alejandro m. manelli, optimal procurement

Econometrics, Vol. 63, No. 3 (May, 1995), 591-620

OPTIMAL PROCUREMENT MECHANISMS

We analyze optimal mechanisms in environments where sellers are privately informed about quality. A methodology is provided for deriving conditions that are necessary and sufficient to determine when two simple trading environments maximize either social or private surplus. The commonly used auction mechanism is frequently inefficient in procurement environments. Often, the optimal mechanism is simply to order potential suppliers and to tender take-it-or-leave-it offers to each sequentially. We completely characterize the environments in which either mechanism is optimal. In doing so, we develop a general methodology that determines when and if a given trading institution is optimal.

KEYWORDS:Auctions, bargaining, incentive compatibility, adverse selection, procurement.

1. INTRODUCTION

INATTEMPTING TO ACQUIRE AN OBJECT,a buyer often must decide whether to exploit competition among a pool of potential sellers. For instance, an individual purchasing a new car, a home owner hiring a contractor, or a government agency procuring goods and services, may seek bids from different suppliers or simply bargain with a given seller. Although economists often tout the benefits of relying on competitive behavior, one can also show that competition may have some highly undesirable self-selection consequences. This article addresses the question of what modes of transacting should be used to acquire goods or services when the buyer does not know the quality ex ante and when a court cannot verify the quality ex post.

"Seeking bids" is naturally associated with auctions or other forms of low-price bidding institution^.^ The efficiency and simplicity of auctions are major reasons for their popularity, both in practice and in research. Trading at auction is a common way to purchase and sell goods and services. The Department of Defense, for example, procures as much as thirty percent of its supplies via what amounts to a first price sealed bid auction. While auctions are efficient and (if properly conducted) profit-maximizing in some environments with incomplete information, it is well known that in others, they can be strikingly inefficient. For example, if the quality of the good to be sold is in doubt, price competition may ensure that only the poorest types of goods are provided.

We are grateful to Kerry Back and Bentley McLeod, who pointed out errors in a previous formulation, and to Morton Kamien, Roger Myerson, John Nachbar, Rafael Repullo, and Jorge Padilla for helpful comments. This paper was revised while the first author was visiting the Instituto de Anglisis Econ6mic0, Universidad Aut6noma de Barcelona, Spain, and the second author was visiting the California Institute of Technology, Pasadena, California. Manelli's research was partially supported by the Spanish Ministry of Education and Science, CICYT Grants PB89-0075 and PB20-0132.

The term "auction" has been used in the literature to refer to any best-price bidding institution and to any general trading mechanism. We will use it with the former meaning.

592 A. M. MANELLI AND D. R. VINCENT

An alternative institution allows the abandonment of competition and the presentation of take-it-or-leave-it offers to selected sellers. We completely characterize "when to seek bids" and "when to bargain with an individual supplier." In doing so, we develop a general methodology that determines when and if a given trading institution is optimal. For example, if the potential gains from trade are large, and if the buyer values marginal quality more than the sellers, then arbitrarily selecting a seller and tendering a take-it-or-leave-it offer maximizes expected social surplus-competition among sellers cannot be ex-ploited to improve upon this outcome even though many potential sellers may be available. On the other hand, if sellers value marginal quality more than the buyer,-an auction is the socially optimal institution.

It has long been recognized that many mechanism design problems are linear programs. Our approach is based on two results, Myerson's (1981) well-known characterization of incentive compatibility, and the duality theory of convex programs. The first result permits an important simplification of the linear program; only the probability of trade with each seller need be considered as a decision variable. Exploiting Myerson's characterization of incentive compatibility requires formulating the problem in infinite dimensional spaces. We solve the infinite dimensional program by solving its dual. Although the approach is familiar from finite dimensional linear programming, a few definitions from functional analysis are also required. The approach allows a complete and explicit characterization of the environments for which a given institution is optimal. Since many other related adverse selection problems are also usually framed as infinite dimensional programs, illustrating how to solve one type of problem may be helpful in studying other problems as well.

This paper is concerned with the adverse selection aspect of procurement and as such is most closely related to work by Myerson (1981) and Samuelson (1984). Myerson characterizes direct revelation mechanisms that maximize buyer's expected profik3 Our results differ from his in two aspects. First, given an economic environment (valuation functions and distributions), Myerson con-structs a family of functions, one for each bidder, that he calls "priority levels." In an optimal mechanism, the transaction takes place with the supplier of highest priority level. Given a trading institution, it is not obvious how to determine the environments that will generate the "priority levels" that in turn will produce the required solution. We select a trading institution and characterize the class of environments for which that institution is optimal in terms of the fundamentals, the valuation of the object and the distributions of private information. Second, Myerson only considers the maximization of expected buyer's surplus. We analyze alternative objectives: maximizing expected social buyer, or aggregate seller surplus. The consideration of the expected social surplus reveals some important consequences. When there is only one seller, for instance, a take-it-or-leave-it offer will always maximize the buyer's expected

Myerson interprets his model as a single, uninformed seller facing potentially many informed buyers. For simplicity, we translate it to our story where the buyer is the uninformed agent.

593 OPTIMAL PROCUREMENT MECHANISMS

surplus; the problem of maximizing expected social surplus typically requires more complicated mechanisms (Samuelson (1984)). Our methodology allows us to characterize exactly when simple take-it-or-leave-it offer mechanisms are sufficient to maximize expected social surplus in environments with a single seller as well as with many sellers.

2. AN EXAMPLE

Although auctions may sometimes be privately suboptimal, one may be tempted to conclude that they are at least socially optimal since they ensure the efficient allocation of goods. This view of auctions stems from a special charac- teristic of the standard single-seller-many-buyer environment. In the typical auction model, the private information of participants on one side of the market is not directly related to the valuation of the single uninformed seller (or buyer) on the opposing side. When a single seller attempts to sell his product to one of many buyers, this formulation may be convincing-a seller may well be ex-pected to understand her opportunity costs of yielding a good in a trade. In the alternative formulation, where a single buyer attempts to obtain an object, it is no longer so obvious that the private information of the provider, which may include the marginal costs of provision or her use value of the good, is independent of the valuation of the buyer. This dependence affects the social efficiency of an auction.

We use the following example to illustrate two points. First, when the buyer's valuation depends on the sellers' private information, an auction may be suboptimal both for the buyer and for society. Second, in environments in which auctions are inefficient, ignoring all possible competition among sellers may dominate mechanisms that exploit that competition.

A potential buyer wishes to purchase a single object of uncertain quality from one of S potential suppliers. Each supplier has privately known opportunity cost q, which is, ex ante, distributed uniformly and independently over I = [O, 11. An object from a seller with cost q, generates a use value for the buyer of v(q,) = 1 / 2 + (3/2)q, and therefore increases social surplus by v(q,) - q, =

(1 + q,)/2. If the allocation mechanism is conducted by one of many possible auction-like institutions which allocate the trade to the lowest bidder, then trade will always occur with the lowest cost and lowest quality seller. A simple computation reveals that any such procedure generates expected social surplus (1 + 1/(S + 1))/2. As S becomes large, then the procedure almost certainly ensures trade with the lowest cost seller, and expected social surplus converges to 1/2.

The poor performance of the auction suggests that we might look for other simple mechanisms which dominate the auction. For example, if a single seller is chosen at random and offered a (credible) take-it-or-leave-it price of one, then it is subgame perfect for the seller to accept and the expected social surplus is 3/4 independent of S. In fact, as Theorem 3 illustrates, this mechanism achieves the highest possible social surplus in this example.


Why are auctions so popular? Consider a variation on the above where v(q,) = 2 for all q,. This is the corresponding many-seller-single-buyer analogue of the typical many-buyer-single-seller auction model. In this case, an auction mechanism is socially optimal. It ensures trade with the lowest cost seller but now such trade generates the highest possible surplus.

