Sequential Auctions of Endogenously Valued Objects

Ž .Games and Economic Behavior 36, 74�103 2001doi:10.1006�game.2000.0802, available online at http:��www.idealibrary.com on

Sequential Auctions of Endogenously Valued Objects1

Ian L. Gale

Department of Economics, Georgetown Uni�ersity, Washington, DC

and

Mark Stegeman2

Department of Economics, VPI, Blacksburg, Virginia 24061-0316

Received October 9, 1995; published online April 26, 2001

Two completely informed but possibly asymmetric bidders buy or sell identical‘‘claims’’ in sequential auctions. They subsequently receive monetary prizes thatdepend upon the final allocation of claims. Iterated elimination of weakly domi-nated strategies leaves a unique Nash equilibrium. For any prize schedule, pricesweakly decline as the auctions progress, and points of strict decline have a simplecharacterization. For one class of prize schedules, which arises naturally if duopolistsbid for a scarce input, the equilibrium is completely characterized; many initial

Ž .allocations generate the same final unequal division of claims, which may beinterpreted as the natural market structure. Journal of Economic Literature Classi-fication Numbers: D43, D44, L11. � 2001 Academic Press

1. INTRODUCTION

The theory of auctions emphasizes one-shot auctions, either auctions ofsingle objects or simultaneous auctions of several objects. In reality,however, unrelated sellers often auction similar objects sequentially. Evena single seller may auction similar objects sequentially, mimicking acollection of unrelated sellers. Examples of goods auctioned sequentiallyinclude art, wine, fish, timber, agricultural products, mineral rights, satel-lite broadcast licenses, and government debt.

In this paper, we assume that two completely informed and asymmetricbuyers bid for N identical objects, called claims, which are auctioned

1 We are grateful to two anonymous referees and to the participants of the UNC�Duketheory seminar for helpful comments.

2 To whom correspondence should be addressed. E-mail: [email protected].

740899-8256�01 $35.00Copyright � 2001 by Academic PressAll rights of reproduction in any form reserved.

SEQUENTIAL AUCTIONS 75

sequentially by N sellers. The sellers are uncoordinated, in the sense thatŽ .each sells her claim in a second-price equivalently, ascending bid auction,

independent of the outcomes of the other auctions. After all claims aresold, each buyer receives a monetary prize that depends upon the finalallocation of claims.3 We pare the set of equilibria by repeated eliminationof weakly dominated strategies and study the price sequence, the order inwhich buyers win claims, and the final allocation of claims. The results arenontrivial because we assume that the value of one claim depends upon

Ž .the number of claims obtained. Krishna 1990, 1993 calls this the problemof endogenous valuations.

Instead of assets, our claims may represent liabilities, so that our two‘‘buyers’’ are actually duopolists selling to fragmented buyers. The formalanalysis is identical. Markets in which two well-informed sellers bid se-quentially for contracts include highway construction, waste disposal, con-sulting services, HMOs, computer procurement by government agencies,milk procurement by school districts, aircraft, and military hardware.

Section 2 defines the general model, Section 3 derives a series ofpreliminary results, and Section 4 states our first major result, which

Ž .concerns equilibrium price sequences. Pitchik and Schotter 1988 predictand experimentally confirm that prices typically decline over the course of

Ž .sequential auctions, and Ashenfelter 1989 provides empirical support forthe proposition that prices decline. We do not impose the budget con-straints introduced by Pitchik and Schotter, but Theorem 1 shows thatdeclining prices are nevertheless a fundamental property of our model. For

Žany specification of terminal prizes i.e., payoffs before deducting prices.paid for claims , prices weakly decrease along any equilibrium path. If the

bidders are sellers, as in procurement problems, then this result impliesthat the procurers pay weakly higher prices as the auctions progress.Theorem 1 also characterizes the points at which the price actually falls

3 In Section 9, we discuss studies that allow bidders to receive prizes before the auctionsare over. Early prizes create a bonus for early acquisition and are likely to reinforce ourresult that prices never increase and often decline during the course of the auctions. In manyrealistic settings auctions occur much more rapidly than prizes are received, so that the earlyacquisition bonus is minimal. Examples include auctions of durable goods or procurementcontracts, agricultural products, corporate consolidation waves, or any circumstance in whichthe auctioning of inputs creates a new industry. We note also that it is common to have fewbidders in sequential auctions, especially in the procurement context. Marshall and RaiffŽ .1998 found an average of fewer than three bids per contract over the course of 2368 milkprocurement auctions in Georgia. In the FCC’s sequential auction of direct broadcastsatellite licenses in 1996, only three bidders participated.

GALE AND STEGEMAN76

and describes a large set of cases for which the price must fall on at leastone equilibrium path.

Sections 5 through 7 narrow the focus to a class of problems in whichthe buyers’ prizes are symmetric quadratic functions of the final allocationof claims, though the buyers may start with an asymmetric initial allocation

Ž .of claims. We assume that the total prize the sum of the buyers’ prizesand each buyer’s valuation of the marginal claim increase in the concen-tration of claims. These conditions arise naturally in duopoly games, whereclaims represent scarce inputs and prizes represent profits.

Given such prize schedules, Theorem 2 characterizes the equilibriumand describes a simple way to calculate it. The equilibrium has interestingproperties. In the early auctions, the leader outbids the follower only when

Ž .necessary to protect her lead i.e., when it falls to one , but then she bidsaggressively to win the final string of auctions. A striking result is that thefinal allocation of claims is largely independent of the initial allocation ofclaims. For instance, in an example below, the equilibrium division of 16claims is always 10 claims for one buyer and 6 for the other if neither

Ž . Ž .buyer starts with more than 6 claims. We call 10, 6 and 6, 10 the modalallocations of claims, and we interpret them to be the ‘‘natural’’ marketstructure induced by a given prize schedule. The modal allocations reflectconflicting influences: anything that causes the total prize to rise morerapidly with concentration tends to increase modal concentration, butanything that causes the follower’s valuation of the marginal claim to risemore rapidly with concentration tends to decrease it. We also show that

Ž .splitting the claims i.e., more auctions of smaller claims generally reducesmodal concentration. In the limit, however, as the claims become in-finitesimal, the modal allocations are bounded away from equal division.

Theorem 2 also shows that if the initial allocation is monopolistic, thenthe monopolist often does not block new entry. Gilbert and NewberyŽ . Ž1982 show that a monopolist preempts entry when a single new claim in

. Ž .their case, an innovation becomes available, but Krishna 1993 showsthat a monopolist facing many potential entrants and opportunities forentry may elect not to preempt entry. Our result covers the intermediatecase of several opportunities for entry by a single potential rival; we showthat the monopolist generally does not preempt entry, although our singleentrant may gain a smaller foothold in the market than Krishna’s manyentrants.

Finally, Section 8 shows that adding a relatively small fixed cost ofproduction to a Cournot game can greatly increase the tendency towardmonopoly. In other words, the first-mover advantage, which has beenexplored extensively in other contexts, becomes quite pronounced. Section9 concludes and discusses some related contributions to the literature.


2. THE MODEL

A fixed number of identical objects, which we call claims, are initiallyallocated among two buyers and an arbitrary number of sellers. The sellerssell their claims individually, in a sequence of second-price auctions. Afterall claims have been sold, each buyer receives a monetary prize, whichdepends on the final allocation of claims. For instance, the prize could bethe profit from a production game, where the claims represent capacity orsome other rationed input. The buyer’s payoff is her prize less whatevershe paid for the claims. The buyers play the sequential auction game withcomplete information: they know the auction rules, the initial allocation ofclaims, and the schedule of prizes.4 The rest of this section definesnotation and adopts a standard refinement which selects a single Nashequilibrium.

At any point during the sequence of auctions, the current allocation ofŽ .claims is an ordered pair, x, y , where buyer 1 owns x claims and buyer 2

owns y claims. The set of possible allocations can be viewed as a map,Ž . Ž . Ž .in which 0, 0 branches to 1, 0 and 0, 1 , which in turn branch to

Ž . Ž . Ž . Ž .2, 0 , 1, 1 , and 0, 2 , etc. The map terminates at the allocations N, 0 ,Ž . Ž .N � 1, 1 , . . . , 0, N , where N � 1 denotes the exogenous total number

�Ž . 2 4of claims. Section 5 depicts such a map. Let A � x, y � � � x � y � N�� 4denote the set of intermediate allocations, where � � 0, 1, 2, . . . . Let�

�Ž . 2 4T � x, y � � � x � y � N denote the set of terminal allocations. The�Ž .final equilibrium allocation is an element of T. For any x, y � A,

Ž .‘‘auction x, y ’’ refers to the auction that occurs when the current alloca-Ž .tion is x, y .

Ž .For each x � 0, 1, . . . , N and i � 1, 2, f x denotes buyer i’s prize if theiŽ .final allocation is x, N � x . If claims are assets, then f is increasing and1

f is decreasing in x. If claims are liabilities, such as promises to build2roads, then f is decreasing and f is increasing in x. Adding any fixed1 2

Ž .c � � to f x , for all x, transfers c to buyer i unconditionally, affectingi i ineither the equilibrium bids nor our results. The buyers and their prizesmay be asymmetric.

We will consider only the unique equilibrium that survives iteratedelimination of weakly dominated strategies. This allows us to confine

4 Complete information is not essential: the model can be interpreted as the limit of aŽ .sequence of models in which payoff noise vanishes cf. fn. 4 . Complete information greatly

Ž . Ž .simplifies the analysis, however. Dudey 1992 and Krishna 1993, 1999 , in their studies ofsequential auctions with endogenous valuations, also abstract away from incomplete informa-tion. If the prize schedules are unrestricted, then our assumption that some claims beallocated among the buyers initially does not generalize the model in any meaningful way. Ifthe prize schedules are symmetric, however, as we assume in Section 5, then the possibility ofinitial allocation reintroduces the possibility of asymmetric bidders.

