Linear Programming for Mechanism Design:

An Application to Bidder Collusion at

First-Price Auctions�

Giuseppe Lopomo

glopomo@duke.edu

Leslie M. Marx

marx@duke.edu

Peng Sun

peng.sun@duke.edu

Fuqua School of Business, Duke University, Durham, NC 27708

March 27, 2008

Abstract

We demonstrate the use of linear programming techniques in the analysis

of mechanism design problems. We use these techniques to analyze the extent

to which a �rst-price auction is robust to collusion when, contrary to some

prior literature on collusion at �rst-price auctions, the cartel cannot prevent its

members from bidding at the auction. In this case, cartels have been shown to

be less pro�table facing a �rst-price auction than facing other common auction

formats, but we show the stronger result that in certain environments collusion

at a �rst-price auction is not pro�table at all. Our results suggest that �rst-

price auctions are more robust to collusion than previously believed.

Subject classi�cations: Economics: mechanism design. Games/group decisions:

Bidding/auctions.

�The authors thank Bob Marshall for helpful comments.

1 Introduction

The economics literature on mechanism design provides techniques for analyzing how

individual preferences can be elicited and used to make collective decisions (see the

foundational paper Hurwicz (1973), surveys in Eatwell, Milgate, and Newman (1989),

especially Myerson (1989) and Ledyard (1989), and the textbook Mas-Colell, Whin-

ston, and Green (1995)). These techniques have been particularly useful in the design

of auctions (Milgrom (1985, 1989), Wilson (1987), McAfee and McMillan (1992),

and McMillan (1994)), but also have applications to matching models (Roth and

Sotomayor (1990)), price discrimination (Varian (1989)), and a variety of other eco-

nomic applications. The operations research/management science literatures see ap-

plications in revenue management (Vulcano, van Ryzin, and Maglaras (2002)), B2B

procurement auctions (Gallien and Wein (2005) and Beil and Wein (2003)) and var-

ious other applications of combinatorial auctions summarized in Cramton, Shoham

and Steinberg (2006). Several computer science applications are described in Rosen-

schein and Zlotkin (1994), and Varian (2000) discusses the role of computerized agents

in mechanism design.

A variety of mechanism design problems can be reformulated as linear programs,

which recent software and hardware innovations allow to be solved quickly. These

innovations open the door for linear programming techniques to be used to inform

the search for analytic results for these problems. In this paper, we direct these

techniques at the problem of auction design in the face of collusive behavior by the

bidders.

Although auction design has bene�ted from the application of mechanism design

techniques, much of the auction design literature assumes noncooperative behavior

by the bidders. Unfortunately, auctions can be susceptible to collusion among the

bidders. Such collusion decreases the revenue obtained by the seller and can distort

the e¢ ciency of an auction. In the U.S., bid rigging is a per se violation of the

Sherman Act. In an attempt to alert the public to the problem of bidder collusion

and other price �xing schemes, the U.S. Department of Justice publishes the primer

�Price Fixing & Bid Rigging �They Happen: What They Are and What to Look

For�(available at http://www.usdoj.gov/atr/public/guidelines/pfbrprimer.pdf). Un-

derstanding the mechanisms by which bidders support collusive agreements can be

valuable for informing the choice of auction format, detecting and prosecuting cartels,

1

and also potentially for understanding within-organization mechanism design, where

an organization might want to encourage �collusion�among its employees.

In this paper, we address an open question in the literature on auction design,

which is the extent to which a �rst-price auction is robust to collusion. As described in

Section 2, results in the literature suggest that a �rst-price auction is more robust to

collusion than a second-price or ascending-bid auction, but that a �rst-price auction is

still vulnerable to collusion (see Section 2 for the underlying intuition). In this paper,

we illustrate how linear programming techniques can be used to study this question

and ultimately to facilitate analytic results showing that in certain environments,

not only is the �rst-price auction more robust to collusion than some other auction

formats, but in fact no pro�table collusion is possible at a �rst-price auction. This

result provides a strong theoretical basis for the existing recommendation that sellers

concerned about collusion use a �rst-price rather than second-price or ascending-bid

auction (see �Practical conclusion 1�of Kovacic et al. (2006)).

As we show in this paper, the mechanism design problem for a bidding cartel at

an auction can be formulated as a linear program. Thus, given su¢ cient computing

power, for discrete environments we can solve for the optimal collusive mechanism.

Using noncooperative bidding as a benchmark, we can calculate the gain to bidders

and loss to the seller from collusion. Furthermore, a study of the solutions generated

by the linear programming techniques to a variety of examples can inform us as to

which constraints are binding and, as in the case of this paper, facilitate analytic

proofs based on duality arguments. Speci�cally, using insights gained through solved

numerical examples, we are able to use linear programming duality arguments to

prove that for a class of symmetric environments pro�table collusion is not possible

at a �rst-price auction. Although we leave analytical results for the asymmetric

case to future research, numerical solutions of linear programs for the asymmetric

case suggest that �rst-price auctions are robust to collusion in certain asymmetric

environments as well.

We believe the techniques illustrated in this paper have potential value in a va-

riety of mechanism design applications. Our concluding section provides additional

discussion related to this.

The paper proceeds as follows. In Section 2, we provide background on bidder

collusion at �rst-price auctions. In Section 3, we describe the model and the mech-

anism design problem of the cartel. In Section 4, we present our main results. In

2

Section 5 we discuss extensions. In Section 6 we provide concluding discussion.

2 Collusion at �rst-price auctions

A common auction form is the �rst-price auction, in which bidders simultaneously

submit bids and the high bidder wins and pays the amount of its bid. Our results

apply equally to sealed-bid procurements, in which the low bidder wins, as long as

the only dimension of the bid is price.

Bidding cartels at �rst-price auctions have been prosecuted in: U.S. v. A-A-A Elec.

Co., Inc. (1986); U.S. v. W.F. Brinkley & Son Construction Company, Inc. (1986);

U.S. v. Raymond J. Lyons (1982) (sheet metal); and U.S. v. Addyston Pipe & Steel

Co. et al. (1897). In each of these cases, colluding bidders met prior to the auction to

discuss their bids and determine transfer payments among the cartel members. For

example, in U.S. v. Addyston Pipe, colluding cast-iron pipe manufacturers met prior

to the auction, determined how the colluding �rms would bid at the auction, and

agreed on transfer payments:

�... the executive committee determines the price at which the bid is to be

put in by some company in the association, and the question to which company

this bid shall go is settled by the highest bonus which any one of the companies

... will agree to pay or bid for the order. ... the company to whom the right

to bid upon the work is assigned sends in its estimate or bid to the city or

company desiring pipe, and the amount thus bid is �protected�by bids from

such of the other members of the association as are invited to bid, and by the

bidding in all instances being slightly above the one put in by the company

to whom the contract is to go. ... Settlements are made at stated times of

the bonus account debited against each company, where these largely o¤set

each other, so that small sums are in fact paid by any company in balancing

accounts.�(U.S. v. Addyston Pipe & Steel Co. et al. (1897) at p.3)

In this paper, we focus on collusion at �rst-price auctions in an environment in

which the cartel can communicate prior to the auction and enforce transfer payments

among cartel members, although because our main results are �negative�in the sense

that they show pro�table collusion is not possible, eliminating the ability of the cartel

to determine transfer payments does not a¤ect these results. In addition, we focus on

3

a cartel operating at a one-shot auction, but with repeated interaction, a number of

authors show that for su¢ ciently large discount factors, a cartel can do better than

noncooperative play (see Oren and Rothkopf (1975), Aoyagi (2003), Skrzypacz and

Hopenhayn (2004), Blume and Heidhues (2006), and Hörner and Jamison (2008)).

Thus, our results suggest bene�ts for sellers concerned about collusion from limiting

repeated interaction among the bidders.

We focus on cartels that operate prior to the auction. Thus, we assume the cartel

cannot condition transfer payments on the outcome of the auction, i.e., a cartel cannot

impose penalties on its members after the auction if the auction outcome does not

correspond to the cartel�s desired actions. This restriction would be imposed on a

cartel if, for example, the auction�s outcome were not observable. Of course, a seller

facing a cartel that relied on the observation of the auction�s outcome to enforce cartel

behavior would have an incentive to suppress information about auction outcomes

beyond notifying the winning bidder that it had won and collecting payment.

