Collusion and Optimal Reserve Prices

in

Repeated Procurement Auctions

Charles J. Thomas

1

July 30, 2001

1Federal Trade Commission. I thank Dave Balan, Ken Chay, Stephen Holland, Dan Hosken, Dave Rei®en,

and Bart Wilson for helpful comments. Send correspondence to Federal Trade Commission, 600 Pennsylvania

Avenue, NW, Washington, DC 20580, or to cjthomas@ftc.gov. This work represents the opinions of the

author and does not necessarily represent the position of the Federal Trade Commission or any individual

Commissioner.

JEL: C72, D44, L1

Keywords: auction, procurement, collusion, optimal reserve prices
Abstract

Previous research on collusion in procurement markets uses static mechanism design theory to address the limitations on collusive activity imposed by asymmetric information, but in most instances

it does not address how to enforce the proposed mechanisms. This paper uses repeated game theory

to examine the sustainability of two commonly reported collusive schemes that have been identi ̄ed

as optimal static mechanisms. If a buyer does not select its reserve price strategically, or if its

value is large relative to the sellers' costs, then collusion may be sustainable for a wide range of

plausible discount factors. However, even mildly sophisticated reserve price selection can dramatically shrink the set of discount factors for which collusion can be sustained. These  ̄ndings provide

a rationale for existing arguments that buyers are vulnerable to collusion, but suggest that buyers

possess tools that may pro ̄tably induce sellers to act competitively. The analysis also reveals that

collusion tends to be more easily sustained if the sellers' costs have a low mean or a high variance,

or, in some instances, if the number of sellers increases.
1 Introduction

While economists have suggested that collusion occurs in many procurement settings,

1

there are

few analyses of it compared to studies of collusion in other strategic environments. In particular,

relatively little is known or has been proposed about either the form that collusion takes, or the

likelihood that it can be sustained. These are important limitations to arguments that collusion

is a pervasive problem in auction markets, because what does seem clear is that colluding sellers

must overcome several obstacles to sustain supracompetitive prices.

One di±culty facing a cartel in a procurement market is that in many instances its members

cannot credibly reveal their production costs to one another. Hence, the cartel cannot directly

determine which member can most pro ̄tably ful ̄ll a particular contract, and therefore the cartel

cannot obtain the maximum possible surplus from each transaction. Economists have developed

theoretical models of auction environments to examine means by which cartels might attempt to

appropriate rents from buyers.

2 The predominant approach taken has cartels construct collusive

bidding schemes as solutions to mechanism design problems. In the mechanism design approach,

each cartel member reports (possibly falsely) its privately known production cost to the mechanism,

which then speci ̄es the price the  ̄rm is to o®er to the buyer. Because the asymmetric information

about costs limits the sellers' ability to collude, the cartel's primary goal is to design optimal

mechanisms that induce truthful revelation of its members' production costs.

3 A common feature

of these mechanisms is rigid-pricing, in which the seller selected to win sets its price at a speci ̄c

level regardless of its production costs, while other sellers either o®er higher prices or do not submit

price o®ers. Treatments vary in whether or not the participants are able to make transfer payments

to non-winning cartel  ̄rms. This distinction is important, because the use of transfer payments

likely increases the probability that competition authorities can prove the existence of price- ̄xing

agreements.

Another di±culty facing a cartel in any market is that its members' adherence to the collusive

arrangement must be enforced. In most instances, the mechanism design approach to collusion

in procurement markets abstracts away from concerns about enforcing the proposed mechanisms.

4

1For example, several allegations of collusion are reported in Scherer and Ross [1990], Porter and Zona [1993],

Baldwin, Marshall, and Richard [1997], and McAfee and McMillan [1992].

2There exists a small, but growing, empirical literature that attempts to expose collusive behavior using bidding

data from past auctions. For example, see Porter and Zona [1993] and Baldwin, Marshall, and Richard [1997]. Some

of the empirical papers use the theoretical papers as a starting point for their econometric modeling.

3Note that in this instance it is the cartel members who design the mechanism, in contrast to the typical mechanism

design setting in which the auctioneer is the designer.

4The exceptions are particular mechanisms that are enforceable without considering repeated interaction.