GALE AND STEGEMAN78

attention to bidding strategies such that each buyer’s bid is a deterministicŽ . Žfunction of the current allocation x, y i.e., not of the history that

.produced it . Let B : A � � denote buyer i’s bidding strategy, whereiŽ . Ž . ŽB x, y denotes buyer i’s bid in auction x, y . In the correspondingi

Ž .ascending-bid auction, equilibrium prices are unchanged but B x, yi.represents the most that buyer i is willing to bid. Given the bidding

Ž .strategies B and B , let V x, y denote buyer i’s valuation of allocation1 2 iŽ .x, y � A � T , meaning her equilibrium payoff if the initial allocation of

Ž . Ž .claims is x, y . V x, y thus does not account for the prices that theiŽ .buyers paid to get to allocation x, y . Valuations can be calculated

through backward induction. The valuation of a terminal allocation is theimplied prize:

V x , N � x � f x , x � 0, 1, 2, . . . , N ; i � 1, 2. 1Ž . Ž . Ž .i i

Ž . Ž . Ž .If B x, y � B x, y for a given intermediate allocation x, y , then buyer1 2Ž .1 wins auction x, y and pays buyer 2’s bid, implying:

V x , y � V x � 1, y � B x , y ; 2aŽ . Ž . Ž . Ž .1 1 2

V x , y � V x � 1, y . 2bŽ . Ž . Ž .2 2

Ž . Ž . Ž .Symmetrically, if B x, y � B x, y , then buyer 2 wins auction x, y ,1 2implying:

V x , y � V x , y � 1 ; 3aŽ . Ž . Ž .1 1

V x , y � V x , y � 1 � B x , y . 3bŽ . Ž . Ž . Ž .2 2 1

Ž Ž . Ž . Ž . Ž . .We show later that 2 and 3 both hold if B x, y � B x, y . Given the1 2Ž . Ž .bidding strategies B and B , recursive application of 1 � 3 yields the1 2

buyers’ valuations of every allocation.Complete information implies the existence of a continuum of equilib-

rium bidding strategies.5 We assume, however, that each bidder, in eachauction, bids the incremental valuation obtained by winning that auction:

B x , y � V x � 1, y � V x , y � 1 ; 4aŽ . Ž . Ž . Ž .1 1 1

B x , y � V x , y � 1 � V x � 1, y . 4bŽ . Ž . Ž . Ž .2 2 2

Ž .This standard refinement e.g., Dudey, 1992; Krishna, 1993 may be justi-fied by appealing to the iterated elimination of weakly dominated strate-

5 � �� Ž Ž . Ž .. Ž � �� .For any sufficiently small b and sufficiently large b , B x, y , B x, y � b , b1 2Ž Ž . Ž .. Ž �� . Ž .and B x, y , B x, y � b , b are both equilibria of any auction x, y such that1 2

Ž .x � y � N � 1. A similar observation applies to every auction x, y � A, implying that anyfinal allocation of claims is supported by a subgame perfect Nash equilibrium commencing

Ž .from allocation 0, 0 .


Ž . Ž .gies. Henceforth, B x, y and V x, y denote bids and valuations in thei iŽ . Ž .unique equilibrium implied by recursive application of 1 � 4 .

The conventional argument for using weak dominance as a criterion forequilibrium selection is that any other equilibrium is sensitive to strategicand exogenous uncertainty. In the present model, it can be shown that ifprizes are slightly noisy and buyers receive private signals for that noise,then the dynamic game has a unique Nash equilibrium, which approachesthe equilibrium we study in the limit as the noise vanishes.6

Ž . Ž . Ž . Ž . Ž .If B x, y � B x, y then, given 4 , 2 and 3 generate the same1 2Ž . Ž .values for V x, y and V x, y . The tie-breaking rule thus does not affect1 2

equilibrium bids and valuations, but it could affect the final allocation oflicenses. The following definition of equilibrium is agnostic about who winsties.

DEFINITION 1. The equilibrium of the auction game comprises theŽ . Ž . Ž .bidding functions B x, y and B x, y , for all x, y � A, as determined1 2

Ž . Ž .recursively by 1 � 4 . An equilibrium path from the arbitrary allocationŽ . Ž . Ž . Ž .x, y � A is any sequence of allocations, x, y , x , y , x , y , . . . ,1 1 2 2Ž .x , y , where x � y � x � y � i for all i, that is consistentN�x�y N�x�y i i

Ž .with a weakly higher bid winning each auction. For any x, y � A, it isŽ . Ž . Ž .convenient to define the following terms: If B x, y � B x, y , then x, y1 2

Ž . Ž . Ž .is a 1-allocation. If B x, y B x, y , then x, y is a 2-allocation. If1 2Ž . Ž . Ž . Ž .B x, y � B x, y , then x, y is a tie allocation; if not, then x, y is a1 2

Ž .pure 1- or pure 2-allocation. The price paid in auction x, y is

P x , y � min B x , y , B x , y . 5Ž . Ž . Ž . Ž .1 2

Our analysis abstracts from questions of auction design, implicitly as-suming that the fragmented sellers are unable to coordinate their actions.We also disregard reserve prices, implicitly assuming that sellers’ valua-tions are so low and their information about the buyers’ valuations is sopoor that any optimal reserve price is nonbinding. Also, if the bidders

6 Suppose that buyer i privately observes � just before the auction of the nth claim andi nreceives a payment of � for each claim held immediately after that auction. Assume thati nthe � are statistically independent and symmetrically distributed random variables, withi nzero mean and arbitrarily small variance. Then the unique Nash equilibrium in any givenauction is for each buyer to bid the expected increment to her valuation if she obtains thenext claim. Since, in any auction, the effect of the current noise upon the current bid isarbitrarily close to zero with probability arbitrarily close to one, it follows by backwardinduction that, except for tie allocations, the equilibrium paths are unchanged with probabil-ity arbitrarily close to one. Since the resolution of ties does not affect valuations, valuations

Žand hence bids are changed arbitrarily little with probability arbitrarily close to one. The.realization of the noise variables will, however, determine which equilibrium path is selected.

GALE AND STEGEMAN80

know a seller’s valuation, then that seller may be unable to make acredible commitment to a reserve price exceeding that valuation.

To apply the concepts of convexity and concavity to functions withdiscrete domains, we amend the standard definitions in the obvious way.For functions of allocations, we define these concepts only on loci repre-senting a fixed number of total claims.

DEFINITION 2. Suppose that the domain of a function g is a subset ofŽ . Ž . Ž .the integers. Then g is: convex at x if 2 g x � 1 g x � g x � 2 ;

Ž . Ž . Ž . Ž .concave at x if 2 g x � 1 � g x � g x � 2 ; quasi-concave at x if g x � 1Ž Ž . Ž .. Ž . Ž Ž .� min g x , g x � 2 ; quasi-convex at x if g x � 1 max g x ,

Ž ..g x � 2 . Suppose that the domain of a function g is a subset of A � T.Ž . Ž .Then g is convex concave, quasi-concave, quasi-convex at x, y if

Ž . Ž .g x � k, y � k is convex concave, quasi-concave, quasi-convex in kat k � 0.

3. PRELIMINARY RESULTS

This section states several preliminary results concerning equilibriumpaths and prices, which lay the foundation for our two main theorems.None of these results requires symmetric buyers. Let:

f x � f x � f x ;Ž . Ž . Ž .1 26Ž .

V x , y � V x , y � V x , y .Ž . Ž . Ž .1 2

Ž .The bidding rule 4 implies directly that any equilibrium path takes,Ž .locally, the direction of higher total valuation. In other words, x, y is a

Ž . Ž .1-allocation if and only if V x � 1, y � V x, y � 1 , and it is a 2-alloc-Ž . Ž .ation if and only if V x � 1, y V x, y � 1 . We call this the principle of

Ž . 7 Ž . Ž .maximizing local �aluation MLV . Given 2 � 4 , MLV yields a formulafor calculating valuations recursively, without reference to bids. For anyŽ .x, y � A:

V x , y � max V x , y � 1 , V x � 1, y � V x , y � 1 ;� 4Ž . Ž . Ž . Ž .1 27Ž .

V x , y � max V x , y � 1 , V x � 1, y � V x � 1, y .� 4Ž . Ž . Ž . Ž .2 1

These equations produce Lemma 1, which states that any buyer’s valuationis weakly decreasing in the other buyer’s initial allocation of claims.Intuitively, increasing buyer 1’s initial allocation deprives buyer 2 of thepotential value of bidding for those claims.

7 Ž .Gilbert and Newbery 1982 and later writers refer to the same principle.


Ž . Ž . Ž . Ž .LEMMA 1. For any x, y � A, V x, y � V x, y � 1 and V x, y �1 1 2Ž . Ž . Ž .V x � 1, y . The first second inequality is strict if and only if x, y is a2

Ž .pure 1-allocation 2-allocation .

Ž . Ž . Ž . Ž .Proof. Equation 7 implies that V x, y � V x, y � 1 � V x, y � 11 2Ž . Ž . Ž .� V x, y � 1 , and the inequality is strict iff V x � 1, y � V x, y � 1 ,1

Ž .which is true iff x, y is a pure 1-allocation. The proof of the remainingstatement is symmetric. �

It is also useful to express equilibrium prices as simple functionsŽ . Ž . Ž .of valuations. Suppose that x, y � A is a 1-allocation. Then 5 , 2b ,

Ž . Ž . Ž . Ž . Ž .and 4b imply that P x, y � B x, y � V x, y � 1 � V x � 1, y �2 2 2Ž . Ž . Ž . Ž . Ž .V x, y � 1 � V x, y , and 2a implies that B x, y � V x � 1, y �2 2 2 1Ž .V x, y . Hence:1

P x , y � V x � 1, y � V x , yŽ . Ž . Ž .1 1

� V x , y � 1 � V x , y . 8Ž . Ž . Ž .2 2

Ž . Ž .If x, y � A is a 2-allocation, then symmetric arguments again yield 8 . Inother words, starting from any initial allocation, giving a buyer one moreclaim increases his equilibrium payoff by the price that it would havebrought in equilibrium, regardless of who would have won that auction.ŽIntuitively, either the buyer would have won and paid that price or he

.would have lost, in which case the price represents his valuation. EquationŽ . Ž .8 produces Lemma 2, which leads directly to Theorem 1A below , whichstates that equilibrium price sequences are nonincreasing.

Ž .LEMMA 2. Consider x, y � A such that x � y � N � 1.

Ž . Ž . Ž .A. If x, y is a 1-allocation, then: P x � 1, y P x, y , andŽ . Ž . Ž .P x � 1, y � P x, y if and only if x, y � 1 is also a 1-allocation.

