bidding with securities: auctions and security designskrz/secauctions.pdfin a standard second-price...

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 biddingwith securities whose payments are contingent on the assetrsquos realized value Informal security-bid auctions the seller restricts the security design to an ordered setand uses a standard auction format (eg first- or second-price) In informalsettings bidders offer arbitrary securities and the seller chooses the most attractivebid based on his beliefs ex post We characterize equilibrium and show that steepersecurities yield higher revenues that auction formats can be ranked based on thesecurity design and that informal auctions lead to the lowest possible revenues(JEL D44 G32 G34)

Auction theory and its applications have be-come increasingly important as an area of eco-nomic research over the last 20 years As aresult we now have a better understanding ofhow the structure of an auction affects its out-come Almost all the existing literature studiesthe case when bidders use cash payments sothat the value of a bid is not contingent on futureevents

In a few cases such as art auctions the real-ized value is subjective and cannot be used as abasis for payment however this is the excep-tion In many important applications the real-ization of the future cash flow generated by theauctioned asset or project can be used in deter-mining the actual payment That is the bids canbe securities whose values are derived from thefuture cash flow We call this setting a security-

bid auction and provide an extensive charac-terization of such auctions

Formal auctions of this type are commonlyused in government sales of oil leases wirelessspectrum highway building contracts and lead-plaintiff auctions Informal auctions of this type(in the sense that formal auction rules are not setforth in advance) are common in the privatesector Examples include authors selling pub-lishing rights entrepreneurs selling their firm toan acquirer or soliciting venture capital andsports associations selling broadcasting rights1

The major difference between a formal andan informal mechanism is the level of commit-ment by the seller In an informal mechanismbidders choose which securities to offer and theseller selects the most attractive offer ex post Inthis case the auction contains the elements of asignaling game because the seller may inferbiddersrsquo private information from their securitychoices when evaluating their offers In a for-mal mechanism the seller restricts bidders to use Stanford Graduate School of Business 518 Memorial

Way Stanford CA 94305 (e-mails DeMarzo pdemarzostanfordedu Kremer ikremerstanfordedu Skrzypaczandygsbstanfordedu) We thank two anonymous refer-ees Kerry Back Simon Board Sandro Brusco NicolaeGarleanu Burton Hollifield John Morgan and seminarparticipants at Harvard University Hebrew University Uni-versity of Iowa Massachusetts Institute of TechnologyNew York University Stanford University University ofNew York at Stony Brook Technion Tel Aviv UniversityUC Berkeley UC Davis UC San Diego USC WashingtonUniversity Yale University WFA 2003 ASSA 2004 theStony Brook International Conference on Game Theory2004 and the Utah Winter Finance Conference 2004 foruseful comments This material is based upon work sup-ported by the National Science Foundation under Grant0318476 We also acknowledge the financial support of theCenter for Electronic Business and Commerce

1 See Kenneth Hendricks and Robert H Porter (1988) fora discussion of oil lease auctions in which royalty rates arecommonly used In wireless spectrum auctions the bids areeffectively debt securities (leading in some cases to default)Highway building contracts are often awarded throughldquobuild operate and transferrdquo agreements to the bidder thatoffers to charge the lowest toll for a pre-specified periodSee Jill E Fisch (2001) for the use of contingency-feeauctions in the selection of the lead plaintiff in class actionsuits In mergers acquisitions and venture capital agree-ments equity and other securities are commonly used (seeKenneth J Martin 1996) John McMillan (1991) describesthe auction of the broadcast rights to the Olympic gameswhere bids contained revenue-sharing clauses Similarlypublishing contracts include advance and royalty payments

936

securities from a pre-specified ordered set suchas an equity share or royalty rate The seller iscommitted to disqualify any offer outside thisset The seller also commits to an auction for-mat such as a first- or second-price auctionOne of our main results is that the revenuesfrom an informal mechanism are the lowestacross a large set of possible mechanisms Inother words the seller benefits from any form ofcommitment Moreover we show how to ranksecurity designs and auction formats in terms oftheir impact on the sellerrsquos revenues and thatthe design of the securities can be more impor-tant than design of the auction itself

In our model several agents compete for anasset or project that requires an upfront invest-ment This investment may correspond to anamount of cash that the seller needs to raise orthe resources required to run the project Bid-ders observe private signals regarding the valuethey can expect if they acquire the asset Ourinitial structure is similar to an independentprivate values model so that different biddersexpect different payoffs upon winning althoughwe also consider correlated and common val-ues The model differs from standard auctionmodels in that bids are securities Bidders offerderivatives in which the underlying value is thefuture payoff of the asset Because the winnermay make investments or take other actions thataffect this payoff we also discuss the possibilityof moral hazard

One might conjecture that the results fromstandard auction theory carry over to security-bid auctions by simply replacing each securitywith its cash value Unlike cash bids howeverthe value of a security bid depends upon thebidderrsquos private information This differencecan have important consequences as the follow-ing simple example demonstrates

Consider an auction in which two biddersAlice and Bob compete for a project Theproject requires an initial fixed investmentthat is equivalent to $1M (we can inter-pret some or all of this amount as a min-imum up-front cash payment required bythe seller) Alice expects that if she un-dertakes the project then on average itwould yield cash flows of $4M Bob ex-pects future cash flows of only $3M As-suming these estimates are private valuesin a standard second-price auction it is adominant strategy for bidders to bid their

reservation values As a result Alicewould bid 4 1 $3M and win theauction paying Bobrsquos bid of 3 1 $2M

Now suppose that rather than biddingwith cash the bidders compete by offer-ing a fraction of the future revenues Aswe discuss later it is again a dominantstrategy for bidders to bid their reserva-tion values Alice offers 3frasl4 of future cashflows while Bob offers 2frasl3 As a resultAlice wins the auction and pays accordingto Bobrsquos bid that is she gives up 2frasl3 of thefuture revenues This yields a higher pay-off for the auctioneer (2frasl3) $4M $267M $2M (in addition to any up-front payment)

This example is based on Robert G Hansen(1985) who was the first to examine the use ofsecurities in an auction setting Hansen showedthat a second-price equity auction yields higherexpected revenues than a cash-based auction Ina related paper John G Riley (1988) considersfirst-price auctions where bids include royaltypayments in addition to cash He shows thatadding the royalty increases expected revenuesThe intuition in both cases is that adding anequity component to the bid lowers the differ-ence between the winnerrsquos valuation and that ofthe second highest bidder Because this differ-ence is the rent captured by the winner reduc-ing it benefits the seller

In this paper we generalize this insight alongseveral dimensions First we consider a generalclass of securities that includes debt equity orroyalty rates options and hybrids of these Sec-ond we consider alternative auction formats(eg first-price versus second-price) Third weconsider informal auctions in which the sellercannot commit to an auction mechanism inadvance

The structure of the paper is as follows Thebasic model is described in Section I We beginour analysis in Section II by examining formalmechanisms which consist of both an auctionformat and a security design There we establishthe following results

We characterize super-modularity conditionsunder which a monotonemdashand hence effi-cientmdashequilibrium is the unique outcome forthe first- and second-price auctions

937VOL 95 NO 4 DEMARZO ET AL AUCTIONS AND SECURITY DESIGN

First we compare security designs holdingfixed the auction format (first- or second-price) We show that for either format thesellerrsquos expected revenues are positively re-lated to the ldquosteepnessrdquo (a notion that wedefine) of the securities As a result debtcontracts minimize the sellerrsquos expected pay-offs while call options maximize it This re-sult generalizes the observations of Hansen(1985) and Riley (1988)

Fixing the security design we then considerthe role of the auction format We define twoimportant classes of sets of securities sub-convex and super-convex sets For sub-convex setsmdashwhich include for example theset of debt securitiesmdashwe show that a sec-ond-price auction yields higher expected rev-enues than a first-price auction Alternativelyif the set is super-convex (eg call options)the reverse conclusion holds and first-priceauctions are superior However we find theeffect of the auction format to be small rela-tive to the security design

We then ask whether the Revenue Equiva-lence principle for cash auctions which statesthat expected revenues are independent of theauction format can be extended to security-bid auctions We show it holds if the orderedset of securities is a convex set This is truefor important classes of securities such asequity

Finally we combine these results to show thatthe first-price auction with call options max-imizes the sellerrsquos revenue while the first-price format with debt minimizes it over ageneral set of auction mechanisms

In Section III we consider the case inwhich the seller is unable to commit ex anteto a formal auction mechanism Instead heaccepts all bids and chooses the security thatis optimal ex post Though often not labeledas ldquoauctionsrdquo because they lack a formalmechanism we believe that these informalauctions represent the vast majority ofauction-like activity in practice since in mosttransactions the seller is unable to commit toa decision rule ex ante In this case the taskof selecting the winning bid is not trivial itinvolves a signaling game in which the selleruses his beliefs to rank the different securitiesand choose the most attractive one Our mainresult is as follows

In the unique equilibrium satisfying standardrefinements of off-equilibrium beliefs bid-ders use the ldquoflattestrdquo securities availableeg cash or debt Moreover the outcome isequivalent to a first-price auction As a resultwe conclude that this ex post maximizationyields the worst possible outcome for theseller

The intuition is that flattest security provides thecheapest way for a high type to signal his qual-ity Thus bidders find it optimal to competeusing these securities

Section IV extends the model by consideringthe effects of relaxing liquidity constraintsmoral hazard regarding the bidderrsquos investmentreservation prices and the introduction of affil-iated as well as common values We demon-strate that the main insights of our analysis carryover to these settings For example we show that

If the bidderrsquos investment in the project isunverifiable and subject to moral hazardthen it is not optimal for the seller to offercash compensation to the winner for thisinvestment

Combining cash payments with bids effec-tively ldquoflattensrdquo the bids and reduces the ex-pected revenues of the seller

Our conclusions regarding the revenue con-sequences of the security design carry over tothe case of affiliated values with both privateand common components The impact of auc-tion formats is more complicated especiallywith common values We identify a new waysecurity-bid auctions affect the winnerrsquoscurse and illustrate the differences in first-and second-price auctions Interestingly rev-enue equivalence fails even with independentsignals and equity bids

Section V concludes and the Appendix con-tains proofs omitted in the text Proofs of lem-mas are available at httpwwwe-aerorgdatasept05_app_demarzopdf

Related LiteraturemdashAs mentioned aboveHansen (1985) and Riley (1988) first demon-strated the potential advantages of equity versuscash auctions In a more recent paper MatthewRhodes-Kropf and S Viswanathan (2000) focuson first-price auctions in a setting that is similarto the model we study in the first part of the

