is a set of equilibrium of a first-price debt auction. beliefs satisfying D1 that support this equilibrium in the unrestricted auction. We construct Our method of proof is to show that any non- the beliefs R using (5) and (7). If (7) does not debt bids can be replaced with an equivalent produce a unique score, we can choose the debt bid without changing the equilibrium. lowest one. We now show that this supports the equilibrium. Step 2a: If S is not a debt contract, then at most one type uses this security. Step 1a: For debt contracts, the score is in- Ͻ creasing in the face value of the debt. That Suppose not, so that v1 v2 are the lowest and is, R(Sd) is increasing in d, where Sd(z) ϭ highest types that use S. Then, by (5), R(S) ϭ Ͻ Ͻ min(d, z). ES(v*) for some v1 v* v2. Consider the 956 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 debt contract Sd with the same cost for type v*, Thus, ES(v) ϭ ESd(v)(v) and ESЈ(v) ϭ Ј i.e., such that ESd(v*) ϭ ES(v*). From Lemma ESd(v) (v). Therefore by Lemma 5, S ϭ Sd(v). 5, types v Ͻ v* find the Sd more expensive than S, so that B(Sd, v) Ͼ P(R(S)). Therefore, arg PROOF OF PROPOSITION 6: d Ն d Ն minv B(S , v) v*. Thus, from (7), R(S ) ESd(v*) ϭ R(S). But this contradicts an equilib- Case 1: First- and second-price auctions d rium, as type v2 finds S strictly cheaper than S with a weakly higher score. Let sЈ be a bid that wins with positive probability such that S(sЈ,0)Ͻ 0. Due to the Step 2b: Suppose type v uses contract S, and let moral hazard problem, submitting this bid d(v) be the debt level that for type v has equal earns strictly positive profits for any type, cost, i.e., ESd(v)(v) ϭ ES(v). Then bidding Sd(v) since any type can simply not invest and has the same payoff as bidding S, and so is also collect ϪS(sЈ, 0) in a first-price auction, or optimal for type v. Because S is an equilibrium even more in a second-price auction (since bid, P(R(Sd(v))) Յ P(R(S)) by (6). However, by the second highest bid is below sЈ). Thus, by the same argument used in the previous step, incentive compatibility, all equilibrium bids lower types find Sd(v) more costly than S so that earn positive profits. B(Sd(v), vЈ) Ͼ P(R(S)) for vЈϽv. Hence, Let sគ be the lowest bid submitted. Then the R(Sd(v)) Ն ESd(v)(v) ϭ R(S). Thus, P(R(Sd(v))) ϭ above implies this bid must win with positive P(R(S)). The result follows since the cost and probability. Since it is the lowest bid, this im- probability of acceptance of bidding debt with plies a tie—that is, sគ is submitted with positive face value d(v) and bidding S are the same for probability. But then raising the bid slightly type v. would lead to a discrete jump in the probability of winning and hence in profits. Incentive com- គ ϭ Step 2c: d(v) is the unique symmetric equilib- patibility therefore implies s s1, the highest Ն rium for a first-price debt auction and it is possible bid. If S(s1,0) 0, this contradicts the Ј Ͻ increasing. From Step 2b, bidding debt d(v)is existence of s .IfS(s1,0) 0, then all types bid optimal and so solves s1. But at s1, all types lose money if they run the project. Therefore, all types bid s1, do not in- ͑ ͒ ϭ ͑ ͑ d͑v͒͒͒͑ Ϫ d͑v͒͑ ͒͒ Ϫ Ͼ U v P R S v ES v vest, and collect S(s1,0) 0 from the seller.

d d ϭ maxd P͑R͑S ͒͒͑v Ϫ ES ͑v͒͒. Case 2: General mechanisms

Using the same logic as in step 1a, R(Sd)is In an efficient mechanism, the lowest type wins increasing in d. Therefore, this maximization with zero probability and so earns zero expected problem is identical to the problem faced by profits. Since lowest type can claim to be any bidders in a debt-only first-price auction. type and not invest, it must be the case that no Uniqueness and monotonicity follow from type with positive probability of winning pays a Lemma 3. security with S(0) Ͻ 0 with positive probability.