Ž . Ž . Ž .B. If x, y is a 2-allocation, then: P x, y � 1 P x, y , andŽ . Ž . Ž .P x, y � 1 � P x, y if and only if x � 1, y is also a 2-allocation.

Ž . Ž . Ž .Proof. Suppose that x, y is a 1-allocation. Equations 8 and 2b andLemma 1 imply, respectively,

P x , y � P x � 1, yŽ . Ž .� V x , y � 1 � V x , y � V x � 1, y � 1 � V x � 1, yŽ . Ž . Ž . Ž .2 2 2 2

� V x , y � 1 � V x � 1, y � 1 � 0.Ž . Ž .2 2

Lemma 1 also implies that the last inequality holds with equality iffŽ . Ž . Ž .x, y � 1 is a 1-allocation. That establishes A . The proof of B issymmetric. �

GALE AND STEGEMAN82

If total valuations are concave in the sense of Definition 2, then MLVŽ . Ž .has two interesting implications. First, every allocation x, y � 0, 0 is on

an equilibrium path from some previous initial allocation. Second, adeviation from equilibrium behavior at any single allocation never affects

Žthe final allocation by more than one claim unless the tie-breaking rule.acts to exaggerate the effect . These results may be seen as implications of

Lemma 3.

Ž . Ž . 8LEMMA 3. Consider any x, y � A such that V is conca�e at x, y � 1 .Ž . Ž . Ž . Ž .If x, y is a pure 2-allocation, then x � 1, y � 1 is a pure 2-alloc-

Ž . Ž . Ž .ation. Equi�alently, if x � 1, y � 1 is a pure 1-allocation, then x, y is aŽ .pure 1-allocation.

Ž . Ž .Proof. If x, y is a 2-allocation then MLV implies that V x � 1, y Ž . Ž . Ž .V x, y � 1 , which with concavity implies V x � 2, y � 1 V x � 1, y ,

Ž . Ž .and MLV then implies that x � 1, y � 1 is a 2-allocation. If x, y is aŽpure 2-allocation, then the inequalities are strict, implying that x � 1,

.y � 1 is a pure 2-allocation. �

The next lemma, which is helpful for identifying tie allocations, employsŽ .two additional concepts. An auction x, y is compensated on a given

Ž .equilibrium path E if the buyer who wins auction x, y on E wouldŽ . Ž .weakly win the next auction should a deviation cause her to lose at x, y .

Ž . Ž .An auction x, y is inessential if x � 1, y � 1 is an equilibrium outcomeŽ .regardless of who wins at x, y . Note that an inessential auction is

compensated on any equilibrium path that passes through that auction.

Ž .DEFINITION 3. Consider any x, y � A, and any equilibrium path EŽ . Ž . Ž . Ž . Ž .from x, y . If a x � 1, y is on E and x, y � 1 is a 1-allocation, or b

Ž . Ž . Ž .x, y � 1 is on E and x � 1, y is a 2-allocation, then auction x, y isŽ . Ž .compensated on E. If x, y � 1 is a 1-allocation and x � 1, y is aŽ .2-allocation, then auction x, y is inessential.

Lemma 4 shows that the equilibrium paths passing from any inessentialŽ . Ž .auction x, y to x � 1, y � 1 also pass though whichever, if any, of the

two intervening auctions would be compensated on that path. In this sense,equilibrium paths ‘‘prefer’’ compensated auctions. They follow, locally, theroute that minimizes the local impact of deviations.

Ž .LEMMA 4. Consider any inessential auction, x, y � A. If eitherŽ . Ž . Ž .x � 2, y is a 2-allocation or x, y � 2 is a 1-allocation, then x, y is a

Ž . Ž .1-allocation if and only if x � 2, y is a 2-allocation, and x, y is aŽ .2-allocation if and only if x, y � 2 is a 1-allocation.

8 ŽAlternatively, it is sufficient to assume strict quasi-concavity defined in the obvious way.on the discrete set .


Ž .Proof. Adding equations 7 yields

V x , y � 2 max V x , y � 1 , V x � 1, y� 4Ž . Ž . Ž .� V x � 1, y � V x , y � 1 .Ž . Ž .1 2

Ž .Substituting for the last two terms from 7 yields an expression in V alone:

V x , y � 2 max V x , y � 1 , V x � 1, y � V x � 1, y � 1� 4Ž . Ž . Ž . Ž .� max V x � 1, y � 1 , V x � 2, y� 4Ž . Ž .� max V x , y � 2 , V x � 1, y � 1 . 9� 4Ž . Ž . Ž .

Ž . ŽSuppose that x � 2, y is a 2-allocation. Then MLV implies that V x � 2,. Ž . Ž .y � 1 � V x � 3, y . Since x, y is inessential, MLV also implies that

Ž . Ž . Ž . Ž .V x � 1, y � 1 � V x � 2, y , V x, y � 2 . After expressing V x � 1, yŽ . Ž .and V x, y � 1 according to 9 , these facts imply that

V x � 1, y � V x , y � 1 � max V x , y � 3 � V x � 1, y � 2 , 0� 4Ž . Ž . Ž . Ž .� 0.

Ž .MLV then implies that x, y is a 1-allocation, and it is a tie allocation ifŽ . Ž .and only if V x, y � 3 V x � 1, y � 2 , which is true if and only if

Ž . Ž .x, y � 2 is a 1-allocation. That establishes the result if x � 2, y is aŽ .2-allocation. The proof if x, y � 2 is a 1-allocation is symmetric. �

Lemma 4 yields simple sufficient conditions for tie auctions. Theseconditions, if satisfied, are typically still satisfied after small perturbationsof the prize schedules. In this sense, ties are robust.

Ž . Ž . Ž .COROLLARY 4.1. Consider any x, y � A. If x � 1, y and x � 2, yŽ . Ž .are 2-allocations, and if x, y � 1 and x, y � 2 are 1-allocations, then

Ž .x, y is a tie allocation.

Proof. Immediate from Lemma 4. �Ž .The final result of this section, Lemma 5, implies that if say buyer 1’s

valuation V is convex in the sense of Definition 2, and a segment of an1equilibrium path starts at allocation A and ends at allocation B, then onthat segment every pure 2-allocation comes before every pure 1-allocation;moreover, there exists an equilibrium path connecting A and B on whichbuyer 2 wins her claims before buyer 1. The intuition is that, given theconvexity of V , buyer 2 can avoid paying higher prices by winning her1claims early.9 One implication is that convex valuations tend to generatetie bids.

GALE AND STEGEMAN84

Ž . Ž .LEMMA 5. A. Consider any x, y � A such that B x � 1, y �1Ž . Ž Ž .. Ž .B x, y � 1 i.e., V is con�ex at x, y � 2 . If x, y is a 1-allocation and1 1

Ž . Ž . Ž .x � 1, y is a 2-allocation, then x, y and x, y � 1 are tie allocations.Ž . Ž . Ž .B. Consider any x, y � A such that B x � 1, y B x, y � 12 2

Ž Ž .. Ž . Ž .i.e., V is con�ex at x � 2, y . If x, y is a 2-allocation and x, y � 1 is2Ž . Ž .a 1-allocation, then x, y and x � 1, y are tie allocations.

Ž .Proof. Assume the premise of A . The first step is to show, byŽ . Ž Ž .contradiction, that x, y � 1 is a 1-allocation i.e., that x, y is inessen-

. Ž . Ž . Ž .tial . Suppose that x, y � 1 is a pure 2-allocation. Then 3a and 4aŽ . Ž . Ž . Ž . Žimply B x, y � V x � 1, y � V x, y � 1 � V x � 1, y � 1 � V x, y1 1 1 1 1

. Ž . Ž . Ž .� 2 � B x, y � 1 ; 3b , 4b , and convexity imply1

B x , y � V x , y � 1 � V x � 1, yŽ . Ž . Ž .2 2 2

� V x , y � 2 � B x , y � 1Ž . Ž .2 1

� V x � 1, y � 1 � B x � 1, yŽ . Ž .2 1

� B x , y � 1 � B x , y � 1 � B x � 1, yŽ . Ž . Ž .2 1 1

� B x , y � 1 .Ž .2

Ž . Ž . Ž .That x, y is a 1-allocation implies B x, y � B x, y , so it follows that1 2Ž . Ž . ŽB x, y � 1 � B x, y � 1 , contradicting the premise. Therefore, x, y �1 2.1 is a 1-allocation.

Ž . Ž . Ž . Ž . Ž .For any inessential allocation x, y : 2a , 3a , and 4a imply B x, y �1Ž . Ž . Ž . Ž . ŽV x � 1, y � V x, y � 1 � V x � 1, y � 1 � V x, y � 1 � B x, y �1 1 1 1 2. Ž . Ž . Ž . Ž .1 . Symmetrically, B x, y � B x � 1, y . Since x, y and x, y � 1 are2 1

Ž . Ž . Ž .1-allocations, these equations yield B x, y � 1 � B x, y � 1 � B x, y1 2 1Ž . Ž .� B x, y � B x � 1, y . Convexity forces the inequalities to equality,2 1

Ž . Ž .implying A . The proof of B is symmetric. �

That completes the statement of the preliminary results. We have shownthat equilibrium paths take, locally, the direction that maximizes total

Ž .valuation and minimizes the impact of deviations MLV and Lemma 4 .We have also established monotonicity properties of prices and valuationsŽ .Lemmas 1 and 2 , described sufficient conditions for tie allocationsŽ .Corollary 4.1 , shown that concave valuations prevent ‘‘orphan’’ paths and

Ž .induce a kind of stability Lemma 3 , and shown that a buyer with convexŽ .valuations wins her claims last Lemma 5 . These results are helpful for

deriving and understanding our two main theorems.

9 There is a tension between the idea that buyer 2 can avoid higher prices by buying herclaims early, and the idea from Lemma 4 that equilibrium price sequences are weaklydeclining. This tension is essentially what produces the tie bids claimed in Lemma 5.


4. EQUILIBRIUM PRICE SEQUENCES

In observed sequential auctions, prices of identical objects often declineas the auctions proceed.10 Theorem 1, below, finds this phenomenon inour model. Gi�en two completely informed buyers, prices ne�er rise as theauctions progress, and the price often drops at least once.

Ž .The intuition for the weakly declining price result is as follows.Ž .Suppose that the equilibrium price is $1 in auction x, y , where y � 0.