In our environment, the cartel would maximize its collusive gain if it could suppress

all within-cartel rivalry and send the only the highest-valuing ring member to the

auction to bid against the non-cartel bidders. This outcome is obtained in the model

of McAfee and McMillan (1992), a seminal paper on collusion at �rst-price auctions.

In that paper, cartel members �rst bid for the right to be the sole cartel member to

attend the auction. The highest bidding cartel member is sent to the main auction,

while other cartel members are prevented from bidding at the main auction. McAfee

and McMillan focus on the case of symmetric bidders and an all-inclusive cartel, in

which case the mechanism is e¢ cient and extracts all surplus from the seller. In

equilibrium, the lone cartel member to attend the auction is the one with the highest

value for the object, and that cartel member wins the object at a price equal to the

seller�s reserve price.

In the McAfee and McMillan (1992) environment, the cartel can extract all the

surplus from the seller, so clearly in that environment using a �rst-price auction

does nothing to protect the seller from collusion among the bidders. However, the

mechanism of McAfee and McMillan relies on the assumption that the cartel can send

one designated cartel member to the auction and prevent all others from submitting

bids. Although this may be possible in some settings, it is not possible in others.

If the cartel cannot prevent its members from submitting bids, the mechanism of

McAfee and McMillan (1992) does not function as intended. To see this, note that if

4

the high-valuing cartel member is sent to the auction and told to submit a bid equal

to the seller�s reserve price, other cartel members with values greater than the reserve

price will have an incentive to enter bids at the auction, eating away at the collusive

gain. To see that this type of cheating can be a concern of a cartel, note that in U.S.

v. Brinkley & Son (1986), the cartel decided that Brinkley would submit the winning

bid and took measures to monitor other cartel members�bids by having at least one

of the other cartel members give its bid form to Brinkley to submit for it.

The threat of cheating by cartel members that are not selected by the cartel

to win the auction does not arise in second-price or ascending bid auctions. Even

without the ability to prevent cartel members from submitting bids, Marshall and

Marx (2007) show that at a second-price or ascending-bid auction a cartel can achieve

the �rst-best collusive outcome. (See also the mechanism of Graham and Marshall

(1987) for environments in which the cartel can condition on the identity of the

winner and the price paid.) To see why this holds, note that in a second-price

or ascending-bid auction, the high-valuing cartel member need not change its bid

relative to non-cooperative bidding in order to obtain a collusive gain. In contrast,

at a �rst-price auction, unless the high-valuing cartel member reduces its bid below

its non-cooperative bid, there is no collusive gain. This lowering of the high-valuing

cartel member�s bid creates incentives for cheating by other cartel members.

Marshall and Marx (2007) formalize the notion that collusion is less pro�table at

a �rst-price auction than a second-price or ascending-bid auction when the cartel can

prevent cartel members from bidding at the auction. They show that the cartel can

suppress all within-cartel rivalry at a second-price or ascending-bid auction, but not

at a �rst-price auction.

Although Marshall and Marx (2007) show that collusion is less pro�table at a

�rst-price auction, they do not show how much less pro�table. They leave as an open

question whether there exists a collusive mechanism that does better than noncooper-

ative play when a cartel cannot prevent its members from bidding. They suggest the

possibility of a pro�table collusive mechanism based on the following intuition. A car-

tel at a �rst-price auction could �rst identify the highest and second-highest-valuing

cartel member. Then if the highest-valuing cartel member�s noncooperative bid is

greater than the second-highest value, then ask the highest-valuing cartel member to

reduce its bid slightly relative to its noncooperative bid. But if the highest-valuing

cartel member�s noncooperative bid is less than the second-highest value, then simply

5

tell all cartel members to bid noncooperatively. At �rst glance, it seems this type

of mechanism should avoid any incentives for cheating by non-highest-valuing cartel

members. However, the logic is �awed� a cartel member who is told to bid noncoop-

eratively will infer that with some probability its value is second-highest but that it

can outbid the highest-valuing ring member and win the object by increasing its bid

slightly.

In this paper, we show that collusion is not possible at a �rst-price auction in the

sense that no collusive mechanism increases bidder surplus relative to noncooperative

play for a set of environments with symmetric bidders if the cartel cannot prevent its

members from bidding at the auction. Much of the auctions literature characterizes

�rst-price auctions as less susceptible to collusion than other auction formats such

as second-price and ascending-bid auctions (see, e.g., Robinson (1985), Marshall and

Meurer (2004), and Marshall and Marx (2007)), but this is the �rst result showing

that in some environments a �rst-price auction is immune to certain types of collusion.

3 Model

3.1 Setup

We focus on a single-object �rst-price auction with a non-strategic seller, although

our results (appropriately adjusted) continue to hold in the presence of a �xed reserve

price. The high bidder wins and pays the amount of its bid. In the case of a tie, we

assume the object is randomly allocated to one of the bidders with the high bid. We

assume bidders have independent private values and are risk neutral.

For tractability, we assume there are only 2 bidders, but the results should extend

to any �nite number of bidders.

Each bidder i 2 f1; 2g independently draws a value vi from a distribution Fi with�nite support Vi. Let V � V1�V2. Let fi(vi) be the probability that bidder i�s valueis vi. Where convenient we assume discrete bids, but with a vanishingly small bid

increment. Let B denote the set of feasible bids.Let ui (bi; b�i; vi) be bidder i�s surplus in a �rst-price auction, given bidder i�s

6

value vi and bid bi and the opponent�s bid b�i. That is,

ui (bi; b�i; vi) �

8><>:vi � bi; if bi > b�i0; if bi < b�i(vi � bi) =2; otherwise:

All bidders are eligible to participate in a cartel. The assumption of an all-inclusive

cartel is common in the literature, particularly because the study of non-all-inclusive

cartels at �rst-price auctions is made di¢ cult by the lack of analytic bid functions (see

Marshall et al. (1994) and Gayle and Richard (2005) for related numerical techniques).

The cartel mechanism operates as follows: Each cartel member makes a report to

a �center,�which is a standard Myerson (1983) incentiveless mechanism agent. Based

on these reports, the center recommends a bid to be made by each cartel member and

requires payments from the cartel members. Cartel members observe only their own

bid recommendations and required payments. We require that the center�s budget

be balanced in expectation.

We focus on incentive compatible collusive mechanisms, which in our environment

has two dimensions: truth-telling and obedience. First, it must be incentive compati-

ble for cartel members to truthfully report their values. Second, it must be incentive

compatible each cartel member to follow the bid recommended by the center. In

addition, we require ex-ante individual rationality, so that bidders�expected payo¤s

from participation in the cartel are at least as great as their expected payo¤s from

noncooperative play. We assume the cartel can compel its members to make their

required payments, but that it cannot prevent cartel members from participating in

the auction.

The assumption of ex-ante individual rationality can be viewed as an assumption

that cartel members must commit to participation in the cartel prior to learning their

values and that the failure of any bidder to join results in the complete dissolution of

the cartel. One might consider the alternative assumption that refusal by one poten-

tial ring member to join the ring implies that the remaining potential ring members

form a ring, leaving one bidder outside the ring, but this alternative assumption

complicates the veri�cation of individual rationality for �rst-price auctions because a

potential ring member may prefer to be outside a ring of n � 1 bidders rather thaninside a ring of n bidders.

Note that in the environment we consider, a cartel at a second-price or ascending-

7

bid auction extracts all the surplus from the seller. The cartel�s surplus is equal

to the highest value among the bidders and the seller receives zero (or the reserve

price in the case of a positive reserve price). See Marshall and Marx (2007) for the

speci�cation of the collusive mechanism.

3.2 Mechanism design problem

By the revelation principle, we restrict attention (without loss of generality) to the

class of all incentive compatible and individually rational direct revelation mecha-

nisms. Any incentive compatible and individually rational direct revelation mecha-

nism speci�es for each bidder i 2 f1; 2g and each pair of reports (v1; v2) 2 V ; (i) aprobability distribution over all feasible bid pairs pi (� j vi; v�i) 2 4B�B, where 4B�B

represents the probability simplex with each pair of bids as the vertex; and (ii) a

monetary transfer mi (vi; v�i).