938 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

paper and show that securities yield higherrevenues than a cash-based auction Howevernone of these papers provides a general meansof comparing sets of nonlinear securities as wedo here Nor do they compare auction formatsor consider informal auctions Finally the re-sults in Riley (1988) and Rhodes-Kropf andViswanathan (2000) are conditional upon theexistence of a separating equilibrium in which ahigher type bids a higher security For examplein Rhodes-Kropf and Viswanathan (2000) therealways exists a pooling equilibrium and in somecases it is the unique outcome This is becausethey assume that the project does not requireany costly inputsmdashthus the lowest type canoffer 100 percent of the proceeds to the sellerand break even Therefore a low type is alwayswilling to imitate the bid of a high type We usea framework that is closer to Hansen (1985) inwhich the project requires costly inputs In thiscase we show conditions under which the first-price auction has a unique equilibrium and it isseparating

One reason security-bid auctions may nothave received greater attention in the literatureis perhaps due to Jacques Cremer (1987) whoargues that the seller can extract the entire sur-plus if he can ldquobuyrdquo the winning bidder Spe-cifically the seller can offer cash to the bidderto cover the costs of any required investmentask all bidders to reveal their type and awardthe project to the highest type while keeping100 percent of the revenues Initially we ruleout this solution by assuming the seller is cashconstrained Then we show in Section IV thateven if the seller is not cash constrainedCremerrsquos approach does not survive if the bid-derrsquos investment is not verifiable In that casemoral hazard forces the seller to offer onlycompensation that is contingent on the outcomeof the project

Yeon-Koo Che and Ian Gale (2000) CharlesZ Zheng (2001) Simon Board (2002) andRhodes-Kropf and Viswanathan (2002) con-sider auctions with financially constrained bid-ders who use debt or external financing in theirbids Hence while bids may be expressed interms of cash they are in fact contingent claimsand are thus examples of the security bids thatwe examine here Mark Garmaise (2001) stud-ies a security-bid auction in the context ofentrepreneurial financing The entrepreneurcommits to a probability distribution over cash

flows that he will use to rank securities Heexamines a common value environment andobtains a partial characterization of the equilib-rium in a binary model (two bidders two typestwo values)

Other related literature includes R PrestonMcAfee and John McMillan (1987) who solvefor the optimal mechanism in a model with amoral hazard problem The optimal mechanismis a combination of debt and equity with themixture depending on the distribution of typesJean-Jacques Laffont and Jean Tirole (1987)examine a similar model Board (2004) ana-lyzes selling real options to competing buyerswith payments possibly conditional on exercis-ing the option (where the exercise decision issubject to moral hazard)

We can interpret our results in a setting inwhich the seller needs to raise a fixed amount ofcash by selling a security backed by an asset Inthis case the security sold is the complement ofthe winning bid and higher auction revenuescorrespond to the seller raising the cash usingldquocheaperrdquo securities Thus our result that calloptions are optimal is equivalent to the sellerraising funds by issuing debt with bidders com-peting on the loan interest rate This resulttherefore extends results in the security designliterature (eg David C Nachman and ThomasH Noe 1994 DeMarzo and Darrell Duffie1999 DeMarzo 2002) which demonstrate debtis optimal when the seller is privately informedto the case in which the bidders have superiorinformation2 Similarly our result for informalauctions suggests that when informed investorsare unconstrained they will prefer to fund thefirm in exchange for equity or option-likesecurities

I The Model

There are n risk neutral bidders who competefor an asset which we think of as the ldquorights to aprojectrdquo The winner is required to make an in-vestment X 0 which we can interpret as re-sources required by the project or as a minimalamount of cash the seller must raise In either caseX is known and equal across bidders

2 Ulf Axelson (2004) derives a related result in a settingin which the seller has a preference for cash and can changethe design of the security in response to the bids


Conditional on being undertaken by bidder ithe project yields a stochastic future payoff ZiBidders have private signals regarding Ziwhich we denote by Vi The seller is also riskneutral and cannot undertake the project inde-pendently The interest rate is normalized tozero We make the following standard eco-nomic assumptions on the signals and payoffs

ASSUMPTION A The private signals V (V1 Vn) and payoffs Z (Z1 Zn) satisfythe following properties

(a) The private signals Vi are iid with densityf(v) with support [vL vH]

(b) Conditional on V v the payoff Zi hasdensity h(zvi) with full support [0 )

(c) (Zi Vi) satisfy the strict Monotone Likeli-hood Ratio Property (SMLRP) that is thelikelihood ratio h(zv)h(zv) is increasing3

in z if v v4

The important economic assumptions con-tained above are first that the private signals ofother bidders are not informative regarding thesignal or payoff of bidder i (We extend ourresults to allow for affiliated andor commonvalues in Section IV C) Second because Zi isnot bounded away from zero the project payoffcannot be used to provide a completely risklesspayment to the seller Finally the private signalVi is ldquogood newsrdquo about the project payoff Ziusing the standard strict version of the affiliationassumption (see Paul R Milgrom and Robert JWeber 1982)

Given the above assumptions we normalize(without loss of generality) the private signalsso that

EZiVi X Vi

Thus we can interpret the signal as the NPV ofthe project which we assume to be nonnegativeTo simplify our analysis we make several ad-ditional technical assumptions

ASSUMPTION B The conditional densityfunction h(zv) is twice differentiable in z and v

In addition the functions zh(zv) zhv(zv) andzhvv(zv) are integrable on z (0 )

These assumptions are weak and allow us totake derivatives ldquothroughrdquo expectation opera-tors An example of a setting that fits our as-sumptions is the payoff structure

(1) Zi X Vi

where the project risk is independent of V andlog-normal with a mean of 15

The focus of this paper is on the case inwhich bids are securities Bidders compete forthe project by offering the seller a share of thefinal payoff That is the bids are in terms ofderivative securities in which the underlyingasset is the future payoff of the project Zi Bidscan be described as function S(z) indicating thepayment to the seller when the project has finalpayoff z We make the following assumptionsregarding the set of feasible bids6

DEFINITION A feasible security bid is de-scribed by a function S(z) such that S is weaklyincreasing z S(z) is weakly increasing and0 S(z) z

The set of feasible securities encompassesstandard designs used in practice However it isnot completely general First S(z) z can beviewed as a liquidity or limited liability con-straint for the bidder only the underlying assetcan be used to pay the seller We assume fornow that bidders do not have access to cash (orother liquid assets) that they can pledge as pay-ment they can transfer only property rights inthe project7 We make this assumption in orderto focus first on pure security bids and simplifythe exposition In Section IV we will generalizethe setting to allow for cash payments

Similarly S(z) 0 corresponds to a liquidityor limited liability constraint for the seller theseller cannot commit to pay the bidder exceptthrough a share of the project payoff For ex-

3 We use increasinghigherlower in the strict sense andexplicitly note weak rankings

4 Equivalently h is log-supermodular which can be writtenas (2zv)log h(zv) 0 assuming differentiability

5 More generally what is required for the SMLRP is thatlog() have a log-concave density function

6 These assumptions are typical of the security designliterature (eg Nachman and Noe 1994 Oliver Hart andJohn Moore 1995 DeMarzo and Duffie 1999)

7 Bidders can invest X in the project but X might corre-spond to an illiquid asset such as human capital


ample the seller may not have the financialresources to do so and is selling the project toraise X Because the seller cannot reimburse thebidder for the upfront investment this assump-tion rules out a solution as in Cremer (1987)We take this constraint as given for now but weshow in Section IV that this constraint can fol-low from an assumption that the bidderrsquos invest-ment X is not verifiable

Finally we require both the sellerrsquos and thebidderrsquos payment to be weakly increasing in thepayoff of the project Monotonicity is a standardfeature of almost all securities used in practiceand so is a natural constraint to consider8 Mostimportantly without monotonicity for the bid-ders equilibria would not be efficient and with-out monotonicity for the seller the seller wouldhave incentives to choose other than the highestbid

Together these requirements are equivalentto S(0) 0 S is continuous and S(z) [0 1]almost everywhere Thus we admit standardsets of securities including

(a) Equity The seller receives some fraction [0 1] of the payoff S(z) z

(b) Debt The seller is promised a face valued 0 secured by the project S(z) min(z d)

(c) Convertible debt The seller is promised aface value d 0 secured by the project ora fraction [0 1] of the payoff S(z) max(z min(z d)) (This is equivalent to adebt plus royalty rate contract)

(d) Levered equity The seller receives a frac-tion [0 1] of the payoff after debt withface value d 0 is paid S(z) max(z d 0) (This is equivalent to a royalty agree-ment in which the bidder recoups somecosts upfront)

(e) Call option The seller receives a call optionon the firm with strike price k S(z)

max(z k 0) Higher bids correspond tolower strike prices (This is equivalent tothe bidder retaining a debt claim)

Given any security S we let

ESv ESZiVi v

denote the excepted payoff of security S condi-tional on the bidder having value Vi v Thusthe expected payoff to the seller if the bid S isaccepted from bidder i is ES(Vi) On the otherhand the bidderrsquos expected payoff is given byVi ES(Vi) Thus we can interpret Vi as theindependent private value for bidder i andES(Vi) as the payment offered The key differ-ence from a standard auction of course is thatthe seller does not know the value of the bidsbut only the security S The seller must infer thevalue of this security Since the security S ismonotone the value of the security is increasingwith the signal Vi of the bidder as we showbelow

LEMMA 1 The value of the security ES(v) istwice differentiable For S 0 ES(v) 0 andfor S Z ES(v) 1

II Formal Auctions with Ordered Securities

In many auctions bidders compete by offer-ing ldquomorerdquo of a certain security For examplethey compete by offering more debt or moreequity We begin our analysis by examiningformal auctions in which the seller restricts thebids to elements of a well-ordered set of secu-rities Bidders compete by offering a highersecurity

There are two main reasons why sellers re-strict the set of securities that are admissible asbids in the auction First it allows them touse standard auction formatsmdashsuch as first- orsecond-pricemdashto allocate the object and to de-termine the payments Without an imposedstructure ranking different securities is difficultand depends on the beliefs of the seller There isno clear notion of the ldquohighestrdquo bid

The second reason a seller may want to re-strict the set of securities is that it can enhancerevenues We will demonstrate this result byfirst (in this section) studying the revenues fromauctions with ordered sets of securities and then(in Section III) comparing them to the revenues

8 A standard motivation for dual monotonicity is that ifit did not hold parties would ldquosabotagerdquo the project anddestroy output (Dual monotonicity is also implied if oneparty could both destroy and artificially inflate output egif S(z0) S(z1) for z0 z1 the bidder could inflate the cashflows from z0 to z1 via a short-term loan to get payoff z0 S(z1)) Whether revenues can be distorted in this way de-pends on the context We do not further defend this assump-tion here but point out that it is a standard one includestypical securities used in practice and guarantees a well-behaved equilibrium


from auctions in which the seller cannot committo a restricted set and bidders can bid using anyfeasible security