We focus on the problem of determining when competition among sellers should be exploited and when it should be avoided by analyzing two specific institutions-a second-price auction4 and what we term a sequential offer institution. In a second-price auction with reserve price k, for seller s, admissi-ble bids are collected (i.e., bids below the respective seller's reserve price) and trade occurs with the seller with the lowest admissible bid at either her reserve price or at the second lowest bid, whichever is the lowest. Alternatively, competition among sellers can be limited by making a sequence of take-it-or- leave-it offers to sellers. In more general environments, offers below the highest possible seller type may be more profitable and, of course, need not be accepted. However, a rejection frees the buyer to bargain with another supplier. Furthermore, when sellers are not identical, it may matter to which seller the offer is tendered. To incorporate these possibilities, we define the sequential offer mechanism as follows: the buyer offers a take-it-or-leave-it price of k, to seller 1. If it is accepted, trade takes place and the game ends. If it is rejected, the buyer makes another offer, k , to seller 2, and so on until all sellers have rejected their offer, in which case no trade takes place.

We characterize necessary and sufficient conditions that the economic environment must satisfy for these institutions to be optimal. There are economic environments, however, where neither an auction nor a sequential offer institution is optimal. Thus, other potential forms of transacting can be studied, including a mechanism which consists of take-it-or-leave-it offers followed by an auction in the event of rejectionss, or a variation on the sequential offer institution in which, in the event of rejections from all sellers, a new round of offers is made. Our methodology can be applied to these and other institutions.

3. MODEL AND METHODOLOGY

This section presents the model, a routine application of the revelation principle, the linear program, and a restatement of Myerson's (1981) characterization of incentive compatibility.

The model consists of one buyer and potentially many sellers. It covers many situations of adverse selection: a government attempting to purchase goods of undeterminable quality from many potential suppliers; a firm hiring from a pool of workers with differing skills; an insurance company purchasing risk from clients with privately known probabilities of accident. In all of these situations, the private information of a supplier affects her reservation value and may be directly relevant for the ex post valuation of the buyer.

4 ~ n ylow bidding institution could be used as well. Second-price auctions have the advantage of having a unique equilibrium in undominated strategies.


We index sellers by s, s = 1,2,. . . ,S, and abuse notation by letting S represent the set of sellers as well; thus s E S. Every seller s observes some private information q, E I = [O, 11. We will often refer to q, as quality. Each individual parameter q, is independently distributed according to a continuous and strictly positice density function f,(q,). The corresponding cumulative distribution is denoted F,(q,). The vector q = ( q , ,.. . ,q,) EIS is a profile of types. Given q E IS, q-, denotes the projection of q to I~\( ' ) . We define f(q) =f,(q,) X

f2(q2)x . . . x fS(qS) and for i E S, f-, = Il,,,f,. Agents' preferences are defined over money and the use of the good in the

standard way. If a seller with quality q, engages in trade and receives a money transfer of m, her net payoff is given by m - u(q,). If no trade occurs, then the net payoff is zero. Ordering indices so that u(.) is strictly decreasing and selecting units appropriately, we may assume, without loss of generality, that u(qs) =q,. Thus, q, represents the opportunity costs to seller s of parting with her good. The buyer has a potential use value for a good of quality q, given by v,(q,).5 If he gives a money transfer of m and receives an object of quality q,, his net payoff is given by u,(q,) -m. No trade yields a payoff of zero for the buyer as well. Agents maximize expected utility.

Since seller types are determined exogenously, this formulation establishes a pure adverse selection problem. Another branch of the literature on procurement has concentrated on moral hazard issues such as the problems of incen- tives for investment in research and development. See, for example, Rob (1986) or Lang and Rosenthal (1991). Consumption of the good allows the buyer to perceive the quality of the object q, and, of course, his own benefit v,(q,), derived from it. We assume that neither the quality of the object nor the actual benefit to the buyer are verifiable by a court. Thus, it is not possible to contract contingent on an object's true quality or on the true benefit that may result from its consumption. We justify this assumption with the observation that for many situations of asymmetric information, even when the contracting space is large, there will often remain some residual elements of private information on which contracting cannot be done. Our model is intended to capture those features of quality that are not easily observable. Gilbert (1977), Thiel (19881, Sinclair- DesgagnC (1990), and Branco (1992) consider models in which private information is verifiable. As a result, agents may submit bids that consist of more than just prices.

In this paper, we consider all outcomes that can be achieved as a Bayesian Nash equilibrium of some game. An optimal institution is a game generating the best outcome within this class. From the revelation principle, for any Nash equilibrium outcome of any trading game, there exists a Nash equilibrium of a direct revelation mechanism generating the same outcome in which agents truthfully report their private information. Since expected payoffs depend only on the expected transfer an agent gives or receives and on the probability that

To take advantage of duality results, we need a space with a nonernpty positive orthant. Unless othenvise specified, all relevant functions lie in L,. In particular, us: I -% (v, E L,(I)).


the agent will end up with the good, a direct mechanism can be expressed as a pair of functions p,: zS -,I and t,: zS -;. 8 ,for each possible seller s, such that

If sellers report q = (q,, . . . , q,), the probability that seller s will trade is p,(q) and the expected transfer payment to selIer s, which can be negative, is t,(q).

If seller s declares her type to be x, and assumes that the rest of the sellers behave truthfully, s would expect to receive a monetary transfer of /t,(x,, q- , )f-,(q-,) dq-,. We denote this expected monetary transfer by E,t,(x,). (We define E,p,(x,) The expected payoff of seller analogo~sly.~) s from announcing x, when her true quality is q, is the difference between her expected monetary transfer E,t,(x,) and her expected cost of giving up the good7 q ,E ,~~(x , ) ,

Note that E,p,(x,) is seller s expected probability of trade when she reports type x,. Agents will truthfully reveal their private information only if the mechanism is incentive compatible (IC): for every seller s,

(IC) . r r , ( q , l q s ) ~ ~ s ( x s ~ q s ) , Vqs€ I , VX,EI .

One may require, in addition, that it be in the agents' interest to participate in a trading institution, i.e., individual rationality ( IR) .~ Without loss of generality, let zero be the payoff for all agents who do not participate in the mechanism. Voluntary participation then requires that

( IR) T, 0 and rs(q,lq,) > 0, vq, €1, V s E S .

There are several reasonable objectives that might be considered in evaluat- ing alternative mechanisms. A mechanism may be best for society as a whole, for the buyer, or for the sellers as a group. Procuring the object from seIler s with quality q,, generates a total surplus of ws(q,) = - q,. Hence, for a given mechanism, {p,, t,},, ,, expected social surplus is

The buyer's expected profits are

Formally, for any p,(q) E L _ ( I ~ ) ,E,: Lm(IS)* Lm(I ) is defined by E,p,(q,) = s c~)p,(q,, q-,If-,(q-,)dq-,. We use this characterization of E, as an operator in our proofs. ' 'A mechanism designer with great authority may force the agents to participate even when

payoffs are negative. In those cases, the IR constraint should be ignored.

597 O P T I M A L PROCUREMENT M E C H A N I S M S

REMARK:Any mechanism that maximizes expected social surplus will also maximize ex ante aggregate seller surplus, provided it is modified by adding additional transfers chosen so that expected buyer surplus is zero. Hence, without loss of generality we shall explicitly consider only the maximization of expected social and buyer surplus.

It has long been recognized that finding an optimal mechanism is formally equivalent to solving a linear program: the maximization of (1)subject to IC and IR, or the maximization of (2) subject to the same constraints. The linear program, however, is simplified considerably with the application of the following result.'

THEOREM1 (Myerson (1981)): Let {p,}, ,,, p,: zS -,I and C" =, P, < 1. (a) There exists a collection of transfer functions {ts(q)}s,s making { P,, t,},, ,

an IC mechanism if and only if for every seller s,

Esp,( q ),is nonincreasing in q,.