Step 3: In a symmetric D1 equilibrium in the Case 3: Informal auctions unrestricted auction, almost every bid is a debt contract. From Step 2 and Lemma 3, the equi- Here the result follows immediately from our librium payoff of type v in the first-price auction result that an equilibrium will always use the with debt is flattest possible securities. If S(0) Ͻ 0, we can “flatten” the security by raising S(0) and flat- U͑v͒ ϭ Fn Ϫ 1͑v͒͑v Ϫ ESd͑v͒͑v͒͒, tening it elsewhere. ʦ and U is differentiable. Suppose v (vL, vH) PROOF OF RESULTS IN SECTION IV C: bids S in equilibrium. By a standard envelope argument, Result 1: Consider a first-price auction and two sets of securities A and B where A is steeper UЈ͑v͒ ϭ Fn Ϫ 1͑v͒͑1 Ϫ ESЈ͑v͒͒. A B than B. Let Sv and Sv be the equilibrium bid VOL. 95 NO. 4 DEMARZO ET AL.: AUCTIONS AND SECURITY DESIGN 957

1 Ͻ for type v using these two sets. Consider the linkage principle, we must show that M1(v, v) 2 2 expected payments of a type v who bids as type M1(v, v). But this follows since (a) Sv,v is in the vЈ: convex hull of A and therefore is steeper than 1 Sv, and (b) afﬁliation and the monotonicity of j͑ Ј͒ ϭ ͓ j ͑ ͉͒ ϭ Յ Ј͔ 2 ͉ M v, v E Sv Ј Zi Vi v, V*Ϫi v bids implies that E[Sv,v Ј(Zi) Vi] is increasing in v. for j ϭ A, B. Result 3: Let ␣j(v) be the equity bid of type v in Suppose MA(v, v) ϭ MB(v, v). Given afﬁliation, a j-price auction. Then a direct generalization of Lemma 5 implies that A Ͼ B 1͑ Ј͒ ϭ ͓␣1͑ Ј͒ ͉ ϭ Յ Ј͔ M1(v, v) M1(v, v). The conclusion then fol- M v, v E v Z Vi v, V*Ϫi v lows from the same linkage principle argument 1 as in the proof of Proposition 1. ϭ ␣ ͑v Ј͒E͓Z͉Vi ϭ v, V*Ϫi Յ v Ј͔

2 2 Now consider a second-price auction. The equi- M ͑v, v Ј͒ ϭ E͓␣ ͑V*Ϫi ͒Z͉Vi ϭ v, V*Ϫi Յ vЈ͔ librium bid satisfies the zero profit condition: 2 Ͼ E͓␣ ͑V*Ϫi ͉͒V*Ϫi Յ v Ј͔ ͓ j ͑ ͉͒ ϭ ϭ ͔ E Sv Zi Vi v, V*Ϫi v ϫ E͓Z͉Vi ϭ v, V*Ϫi Յ v Ј͔ ϭ E͓Zi Ϫ X͉Vi ϭ v, V*Ϫi ϭ v͔. ␣2 The proof follows the same logic as in Propo- where the inequality follows since (V*Ϫi) and sition 1; the seller’s revenues depend on the set Z are positively correlated, and we use the as- of securities through the difference sumption that types are independent. Therefore, M1(v,vЈ)ϭM2(v,vЈ) implies that ͓ j ͑ ͉͒ ϭ ϭ ͔ E Sv Zi Vi v1 , V*Ϫi v2 2 1 2 ␣ ͑v Ј͒ Ͼ E͓␣ ͑V*Ϫi ͉͒V*Ϫi Յ v Ј͔. Ϫ ͓ j ͑ ͉͒ ϭ ϭ ͔ E S Zi Vi v2 , V*Ϫi v2 . v2 ͉ ϭ ¥ But since E[Z V] l Vl, Again, affiliation implies this difference will be 1͑ Ј͒ ϭ ␣1͑ Ј͒ Ͼ 2͑ Ј͒ larger for steeper securities. M1 v, v v M1 v, v

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