Example 1 Comparison of Revenues acrossSecurities and Auction FormatsmdashBefore pre-senting the technical details of the analysis weconsider an example that illustrates our mainresults Two bidders compete for a project thatrequires an upfront investment of X 100 TheNPV of the project if run by bidder i is Viwhere Vi is uniform on the interval [20 110]The project is risky with final value Zi which islognormal with mean X 13 Vi and volatility of 50percent

Total surplus is maximized by allocating theproject to the highest type in this case leadingto an expected value of E[max(V1 V2)] 80This is the maximum expected revenue achiev-able by any mechanism On the other handusing a cash auction the expected revenue isgiven by E[min(V1 V2)] 50 (which is thesame for first- and second-price auctions byrevenue equivalence) Table 1 shows the reve-nues for different security designs and auctionformats

Several observations can be made which co-incide with our main results of this section

(a) Fixing the auction format (first- or second-price) revenues increase moving from debtto equity to call options In Section II B wewill define a notion of ldquosteepnessrdquo for se-curities and show that steeper securities leadto higher revenues and that all securitydesigns yield higher revenues than cashauctions

(b) The auction format is irrelevant for a cashauction and for an equity auction While theformat does make a difference for debt andcall options the rankings are reversed In Sec-tion II C we will generalize these observations

and show precisely when revenue equiva-lence will hold or fail Overall though theimpact of the auction format on revenues isminor compared to the security design

(c) Among the mechanisms examined the first-price auction with debt yields the lowestexpected revenues while the first-price auc-tion with call options yields the highestexpected revenues In Section II D we shallsee that these are the worst and best possiblemechanisms in a broad class of security-bidauctions and that all security-bid auctionsdominate cash auctions

A Securities Auctions and Mechanisms

The first step in our analysis is to formalizethe notion of an ordered set of securities Anordered collection of securities can be definedby a function S(s z) where s [s0 s1] is theindex of the security and S(s ) is a feasiblesecurity That is S(s z) is the payment ofsecurity s when the output of the project hasvalue z As before we define ES(s v) E[S(sZi)Vi v]

For the collection of securities to be orderedwe require that its value for any type is in-creasing in s Then a bid of s dominates a bid ofs if s s We would also like to allow for asufficient range of bids so that for the lowestbid every bidder earns a nonnegative profitwhile for the highest bid no bidder earns apositive profit This leads to the following for-mal requirements for an ordered set ofsecurities

DEFINITION The function S(s z) for s [s0s1] defines an ordered set of securities if

(a) S(s ) is a feasible security(b) For all v ES1(s v) 0(c) ES(s0 vL) vL and ES(s1 vH) vH

Examples of ordered sets include the sets of(levered) equity and (convertible) debt indexedby the equity share or debt amount and calloptions indexed by the strike price Given anordered set of securities it is straightforward togeneralize the standard definitions of a first- andsecond-price auction to our setting

First-Price Auction Each agent submits a se-curity The bidder who submitted the highest

TABLE 1mdashEXPECTED REVENUES FOR DIFFERENT SECURITY

DESIGNS AND AUCTION FORMATS IN EXAMPLE 1

Expected seller revenues

Security type First-price auction Second-price auction

Cash 5000 5000Debt 5005 5014Equity 5865 5865Call option 7453 7449


security (highest s) wins and pays according tohis security

Second-Price Auction Each agent submits asecurity The bidder who submitted the highestsecurity (highest s) wins and pays the second-highest security (second-highest s)9

Are the equilibria in these auction formatsefficient That is does the highest value bidderwin the auction For second-price auctions theanswer is straightforward the standard charac-terization of the second-price auction with pri-vate values generalizes to

LEMMA 2 The unique equilibrium in weaklyundominated strategies in the second-price auc-tion is for a bidder i who has value Vi v tosubmit security s(v) such that ES(s(v) v) vThe equilibrium strategy s(v) is increasing in v

In words similarly to a standard second-priceauction each bidder submits bids according tohis true value We now turn our attention to thefirst-price auction Incentive compatibility inthe first-price auction implies that no biddergains by mimicking another type so that s(v)satisfies

(2) Uv maxvFn 1vv ESsv v

Fn 1vv ESsv v

where U(v) is the expected payoff of type v Thefirst-order condition of (2) then leads to a dif-ferential equation for s However an additionalassumption is required to guarantee optimality

ASSUMPTION C For all (s v) such that thebidder earns a positive expected profit iev ES(s v) 0 the profit function is log-supermodular

2

vslogv ESs v 0

Under this assumption we have the followingcharacterization

LEMMA 3 There exists a unique symmetricequilibrium for the first-price auction It is in-creasing differentiable and it is the uniquesolution to the following differential equation

sv n 1fv

Fv

v ESsv v

ES1sv v

together with the boundary condition ES(s(vL)vL) vL

Assumption C is a joint restriction on the setof securities and the conditional distribution ofZ10 It can be shown to hold generally in thelognormal setting (1) in the case of debt equityand levered equity securities with d X It canbe established numerically for other types ofsecurities such as call options under suitableparameter restrictions for example it holds inthe numerical example computed earlierThroughout our analysis we assume that itholds for all sets of securities underconsideration

The first- and second-price auctions are twostandard auction mechanisms They share thefeatures that the highest bid wins and only thewinner pays The first property is necessary forefficiency and the second is natural in our set-ting since only the winner can use the assets ofthe project to collateralize the payment One canconstruct many other auction mechanismshowever that share these properties For exam-ple one can consider third-price auctions orauctions where the winner pays an averageof the bids etc Below we define a broad classof mechanisms that will encompass theseexamples

DEFINITION A General Symmetric Mecha-nism (GSM) is a symmetric incentive compati-ble mechanism in which the highest type winsand pays a security chosen at random from agiven set S The randomization can depend onthe realization of types but not on the identity ofthe bidders (so as to be symmetric)

9 Note that with private values the second-price auctionis equivalent to an English auction

10 Assumption C is the same condition imposed on util-ity functions in the auction literature eg Eric S Maskinand Riley (1984) use it to show existence and uniqueness ofequilibria with risk-averse bidders The requirements ofsymmetry and Assumption C underscore the fact that effi-ciency is more fragile in first-price than in second-priceauctions


The first-price auction fits this descriptionwith no randomization (the security is a func-tion of the winnerrsquos type) In the second-priceauction the security paid depends upon the re-alization of the second-highest type GSMs alsoallow for more complicated payment schemesthat depend on all of the bids

It will be useful in what follows to derive abasic characterization of the incentive compat-ibility condition for a GSM We show that anyGSM can be converted into an equivalent mech-anism in which the winner pays a security thatdepends only on his reported type without fur-ther randomization

LEMMA 4 Incentive compatibility in a GSMimplies the existence of securities Sv in theconvex hull of S such that11

v arg maxvFn 1vv ESv v

Thus it is equivalent to a GSM in which thewinner pays the non-random security Sv

This observation will allow us to compare rev-enues across mechanisms by studying the rela-tionship between the set of securities S and itsconvex hull

B Ranking Security Designs

Recall from Table 1 that the sellerrsquos revenuesvaried greatly with the security design As wewill show the revenues of different designsdepend upon the steepness of the securities Todo so we need to formalize the notion of steep-ness of a set of securities A simple comparisonof the slopes of the securities is inadequatecomparing debt and equity debt has higherslope for low cash flows and lower slope forhigh cash flows Rather our notion of steepnessis defined by how securities cross each otherIntuitively one security is steeper than anotherif it crosses that security from below Thus weintroduce the following definition

DEFINITION Security S1 strictly crosses se-curity S2 from below if ES1(v) ES2(v)implies ES1(v) ES2(v) An ordered set of

securities S1 is steeper than an ordered set S2 iffor all S1 S1 and S2 S2 S1 strictly crossesS2 from below

The following useful lemma shows thatsteepness is naturally related to the shape of theunderlying securitiesmdashif the payoffs of the se-curities cross from below then their expectedpayoffs strictly cross

LEMMA 5 (Single crossing) A sufficient con-dition for S1 to strictly cross S2 from below isthat S1 S2 and there exists z such thatS1(z) S2(z) for z z and S1(z) S2(z) forz z

Comparing standard securities note that a calloption is steeper than equity which in turn issteeper than debt See Figure 1

Why is steepness related to auction revenuesConsider a second-price auction where the win-ning bidder with type V1 pays the security bidby the second highest type V2 That is thewinner pays ES(s(V2) V1) Since bidders bidtheir reservation value in a second-price auc-tion ES(s(V2) V2) V2 Hence the securitydesign impacts revenues only through the dif-ference

ESsV2 V1 ESsV2 V2

which is just the sensitivity of the security to thetrue type So to compare two securities we needto compare their slopes where the expected pay-ments are the same and equal to V2 By ourdefinition steeper securities are more sensitive

11 S is in the convex hull of S if S(z) yenk kSk(z) for allz with k 0 Sk S and yenk k 1

FIGURE 1 PAYOFF DIAGRAMS FOR CALL OPTIONS EQUITYAND DEBT


at the crossing point and so lead to higherrevenues

More generally steepness enhances competi-tion between bidders since even with the samebid a higher type will pay more This is theessence of the Linkage Principle first used byMilgrom and Weber (1982) to rank auctionformats for cash auctions when types are affil-iated12 Applying the envelope theorem to theincentive condition (2) for a first-price auctionwe get

(3) Uv Fn 1v1 ES2sv v

To compare expected revenues in (efficient)auctions with different sets of securities it issufficient to rank expected payoffs of bidders IfES2(s v) are ranked for every s and v thenU(v) are ranked and hence so are U(v) (usingthe boundary condition U(vL) 0) In generalwe cannot rank ES2(s v) everywhere but it issufficient to rank ES2(s v) at securities with thesame expected payments Our definition ofsteepness again captures that ranking Thisleads to the following main result

PROPOSITION 1 Suppose the ordered set ofsecurities S1 is steeper than S2 Then for eithera first or second-price auction for any realiza-tion of types the sellerrsquos revenues are higherusing S1 than using S2

As a result flat securities like debt lead tolow expected revenues and steep securities likecall options lead to high expected revenues Infact since debt and call options are the flattest13

and steepest possible securities they representthe worst and best designs for the seller We canalso extend the logic of Proposition 1 to cashauctions as a cash bid is flatter than any secu-rity Thus we have the following

COROLLARY For a first- or second-priceauction standard debt yields the lowest possi-

ble revenues and call options yield the highestpossible revenues of any security-bid auctionAll security-bid auctions yield higher revenuesthan a cash auction