1
This is not a trivial concern, because sellers may have incentives either to set prices other than the

ones speci ̄ed by the mechanism, or to not make agreed-upon transfer payments. That is, while

truthfully revealing costs is made incentive compatible, abiding by the mechanism's rules is not.

For example, suppose that two  ̄rms report their costs of 3 and 4 and are told to o®er prices of 6

and 7, respectively. The  ̄rm with cost 4 has incentives to set its price just below 6 in order to

win the contract. As the mechanism design approach is static, it does not address the possibility

or prevention of such deviations.

In this paper I use repeated game theory to evaluate the ability of sellers in  ̄rst-price procurement auctions to enforce adherence to two commonly reported rigid-pricing collusive schemes

identi ̄ed as being optimal static mechanisms.

5 Speci ̄cally, I determine the lowest value that sellers can place on future pro ̄ts such that the short-term gain from defection is outweighed by the

long-term loss from rivals' retaliation to that defection. The lower is the critical discount factor for

which collusion can be sustained, the more likely it is that the sellers' discount factor in a particular

market is high enough to sustain the collusive agreement.

The critical discount factor for which collusion can be sustained depends in a complicated

manner on the number of  ̄rms, the distribution of their production costs, and the reserve price.

Consequently, its theoretical determination o®ers limited insight about either the ease of sustaining

collusion, or how sustaining collusion is a®ected by changes in the strategic environment. To more

clearly identify the conditions under which collusion can occur in procurement markets, I therefore

compute critical discount factors for a large number of cost distributions and market concentration

levels, under various assumptions about the buyer's reserve price strategy. Next I  ̄t to the

generated data a simple function of the strategic environment's primitives. This technique usefully

summarizes the data and permits straightforward numerical evaluation of comparative statics.

The computational results reveal that the reserve price imposed by the buyer has a substantial

impact on the sellers' ability to collude, which is an interesting element that is absent from standard

collusion models in which non-strategic buyers are embodied in the market demand curve. If the

buyer cannot or does not credibly impose a reserve price, or simply has a large value relative to the

sellers' costs, then collusion likely can be sustained for a wide range of discount factors. In contrast,

if the buyer can credibly impose a strategically selected reserve price, then the set of discount factors

for which collusion can be sustained may be quite small. With a well-chosen reserve price, say at

5

In a  ̄rst-price procurement auction, the  ̄rm o®ering the lowest price is designated the winner and is paid the

price it o®ered.

2
or below the level that is optimal versus static Nash equilibrium price-setting, the critical discount

factor for sustaining collusion is high because the short-term gain from defection is large relative to

the per-period loss from rivals' retaliation to that defection. Hence,  ̄rms must highly value future

pro ̄ts in order for the loss of the foregone future collusive pro ̄ts to outweigh the short-term gain

from defection.

The computational results also reveal how the number of sellers and their cost distributions

a®ect the sellers' ability to collude. First, in some instances collusion is more easily sustained as

the number of sellers increases. This results from a widening of the gap between the per-period

cooperative and noncooperative expected pro ̄ts as the number of sellers increases from low levels.

Second, collusion appears to be more easily sustained when sellers have low expected costs, all else

equal. This result likely emerges because sellers with high expected costs have little to gain from

collusion, which exacerbates a seller's temptation to deviate from the collusive agreement when it

has unexpectedly low costs. Third, collusion appears to be more easily sustained when the sellers'

costs are more variable.

To conclude the analysis, I evaluate the sustainability of collusion by comparing the computed

critical discount factors to plausible values of  ̄rms' actual discount factors, which helps one to

assess the economic signi ̄cance of the static mechanism design approach to collusion in auctions.

The comparison suggests that  ̄rms using the optimal static mechanism are unlikely to be able

to collude against sophisticated buyers. However, unsophisticated buyers may be vulnerable to

collusion.