Then either buyer 1 wins the auction and pays $1, or she loses the auctionwhile bidding $1. In either case, she should bid at least $1 in auctionŽ . Ž .x, y � 1 , because if she loses at x, y � 1 , then she will pay or be willing

Ž . Ž . Ž .to pay $1 to get from x, y to x � 1, y , but if she wins at x, y � 1 , thenŽ . Ž .she reaches x � 1, y � 1 and can get to x � 1, y at no additional cost.

Ž .Therefore, if buyer 1 loses auction x, y � 1 , then the price in thatauction must be at least $1. By a symmetric argument, if the price is $1 in

Ž . Ž .auction x, y , and buyer 2 loses auction x � 1, y , then the price in thatauction must be at least $1. Either way, the price in the auction precedingŽ .x, y is at least $1.

Theorem 1 also characterizes the points at which price drops occur. IfŽ .buyer 1 wins auction x, y in equilibrium, then the price falls immediately

Ž .thereafter if and only if x, y is uncompensated on that equilibrium pathŽi.e., if a deviating win by buyer 2 would have been immediately reinforced,

.in the equilibrium of the resulting subgame, by another win by buyer 2 . AŽ .rough intuition is that buyer 2 bids up the price in auction x, y because

winning that auction would increase her surplus if she went on to winŽ . Ž .auction x, y � 1 , an option that vanishes when she loses auction x, y .

To illustrate the strictly declining price result, suppose that bidders aresymmetric and, in equilibrium, the winner of a tie-breaking auction alwayswins the next auction. This first-mover advantage implies that any tie-breaking auction is uncompensated on any equilibrium path, implying thatthe price drops immediately after the lead is established. Alternatively,

Ž . Ž . Ž .suppose that N � 2 and the total prize satisfies f 2 � f 0 � f 1 . MLVŽ . Ž .implies that the final equilibrium allocation is 0, 2 or 2, 0 , implying from

Theorem 1A that the price drops in the second auction. More generally, ifthe total prize increases in terminal concentration, then Theorem 1Bdemonstrates the existence of an equilibrium path on which the pricedrops at least once.

10 Ž .Pitchik and Schotter 1988 show that budget constraints can lead to declining prices,Ž . Ž .given independent private values. McAfee and Vincent 1993 , von der Fehr 1994 , Black and

Ž . Ž .deMeza 1992 , and Gale and Hausch 1994 show, respectively, that risk aversion, participa-tion costs, options to buy more items at the same price, or options to choose from all itemsyet unsold, can have similar effects.

GALE AND STEGEMAN86

THEOREM 1. A. The price paid declines weakly along any equilibriumŽ .path E, from any allocation. The price strictly declines after auction x, y , on

Ž .path E, if and only if x, y is uncompensated on E.B. If f is quasi-con�ex, but not monotonic, then there exists an equilib-

Ž � �.rium path E, from some x , y � A, on which the price strictly declines atleast once.

Ž . Ž . Ž .Proof. If x � 1, y follows x, y on the given path, then x, y is aŽ . Ž .1-allocation, and Lemma 2A implies P x � 1, y P x, y . Lemma 2A

Ž . Ž . Ž .also implies that P x � 1, y � P x, y iff x, y is compensated on P. TheŽ . Ž . Ž .argument is similar if x, y � 1 follows x, y . That establishes A .

Ž . Ž . Ž .To establish B , assume that f x � V x, N � x is not monotonic.Ž .Then MLV implies � x , x such that x , y is a pure 1-allocation and1 2 1 1

Ž .x , y is a pure 2-allocation, where x � y � x � y � N � 1. Since2 2 1 1 2 2Ž . Ž . Ž . Ž .MLV implies f x � 1 � f x , and f x is quasi-convex, f x must be1 1

weakly increasing for x � x , implying, by MLV, that x � x . The proof1 2 1Ž .proceeds by contradiction. Suppose that every x, y � A is compensated

Ž .on every equilibrium path from x, y , implying that P is constant. SinceŽ . Ž .x , y is a pure 1-allocation, the supposition implies that x � 1, y � 11 1 1 1

Ž . Ž .is not on any equilibrium path from x � 1, y , implying that x � 1, y1 1 1 1

Ž . Ž .is a pure 1-allocation. By induction, i every x, y � A such that x x1 1Ž . Ž .is a pure 1-allocation. By an analogous argument, ii every x , y � A2

Ž .such that y y is a pure 2-allocation. Since x � x implies y � y , i2 2 1 1 2Ž . Ž .and ii imply that x , y is a pure 1-allocation and a pure 2-allocation, a2 1

Ž � �.contradiction. Therefore, some x , y � A is uncompensated on someŽ .equilibrium path, and part A implies that a price drop occurs. That

Ž .establishes B . �We conclude this section with an example in which the declining price

effect is so strong that the price changes sign as the auctions progress.Suppose that N � 2 procurement contracts are allocated sequentially, and

Ž . Ž . Ž . Ž .the prizes the negative of costs are: f 0 � 0, f 1 � �4, f 2 � �5,1 1 1Ž . Ž . Ž . Ž .f 0 � �3, f 1 � �2, and f 2 � 0. Then buyer 2 wins auction 0, 02 2 2

Ž .with a bid of 1, after which she wins auction 0, 1 with a bid of �4. Buyer2 is actually offering to pay to provide the first unit, because she knowsthat buyer 1 enjoys pronounced economies of scale and will make arelatively weak bid once she loses the chance to supply both units.

5. CHARACTERIZATION OF EQUILIBRIUM FOR ACLASS OF PROBLEMS

This section studies the equilibrium of a class of problems in whichbuyers are symmetric with respect to their prizes. This class, which in-


cludes interesting economic examples, also illustrates the implications ofTheorem 1 and Lemmas 3�5. Symmetry allows us to confine attention to

� � � �Ž . 4 � �Ž . 4A � T , where A � x, y � A � x � y and T � x, y � T � x � y , theŽ . Ž .allocations such that buyer 1 is the weak leader and buyer 2 is the weak

�Ž . 4follower. Let A� � x, y � A � x � y . Note that the buyers may still beasymmetric with respect to the initial allocation of claims.

We consider the class of problems such that the prize schedules satisfyŽ .10 . For any nonnegative integer n, n denotes the smallest integer weaklyexceeding n�2.

f x � f N � x � x symmetry . 10aŽ . Ž . Ž . Ž .1 2

f x is strictly increasing and strictly convex, for x � N�2. 10bŽ . Ž .1

f x is strictly decreasing and strictly concave, for x � N�2. 10cŽ . Ž .2

f x is strictly increasing, for x � N�2. 10dŽ . Ž .f x is strictly concave, for x � N�2. 10eŽ . Ž .f and f are quadratic, for x � N�2. 10fŽ .1 2

f N � 1 � 2 f N � 1 � 3 f N if N is odd . 10gŽ . Ž . Ž . Ž . Ž .1 2 2

Ž . Ž .Assumptions 10b � 10d mean that the total prize, and each buyer’svaluation of the marginal claim, increase as the terminal concentration ofclaims increases. These conditions arise naturally if claims represent ascarce input in a production game, because higher concentration typicallyleads to higher output prices, increasing both total profits and the value ofmarginal output. These effects depend upon the leader bidding for capac-ity to keep it out of production, but that behavior can be optimal for the

Žleader. Government policy appears to recognize this possibility, by provid-ing that takeoff and landing slots can be removed from airlines that are

Ž Ž ..not using them but refusing to sell them 14 C.F.R. Part 93.223 a . TheFCC until recently had similar minimal use requirements for specialized

. Ž .mobile radio licenses. Assumption 10e means, as is often true in suchgames, that increasing concentration increases total profits at a decreasingrate.

Ž .The quadratic assumption 10f simplifies the analysis. The results arenot sensitive to this assumption, in the sense that small perturbations fromquadratic payoffs do not change the characterization of the equilibriumbelow. We do not know, however, of any other simple class of payofffunctions that necessarily generates this particular characterization.11 We

11 Ž . ŽIf the payoffs are nonquadratic, then � in Eq. 12 must be replaced by B N � y �2 2. Ž . Ž .1, y � B N � y � 2, y � 1 , a measure of the concavity of f . An earlier 1995 version of2 2

this paper describes a weaker and nonquadratic set of conditions sufficient for the characteri-zation of Theorem 2, but the conditions have no simple economic interpretation.

GALE AND STEGEMAN88

parameterize the prize schedules as follows.

f x � � � � x � � x 2�2 for x � N�2;Ž .1 1 1 111Ž .

f x � � � � x � � x 2�2 for x � N�2.Ž .2 2 2 2

Ž . Ž .Assumptions 10b and 10e require � � � � 0. Note that assumptions2 1Ž . Ž . �10a � 10d apply to f , f , and f on the real interval N�2, N , not just1 2on the integers in that set.

Ž .Assumption 10g ensures that the lead never switches on an equilib-Ž . Ž .rium path. If N is even, then 10a � 10f are sufficient for this purpose,

but those assumptions have less bite if N is odd because in that caseŽ .the lead can reverse in the last two auctions. Condition 10g has an

intuitive interpretation, as follows. Suppose that N is odd. Then 2 N � 2Ž . Ž .� N � 1, and 10d implies that N, N � 2 is a 1-allocation, implying

Ž . Ž . Ž . Ž . Ž . Ž . Ž .from 1 , 2 , 4b , and 6 : V N, N � 2 � f N � 1 � 2 f N � 1 �1 2

Ž . Ž .f N . Similarly, since symmetry implies that N � 1, N � 1 is a 1-alloc-2

Ž . Ž . Ž . Ž . Ž .ation: V N � 1, N � 1 � f N � 2 f N � f N � 1 � 2 f N , where1 2 2 2Ž .the last step follows from symmetry. MLV implies that N � 1, N � 2 is a

Ž . Ž .pure 1-allocation iff V N, N � 2 � V N � 1, N � 1 , a statement equiva-Ž . Ž .lent to 10g . Hence, 10 g holds if and only if the leader always bids enough

to maintain her lead when exactly two claims remain to be sold. Thisinitializes a backward induction that ultimately ensures that the lead neverswitches on an equilibrium path.

Ž .The previous paragraph explains why assumption 10g is important, butit does not explain why one should expect it to be satisfied. In Section 6 we

Ž .show that 10g must be satisfied if the claims are divided sufficientlyfinely.