Note that in all cases these rankings are for anyrealization of types and hence are stronger thanthe usual comparison based on an expectationover types

C Ranking Auction Formats

In our setting of symmetric independentprivate values and risk neutrality a well-known and important result for cash auctionsis the Revenue Equivalence Principle Itimplies that the choice of the auction for-mat is irrelevant when the ultimate alloca-tion is efficient14 We now turn to examiningthe revenue consequences of the choice ofauction format in a security-bid auction Aswe have seen from the numerical analysisof Example 1 in Table 1 revenue equiva-lence seems to hold for some security de-signs but not for others To develop somefurther intuition we begin with two simpleexamples

Example 2 Equity Auctions and RevenueEquivalencemdashThere are two bidders with in-dependent types Vi distributed uniformly on[0 1] Upfront investment is X 1 Thedistribution of Zi has full support with meanX 13 Vi

Consider a second-price equity auction Aswe know it is a dominant strategy to bid thereservation value SPA(v) [v(v 13 1)]which is increasing in v In a first-price auc-tion it is an equilibrium strategy for agents tobid FPA(v) 1 [ln(1 13 v)v] which isalso increasing15

Now observe that both auctions yield equalpayoffs to the auctioneer as in both auctionformats the highest type wins and the averagelosing bid in a second-price auction equals thehighest bid in the first-price auction

12 See also Vijay Krishna (2002) The linkage principleis typically used to compare formats when signals are affil-iated In security-bid auctions unlike cash auctions evenwith independent types the winnerrsquos expected payment de-pends on his true type as pointed out by Riley (1988) in thecontext of royalty rates

13 By flattest we mean that all other sets of securities aresteeper

14 William Vickrey (1961) Roger B Myerson (1981)Riley and William F Samuelson (1981)

15 The payoff of a type v who pretends to be v isv[ln(1 13 v)v](1 13 v) 1 (1 13 v)ln(1 13 v) vwhich is maximized by setting v v verifying anequilibrium


ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1

While in this example revenue equivalenceholds as the next example shows this is not thecase for all securities

Example 3 Second-Price Auctions YieldHigher Revenues for DebtmdashConsider a debtauction There are two bidders types Vi areindependent and uniform on [0 1] X 0 andthe distribution of Zi given Vi is uniform on [02Vi]

16 If a bidder wins and pays according to adebt bid with face value b the payoff to theseller is min(b z) which for a type v yields onaverage

(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v

In a second-price auction it is an equilibriumstrategy for agents to bid their reservation val-ues bSPA(v) 2v In a first-price auction it is anequilibrium strategy to bid bFPA(v) 2frasl3 v17

Suppose bidder 1 wins the auction In a first-price auction his payoff is

2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1

while in a second-price auction his payoff is

E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1

We conclude that biddersrsquo welfare is higher inthe first-price auction and since in both auctionformats the highest type wins revenues arehigher in the second-price auction Thus reve-nue equivalence fails

To gain some insight into why revenueequivalence fails note that in the second-priceauction the winner pays a random security (de-termined by the second highest bid) The win-nerrsquos payment is equivalent to paying theldquoexpected securityrdquo

Sv1 z Eminz 2V2V2 v1 z z2

4v1

in a first-price auction18 This security is not adebt security and therefore is steeper thandebt19 As a result of this steepness the sellerrsquosrevenues are enhanced On the other hand inthe case of equity a convex combination ofsecurities is also an equity security Thus thereis no change in steepness and so no change inrevenues

Sub- and Super-Convex Sets of SecuritiesmdashThe previous examples suggest that the revenuedifferences across auction formats will stemfrom the differences in steepness between theset of securities and its convex hull This moti-vates the following formal classification

DEFINITION An ordered set of securities S S(s ) s [s0 s1] is super-convex if it issteeper than any nontrivial convex combinationof the securities in S It is sub-convex if anynontrivial convex combination of the securitiesin S is steeper than S20

16 While this example violates some of our technicalassumptions (X 0 and Z has full support) it provides asimple closed-form solution (and suggests that our resultsare more general) We thank a referee for this example

17 To verify that this is an equilibrium note from (4) thatthe payoff of a type v who pretends to be v is v[(2v 2v3)24v] which is maximized for v v

18 In this example the support of Z1 is bounded by 2v1 so the security is monotone Note that the bidderrsquos expectedpayoff with this security is E[Z1 Sv1

(Z1)V1 v1] E[Z1

24v1V1 v1] v13 as before19 In general a convex combination of debt securities is

steeper than debt For example a 50-50 mix of debt withface value 50 and face value 100 has slope 1frasl2 for z (50100) and so crosses debt from below

20 A nontrivial convex combination puts positive weighton more than one security


Not every set falls into one of the categoriesabove Still there are some important examplesof sub- and super-convex sets

LEMMA 6 The set of standard debt contractsis sub-convex The set of convertible debt con-tracts indexed by the equity share the set oflevered equity contracts indexed by leverageand call options are super-convex sets

Based on the characterization above we canagain use the Linkage Principle to rank theexpected revenues of first- and second-priceauctions Here the proof relies on Lemma 4which allows us to interpret the second-priceformat as a first-price mechanism in the convexhull of the set of securities

PROPOSITION 2 If the ordered set of secu-rities is sub-convex then the first-price auctionyields lower expected revenues than the second-price auction If the ordered set of securities issuper-convex the first-price auction yieldshigher expected revenues than the second-priceauction (This revenue comparison also holdsconditional on the winnerrsquos type for all but thelowest type)

One subtlety in the proof of Proposition 2 isthat the security paid by the lowest type is thesame for both auction formats (and is defined bythe zero-profit condition) Thus neither formatemploys a ldquosteeperrdquo security for that type Weget around this problem by slightly perturbingthe support of the types for one of the auctionformats comparing revenues and taking thelimit

Proposition 2 reveals that the auction formatcan have an impact on revenues As we haveseen however this revenue impact stems fromthe difference in steepness between the set ofsecurities and their convex hull This differenceis always less extreme than the difference insteepness that can be obtained by changing thesecurity design directly In that sense the designof the securities is much more important thanthe design of the auction format in determiningrevenues

Revenue Equivalence for Convex Sets of Se-curitiesmdashWhile revenue equivalence does nothold for general security auctions it does holdfor cash and holds for equity in our examples

Here we ask whether it can be recovered forsome classes of securitiesmdashthat is what is spe-cial about cash

From Proposition 2 revenue equivalencefails in one direction for a super-convex set andin the opposite direction for a sub-convex setHence a natural candidate is a set in the middleie a convex set

DEFINITION An ordered set of securities S isconvex if it is equal to its convex hull

In fact convex sets of securities have a sim-ple characterizationmdasheach security is a convexcombination of the lowest security s0 and thehighest security s1 Thus each security can bethought of as s0 plus some ldquoequity sharesrdquo of thesecurity (s1 s0) and so it can be thought of asa generalization of a standard equity auctionOur main result in this section is that underconvexity the Revenue Equivalence holds

PROPOSITION 3 (Revenue equivalence) Ev-ery efficient equilibrium of a GSM with securi-ties from an ordered convex set yields the sameexpected revenues (This equivalence also holdsconditional on the winnerrsquos type)

Note that this is a stronger statement thanequivalence between a first- and second-priceauction as it holds for any symmetric mecha-nism Also note that the standard envelope ar-gument behind Revenue Equivalence does notextend directly to security auctions For cashthere is no linkage between the true type and thebidderrsquos expected payment when types are in-dependent so revenues depend only upon theallocation21 That is not the case with securitybids as we have seen However when the se-curity set is convex because paying a randomsecurity is equivalent to paying the expectedsecurity drawn from the same set the expectedlinkage across all mechanisms is identical

D Best and Worst Mechanisms

We can combine the results of the previoustwo sections to determine the best and worstsecurity design and format combinations Note

21 That is in the case of cash auctions (3) reduces toU(v) F n1(v)


that since debt is a sub-convex set from Prop-osition 2 the first-price auction is inferior to thesecond-price auction and conversely for calloptions which are super-convex The followingproposition establishes that a first-price auctionwith debt and with call options provides lowerand upper bounds for the sellerrsquos revenue acrossa broad class of mechanisms

PROPOSITION 4 A first-price auction withcall options yields the highest expected reve-nues among all general symmetric mechanismsA first-price auction with standard debt yieldsthe lowest expected revenues among all generalsymmetric mechanisms

Proposition 4 establishes that the design of thesecurity is more important than the specific auc-tion format the revenue consequences of shift-ing from debt to call options in a first-priceauction exceeds the consequences of anychange in the auction mechanism

We remark that Proposition 4 is stated withrespect to the particular set of feasible securitieswe have allowed thus far It can be extended inthe obvious way for any feasible set if there isa steepest set of securities which is (super-)convex then a (first-price) auction using this setyields the highest possible revenues Similarlyif there is a flattest set which is (sub-)convexthen a (first-price) auction using this set yieldsthe lowest possible revenues

For example if bidders can pay cash a cashauction is the worst possible auction for theseller This is because cash which is insensitiveto type is even flatter than standard debt secu-rities Alternatively the seller may be able toincrease revenues by using securities that areeven more leveraged than call options For ex-ample the seller might pay the bidder cash foradditional equity (See Section IV A for a fur-ther discussion of this case)

III Informal Auctions The Signaling Game

In the previous section we considered formalauctions in which bidders are restricted tochoose securities from a specific well-orderedset In reality there is often no such restrictionThat is the seller is unable to commit to ignoreoffers that are outside the set As a result theseller will consider all bids choosing the mostattractive bid ex post In this case the ldquosecurity

designrdquo is in the hands of the bidders who canchoose to bid using any feasible security

Without the structure of a well-ordered setonce the bids are submitted there is no obviousnotion of a ldquohighestrdquo bid In this case the sellerfaces the task of choosing one of the submittedbids Since there is no ex ante commitment bythe seller to a decision rule the seller willchoose the winning bid that offers the highestexpected payoff Since the payoff of the secu-rity depends on the bidderrsquos type the sellerrsquoschoice may depend upon his beliefs regardingthe bid each type submits in equilibrium Thusthis setting has the features of a signaling gamethat takes the form

(a) Bidders submit simultaneous bids that arefeasible securities

(b) The seller chooses the winning bid(c) The winner pays his bid and runs the

project

We consider a sequential equilibrium of thisgame We argue that in the informal auctionsbidders will choose the flattest securities possi-ble That is they will bid with cash if it isfeasible otherwise they will bid with debtThus Proposition 4 implies that the sellerrsquosexpected revenues are the lowest possible fromany general auction mechanism