The center�s problem can be written as follows (the notation below parallels that

of Myerson (1985)):

maxp1;p2;m1;m2

Xi2f1;2g;(v1;v2)2V;b12B;b22B

pi(bi; b�i j vi; v�i)ui(bi; b�i; vi)fi(vi)f�i(v�i); (1)

subject to pi representing a probability distribution, 8i 2 f1; 2g; 8(v1; v2) 2 V ;

pi (� j vi; v�i) 2 4B�B and p1 (b1; b2 j v1; v2) = p2 (b2; b1 j v2; v1) ; (2)

ex-ante budget balance, Xi2f1;2g;(v1;v2)2V

mi (vi; vi) fi(vi)f�i(v�i) = 0; (3)

and incentive compatibility, 8i 2 f1; 2g; 8vi; v0i 2 Vi; 8�i : B ! B;

U�i (pi; �i; v0i j vi)� Ui (pi j vi) � 0; (4)

8

where Ui (pi j vi) is bidder i�s expected surplus under truthtelling and obedience,

Ui (pi j vi) �P

v�i2V�i;bi2B;b�i2Bpi(bi; b�i j vi; v�i)ui(bi; b�i; vi)f�i(v�i)

�P

v�i2V�imi (vi; v�i) f�i(v�i);

(5)

and U�i (pi; �i; v0i j vi) is bidder i�s expected surplus when bidder i�s value is vi; it

reports v0i; and it bids according to a deviation function �i (bi) that depends on the

center�s recommendation bi,

U�i (pi; �i; v0i j vi) �

Xv�i2V�i;bi2B;b�i2B

pi(bi; b�i j v0i; v�i)ui(�i(bi); b�i; vi)f�i(v�i)

�X

v�i2V�i

mi (v0i; v�i) f�i(v�i): (6)

The above formulation allows bidder i�s payment to depend on its opponent�s

report, but it does not allow an inference by bidder i regarding the type of its rival

from its required payment. Given the result of Myerson (1982, Proposition 2), this is

without loss of generality. One can see this from the fact that only bidder i�s expected

payment given its own report enters the problem above.

The incentive compatibility constraint in (4) considers all possible deviation func-

tions �i that map the recommended bid from the center to the actual bid. Such a set

of deviation functions is large (exponential to the number of feasible bids). In order

to formulate the problem as a tractable linear program, we introduce the following

new optimization model which involves the function Ji(vi; v0i; bi), which gives bidder

i�s expected surplus if it has type vi, reports v0i, receives recommendation bi from the

center, and bids optimally; multiplied by Pr(bi j v0i). We are able to simplify fur-ther by noting that it is su¢ cient to consider the expected payment function Mi(vi),

instead of mi(vi; v�i).

Lemma 1 The center�s problem can be written as follows:

max�;M1;M2;J1;J2

2Xi=1

X(v1;v2)2V;b12B;b22B

�(b1; b2; v1; v2)ui (bi; b�i; vi) f1(v1)f2(v2) (7)

subject to � representing a conditional probability distribution, 8(v1; v2) 2 V ; 8b1 2 B;

9

8b2 2 B;�(b1; b2; v1; v2) � 0; (8)

and 8(v1; v2) 2 V ; Xb12B;b22B

�(b1; b2; v1; v2) = 1; (9)

ex-ante budget balance,X(v1;v2)2V

(M1(v1) +M2(v2)) f1(v1)f2(v2) = 0; (10)

incentive compatibility, 8v1 2 V1; 8v01 2 V1;Xv22V2;b12B;b22B

�(b1; b2; v1; v2)u1(b1; b2; v1)f2(v2)�M1 (v1)

�Xb12B

J1(v1; v01; b1)�M1(v

01) ; (11)

and 8v2 2 V2; 8v02 2 V2;Xv12V1;b12B;b22B

�(b1; b2; v1; v2)u2(b2; b1; v2)f1(v1)�M2 (v2)

�Xb22B

J2(v2; v02; b2)�M2(v

02) ; (12)

and the de�nition of J , 8v1 2 V1; 8v01 2 V1; 8b1 2 B; 8b01 2 B;

J1 (v1; v01; b1) �

Xv22V2;b22B

�(b1; b2; v01; v2)u1 (b

01; b2; v1) f2 (v2) ; (13)

8v2 2 V2; v02 2 V2; b2 2 B; b02 2 B;

J2 (v2; v02; b2) �

Xv12V1;b12B

�(b1; b2; v1; v02)u2 (b

02; b1; v2) f1 (v1) : (14)

Proof. See Appendix A.

Using Lemma 1, we can apply linear programming techniques to analyze the

optimal collusive mechanism in the �rst-price auction. In Section 4 we focus on

10

symmetric bidders, and in Section 5 we consider the extension to asymmetric bidders.

4 Results for symmetric bidders

In this section, we show that pro�table collusion is not possible when bidders are

symmetric. Speci�cally, we consider an environment with discrete bids and show

that as the bid increment approaches zero, bidder surplus from the optimal collusive

mechanism approaches that from noncooperative bidding. Thus, although discrete

bids may create an environment in which bidders can pro�tably collude at a �rst-price

auction, the bene�t from such collusion vanishes as the bid increment approaches zero.

Even without the use of transfer payments, a cartel could achieve any correlated

equilibrium (in the sense of Forge�s (2006) communication equilibria for games with

incomplete information). Thus, our result implies that in our environment the best

correlated equilibrium from the perspective of the cartel is a Nash equilibrium (see

Nau, Canovas, and Hansen (2003) on the relation between correlated and Nash equi-

libria).

In this section we assume symmetry, which means that V1 = V2 and that forany value v 2 Vi and bids bi; b�i 2 B, we have f1(v) = f2 (v) and u1(b1; b2; v) =

u2 (b2; b1; v). Therefore, in this section we suppress the subscripts on fi and ui.

The proof of the main result of this section relies on the assumption that each

bidder has two possible values, h and l, h > l > 0. As discussed in Section 6, our

numerical investigations suggest our results continue to hold when this assumption is

relaxed.

4.1 Noncooperative benchmark

For the two-type case, the following proposition describes the surplus from a nonco-

operative equilibrium.

Proposition 1 Assume two bidders, each having value l with probability f (l) andvalue h with probability f (h) = 1 � f (l), where 0 < l < h. The noncooperative

equilibrium of the �rst-price auction game is as follows: a bidder with value l bids l;

11

a bidder with value h bids according to the following cumulative distribution

F (b) =

8><>:0; if b < lf(l)(b�l)f(h)(h�b) ; if b 2 [l;�b]1; otherwise,

where �b � lf (l) + hf (h) :

Proof. See Appendix C.

Using Proposition 1, we can characterize the noncooperative equilibrium as fol-

lows.

Corollary 1 In the environment of Proposition 1, the noncooperative equilibriumoutcome is e¢ cient, expected bidder surplus is 2 (h� l) f (h) f (l) ; and expected sellerrevenue is f (l) (1 + f (h)) l + f (h)2 h:

With this noncooperative outcome as a benchmark, we can consider collusive

outcomes.

4.2 Collusion

The results that follow assume discrete bid increments. We model the bid increment

in the following way. For any positive integer B, the bid increment is de�ned to be

� � �b�l2B. The set of feasible bids can be expressed as

BB � fbj j bj = l +�j for j 2 f0; 1; : : : ; nBgg;

in which n is a su¢ ciently large integer. Therefore, b0 = l and b2B = �b, and the

interval�l; �b�contains 2B + 1 feasible bids. We also allow bids arbitrarily higher

than �b by having n > 2. As the integer B approaches in�nity, the bidding increment

� approaches zero. When it is clear in the context, we omit the subscript B from

BB.We consider the lowest feasible bid to be l, which can be enforced by the seller

setting a reservation price l. Such an assumption is without loss of generality, as

demonstrated by the following lemma, which states that for any given arbitrarily

12

small bid increment �; it is not incentive compatible for the collusive mechanism to

recommend any bid below l � 2�:

Lemma 2 The lowest incentive compatible bid recommended by the center with pos-itive probability, blow, is l � 2�.