That concludes the discussion of the assumptions. The characterizationof equilibrium begins with the following subproblem. Consider an arbitrary

Ž .auction x, y � A, such that, in equilibrium, buyer 1 wins any auctionŽ � �. Ž . Ž � �. Ž . �x , y � x, y , x , y � x, y , such that y y � 2. The subproblem is:

Ž .who wins auction x, y ? If buyer 2 wins, then any subsequent claim thatŽ .she wins will, in equilibrium, be her last claim, implying that at x, y she

Ž . Ž . Žbids the value of that last claim, f N � y � 2 � f N � y � 1 � N � y2 2.� 3�2 � � � , in every remaining auction on the equilibrium path.2 2

Ž . Ž . Ž .Similarly, she bids f N � y � 1 � f N � y � N � y � 1�2 � � �2 2 2 2Ž .in auction x, y and, if she loses that auction, in every remaining auction

Ž .on that equilibrium path. Winning auction x, y thus reduces buyer 2’s bidby � in every remaining auction. Since buyer 1 wins all of those auctions,2

Ž . Ž .regardless of who wins auction x, y , winning auction x, y increasesŽ .buyer 1’s subsequent total payments by N � x � y � 1 � but adds2

Ž . Ž . Ž .f N � y � f N � y � 1 the marginal value of that claim to her final1 1


Ž . Ž .prize. Therefore, buyer 1 weakly wins auction x, y if and only if

B x , y � f N � y � f N � y � 1 � N � x � y � 1 �Ž . Ž . Ž . Ž .1 1 1 2

� f N � y � 1 � f N � y � B x , y ,Ž . Ž . Ž .2 2 2

which can be rewritten:

f N � y � f N � y � 1 � N � x � y � 1 � . 12Ž . Ž . Ž . Ž .2

Ž .In other words, buyer 1 wins auction x, y only if the increment to theŽ Ž ..total prize the left-hand side of 12 exceeds the increment to total prices

Ž Ž ..paid the right-hand side of 12 . Recall that this conclusion rests on theŽ � �. Ž .premise that in equilibrium buyer 1 wins any auction x , y � x, y such

that y� y � 2. That proves to be true for enough auctions to allow us toŽ .use 12 to characterize the region in which buyer 1 wins all remaining

auctions.Ž . Ž .Specifically, let be the smallest integer such that x, y � � 2,

Ž .satisfies 12 , and, for i � 1, 2, . . . , , let be the smallest integer suchiŽ . Ž . Ž .that x, y � , � i satisfies 12 . Theorem 2 shows that, starting at ani

Ž . � Ž .arbitrary allocation x, y � A , buyer 1 wins auction x, y and all subse-quent auctions if y � or x � . In the theorem, this region is called�yA . The remaining allocations in A are partitioned into three regions,1according to who wins the auction at that allocation. We discuss the otherregions below. The proof is sketched in an appendix.

Ž . �Ž . Ž . ŽTHEOREM 2. Assume 10 . Let Z � , � 1 , , � 2 . . . ,1 2

.4 Ž Ž .0 . Then, generically specifically, if e�ery allocation that satisfies 12.satisfies it with strict inequality :

� 4 � � 2, 3, 4 , 13aŽ .1

� 4 � � 1, 2 � i � 1, 2, . . . , � 1, 13bŽ .i�1 i

x , y is a pure 1-allocation � x , y � A � A , 13cŽ . Ž . Ž .1 3

x , y is a pure 2-allocation � x , y � A , and 13dŽ . Ž . Ž .2

x , y is a tie allocation � x , y � A , 13eŽ . Ž . Ž .4

where A � x , y � A � y � or x , y � z for some z � Z ;� 4Ž . Ž .1

A � x , y � A � x � max � 1, � 2 � A ;� 4Ž . Ž .2 1 1

A � x , y � A � x � y � 1 � A � A ;� 4Ž .3 1 2

A � A � A � A � A .4 1 2 3

Note that if � 0, then A � A � A � �. �2 3 4

GALE AND STEGEMAN90

It is easiest to explain Theorem 2 through an example. Suppose thateach claim represents one unit of capacity in a Cournot duopoly game,

Žwhere the aggregate demand curve is P � 24 � Q and production up to.the capacity constraint is costless. In the absence of capacity constraints,

each duopolist produces eight units of output in the Cournot equilibrium.Assume, however, that only N � 16 claims are available. Then it is easy toshow that the capacity constraint binds a duopolist holding fewer than

Žeight claims. The leader’s prize i.e., profit in the capacity-constrained. Ž . Ž .2Cournot equilibrium is f x � 8 � x �4, and the follower’s prize is1

Ž . Ž .Ž .f x � 8 � x 16 � x �2, in each case for x � 8.2Ž .Since these prize schedules satisfy 10 , Theorem 2 describes the equilib-

Ž .rium. Condition 12 reduces to 4 x � 6 y � 59, and it is then trivial tocompute Z and the equilibrium paths. Figure 1 shows these paths, for theallocations such that x � y. The terminal allocations T � lie along theupper-right frontier, where x � y � 16. A 1-allocation is marked by ahorizontal arrow from the allocation, and a 2-allocation is marked by avertical arrow. The six allocations marked by boxes are the thresholdallocations Z. The pure 1-allocations above the Z frontier are region A ,1the pure 2-allocations below the Z frontier are region A , the pure21-allocations to the left of the frontier and adjacent to the midline are

Ž .region A , and the remaining tie allocations are region A . The pattern3 4of equilibrium paths implies, from Theorem 1A, that price drops occurbetween the auctions separated by double crossmarks.

FIG. 1. Equilibrium paths if each bidder’s terminal prize equals its profit in a capacity-con-strained Cournot equilibrium, given the aggregate demand curve P � 24 � Q, N � 16 units

Ž .of capacity auctioned or held ex ante , and production that is otherwise costless. TheŽ .equilibrium paths where y � x not shown are symmetric. Many initial allocations lead to

Ž . Ž .the modal allocation x, y � 10, 6 , at which equilibrium output is 15. The equilibrium pricestrictly drops between nodes separated by double crossmarks. If the initial allocation is theorigin, then equilibrium prices are 10.5 for the first object, 6.25 for the next 12 objects, and5.5 for the last 3 objects, and each bidder’s profit net of bids is 16.5.


Ž .Consider the equilibrium paths from the initial allocation 0, 0 . In theŽ .first auction both bidders bid 10.5 we suppress the calculations , and by

assumption buyer 1 wins. Once the lead is established, the price drops toŽ .6.25 in auction 1, 0 , which buyer 1 wins to preserve the rents that accrue

from the lead. Then the path enters region A , which drains into alloca-4Ž .tion 7, 5 , along with its boundaries in A and A . The tie bids and2 3

indeterminate path in this region follow directly from Corollary 4.1 and areŽthus robust with respect to small changes in the terminal payoffs e.g.,

. Ž .away from quadratic payoffs . Buyer 2 wins a final claim at auction 7, 5before allowing buyer 1 to win the rest. Buyer 1 pays a lower price of 5.5

Ž .for those units; the price drops after auction 7, 5 because buyer 2’sŽ .out-of-equilibrium loss at 7, 5 would not be immediately undone but

would instead be followed by another loss. When the buyers reach themodal allocation, they have paid 102 for prizes totaling 135, and the netsurplus of 33 is divided evenly between them.

6. GENERAL FEATURES OF THE EQUILIBRIUM

A striking feature of the example is that many initial allocations lead toŽ .the same final division of claims: 10, 6 . Such robustness is general:

Ž . Ž .Theorem 2 implies that any initial allocation x, y � 1, leads to1Ž . Ž .N � , or its symmetric counterpart , N � . We call these themodal allocations. In the context of a duopoly, these may be interpreted asthe ‘‘natural’’ market structure implied by the terminal prizes.

To understand why robust modal outcomes exist, first recall that increas-ing returns to concentration make the leader’s payoff convex in capacity,

Ž .as assumed in 10b . Convexity implies that the more capacity the leaderŽ .has or anticipates having , the more he will bid for marginal capacity; the

follower should, therefore, win her units as early as possible, when herrival’s bids are relatively low. Lemma 5 makes this intuition formal. Theimplication that pure 2-allocations generally occur before pure 1-alloc-ations on the equilibrium paths makes even substantial skewness in the

Žinitial allocations irrelevant, because it creates thresholds such as x � 7.in Fig. 1 at which the follower bids aggressively to lock in her anticipated

equilibrium allocation, regardless of how far she trails the leader.ŽHence: although getting the lead is important because it determines

.which modal allocation obtains the size of the lead is relatively unimpor-tant.

Ž . Ž .Since the modal allocations N � , and , N � have such largeŽ .basins of attraction, their determinants are of interest. Condition 12 is

helpful here. Anything that causes the total prize to rise more rapidly withconcentration tends to increase modal concentration, by increasing the left-

GALE AND STEGEMAN92

Ž .hand side of 12 , which decreases . On the other hand, anything thatcauses the follower ’s �aluation of the marginal claim to rise more rapidly withconcentration tends to decrease modal concentration, by increasing the

Ž .right-hand side of 12 . In particular, anything that increases � without2Ž .increasing the left-hand side of 12 tends to decrease concentration.

Aside from the existence of modal allocations, two other properties ofthe equilibrium are noteworthy. First, the equilibrium paths are stable inthe sense that a single deviation from equilibrium bidding affects the final

Žallocation by at most one claim unless the deviation creates a tie alloca-.tion and the tie-breaking rule switches the lead . This follows from the

concavity of the total valuation and Lemma 3. A related point is that,although ties are common, the tie-breaking rule does not affect the finalallocation, except where it determines which of the symmetric biddersbecomes the leader.

Second, Fig. 1 illustrates the general point that an incumbent monopolistoften does not block entry if more claims become available. Suppose that

Ž .the initial allocation is 12, 0 , which effectively makes firm 1 an uncon-strained monopolist. If four new claims become available, then Fig. 1shows that the prospective entrant, firm 2, wins the first two claims before

Ž .firm 1 wins the last two. This situation satisfies all of Krishna’s 1993assumptions, except that Krishna has many potential entrants. In that case,her Lemma 3 implies that entrants win the first three claims before firm 1wins the last one. Reducing the potential entrants to one thus increasesthe monopolist’s ultimate market share without foreclosing entry alto-gether.