(b) Suppose that V s E S, E,ps is nonincreasing in q,. Then there exists a collection of transfer functions {t,(q)},,, making Ips, t,},,s an IC and IR mechanism if and only if

(c) Let { p,, t,}, ,,be an incentive compatible mechanism. Then, the expected profits for any seller are nonincreasing in q,. More precisely,

(d) Let {p,, t,}, ,,be an incentive compatible mechanism. The expected buyer surplus is given by

The proof of this theorem is essentially in Myerson and Satterhwaite (198319

Samuelson (1984) provides a clear explanation of the following theorem for the single seller case.

Myerson and Satterthwaite assume that f, is continuous but their theorem holds as well with piecewise continuity, as do the remainder of our results.


As a result of Theorem 1, the linear program can be set up without explicit consideration of the transfer functions.1° Consider the linear program 51:

E,p,(q,) is nonincreasing in q, Vs,

The program 9summarizes the two optimization problems that concern us. When the objective is to maximize social surplus, h,(q,) becomes

to yield expression (1). Since only one good is to be traded, the decision variables {p,},, ,must not add up to more than one. This is stated by the first constraint. By Theorem l(a), the next S constraints, one for each seller, capture incentive compatibility. To solve 9,it is useful to express these constraints algebraically. If the expected probability of trade functions were differentiable, the incentive compatibility constraint could be stated as a derivative. In general, the functions are not differentiable. We deal with this technical inconvenience in the next section. The last constraint represents individual rationality. Using Theorem l(b), g,(q,) becomes

Consider now the maximization of the buyer's expected profits. Theorem 1 (specifically, parts (c) and (d)) implies that, in the corresponding objective function,

The expected buyer surplus only depends on the transfer indirectly through the term ~ ~ ( 1 ) = 1. For any seller s, the I), the expected profits of sellers of type q, expected profits of type q, are determined by the probability of trade function p, and the profits rS(1( 1) of the highest type q, = 1. If rS(1 11) < 0, seller s's individual rationality constraint is violated. If %(I / 1) > 0, then the buyer's expected profits may be increased by an additional transfer. Thus, any mechanism maximizing the buyer's expected profits must set %(ill) = 0. The sellers'

lo Theorem 1relies on the assumption that agents' types are drawn from the continuum. Without this assumption the buyer's expected profits need not be independent of the transfer functions. Analyses that exploit this result, therefore, must be conducted in infinite dimensional spaces.


individual rationality constraints are satisfied because, from Theorem l(c), sellers profits are nonincreasing in their type. Since the objective is to maximize the buyer's expected profits, and since zero profits are feasible, g,(q,) is defined as

The other constraints remain unchanged. We will use the program 9' in the following manner. First, a candidate

trading institution such as a second price auction or a successive offer mechanism is proposed. The equilibrium outcome of the proposed institution specifies a probability of trade p,(q) and a corresponding transfer t,(q) for each seller. From this outcome we derive the expected profits for any seller s of type 1. In our institutions, .rr,(lll) = 0 for all sellers.'' Second, the revelation principle implies that the same outcome can be obtained as an equilibrium of a direct revelation mechanism. From Theorem 1, any equilibrium of a direct revelation mechanism can be completely characterized by the probability of trade functions, p,, and the expected profits of the sellers of type q, = 1. Thus, any direct revelation mechanism implementing p, and with .rr,(l 11) = 0 will perforce have the same outcome as the proposed trading institution. We may therefore consider the simplified linear program 9'. Third, the probability of trade functions, p,, are considered as the candidate solution to 9'.By solving the dual program, we identify the environments, if there are any, in which the proposed functions p, are optimal. That is, we characterize the class of valuation functions c.,(q,) and distributions F,(q,) for which the candidate probability of trade functions solve 9.These are the conditions under which the proposed institution is optimal.

In Sections 5 and 6 we apply this methodology to the two simple trading mechanisms referred to in the introduction, the sequential offer and the low bidding institution.

4. THE LINEAR PROGRAM AND ITS DUAL

We introduce some definitions from functional analysis, set up the primal program, and state the duality theorem for linear programs. While our proofs rely on the contents of this section, the presentation and discussion of our results do not. The reader not interested in the proofs may proceed to the following section.

To solve the dual 9' we must express the incentive compatibility constraints algebraically. It is useful to state these constraints in terms of derivatives. We use D to represent the differential operator that assigns its derivative to any

11Our previous discussion shows that ~~(111)0 is a necessary condition for any institution =

maximizing expected buyer's profits.


function g: I -8 differentiable almost everywhere. More precisely,

whenever it is defined. Let H, = { p E L,(IS): DEsp E L,(I)}. The set H, is a linear vector space.

Since the expected probability of trade Espsneed not be differentiable everywhere, the incentive constraints cannot be fully expressed in terms of derivatives. To address this inconvenience we define the following convex subset of H, where the optimization will take place:

For an element p, E G,, any "jumps" in the expected probability of trade E,p, have to be decreasing. This rules out mechanisms of the form p, = lqS,which,s

do not satisfy incentive compatibility despite having a nonpositive derivative almost everywhere.12 There is no loss of generality in conducting the maximization on the space G,. This space is pointwise dense in the space of nonincreasing, nonnegative functions in H,. Thus, whenever {ps}s,, attains the maximum in 9over G,, {p,}, ,,also attains the maximum over the space of nonincreasing and nonnegative functions in H,.

The linear program 9can therefore be written

To state the dual program, we must identify the spaces to which the multipliers (i.e., the dual operators) belong. Given a Banach space V, we denote its dual, the space of bounded linear operators on V, by V*. We will write ( . ,. ) to identify the bilinear mapping from V X V* to 8.For instance, when referring to

For any measurable set A c I, lArepresents the indicator function of the set A, that is, the function that takes value one if q, is in A and zero otherwise.


two functions p , h EL,( IS), the dual pairing becomes

The Lagrangian corresponding to 9can be written as

d=CS

(p, , h,f) + CS

( p , , a g , f ) + ( 1 , ~ )- CS

(p s ,y ) s = l s = l s = l

or, rearranging terms,

where the multipliers y, A,, and a all lie in the appropriate dual spaces, i.e., y EL*,(I~),A, EL*,(I) for all s, and a E 8. (Because we cannot guarantee a priori that the notion of integration is well defined for all dual variables, we must use the dual pairing ( . ,. ) instead of the integral representation whenever the multipliers y or As are involved.)

The constraint space, L,, has a positive cone with nonempty interior (denoted by L,,). Also, since h, is in L, and p, is bounded by 1 for all s in S , the optimal value is finite. As is common in optimization problems, in what follows, we require that a regularity condition be satisfied. In infinite dimensional linear programs, the value of the dual program need not coincide with the value of the primal. The regularity condition assures that no such duality gap exists.

REGULARITYCONDITION:For all s, there exists z , E G, such that the constraints in @ are satisfied with strict inequality.14

The following result follows from the duality theory for convex programs (Theorem 1 and Corollary 1, p. 219, and Theorem 1, p. 220, in Luenberger (1969)). Anderson and Nash (1987) discuss in detail the duality of linear programs in infinite dimensional spaces.

THEOREM 2: Suppose the regularity condition is satisfied. A feasible {p,},,, attains the maximum in 9 if and only if there exists y E L*,(I~), As ELz(I)

l3We may consider h EL*,(IS)because, loosely, L*,(IS)contains all elements of LAIS) . l4 In the case of maximizing buyer's surplus, this condition is automatically satisfied. Whether the

regularity condition holds when maximizing social surplus will depend on the functions g,: if there exists an E > 0 such that ws(qs)2 E Vqs< E , the regularity condition will be satisfied.