Proof. See Appendix C.

To take advantage of the assumption of symmetry, it is helpful to simplify the

linear program for the optimal collusive mechanism given in (7)�(14). In particular,

as shown in Lemma 3, we can work with the following formulation:

max�;M;J

Xv;�v2fl;hg

Xbj ;b�j2B

�u(bj; b�j; v) + u(b�j; bj; �v)

��(bj; b�j; v; �v)f(v)f(�v) (15)

subject to � representing a conditional probability distribution, 8v; �v 2 fl; hg ; 8bj; b�j 2B;

�(bj; b�j; v; �v) � 0; (16)

and 8v; �v 2 fl; hg, Xbj ;b�j2B

�(bj; b�j; v; �v) = 1; (17)

ex-ante budget balance, Xv2fl;hg

M(v)f(v) = 0; (18)

incentive compatibility, 8v; bv 2 fl; hg ;P

�v2fl;hg

Pbj ;b�j2B

��(bj; b�j; v; �v) + �(b�j; bj; �v; v)

�u(bj; b�j; v)f(�v)�M(v)

�Pbj02B

J(v; bv; bj0)�M(bv); (19)

and the de�nition of J; 8v; bv 2 fl; hg ; 8bbj; bj0 2 B;J(v; bv; bbj) � X

�v2fl;hg

Xb�j2B

��(bbj; b�j; bv; �v) + �(b�j; bbj; �v; bv)� f(�v)u(bj0 ; b�j; v): (20)

Lemma 3 below establishes that it is su¢ cient for us to work with the linear

program (15)�(20).

13

Lemma 3 The value of the objective function at the optimum is the same in (15)�

(20) and in (7)�(14).

Proof. See Appendix C.

Clearly, for any discrete bidding set BB, the noncooperative equilibrium is a fea-

sible collusive mechanism, and therefore a feasible solution to the linear program

(15)�(20). Consider the dual formulation of the linear program (15)�(20). Fol-

lowing weak duality, the dual objective value from any dual feasible solution is an

upper bound to any primal feasible objective value, in particular the surplus from the

noncooperative equilibrium. In order to prove our result, we construct a sequence

of dual feasible solutions with surplus D(B). We also construct a sequence of lower

bound, NC(B), for the noncooperative equilibrium surplus. Then we show that both

D(B) and NC(B) converge to the surplus from the continuous bid noncooperative

equilibrium given in Corollary 1.

To this end, we �rst present the dual formulation of the linear program (15)�(20):

min�;�;�;�

Xv;�v2fl;hg

�(v; �v)f(v)f(�v) (21)

subject to 8v; v0 2 fl; hg ; 8j 2 f�mB; : : : ; 0; 1; : : : ; nBg ;

f(v0)2BXj0=0

�(v0; v; j; j0) = f(v)�(v; v0); (22)

8v; �v 2 fl; hg ; 8j; �j 2 f�mB; : : : ; 0; 1; : : : ; nBg ;

�(v; �v)�P

v02fl;hg�(v; v0)u(bj; b�j; v)�

Pv02fl;hg

�(�v; v0)u(b�j; bj; �v)

+P

v02fl;hg

2BPj0=0

�(v; v0; j; j0)u(bj0 ; b�j; v0) +

Pv02fl;hg

2BPj0=0

�(�v; v0; b�j; bj0)u(bj0 ; bj; v0)

� u(bj; b�j; v) + u(b�j; bj; �v);

(23)

8v 2 fl; hg ;�f(v) = f (v)

Xv02fl;hg

�(v; v0)�X

v02fl;hg

f (v0)�(v0; v) (24)

14

and 8v; v0 2 fl; hg ; 8j; j0 2 f�mB; : : : ; 0; 1; : : : ; nBg ;

�(v; v0; bj; bj0) � 0 and �(v; v0) � 0: (25)

The following lemma gives properties of a feasible solution to (21)�(25).

Lemma 4 There exists���; ��; ��; ��

�that is feasible in the linear program (21)�(25)

and that generates dual objective value D(B) = (h � l)f (h) f (l)�2 + f(h)2(1+f(l))

2Bf(l)2

�;

where limB!1D(B) = 2(h� l)f (h) f (l) :

Proof. See Appendix B.

As can be seen in Appendix B, the proof of Lemma 4 is quite technical. Indeed,

the linear program (21)�(25) is highly degenerate and has multiple solutions for a

given �nite B. Therefore characterizing its optimal solution is challenging and not

necessary. We are able to construct its feasible solution following insights obtained

from observing the dual optimal solution numerically. Furthermore, the sequence of

dual feasible solutions as constructed in our proof does not converge as B approaches

in�nity. It remains an open question whether one can show the result by directly

working on the in�nite dimensional linear program representing the continuous bid

case.

As shown in Lemma 4, in the limit as the bid increment shrinks to zero (or the

number of bids in the relevant interval increases to in�nity), the dual objective value

associated with our feasible solution converges to 2(h � l)f (h) f (l). For example,with a uniform distribution with f(h) = f(l) = 1

2, this would be (h� l) =2.

The next lemma provides a lower bound for a noncooperative equilibrium surplus.

Lemma 5 For any given integer B and the corresponding discrete feasible bidding

set BB, there exists a symmetric equilibrium such that the total surplus is at least

NC(B) = 2f (h) f (l) (h� l ��) :

Proof. See Appendix C.

Lemmas 4 and 5 imply our main result.

Proposition 2 If bidders are symmetric, n = 2; and bidders have two possible values,in the limit as the bid increment converges to zero, the bidder surplus from the optimal

collusive mechanism converges to the bidder surplus under noncooperative bidding.

15

Proof. The result follows from the fact that D(B) and NC(B) are upper and lower

bounds, respectively, of the optimal collusive bidder surplus for all B, and from

Lemmas 4 and 5 which imply limB!1NC (B) = limB!1D (B). Q.E.D.

Proposition 2 implies that when bidders are symmetric, at least for environments

with two bidders and two possible values, if the bid increment is su¢ ciently small,

pro�table collusion is not possible� a cartel can do no better than to bid noncoopera-

tively. This suggests that when bidders are symmetric, a �rst-price auction is robust

to collusion.

4.3 Symmetric examples

In this section we provide two examples with symmetric bidders and use linear pro-

gramming techniques to solve for the optimal collusive mechanism. The �rst example

illustrates the two-type case. We compare the expected bidder surplus and expected

seller revenue in the optimal collusive mechanism with that of the noncooperative

equilibrium, which is characterized in Proposition 1. By varying the bid increments

we also demonstrate the speed of convergence.

In the second example, we consider n > 2 types. We can construct a feasible

bid set with n bids such that each type of bidder bids a corresponding bid in the

set. We then numerically con�rm that the optimal collusive mechanism coincides

the noncooperative equilibrium. This example suggests that the result that is proved

in the previous section holds more generally in other symmetric bidder settings.

4.3.1 Two types

Assume two symmetric bidders, each with value 40 or 80 with equal probability.

Thus, in the notation above, l = 40; h = 80; f(l) = f(h) = 1=2. With continuous

bids, in the noncooperative equilibrium a bidder with value l bids l and a bidder with

value h bids according to the distribution

F (b) =

8><>:0; if b < lb�lh�b ; if b 2 [l;

h+l2]

1; otherwise.

16

Following Corollary 1, the noncooperative expected bidder surplus with continuous

bids is 20.

Figure 1 shows the lower bound of the noncooperative equilibrium surplus, as

well as the optimal collusive surplus from the linear program (15)�(20) and its upper

bound following Lemma 4. The �gure shows that the optimal collusive surplus, as

well as its upper and lower bounds, approach the noncooperative surplus as the bid

increment approaches zero.

0 5 10 15 20 25 3017

18

19

20

21

22

23

24

B

$

Symmetric Two Type Case

OptSolLowerBoundUpperBoundNonCoop

Figure 1: The optimal collusive bidder surplus (�OptSol�), the upper bound D(B)(�UpperBound�), the lower bound NC(B) (�LowerBound�), and the noncooperativeequilibrium surplus in the continuous bid case (�NonCoop�) as a function of B, whichis inversely related to the bid increment.