We conclude this section by deriving a simple formula for the modalŽ . �allocation, which is subsequently useful. For any allocation x, y � A

Ž . Ž .such that y N�2 � 2, 11 implies that 12 is equivalent to

� � � � N � y � 1�2 � � � � N � x � y � 1 � . 12�Ž . Ž . Ž . Ž .1 2 2 1 2

Ž . Ž .By definition, is the smallest integer such that x, y � � 2, Ž �. Ž � � . �satisfies 12 . Let x , N � x denote the modal allocation with x �

N�2. Theorem 2 implies that x� � N � , implying that x� is the largestŽ . Ž � � . Ž �.integer such that x, y � N � x � 2, N � x satisfies 12 . Equiva-

lently, x� is the largest integer that satisfies:

� � � � N� � � � � �2Ž .1 2 2 1 2�x � 1. 14Ž .3� � �2 1

As expected, the modal concentration of claims increases in � , � , and1 1� . The net effect of � is ambiguous, because higher � implies that the2 2 2total prize and the follower’s valuation of the marginal claim both increasemore rapidly in concentration.


7. THE IMPORTANCE OF CLAIM SIZE

Ž .This section continues the study of the class of problems defined by 10 ,Žby considering the consequences of splitting the claims i.e., the sellers

.auction more and smaller claims . We first justify our earlier assertion thatŽ .10g is satisfied if the claims are sufficiently small. We then show thatsplitting claims generally reduces modal concentration, though the modalallocations do not converge to equal division as the claims become in-finitesimal.

�Suppose that f : 0, N � � is a continuous function expressing buyerii’s prize as a function of buyer 1’s claims, where each claim is split into k

Ž . Ž .identical subclaims. Symmetry requires f N�2 � f N�2 , which implies1 2

8 � � � � 4 � � � N � � � � N 2 � 0. 15Ž . Ž . Ž . Ž .1 2 1 2 1 2

Ž . �Ž .Assumption 10d requires f N � 0, which implies

� � � � � � � N. 16Ž . Ž .1 2 2 1

Ž .Let f x denote buyer i’s prize as a function of buyer 1’s terminaliˆ ˆŽ . Ž .allocation of subclaims. Hence f x � f x�k . Parameterizing f as ini i i

ˆ 2Ž . Ž .11 , it follows that � � � , � � � �k, and � � � �k . Condition 10g ,ˆ î i i i i iapplied to the problem with subclaims, becomes

ˆ ˆ ˆf Nk � 1 � 3 � f Nk � 2 � f Nk � 1 � 0 if Nk is odd . 17Ž . Ž .Ž . Ž . Ž .1 2 2

Ž .If Nk is odd, then Nk� Nk � 1 �2. Making this substitution, and apply-Ž . Ž .ing 15 , routine manipulation reduces 17 to

�4 � � � � 2 � � � N k � 3� � 5� � 0 if Nk is odd . 17Ž . Ž . Ž . Ž .1 2 2 1 1 2

Ž .Equation 16 implies that the expression in brackets is positive, implyingŽ �.that 17 is satisfied for sufficiently large k. In other words, if the original

Ž .claims are split into sufficiently many subclaims, then 10g is satisfied.ŽFor a particular problem illustrated by our earlier example i.e., the

buyers are capacity-constrained Cournot rivals facing linear demand andconstant marginal costs, and the total capacity auctioned equals the

. Ž � .Cournot equilibrium output , it is straightforward to show that 17 andŽ .hence 10g is satisfied for Nk � 3. Since an even number of capacity units

never produces a lead switch, it follows that the lead can switch during thecourse of the capacity auctions only if exactly N � 3 units are auctioned.12

12 Ž . Ž . Ž . Ž .For example, the lead can switch if: N � 3, f 0 � 0, f 1 � 3, f 2 � 4, f 3 � 8, and1 1 1 1Ž . Ž .f x � f 3 � x . Whether it does depends upon the tie-breaking rule.2 1

GALE AND STEGEMAN94

Since selling all claims as a single unit produces monopoly, a crudeintuition suggests that splitting claims reduces modal concentration. We

Ž � � .now show that this intuition is correct. Let x , y denote the modalˆ ˆ� � Ž . �allocation of subclaims that has x � y . Equation 14 implies that x isˆ ˆ ˆ

the largest integer satisfying

� � � � N� � � � � � 2kŽ . Ž .1 2 2 1 2�x � Nk � 1� Nk . 18Ž . Ž . Ž .ˆ3� � � NŽ .2 1

Ž . Ž . � Ž .If k � 1, then 18 is equivalent to 14 . Since x � Nk represents theˆŽ .leader’s modal share of total claims, 18 shows that auctioning claims in

Ž . 13smaller units i.e., increasing k tends to reduce modal concentration. AsŽ .claims are infinitely split i.e., as k � �� , the leader’s share converges to

Ž . � � Ž .� � � �N � � � 3� � � . Equation 16 implies that this expres-1 2 2 2 1sion exceeds one-half. The unequal final allocation of resources thatinvariably obtains in equilibrium is, thus, not merely a manifestation of an‘‘integer problem.’’ In the example of Section 5, the leader’s share drops

Ž . Ž .from 10�16 � 0.625 when k � 1 to 0.60 as k � �� .

8. EXTENSION: PRODUCTION WITH FIXED COSTS

If claims represent capacity in a postauction production game, then afirm owning capacity might bear an internal fixed cost to initiate produc-tion. Such a fixed cost affects the equilibrium of the postauction produc-tion game only if it causes one firm to shut down entirely. The followingexample shows that even relatively small fixed costs can, through the threatof such shutdowns, induce complete preemption.

Reconsider the example of Section 5. In the absence of fixed costs, anŽ .equilibrium allocation is 10, 6 . In the subsequent Cournot game, the price

Žof output is 9, the leader’s profits are 81 she leaves one unit of capacity.idle , and the follower’s profits are 54. Now, suppose that producing any

positive quantity in the Cournot game imposes a fixed cost of 12. Ex post,both firms can easily absorb that cost, but anticipating that cost has astriking effect upon equilibrium bidding. Specifically, a routine computa-

Ž .tion shows that an equilibrium allocation is 16, 0 : whoever gets the firstclaim preempts all further entry. All monopoly profits are dissipated in thefirst auction, because the loser of that auction earns nothing.

13 � Ž . � Ž .If one thinks of x � Nk as a function of continuously increasing k, then x � Nkîncreases whenever x� jumps up to the next integer value. Such increases can also occur fordiscretely increasing k. In other words, the integer constraint can cause modal concentrationto increase for particular discrete increases in k, although it follows a decreasing trend as kincreases.


To understand why such small fixed costs lead to monopolization,Ž .consider the last few auctions. If the final allocation is 15, 1 , then the

leader acts as a monopolist in the unique Cournot equilibrium, and theŽ .follower produces nothing. Moreover, the 15th monopoly-creating claim

is so valuable to the leader that a follower holding one claim bids enoughto win a second only if the leader holds fewer than nine claims. Hence,

Ž .from the allocation 9, 0 , the leader gets the next seven claims free,because it is pointless for the follower to bid more than zero for one claim

Ž .knowing that she will not win another. That makes the allocation 9, 0 sovaluable to the leader that, after winning the tie in the first auction, sheeasily wins the next eight auctions. Consequently, the possible equilibrium

Ž . Ž . Ž Ž .allocations are 16, 0 and 15, 1 at 15, 0 , both buyers bid zero for the.last claim . Both allocations generate the monopoly outcome in the pro-

duction game.14

9. CONCLUDING REMARKS

Most models of oligopoly assume that agents with market power set asingle price or price schedule for the entire market. In fact, prices areoften negotiated case-by-case; in the presence of a vigorous rival, theprice-setting process may resemble an auction. This paper has studied thedetermination of price and market share in a market where well-informedduopolists or duopsonists bid sequentially for the trade of a finite number

Ž .of buyers or sellers. We have shown that the two buyers say are mostaggressive in bidding for the early units, regardless of the schedule ofprizes. In the context of auction theory, this means that empirical observa-tions of declining prices may not require elaborate explanation.

For a specific class of problems, which arise naturally when duopolistsŽ .bid for a scarce input e.g., mineral rights , we have shown that the final

allocation of the auctioned input is surprisingly independent of the initialallocation. Except for extreme initial allocations, the leader’s ultimatemarket share is determined solely by the mapping from terminal alloca-tions to prizes, and we have described the determinants of that share.

The most natural interpretation of our N ‘‘sellers’’ is that they arefragmented, since cooperating sellers could earn more revenue by combin-ing or coordinating their auctions. A government may, however, havenonrevenue objectives that justify entering the market in a fragmented

14 Ž .Rasmussen et al. 1991 describe an exclusion game in which modest returns to scaleŽ .induce monopoly for related reasons, and Fudenberg et al. 1983 and Harris and Vickers

Ž .1985 identify conditions under which a small initial advantage leads to complete preemptionin a model of product innovation.

GALE AND STEGEMAN96

fashion. For instance, if the government is the sole seller in the example ofSection 5, then, although auctioning the claims as a block would maximizeits revenue, auctioning the claims sequentially generates higher totaloutput, which may imply higher social surplus.

As a contribution to the general theory of auctions, this paper general-Ž .izes the models of Gale 1991 , who shows that sequential auctions of

capacity in a Cournot game may or may not produce monopoly, and DudeyŽ .1992 , who characterizes the equilibrium of quantity-constrained Bertrand

Ž .duopolists bidding for customers. Krishna 1990 constructs examples thatillustrate some of our general results, including weakly declining prices,and she shows that one bidder wins every auction if the bidders are

Ž . Ž .symmetric and f x is convex. In Gale et al. 2000 , we study a qualita-itively different class of prize schedules, which arise naturally if duopolistsbid to provide a product or service. That paper extends the analysis byallowing the bidders to recontract after the auctions are over.