To gain some insight consider first a case inwhich bidders can use cash We argue that inequilibrium bidders use only cash The intuitionfor this result is as follows Let Sv be the secu-rity bid by type v When a bidder of type vdecides on his bid he has the option to mimicother types In particular he can mimic a typev v dv just below him Such a deviationhas two effects First it will reduce his proba-bility of winning to that of type v Second itwill lower his expected payment if he wins fromESv(v) to ESv(v) On the margin these twoeffects must balance out (otherwise there is aprofitable deviation)

But now suppose types just below v use se-curity bids rather than cash Consider the devi-ation by type v to a cash bid of amount b(v) ESv(v) Since the seller values this cash bidthe same as the bid Sv by type v the marginaleffect on the probability of winning is the sameas if he deviates to Sv However because theexpected value of the security is increasing inthe true type the expected payment is b(v)


ESv(v) ESv(v) Thus if type v is indifferentto a deviation using securities he will profitfrom a deviation using cash As a result anequilibrium will involve only cash bids

The second step of the logic above can beapplied even when cash bids are not availableWhen mimicking a lower type it is cheaper fora higher type to use a security that is less sen-sitive to the true typemdashie a flatter securitymdashthan that used in the proposed equilibrium Thisreasoning suggests an equilibrium will involvethe flattest securities available

However there is a difficulty with extendingthis result when cash is not available The gainfrom a deviation depends crucially on the sell-errsquos evaluation of the bid The argument wegave above is simplified by the fact that thevalue of a cash bid is unambiguous the sellerdoes not need to rely on his beliefs But whenbidders do not use cash the value of any off-equilibrium bid depends on the sellerrsquos off-equilibrium beliefs As with general signalinggames there are many equilibria of this game ifwe do not impose any restrictions on the beliefsof the seller when an ldquounexpectedrdquo bid is ob-served22 We turn to such restrictions next

A Refining Beliefs The D1 Criterion

To rule out equilibria supported with arbi-trary off-equilibrium beliefs the standard re-finement in the signaling literature is the notionof strategic stability introduced by Elon Kohl-berg and Jean-Francois Mertens (1986) For ourpurposes a weaker refinement known as D1 issufficient to identify a unique equilibrium23

The D1 refinement (see In-Koo Cho and DavidM Kreps 1987 Cho and Joel Sobel 1990) is arefinement commonly used in the security de-sign literature24 Intuitively the D1 refinementcriterion requires that if the seller observes anout-of-equilibrium bid the seller should believe

the bid came from the type ldquomost eagerrdquo tomake the deviation

In order to define the D1 criterion in ourcontext we introduce the following notationFirst let Si be the random variable representingthe security bid by bidder i which will dependon Vi For any feasible security S let Ri(S) bethe scoring rule assigned by the seller repre-senting the expected revenues the seller antici-pates from that security given his beliefsAlong the equilibrium path the sellerrsquos beliefsare correct so that the scoring rule satisfies

(5) RiS EESViSi S

Given the sellerrsquos scoring rule Ri it mustalso be the case in equilibrium that bidders arebidding optimally That is conditional on Vi v Si solves

(6) Uiv maxSPiRiSv ESv

where Pi(r) is the probability that r is the high-est score25 Thus Ui(v) is the equilibrium ex-pected payoff for bidder i with type v

Suppose the seller observes an out-of-equilibrium bid so that the score is not deter-mined by (5) Which types would be most likelyto gain from such a bid For each type v we candetermine the minimum probability of winningBi(S v) that would make bidding S attractive

BiS v minp pv ESv Uiv

Then the D1 criterion requires that the sellerbelieve that a deviation to security S came fromthe types which would find S attractive for thelowest probability of winning26

(7) RiS ESarg minvBiS v

Thus a sequential equilibrium satisfying theD1 criterion for the auction game can be de-scribed by scoring rules Ri and bidding strategySi for all i satisfying (5)ndash(7)

22 If the seller believes all off-equilibrium bids are madeby the lowest type the gain from deviating is minimizedThese beliefs seem unreasonable however since many se-curities would be unprofitable for the lowest type

23 Strictly speaking D1 is defined for discrete typespaces However it can be naturally extended to continuoustypes (see eg Garey Ramey 1996)

24 See eg Nachman and Noe (1994) and DeMarzo andDuffie (1999)

25 If there are ties we require that Pi be consistent withsome tie-breaking rule

26 We have economized on notation here If the set ofminimizers is not unique the score is in the convex hull ofES(v) for v in the set of minimizers


B Equilibrium Characterization

Using the D1 refinement we can now extendthe argument we made for cash deviations toother securities Suppose type v mimics typev v dv The cost of doing so for type v isESv(v) Now suppose v instead deviates to asecurity S that is flatter than Sv and such thatES(v) ESv(v) Because the security is flat-ter it has a lower cost for type v ES(v) ESv(v) How would the seller respond to thedeviation S

Because S is flatter than Sv it is a moreexpensive security for types below v andcheaper for types above v than Sv Thereforethe types that are ldquomost eagerrdquo to deviate to Smust be above v By D1 this implies that theseller will evaluate S as at least as valuable asSv Therefore if type v is indifferent to a de-viation to Sv he will profit from a deviation toa flatter security S As a result an equilibriumwill involve only the flattest possible securities

We now proceed with a formal statement ofour results As is standard in the auction settingwe will focus on symmetric equilibria27 Wemaintain Assumption C for the flattest securi-ties so that existence of an efficient equilibriumof the first-price auction is assured Then wehave

PROPOSITION 5 Given symmetric strategiesthere is a unique equilibrium of the informalauction satisfying D1 This equilibrium is equiv-alent in both payoffs and strategies to theequilibrium of a first-price auction in whichplayers bid with the flattest securities feasibleIn particular if they can bid with cash they willuse only cash if cash is not feasible they willbid with standard debt contracts

Again we can now combine this result withthe result of the previous section to formalizethe value of the sellerrsquos ability to commit to arestricted set of securities

COROLLARY If the seller can commit to aformal auction with an ordered set of securities

other than debt contracts then expected reve-nues are higher than without such commitment

PROOFFollows immediately from Proposition 4 and

Proposition 5

IV Extensions

A Relaxing the Liquidity Constraints

We have assumed that both the seller and thebidders are liquidity constrained We now ex-plore implications of either the seller or biddershaving access to cash

Moral Hazard Non-Contractible Invest-mentmdashThe main setting for our model is thecase in which the seller is liquidity constrainedindeed we can interpret X as an amount of cashthe seller must raise But if X corresponds toresources required for the project and if theseller has surplus cash securities in which theseller reimburses the winner for a portion of theinitial investment (and thus have S(0) 0) arefeasible Importantly these securities can besteeper than call options and so increase reve-nues For example the seller could auction offthe rights to a fraction of the cash flows andreimburse the winner directly for the investment(1 )X By making arbitrarily small theseller can extract the entire surplus While thistheoretical mechanism was proposed by Cremer(1987) it is not observed in practice A likelyreason is moral hazard if the winnerrsquos invest-ment is not fully contractible and if the winnerreceives only a small fraction of future reve-nues then he may underinvest28

For example suppose that after the auction thewinning bidder i can choose whether to invest XIf X is invested the payoff of the project is Zi asbefore and his payment to the seller is S(Zi) If Xis not invested the payoff is 0 and his payment tothe seller is S(0) If S(0) 0 the bidderrsquos payoffis nonpositive without investment and so the op-

27 That is we restrict attention to equilibria in which thebidders use symmetric strategies and the seller uses thesame scoring rule for all players

28 For example in several oil lease auctions run by theUS Department of the Interior in which the bidders bidhigh royalty rates the oil fields were left undeveloped (andthe government received almost no revenues) because bid-ders did not capture enough of the revenues to warrant theirprivate investment (see Ken Binmore and Paul Klemperer2002)


tion not to invest is irrelevant But suppose a bidwith S(0) 0 is accepted with positive probabil-ity Then every bidder including the lowest typecan earn positive profits by making such a bid andnot investing Yet if bidders do not invest andS(0) 0 the seller loses money As a result theseller would choose not to accept such securitiesas shown below

PROPOSITION 6 Suppose that the seller isnot liquidity constrained and the investment X isnot contractible Then

(a) In a first- and second-price formal auction(with an ordered set of securities)(i) If a security without reimbursement is

allowed then with probability 1 thewinning bid satisfies S(0) 0 That iscompetition between bidders rules outreimbursement

(ii) If all securities involve reimbursementthen all bidders bid the highest allowedsecurity and do not invest leading tonegative revenues for the seller

(b) Any mechanism in which bids with S(0) 0win with positive probability cannot beefficient

(c) In an informal auction securities withS(0) 0 are used with probability 0

Thus when X is not contractible even if theseller has cash we can rule out reimbursementfrom the seller it would either not occur inequilibrium or not be in the sellerrsquos best interest

Partial Cash BidsmdashSuppose bidders havecash equal to B where B is known and commonto all bidders29 Cash relaxes the limited liabil-ity restriction for bidders so that the flattestsecurities are now debt claims on the total assetsof the bidder (cash 13 project) defined by

SDd z mind B z

mind B mind B13 z

As the decomposition above reveals we canthink of this security as an immediate cashpayment (up to B) plus a standard debt claim on

the project (for amounts above B) Becausethese securities become flatter as B increasesthe sellerrsquos expected revenues decrease with BA pure cash auction yielding the lowest possi-ble revenues is possible for a second-price auc-tion if B exceeds vH and for a first-price auctionif B exceeds the expected maximum type forn 1 bidders (which is less than vH) Thus afirst-price auction yields lower revenues for theseller as long as B vH These results areconsistent with Board (2002) who considersdebt auctions and shows that they yield higherrevenues than cash auctions with the smallesteffect for first-price auctions

B Reservation Prices

In this section we discuss briefly how reserva-tion prices can be incorporated into our analysisCommitment to a reservation price can improvethe sellerrsquos revenues and even absent commit-ment a reservation price may be relevant if sellingthe project entails an opportunity cost

In the case of formal auctions we assumedearlier that the lowest security s0 was such thatall types earn nonnegative profits that is ES(s0vL) vL A reservation price is equivalent toassuming that s0 restricts that set of types thatcan profitably participate In particular if wechoose s0 so that

ESs0 vr vr

for some type vr [vL vH] then vr is thereservation price and types below vr will not beallocated the project All of our results general-ize to this case In case of informal auctionssince there is no commitment we interpret vr asthe outside opportunity for the seller

C Affiliated Private and Common Values

Our model thus far is based on the classicindependent private values framework We dis-cuss here how our results generalize when val-ues are affiliated (see Milgrom and Weber1982) and may have common as well as pri-vate components30 Formally we assume that