Note that in this example, if the auction used a second-price or ascending-bid

format instead of a �rst-price auction, the expected bidder surplus under noncooper-

ative bidding would be 20 just as in the case of a �rst-price auction, but the optimal

collusive surplus would be 70, which is the expected value of the highest of the two

bidders�values. Under noncooperative bidding, the seller�s expected revenue is 50.

When the bidders do not collude, the seller is indi¤erent between the various auction

formats, but when the bidders do collude, the seller�s expected revenue is 50 with

a �rst-price auction versus zero with a second-price or ascending-bid auction. This

contrast is illustrated in Table 1.

17

Table 1: Comparison of Auction Formats

Noncooperative Bidding Collusive Bidding

First-Price Second-Price First-Price Second-Price

expected bidder surplus 20 20 20 70

expected seller revenue 50 50 50 0

expected total surplus 70 70 70 70

4.3.2 More than 2 types

Now consider two symmetric bidders each with a discrete uniform bid type dis-

tribution with support f1; 2; : : : ; ng. Consider the bid set fb1; b2; : : : ; bng, wherebj =

j(j�1)2j�1 . In this environment, one can show the following result.

Proposition 3 There exists a noncooperative equilibrium in which a type j bidder

bids bj.

Proof. Omitted.

Numerical calculations of the linear program with n = 2; 3; : : : ; 14 show that, in

each case, the noncooperative equilibrium bidding strategy coincides with the bids in

the optimal collusive mechanism. This suggests that as the type support approaches

the continuum, our impossibility result continues to hold. We leave the formal proof

of such a result to later research.

5 Extension to asymmetric bidders

In this section, we demonstrate using simple examples that the results of Section

4 continue to hold in some asymmetric environments, although the assumption of

symmetric helps to simplify the proof. For asymmetric environments, we numerically

solve linear programs with two-type bidders to verify that as bidding increments

approach 0, the collusive surplus converges to the noncooperative equilibrium surplus.

Speci�cally, we focus on asymmetric environments with two bidders, where bidder

i has value li with probability pi = fi (li) and value hi with probability 1 � pi. Weassume l1 � h1 and l2 � h2. Without loss of generality, assume l1 � l2. De�ne

�b � min f(1� p1)h2 + p1l1; (1� p2)h1 + p2l1g

18

and assume the following regularity condition L2 <��b� p1l1

�= (1� p1) ; which is

automatically satis�ed when �b = (1� p1)h2 + p1l1.As in the case of our symmetric example, our benchmark is the noncooperative

equilibrium with continuous bids, which is characterized in the following proposition.

Proposition 4 Assume two bidders, where bidder i has value li with probability piand value hi with probability 1� pi, where 0 � l1 < h1, 0 � l2 < h2. Without loss ofgenerality, let l1 � l2. Assume regularity condition

l2 <�b� l1f1 (l)f2 (h)

; (26)

where �b � min f(1� p1)h2 + p1l1; (1� p2)h1 + p2l1g. The noncooperative equilib-

rium of the �rst-price auction game is as follows: bidder 1 with value l1 mixes aggres-

sively just below l1 (see Hirshleifer and Riley (1993, p.374) for the conditions that must

be satis�ed by the aggressive mixing distribution) bidder 2 with value l2 bids l1; and bid-

der i with value hi mixes on (l1;�b] according to distribution Fi(b) = 11�pi

�h�i��bh�i�b � pi

�;

and mixes aggressively just above l1 with a probability atom 11�pi

�h�i��bh�i�l1 � pi

�if

(1� p�i)hi + p�il1 < (1� pi)h�i + pil1.

Proof. See Appendix C.

Using Proposition 4, in the noncooperative equilibrium, expected total bidder

surplus is �h1 � �b

�(1� p1) +

�h2 � �b

�(1� p2) + (l2 � l1) p1p2 : (27)

We tested a number of cases with various choices of l1; l2; h1; h2; p1 and p2 satisfying

the regularity condition (26). When the bid increment becomes small, we always

observe the optimal collusive surplus being very close to the benchmark (27). We

illustrate the typical �ndings in the computation using the following example.

Consider the following types. l1 = l2 = 4, h1 = 100, and h2 = 50. The

probabilities p1 = p2 = 0:5. In this case the noncooperative equilibrium involves the

low type bidding 4 and the high types randomizing on (l1;�b], or, (4; 27]; with an atom

for bidder 2 with type h2 = 50 on a bid of 4. The expected bidder surplus is 48.

Similar to before, we de�ne the bid increment � to be��b� l1

�= (2B) for some

integer B. In Figure 2, we plot the optimal collusive surplus as a function of B. It

19

depicts the typical behavior of the optimal collusive surplus approaching the noncoop-

erative surplus in all the cases that we tested. Interestingly, in this particular �gure,

the approaching takes place from below. This is because in this example, the nonco-

operative bidder surplus for the discrete-bid case also approaches the continuous-bid

surplus from below.

0 5 10 15 20 25 3046

46.5

47

47.5

48

48.5

49

B

$Asymmetric Two Type Case

OptSolNonCoop

Figure 2: Asymmetric bidder surplus (�OptSol�) as a function of B. The dashedline at 48 (�NonCoop�) is the noncooperative surplus for the continuous bid case.

Note that the pro�tability of collusion at a second-price or ascending-bid auction

is not a¤ected by asymmetries among the bidders. Thus, as in the symmetric example

above, the cartel can extract all of the surplus from the seller if the auction format is

second-price or ascending-bid. So once again, when bidders collude, we have a stark

contrast between the seller�s expected revenue at a �rst-price auction, which is equal

to its noncooperative value, and the seller�s revenue at a second-price or ascending-bid

auction, which is zero.

6 Conclusion

In this paper, we consider �rst-price collusive mechanisms that cannot directly control

the cartel members�bids at the auction, but rather can only coordinate cartel mem-

bers�bids through non-binding recommendations. Thus, the collusive mechanisms we

20

consider di¤er from those of McAfee and McMillan (1992), who assume the cartel can

prevent all but one designated cartel member from bidding at the auction. Focusing

on this type of collusive mechanism, we show that for an environment with two sym-

metric bidders each with two possible values and su¢ ciently small bid increments,

bidders cannot pro�tably collude. Any mechanism that recommends bids that are

less than the noncooperative bids is subject to cheating by ring members. Numerical

calculations suggest that the result extends to more general symmetric environments

and at least some asymmetric environments. This is in stark contrast to the results

for second-price and ascending-bid auctions, where Marshall and Marx (2007) show

that a cartel can achieve the �rst-best collusive outcome, extracting all the surplus

from the seller. Thus, this research strongly supports the recommendation that sellers

that are concerned about bidder collusion should use a �rst-price auction format.

We are able to construct the dual feasible solution (given in Lemma 4) by observing

linear programming computations. In our case, numerical computations were invalu-

able for proving our analytical results. These same linear programming techniques are

potentially useful in developing answers to a number of open questions. For example,

Lopomo, Marshall, and Marx (2005) show that the optimal collusive mechanism for

an ascending-bid auction with post-auction knockout is ine¢ cient (for examples of

this type of mechanism, see U.S. v. Ronald Pook, U.S. v. Seville Industrial Machin-

ery, and District of Columbia v. George Basiliko, et al.), but they do not characterize

or construct an optimal mechanism, so the nature and extent of the ine¢ ciency is

unclear. The collusive mechanisms for second-price auctions studied by Graham and

Marshall (1987) and Mailath and Zemsky (1991) do not work if the cartel can only

collect a transfer from a cartel member if that cartel member wins the auction (for

example if transfers are arranged through subcontracting). Although Marshall and

Marx (2008) provide some results for environments when transfers can only be re-

quired from winning cartel members, a full characterization of what is possible in

this environment remains an open question. The results of McAfee and McMillan

(1992) suggest that for �rst-price auctions, the assumption of an all-inclusive cartel

is important for the existence of a pro�table cartel mechanism that is ex-post budget

balanced, and it remains an open question whether this is possible when the cartel is

not all-inclusive. The issue of individual rationality for cartels at �rst-price auctions

receives some attention in McAfee and McMillan (1992) and in Marshall and Marx

(2007), but open questions remain there as well.