Ž .Krishna’s 1993 model is similar in spirit to our example, but she allowsmany potential entrants to bid against an incumbent monopolist for newcapacity. Although increasing the number of bidders generally complicatesmatters in our model, her assumptions that potential entrants are alwayspresent and that successful entrants always produce to capacity simplifythe analysis by implying that an entrant’s bid is always the equilibriumprice of output if the entrant wins. Her central result is that if marginalcosts are equal and constant and market demand is concave, then themonopolist wins only the last unit of new capacity. For the case of a singlepotential entrant, we have shown that the monopolist may win more

15 Ž .units. Vickers 1986 studies duopolists who bid for a sequence ofcost-reducing innovations, but, unlike the claims sold in our model, theinnovations generate earnings before the auctions are finished. He showsthat Bertrand competition in the product market generates increasingdominance by one of the duopolists, which need not occur under Cournot

Ž .competition. Budd et al. 1993 study a related model, and they show thatat any given moment the state of the game tends to evolve in the directionof highest total valuation, a result echoed in our MLV principle. The‘‘race’’ special case of their model is fairly close to our model, except thatin their model the follower’s profit function is convex in claims obtainedand the race ends only when one firm attains a completely dominantposition. These features cause the leader to ‘‘outbid’’ the follower consis-tently, unlike our model, in which the leader allows the follower to winsome claims.

15 Ž .Rodriguez 1998 examines sequential auctions with more than two bidders but focuseson the incentives to acquire claims that have no productive value other than limitingcompetition.


APPENDIX: PROOF OF THEOREM 2

Much of the intuition behind Theorem 2, as discussed in the text, isstraightforward and comes directly from Lemmas 3�5. The complete proofof Theorem 2 is complicated, however. This appendix provides a fairlydetailed sketch of the proof, including illustrations of how the argumentswork in the example of Fig. 1.

The complications in the proof arise from several sources. First, therepeated application of Lemmas 3 and 5 depends upon certain concavityand convexity properties, which the terminal valuation functions f areiassumed to satisfy, persisting to valuations at earlier stages in the game.Given the turns in the equilibrium paths, keeping track of these propertiesis a bit involved. Second, since the equilibrium paths are determineddifferently in each of four regions, the proof really comprises four proofs,one for each region, and the overall proof must also characterize how theregions fit together. Third, the behavior of the valuation functions near themidline is idiosyncratic, and considerable effort goes into the demonstra-tion that equilibrium paths never return to the midline. Fourth, thediscrete and finite nature of the map requires careful bookkeeping regard-ing indices and their limits, and this complicates all parts of the proof.

The proof of Theorem 2 begins with five preliminary lemmas, which aremostly exercises and stated here without proof. This is followed by acondensed version of the proof of Theorem 2, which shows all of the main

Žsteps but does not provide a detailed proof of every step. The authors will.send a complete proof, including proofs of Lemmas 6�10, upon request.

As a measure of the concavity of valuations, it is useful to define:

B� x , y � B x � 1, y � 1 � B x , yŽ . Ž . Ž .i i i

Ž .for all x, y � A such that y � 0, and i � 1, 2. This definition implies

B� x , y � V x , y � 1 � V x � 2, y � 1 � 2V x � 1, y ;Ž . Ž . Ž . Ž .i 1 1 1

B� x , y � 2V x � 1, y � V x , y � 1 � V x � 2, y � 1 .Ž . Ž . Ž . Ž .2 2 2 2

�Ž . Ž . � Ž .Hence, B x, y 0 if and only if V is concave at x, y � 1 , B x, y � 01 1 2Ž . � Ž . �Ž .if and only if V is concave at x, y � 1 , and B x, y � B x, y � 0 if2 2 1

Ž .and only if V is concave at x, y � 1 .Lemmas 6�10 lay the groundwork for the main proof. Except for

Lemma 9, the lemmas derive general properties of valuations and equilib-rium paths and do not depend upon the assumption of quadratic payoffs.Lemma 6 simplifies the backward induction formulas for computing bids,

Ž .in auctions such that the outcome weakly does not affect who wins thenext auction.

GALE AND STEGEMAN98

Ž .LEMMA 6. Consider any x, y � A, such that x � y � N � 1.

Ž . Ž . Ž . Ž .A. If x, y � 1 and x � 1, y are 1-allocations, then i B x, y �1Ž . � Ž . Ž . Ž . Ž .B x � 1, y � B x, y � 1 and ii B x, y � B x � 1, y .1 2 2 2

Ž . Ž . Ž . Ž .B. If x, y � 1 and x � 1, y are 2-allocations, then i B x, y �2Ž . �Ž . Ž . Ž . Ž .B x � 1, y � B x, y � 1 and ii B x, y � B x � 1, y .2 1 1 1

Ž .Lemma 7 strengthens the argument in the text that condition 12 mayunder certain circumstances be used to determine which buyer wins an

Ž � � .auction. The discussion in the text considers an arbitrary auction x , yŽ . Ž � � .� A, such that in equilibrium buyer 1 wins any auction x, y � x , y ,

Ž . Ž � � . �x, y � x , y , such that y y � 2, and argues that buyer 1 shouldŽ � � . Ž . Ž � � .win at x , y if and only if 12 is satisfied at x , y . Lemma 7 shows

that assumptions about which buyer wins may be dropped for the auctions� � Ž .at which y � y , because at y � y condition 12 is satisfied ‘‘more

easily’’ as x increases. This makes it possible, in the main proof, to applyŽ . �the condition 12 test recursively, starting from high values of y and

decrementing to lower values of y�. Lemma 7 is also used to show that thepattern described in Theorem 2 is generic in the sense that violations can

Ž .occur only when 12 happens to be satisfied with equality.

Ž . Ž .LEMMA 7. Assume that f and f satisfy 10c and 10d . Consider1 2Ž � � . � �Ž . �arbitrary x , y such that y N�2 � 2 and the allocations x, y � A �

� � � 4x x, y � 1 y y � 2 are all 1-allocations.

Ž � � . Ž . � Ž � . Ž .A. If x , y satisfies 12 for some x , then x, y satisfies 12 as astrict inequality � x � x�.

Ž � . Ž . � Ž � � .B. x, y is a pure 1-allocation � x � x if and only if x , yŽ . Ž .satisfies 12 as a strict inequality .

Ž .Assumption 10b states that f , player 1’s valuation function at the final1stage of the game, is convex. Lemma 8 is useful for showing that thisconvexity extends to earlier stages of the game, though it does not extendacross the x � y midline. This will imply, through Lemma 5A, that there isalways an equilibrium path on which buyer 1 wins her claims after buyer 2,subject to the constraint that buyer 1 remains the leader.

Ž � � . �LEMMA 8. Consider any x , y � A .

Ž � � . Ž � � . Ž � �A. If x , y and x � 2, y � 1 are 2-allocations, and x , y �. �Ž � � .1 is a 1-allocation, then B x , y � 0.1

Ž � � . Ž � � . Ž � � .B. If x , y � 1 , x � 1, y , and x � 2, y � 1 are 2-alloc-�Ž � � . �Ž � � .ations, then B x , y � 1 � B x , y .1 1

Ž .Assumption 10e states that f � f � f , the aggregate valuation at the1 2final stage of the game, is concave. Lemma 9, in a fashion similar to that of


Ž . Ž Ž ..Lemma 8, is useful for extending 10e and 10b to earlier stages of theŽ .game; this in turn allows the use of Lemmas 3 and 10 stated below at

earlier stages. The main proof depends on the quadratic assumption onlythrough Lemma 9.

Ž � � . � �LEMMA 9. Consider any x , y � A. Assume that x � y � 0 and:

x , y� � 1 is a 1-allocation � x � x� � 2;Ž .x , y� is a 1-allocation � x � x� � 1;Ž .x , y� � 1 is a 1-allocation � x � x� ;Ž .x , y� � 2 is a 1-allocation � x � x� � 1.Ž .

� Ž � � . �Ž � � . �Ž � � .Then B x , y � � and B x , y � , implying B x , y �2 2 1 1 1� Ž � � . Ž � � . �Ž � � .B x , y , implying that V is conca�e at x , y � 1 . Moreo�er, B x , y2 1

Ž . � � Ž . � �� 0 iff one of the following is true: a x � y � N � 1, b x � y1� Ž . Ž � � .� N � 2 and x � N�2, or c x , y � 2 is a 1-allocation.

Finally, Lemma 10 describes special features of the bidding functionsnear the midline. Its role is to support the demonstration that bidder 1Ž . Ž .the leader wins any auction of the form y � 1, y ; in other words, anequilibrium path never returns to the midline after leaving it. This isestablished recursively, using Lemma 10, from the last stages to the firststages of the game.

ŽLEMMA 10. Consider any y such that 2 y � 3 � N. Assume that y �. Ž .1, y � 1 and y � 2, y � 1 are 1-allocations.

Ž . Ž .A. If y � 2, y is a 2-allocation, y � 3, y is a 1-allocation, and�Ž . � Ž . Ž .B y � 2, y � 1 � B y � 2, y � 1 , then y � 1, y is a pure 1-allocation.1 2

Ž . Ž . Ž .B. If y � 2, y and y � 3, y are 2-allocations, and y � 2, y � 1Ž .is a pure 1-allocation, then y � 1, y is a pure 1-allocation.

Ž . Ž . �ŽC. If y � 2, y and y � 3, y are 1-allocations, and B y � 2, y �1. � Ž . Ž .1 � B y � 2, y � 1 , then y � 1, y is a pure 1-allocation.2

The remainder of this appendix sketches the proof of Theorem 2 anduses Fig. 1 to illustrate some of the arguments.

Ž .Sketch of proof of Theorem 2. Assumption 10d and MLV imply thatŽ . Ž .buyer 1 wins the last auction: i x, N � 1 � x is a pure 1-allocation,

� x � N�2.The next several paragraphs show that the threshold values Z follow the

pattern described in Theorem 2 and that buyer 1 generically wins everyŽ .auction in region A i.e., above the threshold . The argument begins for1

allocations with high y values and proceeds recursively to lower y values.