29 Che and Gale (2000) Zheng (2001) and Rhodes-Kropf and Viswanathan (2000) consider models where thecash amount is heterogeneous and privately known

30 Affiliation (essentially the log-supermodularity of thejoint density function) implies that ldquogood newsrdquo about oneof the variables (learning it lies in a higher interval) raises


ASSUMPTION D The private signals V (V1 Vn) and payoffs Z (Z1 Zn) satisfythe following properties

(a) The private signals Vi are affiliated anddistributed symmetrically with full support[vL vH]

(b) Conditional on V v the payoff Zi hasdensity h(zvi vi) with full support [0 )The distribution is symmetric in the last n 1 arguments

(c) (Z V) are strictly affiliated

First we consider formal auctions with a fixedauction format Under appropriate conditionsthere exists a unique symmetric increasing equi-librium for both the first- and the second-priceauction31 Given an efficient equilibrium wecan generalize our result regarding the impact ofthe security design on the sellerrsquos revenues32

Given a symmetric increasing equilibriumthen fixing the mechanism (first- or second-price) steeper sets of securities yield higherrevenues for the seller33

The intuition for this result is the same as be-foremdashsteeper securities increase the effectivecompetition between bidders since they aremore costly for higher types

What about the comparison of auction for-mats Here it is useful to consider first the caseof affiliated private values (ie h(zvi vi) doesnot depend on vi) In this setting revenueequivalence fails even for cash auctions asshown by Milgrom and Weber (1982) In oursetting

With affiliated private values for both convexand sub-convex sets of securities second-price auctions generate higher expected rev-enues than first-price auctions

This result again follows from the linkage princi-ple since the second-highest bid is affiliated withor ldquolinkedrdquo to the winnerrsquos value This linkagecreates an advantage for the second-price auction

On the other hand if there is a common valuecomponent to the asset this can have an oppos-ing effect regarding the optimal auction formatand create an advantage for a first-price auctionWe show this below for a case of independentsignals and common value

Suppose Vi are independent and E[ZiV] yenj Vj

34 Then for an equity auction thefirst-price auction generates higher expectedrevenues than a second-price auction

The intuition for this result is that since theequity-share is increasing with the bidderrsquostype and therefore correlated with the assetrsquosvalue to generate the same expected revenuesthe expected equity-share in the second-priceauction is lower than in the first-price auctionBut the lower average equity share reduces thelinkage to the winnerrsquos own type reducing rev-enues in the second-price auction

Finally consider the setting of an informalauction With affiliated private values our con-clusions regarding the informal auction holdmdashhigh types prefer to use flat securities toseparate from lower types

In the unique D1 equilibrium bidders use theflattest possible securities leading to the low-est revenues for the seller35

With common values the signaling gamebecomes much more complex Now after ob-serving the bids the seller is potentially moreinformed than the bidder Thus the bidderfaces an adverse selection problem This ad-verse selection is mitigated by making thesellerrsquos payoff sensitive to revenues poten-tially leading bidders to bid using steepersecurities We leave the analysis of this casefor future research

the expectation of any monotone function of the variablesWith two random variables it is equivalent to the MLRP

31 For a second-price auction we do not need extra con-ditions For a first-price auction we again need log-super-modularity of the winnerrsquos profit which becomes morecomplicated in this case

32 See Appendix for proofs33 In this case the notion of steepness is that given in

Lemma 5 which combined with affiliation will imply thatsecurities strictly cross in terms of their expected costs

34 This is the so-called ldquoWallet Gamerdquo see Jeremy Bu-low and Klemperer (2002)

35 The proof is the same as that of Proposition V with aminor modification to step 3 Intuitively affiliation providesthe seller with information about the bidderrsquos type afterobserving other bids but this plays no role in a separatingequilibrium


V Conclusion

We have examined an aspect of bidding rel-atively ignored in auction theorymdashthe fact thatbiddersrsquo payments often depend on the realiza-tion of future cash flows This embeds a securitydesign problem within the auction setting Firstwe analyzed formal auctions in which the sellerchooses the security design and restricts biddersto bid only using securities in an ordered setThis enables a simple ranking of the securitiesand the use of standard auction formats Weshowed conditions for which Revenue Equiva-lence holds and determined the optimal andworst format and security design combinationsIn particular we showed that revenues are in-creasing in the steepness of the securities anddemonstrated that the first-price debt auctionyields the lowest revenues whereas a first-priceauction with call options yields the highestrevenues across a broad class of possiblemechanisms

Next we considered informal auctions inwhich the seller does not restrict the set ofsecurities or the mechanism ex ante butchooses the most attractive bid ex post In thiscase security design is in the hands of thebidders We show that this yields the lowestpossible expected revenues for the seller and isequivalent to a first-price auction using the flat-test feasible securities such as debt or cashThus there are strong incentives for the seller tobe actively involved in the auction design andselect the securities that can be used

Finally we generalized our results to includerelaxed liquidity constraints (incorporating as-pects of moral hazard) partial cash bids reser-vation prices and affiliated and commonvalues All of our main insights and results arerobust to these features

There are a number of natural extensions toour framework For example the role of thesecurity design often extends beyond the auc-tion to determine the winnerrsquos and sellerrsquos in-centives ex post While we discuss a simplemoral hazard setting in Section IV A moregeneral settings can be considered Some ofthese can be modeled as further restrictions onthe set of feasible securities For example con-sider the case in which the winner has the op-portunity to divert cash flows from the projectwith each dollar diverted generating a privatepayoff of 13 1 In this case by the usual

revelation principle argument we can restrictattention to securities that do not induce diver-sion ie such that 1 S(z) 13 Because thislimits the steepness of the security it lowers therevenues the seller can achieve using the opti-mal formal auction On the other hand it alsorules out debt So if bidders are cash con-strained the flattest possible securities have theform S(d z) min(d (1 13)z) Because thesesecurities are not as flat as standard debt therevenues of the seller are in fact enhanced bythis restriction36

More generally our analysis provides clearintuition for the way in which moral hazardconcerns will interact with competition and rev-enues in the auction For example if the reve-nues of the project are costly to verify we knowfrom Robert Townsend (1979) that it is optimalfor the party who observes the cash flows to bethe residual claimant Thus if the seller ob-serves the cash flows the optimal agency con-tract is a call option which also maximizes theauction revenues If the winner observes thecash flows there is a tradeoff between verifica-tion costs which are minimized with debt se-curities and auction revenues

For mergers and acquisitions tax implica-tions and accounting treatment are likely to beimportant For example the deferral of taxespossible with an equity-based transaction maygive rise to the use of equity bids even in anunrestricted setting Our results imply that thistax preference can also lead to enhanced reve-nues for the seller

It would be useful to allow for more compli-cated information structures For example bid-ders may have private information not onlyabout V but also about X or the seller may haveprivate information Another extension of ourmodel that would be useful in applicationswould be to allow for asymmetries in biddersrsquovaluations and costs One difficulty is that insuch a setting it may be impossible to ensure anefficient outcome (see for example recent work

36 Similarly if the seller can divert cash flows the con-straint becomes S(z) 13 which rules out cash or debt andenhances revenues in an informal auction Other moralhazard settings can also be considered If the winner can addarbitrary risk (see eg S Abraham Ravid and Matthew ISpiegel 1997) then bidders are restricted to using convexsecurities See also Samuelson (1987) for a discussion ofother potential constraints on the security design


by Sandro Brusco et al 2004 in the context ofmergers)

APPENDIX

Proofs of all technical lemmas are availablefrom the AER Web Appendix We provide herethe proofs of all propositions in the text

PROOF OF PROPOSITION 1The reasoning for the second-price auction

was provided in the text For the first-priceauction let Uj(v) be the equilibrium payoff fortype v and Sv

j the security bid with the set SjEfficiency implies U1(vL) U2(vL) 0 Bycondition (2) if U1(v) U2(v) we haveESv

1(v) ESv2(v) If S1 is steeper than S2 then

ESv1(v) ESv

2(v) Thus from the envelopecondition

U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v

Hence U1(v) U2(v) for v vL Since bid-dersrsquo payoffs are lower the sellerrsquos expectedrevenue is higher for each realization of thewinning type under S1

PROOF OF COROLLARY TOPROPOSITION 1

Since debt has slope 1 and then 0 it strictlycrosses any other nondebt security from aboveCall options have slope 0 and then 1 and sostrictly cross any other nonoption security frombelow The result then follows directly fromProposition 1

What about the comparison of debt securi-ties to other sets that may include debt Theproof for second-price auctions is unchangedBut for first-price auctions we have the minordifficulty that a debt security does not strictlycross another debt security In that case wemodify the set of debt securities by adding in cash ie the security payoff is S( z) min( z d) 13 This set strictly crosses anyother set from above and so the revenues canbe ranked as in Proposition 1 The result thenfollows from the continuity of the equilibriumstrategies and payoffs in the first-price auc-tion as we take the limit as 3 0 We can dothe same for call options by subtracting cash from each security

PROOF OF PROPOSITION 2Consider the direct revelation game corre-

sponding to the two auctions Let Sv1 be the

security bid in the first-price auction and let Sv2

be the expected security payment in the second-price auction for a winner with type v definedas in Lemma 4 Then if the set of securities issuper-convex Sv

1 crosses Sv2 from below and a

nearly identical argument to that used in theproof of Proposition 1 for first-price auctionscan be applied to prove that the sellerrsquos ex-pected revenues are higher in the first-priceauction

The only complication is that the securitiesissued by the lowest types are identical in thefirst- and second-price auctions so that thesecurities do not strictly cross and U1(vL) U2(vL) To resolve this we can change thesupport in the first-price auction to [vL 13 vH] with an atom at vL 13 with mass equalto that originally on the set [vL vL 13 ]Now U1(vL 13 ) 0 U2(vL 13 ) and bythe same argument as in the proof of Propo-sition 1 U1(v) U2(v) for all v vL 13 Then by the continuity of the strategies andpayoffs in the boundary of the support vL 13 the first-price auction has weakly higherrevenues However since the securitiesstrictly cross for higher types the revenuescannot be equal

The proof for sub-convex sets is identicalwith the inequalities reversed (and taking thelimit of the support for the second-priceauction)

PROOF OF PROPOSITION 3In a GSM the winner pays according to a

random security From Lemma 4 the expectedpayment by type v reporting v can be written asESv(v) where Sv is in the convex hull of theordered set of securities S Since S is convex wecan define s(v) such that

Ssv ) Sv ()

Because S is ordered incentive compatibilityimplies s(v) must be increasing otherwise abidder could raise the probability of winningwithout increasing the expected payment Thuss(v) defines an efficient equilibrium for thefirst-price auction The result then follows fromthe uniqueness of equilibrium in the first-priceauction