21

A Proof of Lemma 1

First, consider a feasible solution (p1; p2;m1;m2) of linear program (1)�(6). We con-

struct a solution (�;M1;M2; J1; J2) to the linear program (7)�(14) and then show

that it is feasible with the same objective function value. Let � (b1; b2; v1; v2) =

p1 (b1; b2 j v1; v2) ; for i 2 f1; 2g; let Mi(vi) =P

v�i2V�imi (vi; v�i) f�i(v�i); and for

i 2 f1; 2g; let

Ji (vi; v0i; bi) = max

b0i2B

Xv�i2V�i;b�i2B

pi(bi; b�i j v0i; v�i)ui (b0i; b�i; vi) f�i (v�i) :

It is straightforward to show that constraints (9), (10), (13) and (14) are satis�ed.

Furthermore, for i 2 f1; 2g; consider the deviation function ��i (bi) de�ned by

Ji (vi; v0i; bi) =

Xv�i2V�i;b�i2B

pi(bi; b�i j v0i; v�i)ui (��i (bi); b�i; vi) f�i (v�i) ;

Constraint (4) implies that for i 2 f1; 2g;

Ui (pi j vi) � U�i (pi; ��i ; v0i j vi) =Xbi2B

Ji (vi; v0i; bi)�Mi(v

0i) :

Therefore, (11) and (12) are satis�ed. It can be veri�ed that the corresponding

objective values are the same.

Second, consider a feasible solution (�;M1;M2; J1; J2) to (7)-(14). Letting

p1 (b1; b2 j v1; v2) = p2 (b2; b1 j v2; v1) = � (b1; b2; v1; v2; )

and for i 2 f1; 2g; mi (vi; v�i) = Mi (vi) ; then (p1; p2;m1;m2) is a feasible solution

to (1)�(6) and has the same objective function value. It is obvious that constraints

(2) and (3) are satis�ed. To see that the incentive compatibility constraints are also

satis�ed, note that for any vi 2 Vi; v0i 2 Vi; bi 2 B; and �i(bi) 2 B, (13) and (14)imply

Ji (vi; v0i; bi) �

Xv�i2V�i;b�i2B

pi(bi; b�i j v0i; v�i)ui (�i(bi); b�i; vi) f�i (v�i) :

Furthermore, (11) and (12) imply (4). Q.E.D.

22

B Proof of Lemma 4

We show that���; ��; ��; ��

�de�ned as follows is feasible to the dual linear program

(21)�(25):

��(h; h; j; j0) =

8>>>>>>><>>>>>>>:

2f(h)f(l); if j0 > j and j0 2 f1; 3; : : : ; 2B � 1g

j f(h)f(l); if j = j0 2 f0; 1; 2; : : : ; 2Bg

f(h)f(l); if j0 = j + 1 2 f2; 4; : : : ; 2Bg

2B f(h)f(l); if j = j0 > 2B

0; otherwise

; ��(h; h) = 2Bf (h)

f (l)

��(l; h; j; j0) =

8>>>>>>>>><>>>>>>>>>:

2f(h)f(l); if j0 > j and j0 2 f3; : : : ; 2B � 1g

(j � 1)f(h)f(l); if j = j0 2 f2; 4; : : : ; 2Bg

j f(h)f(l); if j = j0 2 f3; 5; : : : ; 2B � 1g

f(h)f(l); if j 2 f0; 1g and j0 = 2

(2B � 1) f(h)f(l); if j = j0 > 2B

0; otherwise

; ��(h; l) = 2B�1

��(h; l; j; j0) =

(2B � 1; if j0 = 00; otherwise

; ��(l; h) =f (h)

f (l)(2B � 1)

��(l; l; j; j0) =

(2Bf(l); if j0 = 0

0; otherwise; ��(l; l) =

2B

f (l)

��(h; h) = ��(h; l) = ��(l; h) = (h� l) 2Bf (l) +

f (h)2

2Bf (l)

!

��(l; l) = 2 (h� l) fhfl(B (fh � 2) + 1)

�� = 0

It is easy to verify that���; ��; ��; ��

�is feasible to constraints (22), (24), and (25).

23

We focus on (23), or � (v; �v) � �(v; �v; j; �j); where

�(v; �v; j; �j) � u(bj; b�j; v) + u(b�j; bj; �v)

�Xv0

Xj02B

�(v; v0; j; j0)u(bj0 ; b�j; v0)�

Xv0

Xj02B

�(�v; v0; �j; j0)u(bj0 ; bj; v0)

+Xv0

�(v; v0)u(bj; b�j; v) +Xv0

�(�v; v0)u(b�j; bj; �v):

We do this by considering each one of the four possible combinations of v and �v:

First consider v = �v = h. Without loss of generality, we only need to consider

cases when j � �j.1. 2B � j > �j + 1 and j is an even number.

�(h; h; j; �j) = u(bj; b�j; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))� �(h; h; �j; j)u(bj; bj; h)�Xj0:j0>j

�(h; h; j; j0)u(bj0 ; b�j; h)�Xj0:j0>j

�(h; h; �j; j0)u(bj0 ; bj; h)

= (h� l)

0@�1� f (h)2B

j

��2B

f (l)� f (h)f (l)

j

�� 4f (h)

f (l)

B�1Xk=j=2

�1� 2k + 1

2Bf (h)

�1A= 2(h� l)Bf (l) < � (h; h) :

2. j = �j + 1 � 2B and j is an even number.

�(h; h; j; j � 1) = u(bj; bj�1; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))� �(h; h; j � 1; j)u(bj; bj; h)�Xj0:j0>j

�(h; h; j; j0)u(bj0 ; bj�1; h)�Xj0:j0>j

�(h; h; j � 1; j0)u(bj0 ; bj; h)

��(h; h; j; j � 1)u(bj�1; bj�1; h)

= (h� l)�2Bf (l)� 1

2

�1� f (l)

2Bj

�f (h)

f (l)

�< � (h; h) :

24

3. j = �j � 2B and j is an even number.

�(h; h; j; j) = 2u(bj; bj; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))�2

Xj0:j0>j

�(h; h; j; j0)u(bj0 ; bj; h)

= (h� l)

0@�1� f (h)2B

j

��2B

f (l)� f (h)f (l)

j

�� 4f (h)

f (l)

B�1Xk=j=2

�1� 2k + 1

2Bf (h)

�1A= 2(h� l)Bf (l) < � (h; h) :

4. 2B > j > �j and j is an odd number.

�(h; h; j; �j) = u(bj; b�j; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))� �(h; h; �j; j)u(bj; bj; h)�

Xj0:j0>j+1

�(h; h; j; j0)u(bj0 ; b�j; v0)�

Xj0:j0>j+1

�(h; h; �j; j0)u(bj0 ; bj; h)

��(h; h; j; j + 1)u(bj+1; b�j; v0)

= (h� l)��1� f (h)

2Bj

��2B

f (l)� (j + 1) f (h)

f (l)

�

�4f (h)f (l)

B�1Xk=(j+1)=2

�1� 2k + 1

2Bf (h)

���1� (j + 1) f (h)

2B

�f (h)

f (l)

35= (h� l)2Bf (l) < � (h; h) :

5. j = �j < 2B and j is an odd number.

�(h; h; j; j) = 2u(bj; bj; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))�2

Xj0:j0>j+1

�(h; h; j; j0)u(bj0 ; bj; h)� 2�(h; h; j; j + 1)u(bj+1; bj; h)

= (h� l) 2Bf (l) +

f (h)2

2Bf (l)

!= � (h; h) :

6. j > 2B and j > �j:

�(h; h; j; �j) = u(bj; b�j; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))= u(bj; b�j; h) (1 + �(h; l))

= (h� l) (2B � jfh) < 2B (h� l) fl < � (h; h) :

25

7. j = �j > 2B:

�(h; h; j; j) = 2u(bj; bj; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))= (h� l) (2B � jfh) < 2B (h� l) fl < � (h; h) :

Now consider v = h and �v = l.

1. 2B � j > �j, j > 0 and j is an even number.