The following three papers are representative of the mechanism design approach. McAfee and

McMillan [1992] identify optimal static mechanisms in single-object  ̄rst-price auctions with an

exogenously determined reserve price and revelation of the winning bid, both with and without

the ability to make transfer payments. Announcing the winning bid makes it immediately clear if

a winning  ̄rm has deviated from its prescribed bid. The authors do not model the enforcement

device, but appeal to the repeated game approach I examine in this paper. Without transfer

payments, they  ̄nd that the optimal mechanism speci ̄es a price o®er at the reserve price by all

sellers whose production costs are below the reserve price. With transfer payments, they  ̄nd that

the low-cost seller wins at the reserve price and pays an equal amount to each losing seller. I

examine both of these mechanisms in this paper.

Graham and Marshall [1987] identify static mechanisms in single-object second-price and English auctions with an endogenously determined reserve price. Second-price and English auctions

3
have the attractive feature that, once costs are truthfully announced, no  ̄rm can achieve a shortterm gain by deviating from the collusive scheme. Therefore, there is no need to consider repeated

interaction to enforce adherence to the mechanism, because, even in a static setting, abiding by

the mechanism is incentive compatible. Consequently, the buyer cannot select the reserve price to

deter collusion, in contrast to my  ̄nding in  ̄rst-price auctions.

6

Cramton and Palfrey [1990] examine collusive mechanisms in a quantity-setting oligopoly model,

and they explicitly consider outside options that might arise if the sellers cannot agree on a mechanism. Like McAfee and McMillan [1992], they do not formally address the incentive to deviate

from the mechanism, but appeal to repeated game concepts to justify the enforceability of the

mechanism's quantity prescription after the sellers reveal their costs.

A new approach is taken by Athey, Bagwell, and Sanchirico [2000], who use the same basic

structure as in the present paper, with the exceptions that all sellers' price o®ers become public

information and that sellers cannot make explicit transfer payments. The authors show that

implementing the optimal static mechanism without transfer payments is the optimal collusive

device in the repeated game, provided that the sellers have su±ciently high discount factors. When

sellers' discount factors are below the critical level, the authors partially characterize more pro ̄table

collusive mechanisms that do not use rigid-pricing. They do not address the two central goals of this

paper, a systematic appraisal of the discount factors for which rigid-price collusion can occur, and

the e®ect on such collusion of the buyer's reserve price selection and the sellers' cost distributions.

The rest of this paper is structured as follows. Section 2 presents a standard one-shot procurement auction and states known results relevant to the succeeding analysis. Section 3 uses the

one-shot procurement auction as the stage game of an in ̄nitely repeated game, and determines

conditions necessary for the sustainability of two collusive schemes. Section 4 uses speci ̄c assumptions on the number of  ̄rms and their cost distributions to numerically determine how patient the

sellers must be to successfully employ the proposed collusive schemes, under various assumptions

about the buyer's strategic sophistication in selecting the reserve price. Section 5 brie°y concludes,

6The susceptibility to collusion of English and second-price auctions is o®ered as one reason why  ̄rst-price auctions are more frequently employed in procurement settings. However, a countervailing factor is that the identi ̄ed

mechanism in English and second-price auctions requires the sellers to communicate about each contract, which

increases the risk of successful antitrust prosecution. Therefore, a buyer's use of the second-price format might make

it easier to prove the existence of price- ̄xing agreements, even though it simultaneously increases the likelihood of

the sellers' implementing an agreement. If the sellers chose to forego the communication necessary to implement

collusion using the second-price mechanism, then they could employ the mechanism identi ̄ed by McAfee and McMillan [1992]. However, that mechanism's speci ̄ed bidding behavior would lead to outcomes identical to those in a

 ̄rst-price auction.

4
while the Appendix contains all proofs.

2 The One-Shot Procurement Auction

A prospective buyer of a product solicits price o®ers simultaneously from each of N sellers. Prior

to the making of o®ers to the buyer, each seller i draws its production cost, ci, independently from

the cumulative distribution F(c). Assume that F has a di®erentiable density f with support [c; c].

The buyer purchases from the low-priced seller at the o®ered price. In auction terminology, this is a

symmetric independent private value (IPV)  ̄rst-price auction. Assume further that the buyer and

sellers all are risk neutral, the number of  ̄rms is exogenous, entry is blockaded, and it is costless

for sellers both to learn their production cost and to participate in the procurement process. The

buyer's next best supply alternative costs cB  ̧ c, and the buyer's pro ̄t from purchasing from one

of the N sellers at price p is cB ¡ p. Seller i's pro ̄t from winning with price piis pi ¡ ci. Prior to

the submission of o®ers, the buyer imposes a commonly known reserve price, r, that is less than or

equal to c, and above which price o®ers will be rejected.