Vs E S, and a E % such that

(5) (x , y ) > 0 Vx E L,(rS)+, and

(6) ( y , A,) 2 0 Vy E L,(I) + and Vs E S.

Complementary slackness implies, in addition, that if {p,}, ,,is a solution to 9 then

(9) (~,,(h,+ag,)f-y)-(DE,p,,A,)=O VSES, and

Equation (3) states that the value of the primal and dual program are the same. The first order condition on the Lagrangean or, equivalently, the constraint of the dual program is given by (4). The nonnegativity constraints for the dual variables are given in (5) and (6).

5. SEQUENTIAL OFFER INSTITUTION

We describe the sequential offer institution in detail and characterize the environments in which such a mechanism is optimal.

For a fixed order of sellers and for a given set of prices {k,}, ,,, the buyer is given the choice of participating or not in the sequential offer mechanism. (In order to ensure rS ( l 11) = 0, k, must be less than or equal to 1.) If he accepts, a take-it-or-leave-it offer of k, is tendered to the first potential seller. If accepted, the transaction takes place and the game ends. If it is rejected, then another offer, k,, is made to the next seller. This process continues until one seller accepts the offer. If no seller accepts, then no trade takes place.

This game has a unique subgame perfect equilibrium outcome. Subgame perfection requires that the buyer participate whenever his expected payoff is strictly positive.15 In any subgame in which a seller s receives an offer k,, the seller's response is to accept provided q, < k, and reject when q, > k,. (When q, = k,, the seller is indifferent but this only occurs with zero probability.) Thus, this indirect mechanism generates a unique subgame perfect equilibrium out-

's We ignore the trivial case in which the buyer does not participate when he receives a zero expected payoff.


come characterized by trade with the first seller if her type lies below her reserve price, with the second if the first seller's type lies above k , and the second lies below k , and so on. For any seller s, r S ( 1( 1 ) = 0 and the probability of trade functions implemented by this institution are

1 if q, < k , and q, > k , Vi < s ,5 , ( 4 ) =

0 otherwise.

There are two types of trivial offers in the institution described. First, an offer of k , = 0 implies that seller s trades with probability zero and, therefore, seller s could receive her offer at any time during the procedure. To avoid this indeterminacy in the position of the traders, we relabel sellers so that those who never trade come last. Second, any seller s who receives an offer of one will accept it. Therefore, all potential offers to be made to suppliers who follow are irrelevant. Any potential offer to s t > s will produce the same outcome. Hence, without loss of generality we set k,, = 0 for all s t > s. Finally, if no seller accepts an offer, no trade takes place. To represent this possibility conveniently, we introduce an artificial seller with zero value who, if called to trade, receives an offer of one and sells the good for certain. Define the artificial player s' with-hs= = 0, f j (q j )= 1, and k j = 1. We abuse notation denoting by S the augmented set of sellers. Without loss of generality, we relabel the potential suppliers. After relabeling, there is a seller i with k , = 1 such that

k , = 0 if and only if s > i.

Notice that i is the artificial seller s' only when no "real" seller receives an offer of 1.

Theorem 3 provides necessary and sufficient conditions for the optimality of a given sequential offer mechanism. The following informal discussion of the necessity of those conditions is provided as an aid to the interpretation of the Theorem.

Let {j,},,,be a solution to 9'.Then the take-it-or-leave-it offer k,, made to seller s t must result in a better payoff than any lower offer r ' < k,, provided the lower offer does not violate any constraints. It is immediate that offering a lower price will not violate incentive compatibility or the one-good constraint. Sup- pose, in addition, that it does not violate individual rationality. Then it must be the case that

or equivalently, that

The left side of the inequality represents the contribution to the optimal value of making the offer k,, to seller st. The right side of the inequality represents the contribution to the total value of making the offer r'.


The maximization problem, however, is complicated by the presence of the buyer's individual rationality constraint. Even though the above inequality may favor the smaller offer r', tendering that offer may violate individual rationality. Only feasible take-it-or-leave-it offers can be meaningfully compared. Duality theory provides a relatively "simple7' method of comparing alternative mechanisms without the need to verify first whether they are feasible: it incorporates in the evaluation the costs of satisfying the IR constraint. The multiplier assdciated with this constraint is (Y 2 0 . When comparing alternative take-it-or- leave-it offers r' and k,, (or any other mechanism), the costs of satisfying the individual rationality constraint can be included. The "pseudo-objective function" h,(q,) +ag,(q, ) is constructed to account for those potential costs; the take-it-or-leave-it k,, made to seller st must result in a better payoff, accounting for potential violations of individual rationality, than any lower offer r' < kSv:

~ [ ( h , ,+ ( * g S , ) 1 , k , . ]2 ~ [ ( h , , v r l <k, , .+ a g s , ) l q s , s , , ]

When referring to expected gains or losses we will be referring to the "pseudo- objective function." In a first reading, the reader may wish to ignore the buyer's individual rationality constraint by assuming g, = 0 .

We now compare the offers tendered to two different sellers. Increasing the offer made to a given seller increases the chances that that supplier will accept the trade. In turn, this reduces the trading chances of the following sellers.

First, consider any seller s that follows s f .Reducing the offer to s' from k,, to r' will only have an effect on payoffs if s' rejects r' when she would have accepted k,,, that is, r' <q,?<k,!. In this case the losses are E[h,,+ ag,,Irl< g,, < kS,]. l6Reducing the offer to r' increases the chances of trading with s. The loss must then be compared with the potential benefits derived from trading with s. If the expected gains from tendering any offer r to s, E[h, + cug,lO < q , < r ] ,dominate the expected losses of reducing the offer to s', then the initial offers are not optimal. Formally,

~ [ h , ~ + ( ~ g , ~ I r ' ~ q , ~ < k , ~ ]> ~ [ h , + ( ~ g , l O < q , < r ]

V r > 0 and Vr' < k, , .

Second, let seller s precede s'. If the expected gains from increasing k , to r dominate the expected losses of reducing the k,, to r', then the initial offers are not optimal. Formally,

~ [ h , ~ + ( ~ g , ~ l r ' ~ q , ~ ~ k , ~ ]~ ~ [ h , + ( ~ g , l k , ~ q , ~ r ]

V r > k , and Vr' <k , , .

We have derived three necessary conditions for the optimality of a given set of take-it-or-leave-it offers by comparing them to other take-it-or-leave-it offers. Since take-it-or-leave-it offers constitute a very small subset of all possible mechanisms, the reader would be justified in doubting the usefulness of this

l6 The expression, E[h(q) lq€ A ] , represents the expected value of h with respect to the density f of q conditional on q lying in the set A.


exercise. To characterize the environments where sequential offer mechanisms are optimal, the previous exercise would have to be repeated with respect to mechanisms other than take-it-or-leave-it offers. Theorem 3 characterizes the environments in which a sequential offer mechanism is optimal. It shows that the necessary conditions described above are, essentially, sufficient as well.

THEOREM3 (Sequential Offer): Let {j,}, ,, be a sequential offer mechanism. Then { fi,}, ,, is a solution to 9if and only if there exists a > 0 such that for any s' in S with a nontrivial offer (i.e., such that for all i <s t , ki < 1)

Vs < s t , Vr > k,, and Vr' < k,,;

Vs >s', Vr > 0, and Vr' <k,,;

The proof is provided at the end of this section. The necessity of 6 1 ) through (S3) has already been discussed. The expression

(S4) is a constraint of the problem (and a complementary slackness condition) and is therefore necessary. That these conditions are also sufficient forms the substance of the theorem. Condition 6 3 ) shows that sellers who contribute the higher conditional expected gains to the objective function are made offers earlier. Thus, (S2) and (S3) both help to determine the order in which offers are made and establish when the simple sequential offer mechanism is the best way to procure the object.