�(h; l; j; �j) = u(bj; b�j; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))�Xj0:j0>j

�(h; h; j; j0)u(bj0 ; b�j; h)�Xj0:j0>j

�(l; h; �j; j0)u(bj0 ; bj; h)

= (h� l)

0@�1� f (h)2B

j

��2B

f (l)� f (h)f (l)

j

�� 4f (h)

f (l)

B�1Xk=j=2

�1� 2k + 1

2Bf (h)

�1A= 2 (h� l)Bf (l) < � (h; l) :

2. 2B > j > �j, j � 3 and j is an odd number.

�(h; l; j; �j) = u(bj; b�j; h) (1 + � (h; h) + � (h; l)� �(h; h; j; j))�

Xj0:j0>j+1

�(h; h; j; j0)u(bj0 ; b�j; h)�X

j0:j0>j+1

�(l; h; �j; j0)u(bj0 ; bj; h)

��(l; h; �j; j)u(bj; bj; h)� �(h; h; j; j + 1)u(bj+1; b�j; h)

= (h� l)��1� f (h)

2Bj

��2B

f (l)� f (h)f (l)

(j + 1)

�

�4f (h)f (l)

B�1Xk=(j+1)=2

�1� 2k + 1

2Bf (h)

���1� f (h)

2B(j + 1)

�f (h)

f (l)

35= 2 (h� l)Bf (l) < � (h; l) :

3. j = 1 and �j = 0:

�(h; l; 1; 0) = (h� l) 2Bf (l) +

f (h)2

2Bf (l)

!= � (h; l) :

26

4. 2B � j = �j � 2 and j is an even number.

�(h; l; j; j) = u(bj; bj; h) (2 + � (h; h) + � (h; l)� �(h; h; j; j)� �(l; h; j; j))�Xj0:j0>j

�(h; h; j; j0)u(bj0 ; bj; h)�Xj0:j0>j

�(l; h; j; j0)u(bj0 ; bj; h)

+u(bj; bj; l) (� (l; l) + � (l; h))

= (h� l)��1� f (h)

2Bj

��B

f (l)+1

2� f (h)f (l)

�j � 1

2

��

�4f (h)f (l)

B�1Xk=j=2

�1� 2k + 1

2Bf (h)

���2B

f (l)+f (h)

f (l)(2B � 1)

�j

4Bf (h)

35= (h� l)

�2Bf (l)� 1

2f (l)

�2B � 1 + jf (h)2

�� 1

4Bjf (h)

�< � (h; l) :

5. 2B > j = �j � 3 and j is an odd number.

�(h; l; j; j) = u(bj; bj; h) (2 + � (h; h) + � (h; l)� �(h; h; j; j)� �(l; h; j; j))�Xj0:j0>j

�(h; h; j; j0)u(bj0 ; bj; h)�Xj0:j0>j

�(l; h; j; j0)u(bj0 ; bj; h)

+u(bj; bj; l) (� (l; l) + � (l; h))� �(h; h; j; j + 1)u(bj+1; bj; h)

= (h� l)��1� f (h)

2Bj

��B

f (l)+1

2� f (h)f (l)

j

���1� f (h)

2B(j + 1)

�f (h)

f (l)

�4f (h)f (l)

B�1Xk=(j+1)=2

�1� 2k + 1

2Bf (h)

���2B

f (l)+f (h)

f (l)(2B � 1)

�j

4Bf (h)

35= (h� l)

�2Bf (l)� f (h)

4Bf (l)j � 2B + jf

2 � f � 12f (l)

�< � (h; l) :

27

6. j = �j = 0.

�(h; l; 0; 0) = u(b0; b0; h) (2 + � (h; h) + � (h; l))� �(l; h; 0; 2)u(b2; b0; h)�Xj0:j0>1

�(h; h; 0; j0)u(bj0 ; b0; h)�Xj0:j0>3

�(l; h; 0; j0)u(bj0 ; b0; h)

= (h� l)"�

B

f (l)+1

2

�� 4f (h)

f (l)

B�1Xk=1

�1� 2k + 1

2Bf (h)

��2f (h)f (l)

�1� 1

2Bf (h)

�� f (h)f (l)

�1� 2

2Bf (h)

��= (h� l)

�2Bf (l)� 2B � f (h)� 1

2f (l)

�< �(h; l):

7. j = �j = 1.

�(h; l; 1; 1) = u(b1; b1; h) (2 + � (h; h) + � (h; l))� �(l; h; 1; 2)u(b2; b0; h)�Xj0:j0>1

�(h; h; 1; j0)u(bj0 ; b1; h)�Xj0:j0>3

�(l; h; 1; j0)u(bj0 ; b1; h)

= (h� l)"�1� 1

2Bf (h)

��B

f (l)+1

2

�� 4f (h)

f (l)

B�1Xk=1

�1� 2k + 1

2Bf (h)

��2f (h)f (l)

�1� 1

2Bf (h)

�� f (h)f (l)

�1� 2

2Bf (h)

��= (h� l)

�2Bf (l)� 4B

2 � 2B + f (h) f (l)4Bf (l)

�< �(h; l):

8. j > 2B and j > �j:

�(h; l; j; �j) = u(bj; b�j; h) (1 + �(h; h) + �(h; l)� �(h; h; j; j))

= (h� l)�1� f (h)

2Bj

��2B

f (l)� f (h)f (l)

2B

�= (h� l) 2B

�f (l)� j � 2B

2Bf (h)

�< � (h; l) :

28

9. j = �j > 2B:

�(h; l; j; j) = u(bj; bj; h) (2 + �(h; h) + �(h; l)� �(h; h; j; j)� �(l; h; j; j))+u(bj; bj; l) (� (l; l) + � (l; h))

= (h� l)��1� f (h)

2Bj

��B

f (l)+1

2� f (h)f (l)

�2B � 1

2

����2B

f (l)+f (h)

f (l)(2B � 1)

�j

4Bf (h)

�=

1

4B (1� f)��8B2f + 4B2 + 2B + jf (2Bf � 4B + f � 1)

�<

1

4B (1� f)��8B2f + 4B2 + 2B + 2Bf (2Bf � 4B + f � 1)

�= (h� l)

"2Bf (l)�

(2B � 1)�1 + f (h)2

�+ f (h)

2f (l)

#< (h� l) 2Bf (l) < � (h; l) :

10. j < �j:

�(h; l; j; �j) = u(b�j; bj; l) (1 + �(l; h) + �(l; l))� �(h; h; j; �j)u(b�j; b�j; h)�Xj0:j0>�j

�(h; h; j; j0)u(bj0 ; b�j; h)�Xj0:j0>�j

�(l; h; �j; j0)u(bj0 ; bj; h)

< 0 < �(h; l):

Finally, consider v = �v = l. Without loss of generality, we only consider j � �j.1. 2B > j > �j, j > 2 and j is an even number.

�(l; l; j; �j) = u(bj; b�j; l)(1 + �(l; l) + �(l; h))�Xj0:j0>j

�(l; h; j; j0)u(bj0 ; b�j; h)

�Xj0:j0>j

�(l; h; �j; j0)u(bj0 ; bj; h)� �(l; h; j; j)u(bj; b�j; h)

= (h� l)

0@�f (h)2B

j

�1 +

f (h)

f (l)(2B � 1) + 2B

f (l)

�� 4f (h)

f (l)

B�1Xk=j=2

�1� 2k + 1

2Bf (h)

�1A� (h� l)

�1� f (h)

2Bj

�(j � 1) f (h)

f (l)

= (h� l)�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + f (h) j +

j

2Bf (l)

���< �(l; l):

29

2. j = 2 > �j

�(l; l; 2; �j) = u(b2; b�j; l)(1 + �(l; l) + �(l; h))�Xj0:j0>2

�(l; h; 2; j0)u(bj0 ; b�j; h)

�Xj0:j0>2

�(l; h; �j; j0)u(bj0 ; b2; h)� �(l; h; 2; 2)u(b2; b�j; h)� �(l; h; �j; 2)u(b2; b2; h)

< �(l; l; j; �j) for j in case 1.

< �(l; l):

3. 2B > j > �j and j � 3 is an odd number.