GALE AND STEGEMAN100

� Ž . Ž .For arbitrary y , define the statement � x, y is a 1-allocation � x � y,� Ž . Ž .y � 1, . . . , N � 1 � y, � y � y . Bidder symmetry, i and 10g imply that

� Ž . Ž .y � y � N � 1 � 1 � 7 in Fig. 1 satisfies � . Hence, for the highest0ˆvalues of y, buyer 1 wins regardless of the value of x. Let be the

� ˆ ˆŽ . Ž .smallest integer such that y � satisfies � � 6 in Fig. 1 ; thenˆ ˆ ˆŽ . y implies ii N�2 � 1. For each i � 1, 2, . . . , , let be theˆ0 i

ˆŽ .smallest integer such that x, � i is a 1-allocation � x � , � 1, . . . ,ˆ î iˆ Ž .N � 1 � � i. In Fig. 1, � 8, � 9, � 11, etc. Lemma 7 willˆ ˆ ˆ1 2 3ˆshow that and are equal to and , as defined in Theorem 2.i i

� � � ˆŽ . Ž .Lemma 9 implies: iii for arbitrary x , y � A , if y � , or if� � ˆ � � � � � �Ž . Ž .x � � 2 and y � , then B x , y � and B x , y � � ;ˆ1 1 1 2 2

ˆ ˆ ˆŽ . Ž .given this and ii , Lemma 10C implies that if � 0 then , � 1 is aˆ ˆŽ . Ž . Ž .1-allocation; hence iv if � 0, then � � 1. Given ii and iv ,ˆ1

ˆLemma 7B implies that � .To use Lemma 7 to show that � , we will first show by inductioni i

Ž .that � � i � 1. Suppose that . Given iv , Lemma 9 can beˆ ˆ ˆ î i�1 2 1Ž . Ž .used to show that V is concave at � 1, . Since � 1, � 1 is aˆ î 1

Ž .pure 2-allocation by construction, Lemma 3 implies that , � 2 is aˆ1Žpure 2-allocation, contradicting . In the example of Fig. 1: ifˆ ˆ2 1

Ž . � 8 then Lemma 9 shows that V is concave at 7, 6 ; since theˆ ˆ2 1Ž .construction of implies that 7, 5 is a pure 2-allocation, Lemma 3ˆ1

Ž . .implies that 8, 4 must also be a 2-allocation, contradicting 8.ˆ2Therefore, it must be that � , and repeating this argument estab-ˆ ˆ2 1

Ž .lishes: v � � i � 2, 3, . . . , . Lemma 3 is here requiring thatˆ î i�1player 2 win at sufficient allocations to ensure that increasing player 1’sinitial allocation by a given number never increases player 1’s final alloca-tion by a greater number; this gives the frontier between regions A and2

Ž . Ž .A its negative slope of magnitude less than one. Given ii and v , Lemma37 shows that � for all i.i i

It has thus been established that every allocation in A is a 1-allocation,1but it remains to show that they are, generically, pure 1-allocations.Reversing the order of the previous argument, this is shown first forallocations such that y , and we can then use Lemma 3 ‘‘in reverse’’ to

Ž . Ž .extend these results to y � . Given iv and v , Lemma 7B implies that,Ž . Ž .generically: vi x, � i is a pure 1-allocation � x � , � 1, . . . , N � 2i i

Ž . Ž .� � i, � i � 0, 1, . . . , , where � � 2. Given ii , statement i0extends this claim to x � N � 1 � � i. For the rest of the proof ofTheorem 2, we assume the generic case.

Ž . Ž .Given iii and vi , Lemma 3 may be applied repeatedly to show thatŽ .x, y is a pure 1-allocation � x � y � 2, y � 3, . . . , N � y � 1, � y � .ŽThis part of the argument is not necessary for the example of Fig. 1, but it


Ž . Ž .works in this fashion: since vi implies that 9, 6 is a pure 1-allocation,Ž . Ž .and V is concave at 8, 8 , Lemma 3 implies that 8, 7 is a pure 1-alloc-

. Ž . Žation. Finally, since y � and y � 1, y is a pure 1-allocation from0 0 0Ž .. Ž . Ž .i , Lemma 10C and the concavity implied by iii imply that any y � 1, y

Ž . Ž .� A, y � , is a pure 1-allocation. It is thus shown that generically : viiŽ .every x, y � A is a pure 1-allocation. That establishes Theorem 2 for1

the case � 0. Henceforth assume � 0.A straightforward application of Corollary 4.1 establishes the upper

Žbounds on � and � claimed by Theorem 2. For instance, in1 i�1 iŽ . Ž .Fig. 1, � 9 and � 11. If � 12 instead, then 10, 3 and 11, 32 3 3

Ž . Ž .would be 2-allocations, 9, 4 and 9, 5 would be 1-allocations, and Corol-Ž . Ž .lary 4.1 would imply that 9, 3 is a tie allocation, contradicting ix , which

Ž . . Ž .requires 9, 3 to be a pure 2-allocation. That establishes claims 13a andŽ .13b .

The next step is to show that every allocation in A is a pure 2-alloc-2ation. The argument is based on Lemma 5, but to use Lemma 5 we need

Ž . Ž .the following preliminary result. Given v and vii , Lemma 9 can be usedŽ . �Ž � . � Ž � .to show: viii B x , � i � � for any x and i such that x , � i �1 1

A, x� � � i � 2, x� � � 2, and 0 i � .i�1The argument proceeds as an induction from higher to lower values of

y, with an inside induction from higher to lower values of x. For arbitraryŽ . Ž .i � 0, define the statement � x, � i is a pure 2-allocation and

�Ž . Ž .B x, � i � 1 � 0 for x � � 2, � 3, . . . , � 1. For i � 0, � is1 iŽ . Ž .vacuously true. From above, � � 2. Suppose � holds for i �0

� � Ž .0, 1, . . . , i for some given i � 0, 1, . . . , � 1. We will show that � holdsfor i� � 1. This is immediate if i� � 0 and � � 2. For all other cases,1

� Ž . Ž � .define for arbitrary x the statement �� x, � i � 1 is a pure 2-alloc-�Ž �. � � Ž .�ation and B x, � i � 0 for x � x , x � 1, . . . , � 1. Given 13a1 i �1

Ž . Ž . Ž . ��and 13b , viii implies that �� holds for x � � 1. That initializesi �1

Ž . Ž . �the interior induction on x . Suppose �� holds for some x � � 3,Ž . Ž . �

� � 4, . . . , � 1. Statements vii and � at i � 1, 2, . . . , i imply that,i �1Ž �.for each x � � 2, x, � i is either a pure 2-allocation or a pureŽ � �.1-allocation. Therefore, x � 1, � i is not a tie allocation. An involved

Ž . Ž . Ž . Ž .argument, based upon statements 13a , 13b , vii , and viii , and employ-Ž . �Ž �ing Lemmas 8 and 9 and Corollary 4.1, next shows that a B x � 1,1

�. Ž . � i � 0. That sets up the application of Lemma 5. Since �� holds for� Ž � �.x , and x � 1, � i is not a tie allocation, Lemma 5A implies thatŽ � � . Ž . Ž .x � 1, � i � 1 is a pure 2-allocation, implying with a that ��

� Žholds for x � 1. To see the application of Lemma 5A on Fig. 1, supposethat i� � 2 and x� � 10. At this point in the proof it has been shown thatŽ . Ž .9, 4 is a pure 1-allocation, 10, 3 is a pure 2-allocation, and V is convex1

GALE AND STEGEMAN102

Ž . Ž . Ž .at 9, 5 . If 9, 3 is a 1-allocation, then Lemma 5A implies that 9, 4 is a tie.allocation, a contradiction. In the absence of subsequent tie allocations,

Lemma 5 is essentially precluding any equilibrium path on which player 2wins a claim after player 1. Completing the nested inductions establishes:Ž . Ž .ix x, � i is a pure 2-allocation � x � � 2, � 3, . . . , � 1, � i �i1, 2, . . . , .

Ž .Let � max � 1, � 2 , the value taken by x in the first column of1Ž .region A . If � � 4, then � � 2 and ix shows that every2 1

allocation in A is a pure 2-allocation. If � � 4, then A includes2 1 2Ž .allocations x, y for which x � � 1, and it remains to be shown that

Ž Ž .these are pure 2-allocations. In Fig. 1 these allocations comprise 7, 0Ž . . Žthrough 7, 5 . This is done by induction, beginning with allocation �. Ž .1, � 1 . If � � 2, then � 1, � 1 is a pure 2-allocation by1

Ž . Ž .construction; if not, then 13a implies � � 3, and, given vii and1Ž . Ž .ix , Lemma 4 implies that � 1, � 1 is a pure 2-allocation. From thatpoint, an induction similar to the induction in the previous paragraph,

Ž . Ž . Ž . Ž . Ž .using iv , v , vii , and ix , and Lemmas 8 and 5A shows that � holds fori � 1, 2, . . . , , implying that every allocation in A is a pure 2-allocation.2

Ž .That establishes claim 13d .Ž .The last step of the proof is to establish that every allocation x, y in

A is a pure 1-allocation and every allocation in A is a tie allocation.3 4These facts are shown simultaneously through an induction that proceedsfrom high to low values of y.

Ž . Ž . Ž .For given i, define the statements � a x, y is a tie allocationŽ . Ž . Ž .� x � y � 2, y � 3, . . . , � 1 and � b y � 1, y is a pure 1-allocation,

both statements � y � � i, � i � 1, . . . , � 1. Consider i � 1. If 1Ž . Ž . Ž� � 4, then � � 1 and � a holds vacuously. In this case, as in

Ž . Ž . .Fig. 1, the allocation y � 2, � 1 � � 1, � 1 is in region A . If2Ž . � � 4, then � � 2 and Lemma 4 shows that � 1, � 1 is a1

Ž .tie allocation, much as it showed above that � 1, � 1 is a pureŽ . Ž .2-allocation in the case � � 3. That establishes � a for i � 1. If1

Ž . Ž . Ž . � � 1, then 13a and 13d imply that � 1, � 1 is a 2-allocation;Ž . Ž . Ž .if not, then � � 2 and � a implies that � 1, � 1 is a 2-alloc-

Ž . Ž .ation. If � � 2, then, given 13a and 13d , Lemma 10B shows that1Ž . Ž . Ž . , � 1 is a pure 1-allocation; if not, then, given iii and vii , Lemma

Ž . Ž . Ž .10A shows that , � 1 is a pure 1-allocation. That establishes � bfor i � 1.

Ž .Completing the induction on � , from i � 1 to i � , requires routineapplications of Corollary 4.1 to establish the tie allocations in region A4Ž .using, for each y, an interior induction from high to low values of x and

Ž .Lemma 10B to establish the pure 1-allocations in region A . Given vii ,3Ž . Ž .this establishes 13c and 13e . �


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Sequential Auctions of Endogenously Valued Objects