Theorem 3 applies to the maximization of either buyer or seller's surplus. In the former, g, equals 0, and h, equals w,(q,) -F,(q,)/f,(q,). As a result, Theorem 3 is simplified considerably: the existence of a is guaranteed, and 6 4 ) can be eliminated from 'the statement. Given his valuation v,(.) and the a priori distributions of sellers7 types, a buyer need only verify that (S1) to (S3) hold to determine whether his proposed offers are indeed optimal.

To further illustrate Theorem 3 we apply it to the maximization of social surplus in the simpler case in which sellers are symmetric, as in the example in Section 2. In the following Corollary, we ignore the buyer's individual rationality constraint, that is, g, = 0. Individual rationality is discussed afterwards.

COROLLARY1: Consider an environment with at least two sellers. Suppose all suppliers are ex ante identical, that is, us = v, and f, =f, for all sellers s, and suppose g, = 0. Then, ordering suppliers arbitrarily and making take-it-or-leave-it offers of k to each seller in turn is an ex ante socially optimal procedure if and only


if k is the offer that maximizes the expected surplus,

E[v,(q,) - q1lq1 < k] >E[u,(q1) -q1Iq1 < r l Vr,

and increasing the offer yields a negative expected surplus, that is,

E[v,(ql) - q l l k < q l < r ] G O V r > k .

PROOF:Since social surplus is the objective, sellers are identical, and the IR constraint is ignored, we let h, = w, = w, g, = 0, and k, = k for all s E S. To apply Theorem 3, we must also consider the artificial player ?, with hi = gi = 0, f,-(9,) = 1, and k, = 1.

We first show sufficiency. We must prove that (S1)-64) hold. Condition 6 1 ) follows by setting k, = k and (S4) is trivially satisfied. It remains to prove 6 3 ) and 62) . Let 0 < a < b g 1. For any seller st,

Let b = k and a = r'. Since E[wlq,, G k] > E[wlq,?< r'] and the expression above is a convex combination, we have

Since E[wlqSf g k] 2 E[wlq, < r ] Vr, (S3) follows. We now prove 6 2 ) . First, consider the case of s f = S:

~ = E [ w ~ l r ' < ~ ~ < k , ] ~ r ] ,> E [ w l k , ~ ~ , V r > k,,

where the equality follows by definition of S and the inequality follows by assumption. This proves 6 2 ) in this case.

Second, let s t + 5. Consider (11) for seller s with a = k, and b = r. Convexity and the fact that E[wlq, < k,] >, E[wlq, < r ] imply that

This expression combined with (12) proves (b) and completes the sufficiency part of the proof.

We now prove necessity. Suppose (Sl) to (S3) hold. That k maximizes social surplus is immediate from (S3) and the fact that sellers are identical. Consider once more equation (1 1) for s = 1, b = r, and a = k,. Then

where the equality follows by definition of s' and the inequality follows by 6 2 ) for s t = 5. Q.E.D.

REMARK:When maximizing the expected buyer's surplus, a similar result is obtained by replacing u,(q1) -q, in the statement of the Corollary with ul(ql) -q1- (F(q,)/f(q,)).


It follows from Corollary 1that if tendering the same take-it-or-leave-it offer k to all sellers is an optimal mechanism, then k maximizes the unconditional total surplus obtained from each consumer,

(13) ~ [ ( ~ ' ( q l )q1)lqs.k,] > ~ [ ( ~ 1 ( q 1 ) Vr.- - q l ) l q s ~ r ]

For any alternative offers r G k, (13) follows from the first condition in Corollary 1noticing that l /F(k) G 1/F(r). For offers r > k, (13) follows using the second condition in Corollary 1. Condition (131, however, does not suffice to guarantee that tendering the offer k is an optimal mechanism when there is more than one seller since a buyer must weigh both the alternative of no trade and the potential of trade with other sellers against trade with the current seller. It is for this reason that conditional expected surplus plays a role in general in the Theorem and in the Corollary. Of course, when there is only one seller, the option of trade with other sellers is not available. As a result, both the sequential offer mechanism and the auction have the same form: a take-it-or- leave-it offer in the former and a reservation price in the latter. In this case, (13) is both necessary and sufficient to guarantee that an offer of k is optimal.17

Consider now the role of the individual rationality constraint." A take-it-or-leave-it offer ensures that the sellers' expected surplus is always positive since suppliers can decide whether to accept a given offer. Therefore, to verify that the constraint is satisfied we only need to show that the buyer's expected profits are nonnegative. If a take-it-or-leave-it offer of k to a given seller is accepted, the buyer profits are

The offer k, however, is only accepted with probability Fl(k), the probability that the supplier has a quality below k. The probability that at least one supplier will accept the given offers is [I - (1 - Therefore, the buyer's ~ , ( k ) ) ~ ] . expected profits in the symmetric environment are

Thus the IR constraint is satisfied if and only if this expression is nonnegative, and this will occur if and only if

that is, the buyer's expected profits (from an offer of k) are positive. In the particular case of an offer of k = 1, the expression above becomes

E[ul(ql)] 2 1.These observations and Corollary 1yield the following result.

COROLLARY = and2: Suppose all suppliers are ex ante identical, that is, L), v, f, =f, for all sellers s. Suppose the buyer's valuation vl(ql) is differentiable, with

l7 We are grateful to a referee for highlighting this issue. " ~ nexample with a sequential offer game in which the IR constraint binds is f,= 1 Vs E S,

w(q,) = 1/2q, + 1 / 8 if q, < 1/2 and 1 / 4 otherwise. This yields sequential offers of k = 1 / 2 and a = 1.


d ~ , ( ~ ~ ) / d q ~ ]> 1for all q,, and that there are large gains from trade, E[r;,(q,)l 2 1. Then, randomly selecting a supplier and tendering her a take-it-or-leaue-it offer of 1 maximizes expected social surplus.

The corollary identifies a large class of simple environments where it is socially optimal to trade with an arbitrarily selected supplier ignoring all other potential sellers. The example in Section 2 satisfies all the assumptions of this corollary. Thus, a take-it-or-leave-it offer of k = 1 to the first (arbitrarily selected) seller is the socially optimal mechanism in that example.

When suppliers are not identical (but dv,(q,)/dq, 2 I), potential sellers will be ordered according to their contribution to the expected social surplus E[U,(~,)-q,] and a take-it-or-leave-it offer of 1 will be made to the supplier offering the largest contribution (provided that individual rationality is satisfied, i.e., the expected buyer's profits of dealing with the first individual must be positive, E[ul(ql)l 1).

Theorem 3 can be used both to test whether a candidate sequential offer mechanism is optimal and to construct the optimal offers when such a mechanism is optimal. Since (S1)-(S4) are necessary conditions, the failure of any of these conditions for a given set of k,'s immediately implies the failure of the candidate mechanism as a solution to the problem. The conditions may also be used to construct an optimal mechanism for a given environment, for instance, a symmetric environment. Corollary 1 illustrates how such a mechanism can be obtained in the relatively simple case of maximizing expected buyer surplus or expected social surplus with no IR constraint.

The discussion leading to Corollary 2 also suggests a strategy for computing the optimal sequential offer mechanism in the symmetric case including the IR constraints. Complementary slackness implies that a is nonzero only if the buyer's IR constraint binds, that is, (14) holds with equality. Since (14) is also a necessary condition, only k's satisfying (14) with equality are potential solutions to the given optimization problem. To determine whether a specific k is optimal, we use a simple extension of Corollary 1. For any k satisfying (14) with equality, a search over a , a 2 0, can be conducted to see if there exists an a such that

and increasing the offer yields a negative expected surplus, that is,

It is worth emphasizing two additional consequences of Theorem 3. First, the actual number of potential sellers does not play a role in the example (or in the conditions of Corollary 2). Even if there is a large number of sellers, so that the ex ante probability of a high quality seller is very high, and even though high quality is very valuable, the best that can be achieved is the unconditional average quality. Second, Theorem 3 answers a question left open by Samuelson

OPTIMAL PROCUREMENT MECHANISMS 609

(1984). Samuelson's model can be considered as a special case of our model with only one seller. Samuelson proves that take-it-or-leave-it offers maximize the buyer's expected surplus. Take-it-or-leave-it offers correspond to probability of trade functions with only one step. This result is directly implied by Theorem 3. Samuelson also shows that maximizing social surplus requires probability of trade functions with at most two steps but does not determine when one-step mechanisms, i.e., take-it-or-leave-it offers, are optimal and when two-step mechanisms are optimal. Theorem 3 indicates when single or double stepped mechanisms are required.