7 Following most of the auction literature,

I assume that the buyer can commit to the procurement format.

The one-shot procurement process has been extensively discussed elsewhere, so here I simply

state relevant known results. Proofs can be found in, for example, McAfee and McMillan [1992].

8

The  ̄rst result describes the sellers' static Nash equilibrium price-setting strategies and expected

pro ̄ts, while the second describes the buyer's expected pro ̄t and optimal reserve price versus

static Nash equilibrium price-setting.

Result 1 In the symmetric IPV one-shot procurement auction with reserve price r · c, for c 2 [c; r]

the sellers' static Nash equilibrium price-setting function is

p

NE(cjr) = c +

Z r

c

·

1 ¡ F(s)

1 ¡ F(c)

 ̧N¡1

ds:

A seller's interim expected pro ̄t in the static Nash equilibrium, when it has drawn cost c · r, is

1⁄4

NE

S

(cjr) =

Z r

c

[1 ¡ F(s)]

N¡1

ds;

7Technically speaking, the buyer might select a reserve price that exceeds c, but in the symmetric equilibrium

sellers will never set a price greater than c.

8While their proofs all are for auctions where bidders are buyers rather than sellers, all of the proofs have analogues

in the procurement model.

5
and its ex ante expected pro ̄t in the static Nash equilibrium, when it has not yet drawn its cost, is

1⁄4

NE

S

(r) =

Z r

c

F(c)[1 ¡ F(c)]

N¡1

dc:

Result 1 illustrates that sellers shade their price above their cost by an amount that depends on the

reserve price and the number of rivals. As the reserve price increases, the amount of shading and

expected pro ̄ts increase. As the number of rivals increases, the amount of shading and expected

pro ̄ts decrease.

Result 2 In the symmetric IPV one-shot procurement auction, if the sellers employ their static

Nash equilibrium price-setting strategies, then the buyer's expected pro ̄t from imposing reserve

price r is

1⁄4

NE

B (r) =

Z r

c

£

cB ¡ p

NE(cjr)

¤

N [1 ¡ F(c)]

N¡1

f(c)dc:

The buyer's optimal reserve price versus static Nash equilibrium price-setting, r

NE, solves

cB = r

NE +

F

¡

r

NE

¢

f (rNE)

;

provided the resulting r

NE · c. Otherwise, rNE = c, and the reserve price is said to be non-binding.

The reserve price presented in Result 2 has some interesting features. First, the buyer's optimal

reserve price versus static Nash equilibrium price-setting is independent of the number of sellers.

Second, for a certain range of reserve prices, both the sellers' and the buyer's expected pro ̄t

increase as r increases. This occurs because social welfare increases as the probability of a sale

increases, so parties on both sides of the transaction can gain as r increases. In many instances,

the buyer's expected pro ̄t eventually decreases in the reserve price, leading to a binding optimal

reserve price strictly less than c. However, if cB is su±ciently large, then the buyer optimally sets

r at c, because its outside option is so costly relative to the most costly seller. Therefore, the cost

to the buyer of possibly losing a purchase from one of the sellers outweighs the gain from leading

the sellers to set their prices more aggressively.

3 The Repeated Procurement Auction

Suppose that the same buyer and sellers play the preceding one-shot game in each of an in ̄nite

number of discrete periods, with the sellers' costs redrawn each period and with future pro ̄ts

discounted by the factor ± 2 [0; 1). While employing the static Nash equilibrium price-setting

6
strategies each period is a subgame perfect equilibrium, of greater interest is the characterization

of subgame perfect equilibria yielding the sellers higher expected pro ̄ts. In what follows, I assume

that the buyer does not reveal the winning price, as is common in private sector procurement.

9

Additionally, I assume that the winner in each period is revealed, and that the buyer commits at

the outset to a reserve price to be used throughout the game.

10 Subject to the reserve price, the

sellers are presumed to select the most collusive outcome possible.