�(l; l; j; �j) = u(bj; b�j; l)(1 + �(l; l) + �(l; h))�Xj0:j0>j

�(l; h; j; j0)u(bj0 ; b�j; h)

�Xj0:j0>j

�(l; h; �j; j0)u(bj0 ; bj; h)� �(l; h; j; j)u(bj; b�j; h)� �(l; h; �j; j)u(bj; bj; h)

= � (h� l) f (h)2B

j

�1 +

f (h)

f (l)(2B � 1) + 2B

f (l)

�

� (h� l) f (h)f (l)

0@4 B�1Xk=(j+1)=2

�1� 2k + 1

2Bf (h)

�+

�1� f (h)

2Bj

�(j + 1)

1A= (h� l)

�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + f (h) j +

f (l) j + f (h)

2B

���< �(l; l):

4. 2B � j = �j � 2, and j is an even number.

�(l; l; j; j) = 2u(bj; bj; l)(1 + �(l; l) + �(l; h))� 2Xj0:j0>j

�(l; h; j; j0)u(bj0 ; bj; h)

�2�(l; h; j; j)u(bj; bj; h)

= (h� l)�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + f (h) j +

j

2Bf (l)

���< �(l; l):

30

5. 2B > j = �j � 3 and j is an odd number.

�(l; l; j; j) = 2u(bj; bj; l)(1 + �(l; l) + �(l; h))� 2X

j0:j0�j+2

�(l; h; j; j0)u(bj0 ; bj; h)

�2�(l; h; j; j)u(bj; bj; h)

= � (h� l) f (h)2B

j

�1 +

f (h)

f (l)(2B � 1) + 2B

f (l)

�

� (h� l) f (h)f (l)

0@4 B�1Xk=(j+1)=2

�1� 2k + 1

2Bf (h)

�+

�1� f (h)

2Bj

�(j + 1)

1A= (h� l)

�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + f (h) j +

f (l) j + f (h)

2B

���< �(l; l):

6. j = �j = 1:

�(l; l; j; j) = 2u(bj; bj; l)(1 + �(l; l) + �(l; h))� 2X

j0:j0�j+2

�(l; h; j; j0)u(bj0 ; bj; h)

�2�(l; h; j; j)u(bj; bj; h)� 2�(l; h; j; 2)u(b2; bj; h)< �(l; l; j; �j) for j in case 5.

< �(l; l):

7. j = �j = 0:

�(l; l; 0; 0) = �2Xj0:j0�3

�(l; h; 0; j0)u(bj0 ; b0; h)� 2�(l; h; 0; 2)u(b2; b0; h)

= � (h� l) f (h)f (l)

4B�1Xk=1

�1� 2k + 1

2Bf (h)

�+ 2

�1� f (h)

2B2

�!

= (h� l)�f (h)

f (l)(2B (f (h)� 2) + 2)

�= �(l; l):

31

8. j > 2B and j > �j:

�(l; l; j; �j) = u(bj; b�j; l)(1 + �(l; l) + �(l; h))� �(l; h; j; j)u(bj; b�j; h)

= (h� l)��f (h)2B

j

�1 +

f (h)

f (l)(2B � 1) + 2B

f (l)

��� (h� l)

�1� f (h)

2Bj

�(2B � 1) f (h)

f (l)

= (h� l)�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + j � 2Bf (l) + jf (l)

2B

���< �(l; l):

9. j = �j > 2B.

�(l; l; j; �j) = 2u(bj; bj; l)(1 + �(l; l) + �(l; h))� 2�(l; h; j; j)u(bj; bj; h)

= (h� l)�f (h)

f (l)

�2B (f (h)� 2) + 2�

�1 + j � 2Bf (l) + jf (l)

2B

���< �(l; l):

This completes the demonstration that � (v; �v) � �(v; �v; j; �j) and proves that���; ��; ��; ��

�is feasible in the linear program (21)�(25). Substituting

���; ��; ��; ��

�into

the dual objective function gives D(B) = (h� l)f (h) f (l)�2 + f(h)2(1+f(l))

2Bf(l)2

�. Q.E.D.

C Online Appendix �Additional proofs

Proof of Lemma 1. It is clear that a bidder with value l can do not better than tobid l. If bidder i has value h; its expected payo¤ from bid bi 2 [l;�b] is

�i = (1� p)Z bi

l

(h� bi)dF (b�i) + p(h� bi) = p(h� l);

which is constant for all bi 2 [l;�b]. A bid below l has expected payo¤ of zero, and abid bi > �b has expected payo¤ of h� bi < h� �b = p(h� l). So the bidding strategygiven in the proposition is a best reply. Q.E.D.

Proof of Lemma 2. It is only incentive compatible such that a bidder receivingblow to have a chance to win the object by a tie. Assume a bidder with type l has

probability plow to win by tie, and therefore receive surplus (l � blow) plow=2. Supposeblow = l�t� < l�2� for some t > 2. Whenever the bidder receives the recommended

32

bid blow = l � t� from the center, by bidding l � (t� 1)� instead, she wins with

probability at least plow, which generates surplus (t� 1)�plow, which is higher thant�plow=2 = (l � blow) plow=2, which is the surplus from following the recommended

bid blow. Similar logic indicates that a type h bidder also has no incentive to bid

below l � 2�. Q.E.D.

Proof of Lemma 3. For any feasible solution (�;M1;M2; J1; J2) to the linear

program (7)�(14), we can construct a feasible solution (~�; J;M) to the linear pro-

gram (15)�(20) with the same objective function value as follows: ~�(b;�b; v; �v) =12

��(b;�b; v; �v) + �(�b; b; �v; v)

�; M = M1 + M2; and J = J1 + J2. Similarly, for any

feasible solution (~�;M; J) to the linear program (15)�(20), we can construct a feasi-

ble solution (�; J1; J2;M1;M2) to (7)�(14) with the same objective function value as

follows: �(b;�b; v; �v) = �(�b; b; �v; v) = 12

�~�(b;�b; v; �v) + ~�(�b; b; �v; v)

�; M1 =M2 =

M2; and

J1 = J2 =J2. Q.E.D.

Proof of Lemma 5. Let the type l bidder bid l and the type h bidder mix on bidsabove l. Type l bidder has surplus 0 and no incentive to deviate from l. Given the

type l bidder�s strategy, one can show that a symmetric mixed strategy equilibrium

exists. Now focus on the bid b1 = l+�. The high type�s surplus from bidding b1 must

be no less than f(l)(h�b1) because if he bids b1, his surplus is at least f(l)(h�b1). Ifbid b1 is not in an equilibrium bid, the surplus can only be higher than f(l)(h� b1).Therefore, for each bidder, f(h)f(l)(h�b1) is a lower bound for the expected surplus.Q.E.D.

Proof of Proposition 4. One can show that F1(b) and F2(b) are well de�ned CDFswith support (l1;�b]. If bidder 1 has value h1; its expected payo¤ from bid b 2 (l1;�b] is

u1 (h1; b) = (h1 � b) (p2 + (1� p2)F2(b)) = h1 � �b;

which is a constant with respect to b. Similarly, if bidder 2 has value h2; its expected

payo¤ from bid b 2 (l1;�b] is

u2 (h2; b) = (h2 � b) (p1 + (1� p1)F1(b)) = h2 � �b;

which is also a constant with respect to b. Now we consider bidder 2 with type l2. It

33

has no incentive to bid below l1. If it bids b 2 (l1;min��b; l2], the expected surplus

is

u2 (l2; b) = (l2 � b) (p1 + (1� p1)F1 (b))

= (l2 � b)�h2 � �bh2 � b

�=

�h2 � �b

��1� h2 � l2

h2 � b

�;

which is a decreasing function of b, taking maximum value at l1. Thus, when l2 � �b,it has no incentive to bid above l1. If �b < l2; the surplus from bidding b 2 (�b; l2] is

u2 (l2; b) = l2 � b< p1l2 + (1� p1) l2 � �b

< p1l2 + (1� p1)�b� l1p11� p1

� �b

= p1 (l2 � l1) = u2 (l2; l1) ;

where the second inequality uses the regularity condition l2 <�b�l1p11�p1 . Thus, when

�b < l2 there is again no incentive to bid above L1. Q.E.D.

34

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