PROOFOF THEOREM3: (*) The complementary slackness conditions, and the constraints of the dual program (Theorem 2) are manipulated to obtain (S1) through 64).

We begin by establishing an inequality for y from which we derive (S1)-64). Let z E G, be such that z G 1 and E,(fi,z) is constant over [O, k,]. Let r' E [0, k,]. From equation (9) and linearity we obtain that

(lqs Y) ( ~ ~ ~ ( 1 q ~As)< r,fisz, (hs + ags ) f - - < r z f i s ~ ) ,

+(1r'<q2<kxfisz> - ( ~ ' s ( ~ r ' < q ~ < k ~ f i s ~ ) 7 ~ s )( h s + a g s ) f -7)

+($,(I - z ) , (h, +%,If - Y) -(DE,(fi,(l -z) ) ,h , ) = 0. Since E,fi, -E,(fi,z) is constant over [0, k,] and zero elsewhere, ;,(I -z) E

G,. Then (4), the constraint in the dual program, implies that the first and third lines above are nonpositive. The second line must then be nonnegative; therefore, we have

(lr '<q,<k,fisz,~)~( l r '<q,<k,f isz ,(hs + ~ ~ g , ) f )

- ( D E S ( ~ ~ ' < ~ ~ ~ ~ ~ ~ ~ S ~ ) ~AS).

By complementary slackness (81, we have

( ~ ~ , ( l , < , ~ f i , ~ ) ,+ (~~ , ( l , .< . .+ f i~z ) , A,) 0.A,) =

The first term is nonpositive because A, is nonnegative; the second term, then, must be nonnegative. Thus.

(15) ( l r t < q s < k , f i s ~ , ~ ) (hs + a g s ) f ) . ~ ( l r ' < q , < k S f i s z ~

Choose any x E G, such that DESx G 0. The first order condition (4) and the nonnegativity of A, imply

(16) ( x , ( h , +ag,)f) G (x ,y ) . We will prove (Sl) through (S3) by combining (15) and (16) for different

functions x. First, to prove (Sl) let x = lq3< ,fi, for any r E [0, k,]. Then (16) becomes

(1qSGrfis,( h s + a g s ) f ) ~ ( l q ~ < r f i s , ~ ) G ( f i s , ~ ) ,

where the second inequality follows by the nonnegativity of y. By (15) with


r ' = 0 and z = 1the previous inequality implies

Using the definition of the dual pairing ( . ,.) and of fi,, (Sl) follows. Second, to show 6 2 ) and 6 3 ) let x = k,, j , for any lq,Crlr,<q,,< r E [O,l].

Using (15) for seller s' with z = lqs,,., (15) and (16) yield

Vr E [O,1] and r ' E 10, k,,].

Using the definition of conditional expected value, the expression above implies (S2) and 63).

Finally, 6 4 ) is formed by a constraint of 9and its complementary slackness condition. It is therefore necessary.

(*) Using (S4), it is clear that {fi,},,, is feasible. We will show that if 61)-(S4) hold, then there exist nonnegative a , y, and {A,},,, that satisfy (31, (41, (5) , and (6) in Theorem 2.

We begin with a Lemma establishing the existence of certain functions that will be used to define the variables of the dual problem.

- LEMMA 1: Assume (S3) holds. Then, for any s E S there exists a function h, EL,( I ) such that

- (lr'<q,<k,, (hs +ags)fs )(19 ) h,(k,) = inf

r '<k , Fs(ks) -Fs(rl) f

The proof of the Lemma is provided in the Appendix. We now define the dual variables and check that they satisfy Theorem 2,

conditions (5), (31, (6), and (4) in that order. Let a be as in the hypothesis of Theorem 3 and define y by

where h, satisfies the conditions of Lemma 1. By construction, y EL,(I~) and therefore it may be regarded as an element of L:(I~). Since h, is nonnegative, the dual variable y is nonnegative. This establishes (5).

We now verify that the value of the dual program equals the value of the primal. By Lemma 1 (18), (1, fish,f ) = (fi,, (h, + ag,)f ). Then, using the


definition of y and condition (S41, we obtain S S

( 1 , ~ )= C (5,,( h , + w , ) f ) = C (fis7h,f ). s = 1 s = l

Thus, the value of the dual equals the value of the primal, equation (3). We define a dual variable A, for each seller s. Recall that s is the last seller

to trade with positive probability. Such a seller always exists because of the introduction of the artificial seller. Define

i; =Z,(l).

Observe that h takes on the lowest value that y ever acquires. We use h to define A,. Let

We now verify that the variables, A,, are nonnegative. The nonnegativity of A, for q, E [0, k,] follows from Lemma 1,expression (17). To check the nonnegativity of A, for q, > k,, replace s' with s in (S2). First, consider any s <3. Since 6 2 ) requires that

for all r ' < k, and for all r > k,, the above inequality holds for its infimum. Using (19)

Multiplying both sides by (F,(r) -F,(k,)), we have

(lks<qs<r,( A - ( h s + a g s ~ ) f s ) ~ o ,

which is what we set out to prove. Second, consider any s > s . Then k, = 0 and 6 3 ) directly implies that the second part of the definition of A, is nonnegative. Therefore, (6) is satisfied.

Finally, we must prove that the dual variables satisfy the constraint of the dual, that is, that A,, a, and y satisfy (4) in Theorem 2 for z E G,. To verify this inequality we first find an expression for (DE,z, As) that does not directly use A,.

LEMMA2: For any z E G,, there is a constant K 2 0 such that -

(DEs~7~s)=~+(lqs<kr,(hx+ags-hs)f)

+ (I,, k ~ ,(4+ "9. -h)f ).

612 A. M. MANELLI AND D. R.VINCENT

The proof of the Lemma is provided in the Appendix. We verify that A,, a , and y satisfy (4). Choose z E G,. We have

"0, where the second equation follows using Lemma 2. The first inequality uses the definition of y and the fact that K 2 0. The last equation is obtained using the fact that fis,lqs<0 for any st >s and that lq3 = lq,> k, . The final = > k,CS= fii inequality follows from the definition of h, and h because fi, = 0 for s > s and by (20) applied also to h. Q.E.D.

6. AUCTIONS

A common institution for determining trade between a buyer and many sellers is to invite bids from the sellers. We concentrate on the many seller-single buyer analogue of a Vickrey or second price auction. Each seller s is assigned a reservation price k,. Sellers submit sealed bids simultaneously. Bids above the corresponding reserve price are discarded. Of the remaining bids, the lowest bidder wins the contract and receives either the lowest of the other S - 1bids in payment or her reserve price, whichever is lowest. It is well known that this indirect mechanism has a "unique" equilibrium in undominated strategies (Vickrey (1961)). Each seller truthfully "reveals" her type in that a seller s of type q, bids exactly q,.19 The buyer only chooses whether to participate in the auction. We assume that the buyer participates in any auction which yields a nonnegative surplus.

The outcome of a Vickrey auction can also be obtained as a Nash equilibrium in other types of auctions, for instance a "first price" auction with reserve prices.20 When all the bidders are symmetric, first price auctions have in general a unique symmetric Bayesian Nash equilibrium in pure strategies. When bidders differ, however, there may also exist other equilibria. Multiplicity of equilibria is

19 .B ~ d s of sellers with types above their reserve prices are not determined. Since these sellers never trade, the equilibrium outcome (in undominated strategies) remains unique.