I examine two rigid-pricing collusive schemes in which the seller selected to win the contract

o®ers a price equal to the reserve price. Both schemes are all-inclusive, in the sense that all sellers

are members of the cartel. The sellers not selected to win the current contract either o®er higher

prices or do not submit price o®ers. The  ̄rst scheme assumes that the sellers cannot credibly

reveal their costs to each other prior to submitting price o®ers, and that they cannot or do not

make side payments. It uses a form of bid rotation in which the randomly selected winner has the

option of declining its position. This scheme requires an initial meeting for the participants to agree

on the signals to be employed, but this is not too troubling from the perspective of applications.

As McCutcheon [1997] argues, existing competition policy might enhance collusion by promoting

initial meetings but not subsequent ones. This bid rotation scheme is identical, in terms of the

sellers' expected pro ̄t, to the optimal mechanism without transfer payments identi ̄ed by McAfee

and McMillan [1992]. Following their nomenclature, I refer to a cartel using this scheme as a weak

cartel. Scherer and Ross [1990] report that a version of this scheme was used in the electrical

manufacturers' conspiracy. While bid rotation is unappealing because the low-cost seller does not

necessarily win the contract, McAfee and McMillan [1992] prove that earning supracompetitive

pro ̄ts via their mechanism is incompatible with the e±cient allocation of contracts.

The second scheme assumes that the sellers cannot credibly reveal their costs to each other

prior to submitting price o®ers, but that they can make side payments. McAfee and McMillan

[1992] illustrate that the sellers can implement this scheme by holding their own  ̄rst-price auction

9With slight modi ̄cation, all of the results in this section hold if the buyer reveals the winning price, as is common

in public sector procurement.

10Perhaps surprisingly, the buyer can more successfully deter collusion if he cannot commit to a single reserve price

at the outset of play. Speci ̄cally, if the sellers revert to static Nash equilibrium price-setting following a deviation,

and if the buyer switches from the optimal reserve price versus collusive price-setting to the optimal reserve price

versus static Nash equilibrium price-setting, then the sellers' per-period ex ante expected pro ̄ts will be higher than if

the buyer had not changed the reserve price. This higher pro ̄t level in the punishment regime makes collusion more

di±cult to sustain. In fact, in some instances the sellers' ex ante expected pro ̄t in the punishment phase exceeds

their ex ante expected pro ̄t in the collusive phase, implying that collusion cannot be sustained for any discount

factor. Therefore, assuming a one-time commitment to the reserve price provides a bound on the sellers' ability to

sustain collusion.

7
prior to the buyer's  ̄rst-price auction. They refer to a cartel using this scheme as a strong cartel.

Strong cartels require more frequent contact than do weak cartels, which increases the likelihood

that the antitrust authorities may discover and be able to prove the existence of the agreement.

However, this concern is balanced against the fact that the strong cartel  ̄nds collusion easier to

sustain than does the weak cartel.

11

Four potential stumbling blocks to successful collusion are the selection of the winning  ̄rm, the

selection of a price, the detection of deviations, and the punishment of deviations. The schemes

used by the weak cartel (WC) and the strong cartel (SC) clearly overcome the  ̄rst three. The

winner and the price are predetermined, and deviation can be recognized after the current round

of o®ers and dealt with in the next period. To punish deviations, I employ perpetual reversion to

the static Nash equilibrium. Though there exist more severe punishments, imposing them su®ers

from di±culties similar to those preventing e±cient collusion. For example, suppose all  ̄rms i 6= j

are supposed to punish  ̄rm j in each period by setting pi = ci. Because no prices are revealed and

the sellers' costs are private information, a punishing  ̄rm could deviate to a price pei > ci without

being detected if it lost when it should have won, and earn a positive expected pro ̄t. Thus, it

seems such a severe punishment could not be implemented credibly. In contrast, by de ̄nition

Nash reversion can be implemented credibly.