PROOF OF PROPOSITION 4The proof follows that of Proposition 2

except that instead of the second-price auc-tion we consider a general symmetric mech-anism over some subset of the feasiblesecurities The result follows as a call op-tion contract is steeper and a standard debtcontract is flatter than any convex combina-tion of feasible securities (where we use thesame trick as in the proof of the Corollaryto Proposition 1 if the sets of securitiesintersect)

PROOF OF PROPOSITION 5We focus on the no-cash case as the argu-

ment with cash is similar (and even simpler asoff-equilibrium beliefs do not play any majorrole) In step 1 we show existence of a debt-based equilibrium that survives the D1 refine-ment In steps 2 and 3 we show that no otherequilibrium exists In step 2 we prove thatany equilibrium is equivalent to the equilib-rium of a first-price auction in debt contractsIn step 3 we use the envelope condition toargue that the securities used in the originalequilibrium must have been debt as well

The proof differs somewhat from the intuitionin the text which considers deviations by type vto mimic type v v dv This ldquolocal devia-tionrdquo is more intuitive but is not precise sincetypes are not discrete and therefore there is nonearby type v that type v is truly indifferenttoward mimicking

Step 1 The equilibrium from a first-pricedebt auction is a D1 equilibrium in the unre-stricted auction (with appropriate off-equilibriumbeliefs)

We need to demonstrate that given the strate-gies from the debt auction there is a set ofbeliefs satisfying D1 that support this equilib-rium in the unrestricted auction We constructthe beliefs R using (5) and (7) If (7) does notproduce a unique score we can choose thelowest one We now show that this supports theequilibrium

Step 1a For debt contracts the score is in-creasing in the face value of the debt Thatis R(Sd) is increasing in d where Sd(z) min(d z)

Note that

arg minvBSd v arg maxv

v ESdv

Uv

Because the objective function is strictly log-supermodular by Assumption C from DonaldM Topkis (1978) we know that v is weaklyincreasing with d Thus (7) implies R(Sd) isincreasing in d

Step 1b R supports an equilibrium in the un-restricted auction

Consider any deviation to a debt contract FromStep 1a the probability of winning the auctionis the same as in a first-price auction Since wehave a first-price equilibrium there is no gain tothe deviation Consider a deviation to a nondebtcontract S To show it is not profitable for anytype we must show that P(R(S)) minvB(S v)Let v be the highest type in the set that mini-mizes B(S v) Then ES(v) R(S) It is suffi-cient to show that type v does not find thedeviation to S profitable ie to show thatP(R(S)) B(S v)

Find d such that ESd(v) ES(v) Then B(Sdv) B(S v) From Lemma 5 types v v findSd more expensive than S so that B(Sd v) B(S v) B(S v) Therefore arg minvB(Sdv) v and so from (7)

RSd ESdv RS

Thus if a deviation to S is profitable so is adeviation to Sd But this contradicts the fact thatno deviation to a debt contract is profitable

Step 2 A symmetric D1 equilibrium in theunrestricted auction has the same payoffs as theequilibrium of a first-price debt auction

Our method of proof is to show that any non-debt bids can be replaced with an equivalentdebt bid without changing the equilibrium

Step 2a If S is not a debt contract then at mostone type uses this security

Suppose not so that v1 v2 are the lowest andhighest types that use S Then by (5) R(S) ES(v) for some v1 v v2 Consider the


debt contract Sd with the same cost for type vie such that ESd(v) ES(v) From Lemma5 types v v find the Sd more expensive thanS so that B(Sd v) P(R(S)) Therefore argminv B(Sd v) v Thus from (7) R(Sd) ESd(v) R(S) But this contradicts an equilib-rium as type v2 finds Sd strictly cheaper than Swith a weakly higher score

Step 2b Suppose type v uses contract S and letd(v) be the debt level that for type v has equalcost ie ESd(v)(v) ES(v) Then bidding Sd(v)

has the same payoff as bidding S and so is alsooptimal for type v Because S is an equilibriumbid P(R(Sd(v))) P(R(S)) by (6) However bythe same argument used in the previous steplower types find Sd(v) more costly than S so thatB(Sd(v) v) P(R(S)) for v v HenceR(Sd(v)) ESd(v)(v) R(S) Thus P(R(Sd(v))) P(R(S)) The result follows since the cost andprobability of acceptance of bidding debt withface value d(v) and bidding S are the same fortype v

Step 2c d(v) is the unique symmetric equilib-rium for a first-price debt auction and it isincreasing From Step 2b bidding debt d(v) isoptimal and so solves

Uv PRSdvv ESdvv

maxdPRSdv ESdv

Using the same logic as in step 1a R(Sd) isincreasing in d Therefore this maximizationproblem is identical to the problem faced bybidders in a debt-only first-price auctionUniqueness and monotonicity follow fromLemma 3

Step 3 In a symmetric D1 equilibrium in theunrestricted auction almost every bid is a debtcontract From Step 2 and Lemma 3 the equi-librium payoff of type v in the first-price auctionwith debt is

Uv Fn 1vv ESdvv

and U is differentiable Suppose v (vL vH)bids S in equilibrium By a standard envelopeargument

Uv Fn 1v1 ESv

Thus ES(v) ESd(v)(v) and ES(v) ESd(v)(v) Therefore by Lemma 5 S Sd(v)

PROOF OF PROPOSITION 6

Case 1 First- and second-price auctions

Let s be a bid that wins with positive prob-ability such that S(s 0) 0 Due to themoral hazard problem submitting this bidearns strictly positive profits for any typesince any type can simply not invest andcollect S(s 0) in a first-price auction oreven more in a second-price auction (sincethe second highest bid is below s) Thus byincentive compatibility all equilibrium bidsearn positive profits

Let s be the lowest bid submitted Then theabove implies this bid must win with positiveprobability Since it is the lowest bid this im-plies a tiemdashthat is s is submitted with positiveprobability But then raising the bid slightlywould lead to a discrete jump in the probabilityof winning and hence in profits Incentive com-patibility therefore implies s s1 the highestpossible bid If S(s1 0) 0 this contradicts theexistence of s If S(s1 0) 0 then all types bids1 But at s1 all types lose money if they run theproject Therefore all types bid s1 do not in-vest and collect S(s1 0) 0 from the seller

Case 2 General mechanisms

In an efficient mechanism the lowest type winswith zero probability and so earns zero expectedprofits Since lowest type can claim to be anytype and not invest it must be the case that notype with positive probability of winning pays asecurity with S(0) 0 with positive probability

Case 3 Informal auctions

Here the result follows immediately from ourresult that an equilibrium will always use theflattest possible securities If S(0) 0 we canldquoflattenrdquo the security by raising S(0) and flat-tening it elsewhere

PROOF OF RESULTS IN SECTION IV C

Result 1 Consider a first-price auction and twosets of securities A and B where A is steeperthan B Let Sv

A and SvB be the equilibrium bid


for type v using these two sets Consider theexpected payments of a type v who bids as typev

Mjv v ESv j Zi Vi v Vi v

for j A B

Suppose MA(v v) MB(v v) Given affiliationa direct generalization of Lemma 5 implies thatM1

A(v v) M1B(v v) The conclusion then fol-

lows from the same linkage principle argumentas in the proof of Proposition 1

Now consider a second-price auction The equi-librium bid satisfies the zero profit condition

ESvj Zi Vi v Vi v

EZi XVi v Vi v

The proof follows the same logic as in Propo-sition 1 the sellerrsquos revenues depend on the setof securities through the difference

ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2

Again affiliation implies this difference will belarger for steeper securities

Result 2 Consider a set A of securities that isconvex or sub-convex Let Sv

j be the bid oftype v in a j-price auction The expectedpayment for type v who bids as v in a first-price auction is

M1v v ESv 1 Zi Vi v Vi v

ESv1 Zi Vi v

where the second equality follows from the pri-vate value assumption For a second-price auction

M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi

2 zVi v Vi v] forall z the security Svv

2 is in the convex hull of ASuppose M1(v v) M2(v v) To apply the

linkage principle we must show that M11(v v)

M12(v v) But this follows since (a) Svv

2 is in theconvex hull of A and therefore is steeper thanSv

1 and (b) affiliation and the monotonicity ofbids implies that E[Svv

2 (Zi)Vi] is increasingin v

Result 3 Let j(v) be the equity bid of type v ina j-price auction Then

M1v v E1v ZVi v Vi v

1v EZVi v Vi v

M2v v E2Vi ZVi v Vi v

E2Vi Vi v

EZVi v Vi v

where the inequality follows since 2(Vi) andZ are positively correlated and we use the as-sumption that types are independent ThereforeM1(vv)M2(vv) implies that

1v E2Vi Vi v

But since E[ZV] yenl Vl

M11v v 1v M1

2v v

E2Vi Vi v Vi v

Thus the result follows from the linkageprinciple

REFERENCES

Axelson Ulf ldquoSecurity Design with Investor Pri-vate Informationrdquo University of ChicagoCenter for Research in Security Prices Work-ing Paper No 539 2004

Binmore Ken and Klemperer Paul ldquoThe Big-gest Auction Ever The Sale of the British 3GTelecom Licensesrdquo Economic Journal 2002112(478) pp C74ndash96

Board Simon ldquoBidding into the Redrdquo Unpub-lished Paper 2002

Board Simon ldquoSelling Optionsrdquo UnpublishedPaper 2004

Brusco Sandro Lopomo Giuseppe and Viswana-than S ldquoMerger Mechanismsrdquo Fondazione


Eni Enrico Mattei FEEM Working PaperNo 7 2004

Bulow Jeremy and Klemperer Paul ldquoPrices andthe Winnerrsquos Curserdquo RAND Journal of Eco-nomics 2002 33(1) pp 1ndash21

Che Yeon-Koo and Gale Ian ldquoThe OptimalMechanism for Selling to a Budget-Constrained Buyerrdquo Journal of EconomicTheory 2000 92(2) pp 198ndash233

Cho In-Koo and Kreps David M ldquoSignalingGames and Stable Equilibriardquo QuarterlyJournal of Economics 1987 102(2) pp179ndash221

Cho In-Koo and Sobel Joel ldquoStrategic Stabilityand Uniqueness in Signaling Gamesrdquo Jour-nal of Economic Theory 1990 50(2) pp381ndash413

Cremer Jacques ldquoAuctions with ContingentPayments Commentrdquo American EconomicReview 1987 77(4) p 746

DeMarzo Peter M ldquoPortfolio Liquidation andSecurity Design with Private InformationrdquoUnpublished Paper 2002