20 In this auction the lowest bidder receives the lowest of her bid and her reserve price.


the basis of a well-known critique of the use of the revelation prin~iple.~' We focus on the Vickrey form of an auction to avoid this problem (Wilson (1992)).

We order potential sellers so that k , 2 k , > . . . k,. The probability of trade functions, {c,},,,, generated by the Vickrey auction are

1 if q, < min {q,,, k,} Vs'with q,. ,< k S t ,a , (q ) =

0 otherwise.

Notice that as long as k , 9 1 Vs, then 7 ~ J l l l )= 0; thus the buyer's expected profits can still be expressed as C ~ = , ~ [ w , ( q , )F,(q,)/f,(q,)l.-

Theorem 4 characterizes the environments in which an auction is optimal. For conciseness, we assume that 1 > k , = k,. When this is not so ( k , > k,), there are additional necessary and sufficient conditions which resemble those in Theorem 3. We leave this extension to the reader.

THEOREM4 (Auction): An auction {fi,},, ,with k , = k , < 1 is a solution to 9 if and only if there exists a > 0 such that for all s in S

( A 2 ) h,(q,) + ag,(q,) is nonincreasing and nonnegatirje a .e. in [0, k,] ;

(A3) E [ h ,+ ag,lk, < q, ,< r ] ,< 0 V r > k,;

The proof is provided in the Appendix. Conditions (Al) and (A2) illustrate how special the environments are in

which auctions are optimal. When maximizing buyer surplus, condition (A21 requires that w,(q,) -F,(q,)/ f,(q,) be positive and decreasing functions in q, over the relevant range. When maximizing social surplus, w,(q,) + a(w,(q,) -F,(q,)/f,(q,)) (for some positive a ) must be positive and decreasing functions in q, over the relevant range. Condition (Al)indicates that these functions must be identical over the range in which types have a nonzero probability of trade; for an auction to be optimal, sellers must be essentially identical.

The third condition basically determines the reservation prices k , assigned to each seller s. If trading with a seller s when her quality q, is above the reservation price k , resulted in expected gains, then k , would not be the optimal reservation price. Thus, (A3) determines the seller types q, > k , who never trade, and helps to decide whether an auction is optimal.

As in Theorem 3, condition (A4) is the buyer's participation constraint. Myerson (1981) shows that w,(q,) -F,(q,)/f,(q,) symmetric and decreasing

everywhere is sufficient for a reserve price auction to maximize buyer surplus. It follows from Theorem 4 that these conditions are also necessary although only over the range [0, k,]. Condition (A3) needs to be satisfied only over the set of

21 See, for instance, Demski and Sappington (1984).

614 A. M.MANELLI AND D. R. VINCENT

nontrading seller types. Theorem 4 thus implies the following

COROLLARY3: Suppose that E[u,(q,)] 2 1. An auction with no reserve prices maximizes social surplus if and only if:

(a) all sellers are identical, that is, u s ( . )= v,(.) a.e. for all s E S , (b) u,(q,) -q, is nonincreasing and nonnegative a. e.

Corollaries 2 and 3 imply that when the gains from trade are large enough, an auction is optimal if the marginal value of quality for the buyer is less than the marginal value of quality for the sellers. If, on the contrary, the buyer values marginal quality more than the sellers, then a take-it-or-leave-it offer is the optimal institution from a social perspective.

In the analogue to the typical many buyer-single seller symmetric auction environment, w(q) = E -q. Conditions (Al) to (A4) are automatically satisfied as long as G >, 1. The common view that auction-like mechanisms are at least Pareto efficient stems from these special features of the environment. Notice that it is not simply the introduction of the quality variable which reverses the common wisdom but also the degree to which the buyer's valuation depends on quality.

To illustrate further the relationship between Corollaries 2 and 3, consider the problem of maximizing social surplus when du(q,)/dq, = 1(i.e., w(q) = c is a constant) for all q. In this case, trade with any seller type yields the same social surplus. Furthermore, if c is high enough, the buyer's individual rationality constraint does not bind. Either a no reserve price auction or a sequential offer mechanism (trading with the first seller with probability one) is optimal. However, for low values of c, the constraint is more likely to bind. Since allocation of trade is not an issue in this example, an auction is typically better because this mechanism is more likely to satisfy the participation constraint-competition from the sellers in the auction can be used to yield a high enough surplus to induce the buyer to participate.

7. CONCLUSIONS

It is fairly intuitive that the optimality conditions for sequential offer mechanisms (Theorem 3) are necessary. That these conditions are also sufficient is the main content of the Theorem. The conditions in Theorem 4, on the other hand, are clearly sufficient for the optimality of an auction. It is the fact that symmetry and monotonicity are also necessary that provides the substance of that Theo- rem. Our results also suggest that in some sense, the class of environments for which reserve price auctions are optimal is "smaller" than that for sequential offer mechanisms. For an auction to be optimal, the objective function must be monotonic. There is no such monotonicity requirement for the optimality of

22 Strictly the Corollary does not follow from Theorem 4 because there the reserve prices must be less than 1. It is straightforward, however, to adapt the proof appropriately.


sequential offer mechanisms even in symmetric environments. More generally, there is no symmetry requirement for the take-it-or-leave-it game. On the other hand, even auctions with asymmetric reserve prices require some symmetry across seller types.

The duality approach that we employed can be potentially used with any implementable mechanism. For example, as long as the participation constraints are satisfied, a mechanism which consists of sequential offers followed by an auction in the event of rejections is also incentive compatible and easily represented as a direct revelation mechanism. So, too, is a variation on the sequential offer game in which, in the event of rejections from all sellers, a new, higher round of offers is made. In Manelli and Vincent (1994) we show how the techniques employed here can be used to construct the dual variables that enable a designer to determine if either of these alternative institutions is optimal. Of course, other institutions could be examined as well. The particu- larly simple form of the dual program provides a guide to how other dual variables can be constructed to yield tests of a wide range of potential solutions to the original primal problem.23 Given the advantages of modeling agents' types in the continuum, we believe that the infinite dimensional linear programming techniques used here will prove useful in other situations in which the natural optimization problem can be expressed as a linear program.

Department of Managerial Economics and Decision Sciences, J. L. Kellogg Graduate School of Management, Northwestern University, Evanston, Illinois 60208, U. S.A.

Manuscript receiued September, 1992; final revision receiued October, 1994.

APPENDIX

PROOFOF LEMMA1: Choose any s and let

(21) b = sup ( I q s < r , (hs + ags) fs ) > inf

( l r , < q S < ks, (hs + W s ) f s ) = a > 0 ,

r < k , r 8 < k , Fs(k s )-F S ( r f )

where the first inequality follows by letting r = k , and r' = 0. The second inequality follows from 6 3 ) for s = s because h,-= 0 =g,-. Since h , +a g s is in L,, it can be shown that b is finite.

Let i satisfy

Dividing both sides by F,(k,), it is immediate that such an 0 < i < k , exists. Define

Then A, is nonnegative and, by definition of i,(18) holds. It is also immediate that (19) holds.

23 Gale and Holmes (1993) solve a specific airline pricing problem by constructing dual variables in a more restrictive adverse selection problem.


We now prove that h, satisfies (17). For any r E [0, F], we have, by definition of b, that

For any r E [ f , k,], we have

( l q , < r > h s f , ) =F, i f )b + (Fs(r) -FsiF))a =Fsi f ) (b -0 ) +Fsi r )a

= ( lq ,<k , , i h s +ags)fs ) - (Fs(k,) -Fs i r ) )a

= ( l q S s r >ihs+ags)fs) + ( l r<qA < k s > i h s +ags)fs) - (Fsiks) - Fsir))'

=(I,,,,, ih,+ag,)f ,)

where the first three lines follow by definition of h,, F, and equation (22); the inequality follows by definition of a . This completes the proof of (17).

It remains to prove (20). For s' < s, x,. E [0, k,.] and x, E [O, k,],

(I,.< qs,< k,.r ih,, + ffg,-)fs,) ( l q , < r > ih ,+ags ) f s ) -h,.(x,.) inf > SUP 2 hs(xs)>

r ' < k , . Fs,(ks8)-Fsp(r') r < k s Fsir)

where the first and last inequalities follow from the definition of h, and h,, and the intermediate one from (S3). Q.E.D.

PROOFOF LEMMA2: Let z be in G, and let {c ,}~ -~ ,{b , )~ :~ denote the points of discontinuity of E,z (if any) in (0, k,) and (k,, 1) respectively. Also, let co =0, c, = k,, bo = k,, and bM = 1. For x E (O,1] define z+(x ) = lim,,oE,z(x - E). For x E [0, I), define z-(x) = lim,,oE,z(x + E).

Since E,z is piecewise-c', we have

Integrating by parts in each interval (c,-,, c,) and (b,-,, b,),

Noticing that A,(co) = A,(c,) = A,(bo) = 0, rearranging terms, and using the definition of A,, we obtain that


where A(x ) = z + ( x ) - z- (x) . Since z E G,, A(c,) and A(b,) are nonnegative. Then, the nonnegativity of A, implies that the first two lines are nonnegative. We call them K. Q.E. D.

PROOFOF THEOREM4: (-1 TO prove ( A l l and (A2),we first show that for any x E G,,

(23 ) ( D E , ( ~ , $ . ~ ~ ) , A , )= 0.

Rewrite j,iq) as follows (if s = 1, replace n,,,(.)with 1):

Thus,

Note that f,(q,) > 0 Vq, implies DE,fi,(q,) < 0 provided q, G k,. Then complementary slackness (8) implies (23).

We now prove that h,(q,) + ag,(q,) is positive, the second part of (A2). Let z be such that E,B,z E C 1 ( I )and 1 g z g 0. Then (9) implies

( B , z , ( h , + ag , ) f - Y ) - (DE , ( f i s z ) , A,)

+ ( B , ( I - z ) , ( h , + a g , ) f - y ) - (DE , (B , (~ - z ) ) , A , ) = o .

It follows from (4)(as it did in Theorem 3) that both terms in the expression above are nonpositive. Therefore, (DE,i$,z), A, ) = ($ ,z , (h , + ag,)f - y ) . Using (23)with x =$,z we have

The nonnegativity of y and the Lebesgue Convergence Theorem imply

( lqS.k, f isz , ( h , + a g s ) f ) 2 0.

In turn, this implies that h, +ag , is nonnegative a.e. in [0,k,], the second part of (A2). We now prove that V z EL,(IS),

( 25 ) ( lq , .k , lq ,<k,Bt~ , ( h , + a g s ) f- ( h i + a g i ) f ) G 0

and then we indicate how ( A l l and the rest of (A21 can be obtained from this expression. Let z be such that E,( lqs .k , lqr . ,k , .~s~) and E,.il,,k,l q s , , E C1(Z). BY (231,k , , f i s ~ )

(DE,.(lqSG k , lq3 , ,k , , ix~) , A,.) = 0. Then the first order condition (4) applied to s' implies

Combining (24) and (261, we have

The Lebesgue Convergence Theorem implies that the expression above holds for all z E L,(zS), completing the proof of (25).

We have already established that h,(q,) +ag,(q,) is nonnegative a.e. in [O,k,]. The following lemma proves that (25) implies ( A l ) and the rest of (A2).

LEMMA3: Suppose (25) holds. Then h,(q,) +ag,(q,) satisfies ( A l ) and ( A 2 ) .

PROOFOF LEMMA3: For any s, choose i < s (and for s = 1, let i = 2). Then [O, k,] is a subset of [0, k , ] since k , 2 k,. By (251, we have

and the inequality is reversed (also a.e.) when qi G q,.


We first prove that h,(q,) + ag,(q,) is nonincreasing in q,. Suppose h,(q,) + ag,(q,) is strictly increasing on a set A c [0, k,] of positive measure.24 Take a <b < c in A c[0, k,] so that

h,(a) +ag,(a) a hi (b ) +a g i ( b ) and h,(b) + agi (b )2 h,(c) +ag,(c).

Therefore, h,(a) + ag,(a) 3 h,(c) + ag,(c), which is a contradiction. This proves that h,(qs)+ ag,(q,) is nondecreasing a.e. in [O, k,].

Second, we prove that (25) implies (All . Let B c [O, k,] be such that h,(q,) + ag,(q,) and hi(qi)+agi(q,)are nonincreasing in B and B has full measure. We proceed by contradiction. Let B' = {q, E B 1 hi(qs)+cugi(qs)>h,(q,) +ag,(q,)} and suppose B' has positive measure. Pick any qi, qs f? B' with qi>q,. By (271,we have

hs(qs) + ags(qs )a hi(qi) + agi(qt)

for almost all such q, and q,. Then

hi(4s) +agi(qs)- (hi(4i) + ag , (q , ) ) hi(qs)+agi(qs)- ( h s ( q s )+a g s ( q s ) )> 0, where the second inequality follows because q, E B'.

Letting q, approach q,, it follows that q, must be a discontinuity point of hi(qs) restricted to B. (Note that almost all points q, can be approximated by points qi in B'.) Since this is valid a.e., hi +as i must have a positive measure of discontinuity points when restricted to B. This is a contradiction because hi + agi is monotonic in B. Q.E.D.

We now prove ( ~ 3 ) .Since 5, + ,(I - Ef= E G,, by (4) we have

Complementary slackness (9) yields the first equality and (10) the second. The final inequality is established by observing that, from (8) and (61,

since Ex($,+ 1, <,..(I - Ef=,i , ) ) is decreasing. Finally, (A4) :s formed by a constraint of 9 and its complementary slackness condition. It is

therefore necessary. (*) As in Theorem 3, it follows from (A41 that {js},,, is feasible. We will show that if

(All-(A41 hold, then there exist a ,y, and a A, for each seller s that satisfy (3), (4),(5) and ( 6 ) in Theorem 2.

Define S

Y = C b,(h, + % ) f . s=l

24 .Since F, has a strictly positive density f,, statements hold <-almost everywhere if and only if they hold a.e. with respect to the Lebesgue measure.


The nonnegativity of y, expression (51, follows from (A2). It follows from (A4) that y satisfies (3). Define

if q ,<k, ,

As(qs) = (;Ik,. x, . a,, - (h, +ag,) f,) otherwise.

The nonnegativity of A,, expression (61, also follows from (A2). We must prove that the first order condition, (41, is satisfied. For any z E G,, there exists K > 0

such that

The proof of (28) follows that of Lemma 2. We leave the details to the reader. We must show that the following expression is negative. For any z E G,, we have

<( lqAsk,b ,z , (h, +ags ) f - Y) +(lqAsk,(l -b,)z, (h s +ags)f -Y)

where the equality follows from (281, noticing that z = I,$, ,,z+ l,, > ksz. The inequality follows because K a 0.

The first term in (29) is zero by definition of y and the last term is negative because y is nonnegative. To determine the sign of the second term, notice that lqAGk$l =-b,) 1, Ci + r , i sfii and (El,+,ai) X (Cf=l$JxJ)= Ci+,$,xi, for any set of functions x,. Therefore, rebit ing the middle term In (29),

As we set out to prove, this term is nonpositive. By (Al), h, + ag, = h, +a g , a.e, and, by (A21, h, + ag, is nonincreasing. Since the term is evaluated for q, <qs (i.e., on the support of fii) the result follows. Q.E.D.

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