DeMarzo Peter M and Duffie Darrell ldquoALiquidity-Based Model of Security De-signrdquo Econometrica 1999 67(1) pp 65ndash99

Fisch Jill E ldquoAggregation Auctions andOther Developments in the Selection ofLead Counsel under the PSLRArdquo Law andContemporary Problems 2001 64(2-3)pp 53ndash96

Garmaise Mark J ldquoInformed Investors and theFinancing of Entrepreneurial Projectsrdquo Un-published Paper 2001

Hansen Robert G ldquoAuctions with ContingentPaymentsrdquo American Economic Review1985 75(4) pp 862ndash65

Hart Oliver and Moore John ldquoDebt and Se-niority An Analysis of the Role of HardClaims in Constraining ManagementrdquoAmerican Economic Review 1995 85(3)pp 567ndash 85

Hendricks Kenneth and Porter Robert H ldquoAnEmpirical Study of an Auction with Asym-metric Informationrdquo American Economic Re-view 1988 78(5) pp 865ndash83

Kohlberg Elon and Mertens Jean-Francois ldquoOnthe Strategic Stability of Equilibriardquo Econo-metrica 1986 54(5) pp 1003ndash37

Krishna Vijay Auction theory San Diego Ac-ademic Press 2002

Laffont Jean-Jacques and Tirole Jean ldquoAuc-

tioning Incentive Contractsrdquo Journal of Po-litical Economy 1987 95(5) pp 921ndash37

Martin Kenneth J ldquoThe Method of Payment inCorporate Acquisitions Investment Opportu-nities and Management Ownershiprdquo Journalof Finance 1996 51(4) pp 1227ndash46

Maskin Eric S and Riley John G ldquoOptimalAuctions with Risk Averse Buyersrdquo Econo-metrica 1984 52(6) pp 1473ndash1518

McAfee R Preston and McMillan JohnldquoCompetition for Agency Contractsrdquo RANDJournal of Economics 1987 18(2) pp296 ndash307

McMillan John ldquoBidding for Olympic Broad-cast Rights The Competition before theCompetitionrdquo Negotiation Journal 19917(3) pp 255ndash63

Milgrom Paul R and Weber Robert J ldquoATheory of Auctions and Competitive Bid-dingrdquo Econometrica 1982 50(5) pp1089 ndash1122

Myerson Roger B ldquoOptimal Auction DesignrdquoMathematics of Operations Research 19816(1) pp 58ndash73

Nachman David C and Noe Thomas H ldquoOpti-mal Design of Securities under AsymmetricInformationrdquo Review of Financial Studies1994 7(1) pp 1ndash44

Ramey Garey ldquoD1 Signaling Equilibria withMultiple Signals and a Continuum of TypesrdquoJournal of Economic Theory 1996 69(2)pp 508ndash31

Ravid S Abraham and Spiegel Matthew I ldquoOp-timal Financial Contracts for a Start-up withUnlimited Operating Discretionrdquo Journal ofFinancial and Quantitative Analysis 199732(3) pp 269ndash86

Rhodes-Kropf Matthew and Viswanathan SldquoCorporate Reorganizations and Non-cashAuctionsrdquo Journal of Finance 2000 55(4)pp 1807ndash49

Rhodes-Kropf Matthew and Viswanathan SldquoFinancing Auction Bidsrdquo RAND Journal ofEconomics (forthcoming)

Riley John G ldquoEx Post Information in Auc-tionsrdquo Review of Economic Studies 198855(3) pp 409ndash29

Riley John G and Samuelson William F ldquoOpti-mal Auctionsrdquo American Economic Review1981 71(3) pp 381ndash92

Samuelson William ldquoAuctions with ContingentPayments Commentrdquo American EconomicReview 1987 77(4) pp 740ndash45


Topkis Donald M ldquoMinimizing a SubmodularFunction on a Latticerdquo Operations Research1978 26(2) pp 305ndash21

Townsend Robert M ldquoOptimal Contracts andCompetitive Markets with Costly State Ver-ificationrdquo Journal of Economic Theory1979 21(2) pp 265ndash93

Vickrey William ldquoCounterspeculation Auc-tions and Competitive Sealed TendersrdquoJournal of Finance 1961 16(1) pp 8 ndash37

Zheng Charles Z ldquoHigh Bids and Broke Win-nersrdquo Journal of Economic Theory 2001100(1) pp 129ndash71


securities from a pre-specified ordered set suchas an equity share or royalty rate The seller iscommitted to disqualify any offer outside thisset The seller also commits to an auction for-mat such as a first- or second-price auctionOne of our main results is that the revenuesfrom an informal mechanism are the lowestacross a large set of possible mechanisms Inother words the seller benefits from any form ofcommitment Moreover we show how to ranksecurity designs and auction formats in terms oftheir impact on the sellerrsquos revenues and thatthe design of the securities can be more impor-tant than design of the auction itself

In our model several agents compete for anasset or project that requires an upfront invest-ment This investment may correspond to anamount of cash that the seller needs to raise orthe resources required to run the project Bid-ders observe private signals regarding the valuethey can expect if they acquire the asset Ourinitial structure is similar to an independentprivate values model so that different biddersexpect different payoffs upon winning althoughwe also consider correlated and common val-ues The model differs from standard auctionmodels in that bids are securities Bidders offerderivatives in which the underlying value is thefuture payoff of the asset Because the winnermay make investments or take other actions thataffect this payoff we also discuss the possibilityof moral hazard

One might conjecture that the results fromstandard auction theory carry over to security-bid auctions by simply replacing each securitywith its cash value Unlike cash bids howeverthe value of a security bid depends upon thebidderrsquos private information This differencecan have important consequences as the follow-ing simple example demonstrates

Consider an auction in which two biddersAlice and Bob compete for a project Theproject requires an initial fixed investmentthat is equivalent to $1M (we can inter-pret some or all of this amount as a min-imum up-front cash payment required bythe seller) Alice expects that if she un-dertakes the project then on average itwould yield cash flows of $4M Bob ex-pects future cash flows of only $3M As-suming these estimates are private valuesin a standard second-price auction it is adominant strategy for bidders to bid their

reservation values As a result Alicewould bid 4 1 $3M and win theauction paying Bobrsquos bid of 3 1 $2M

Now suppose that rather than biddingwith cash the bidders compete by offer-ing a fraction of the future revenues Aswe discuss later it is again a dominantstrategy for bidders to bid their reserva-tion values Alice offers 3frasl4 of future cashflows while Bob offers 2frasl3 As a resultAlice wins the auction and pays accordingto Bobrsquos bid that is she gives up 2frasl3 of thefuture revenues This yields a higher pay-off for the auctioneer (2frasl3) $4M $267M $2M (in addition to any up-front payment)

This example is based on Robert G Hansen(1985) who was the first to examine the use ofsecurities in an auction setting Hansen showedthat a second-price equity auction yields higherexpected revenues than a cash-based auction Ina related paper John G Riley (1988) considersfirst-price auctions where bids include royaltypayments in addition to cash He shows thatadding the royalty increases expected revenuesThe intuition in both cases is that adding anequity component to the bid lowers the differ-ence between the winnerrsquos valuation and that ofthe second highest bidder Because this differ-ence is the rent captured by the winner reduc-ing it benefits the seller

In this paper we generalize this insight alongseveral dimensions First we consider a generalclass of securities that includes debt equity orroyalty rates options and hybrids of these Sec-ond we consider alternative auction formats(eg first-price versus second-price) Third weconsider informal auctions in which the sellercannot commit to an auction mechanism inadvance

The structure of the paper is as follows Thebasic model is described in Section I We beginour analysis in Section II by examining formalmechanisms which consist of both an auctionformat and a security design There we establishthe following results

We characterize super-modularity conditionsunder which a monotonemdashand hence effi-cientmdashequilibrium is the unique outcome forthe first- and second-price auctions






















I The Model









in z if v v4



EZiVi X Vi





(1) Zi X Vi






















ESv ESZiVi v




































Fn 1vv ESsv v



2

vslogv ESs v 0



sv n 1fv

Fv

v ESsv v

ES1sv v






















ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES
































































I The Model









in z if v v4



EZiVi X Vi





(1) Zi X Vi






















ESv ESZiVi v




































Fn 1vv ESsv v



2

vslogv ESs v 0



sv n 1fv

Fv

v ESsv v

ES1sv v






















ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES

















































in z if v v4



EZiVi X Vi





(1) Zi X Vi






















ESv ESZiVi v




































Fn 1vv ESsv v



2

vslogv ESs v 0



sv n 1fv

Fv

v ESsv v

ES1sv v






















ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES






















































ESv ESZiVi v




































Fn 1vv ESsv v



2

vslogv ESs v 0



sv n 1fv

Fv

v ESsv v

ES1sv v






















ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES







































































Fn 1vv ESsv v



2

vslogv ESs v 0



sv n 1fv

Fv

v ESsv v

ES1sv v






















ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES


























































ESsV2 V1 ESsV2 V2

























ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES
































































ESPAV2V2 v1 E V2

V2 1V2 v1

1 ln1 v1

v1




(4) EZi minb ZiVi v

1

2vb

2v

z b dz 2v b2

4v



2v1 bFPAv12

4v1

2v1 2v1 32

4v1

4

9v1


E 2v1 bSPAV22

4v1V2 v1

E2v1 2V22

4v1V2 v1

1

v1

0

v1 v1 v22

v1dv2

1

3v1




4v1







(Z1)V1 v1] E[Z1

































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES








































































project













(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES




















































(5) RiS EESViSi S
























Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES





















































Proposition 5

IV Extensions

















SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES





















































SDd z mind B z

mind B mind B13 z






ESs0 vr vr































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES




































































V Conclusion












APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES













































APPENDIX






ESv1(v) ESv


U1v Fn 1v1 ESv1v

Fn 1v1 ESv2v U2v















Ssv ) Sv ()











Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES




















































Note that


v ESdv

Uv





RSd ESdv RS











Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES
















































Uv PRSdvv ESdvv

maxdPRSdv ESdv



Uv Fn 1vv ESdvv


Uv Fn 1v1 ESv
















for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES














































for j A B





ESvj Zi Vi v Vi v

EZi XVi v Vi v


ESv2

j ZiVi v1 Vi v2

ESv2

j ZiVi v2 Vi v2





ESv1 Zi Vi v


M2v v ESVi

2 Zi Vi v Vi v

ESvv2 Zi Vi v

where Svv2 (z) ESVi










1v EZVi v Vi v


E2Vi Vi v

EZVi v Vi v


1v E2Vi Vi v


M11v v 1v M1

2v v

E2Vi Vi v Vi v


REFERENCES












































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