Collusion can be sustained provided that the pro ̄t from colluding exceeds the pro ̄t from defecting today and facing punishment in the future, for any current cost realization. It is important

that the incentive constraints be satis ̄ed for all possible cost draws. If they were not, then the positive probability of the designated seller's being undercut by sellers with particular cost realizations

will prompt the designated seller to undercut as well. Such a response precludes the existence of

an equilibrium in which the selected seller o®ers a price of r. I de ̄ne the critical discount factor,

±

X

(r), to be the smallest discount factor for which the incentive constraints are satis ̄ed for all

possible costs, for X 2 fWC; SCg and for reserve price r.

3.1 Weak Cartels

Suppose that the sellers do not know each other's costs before prices are o®ered and cannot

make side payments, but that they are able to engage in some signaling after drawing their costs.

Speci ̄cally, the sellers construct an ordering, say alphabetically by name, then use the outcome of

11Perhaps surprisingly, the strong cartel scheme is identical, in terms of expected pro ̄t, to a rigid-pricing scheme in

which the sellers credibly can reveal their costs to each other prior to submitting price o®ers, without a need to meet

or otherwise explicitly communicate, but cannot make side payments. Therefore, the theoretical and computational

results regarding strong cartels apply equally well to the credible revelation setting.

8
a public randomization device to select a winner for each contract as it is o®ered. The designated

winner competes unopposed and o®ers a price equal to the reserve price. The seller that is selected

through the randomization process is able to costlessly decline the invitation to be the winning  ̄rm,

which it will do if its cost exceeds the reserve price. The next seller on the list is then designated

to be the winning  ̄rm, it also has the option of declining, and so on. The remaining sellers either

do not submit price o®ers, or submit price o®ers that exceed the reserve price. This is a form of

bid rotation, though the designated seller is determined randomly each period rather than selected

sequentially from a predetermined ordering of the  ̄rms.

Lemma 1 in the Appendix proves that the outcomes using bid rotation are identical in expectation to those using the optimal mechanism, without transfer payments, determined by McAfee

and McMillan [1992]. Their mechanism speci ̄es that each seller with a cost less than the reserve

price submits a price equal to the reserve price. Other sellers either do not bid or set a price equal

to their cost.

12 However, for several reasons the bid rotation scheme I have constructed may be

easier to implement than the optimal mechanism. First, while bid rotation does not require that

the winning price be revealed, enforcing the optimal mechanism is di±cult unless the buyer announces the winning price; otherwise, sellers can deviate by slightly undercutting the reserve price.

Such a deviation cannot be detected immediately and dealt with in the next period, which impedes

collusion. Second, bid rotation can be helpful in avoiding investigation by the antitrust authorities,

as opposed to other schemes, such as the optimal mechanism, that have several  ̄rms submitting

the same price o®er.

13 Third, bid rotation is immune to the buyer's disrupting the scheme, say by

using a non-random tie-breaking procedure.

14

For a weak cartel to sustain supracompetitive prices in an auction, a seller with cost c designated

to lose must not prefer to slightly undercut the designated winner's price and precipitate perpetual

reversion to static Nash equilibrium price-setting. Therefore, it must be the case that for all c, for

a given reserve price r,

μ

±

1 ¡ ±

¶

1⁄4

WC

S

(r)  ̧ (r ¡ c) +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r): (1)

12On a technical note, a weak cartel can pro ̄tably implement the optimal mechanism only if [1 ¡ F(v)] =f(v)

decreases in v. For a procurement auction, the equivalent condition is ¡F(c)=f(c) decreases in c. This condition is

equivalent to requiring that F be log-concave, a restriction that is employed in the simulations.

13However, this scheme still su®ers from the weakness that the winning  ̄rm in each period sets a price of r. Under

noncooperative bidding, that outcome occurs with probability zero. Despite this result, there exist many procurement

settings in which the government has received several identical bids. See Scherer [1967], Comanor and Schankerman

[1976], and McAfee and McMillan [1992] for details.

14See McAfee and McMillan [1992] for elaboration on this point.

9
1⁄4WC

S

(r) denotes a seller's ex ante expected pro ̄t per-period from participating in the weak cartel

when the reserve price is r.

15 The left hand side of (1) is the net present value of future expected

collusive pro ̄t, while the right hand side is the short-term gain from defecting plus the net present

value of future expected static Nash equilibrium pro ̄t.

The incentive constraint can be rearranged to solve for the discount factor, ±

WC

(c; r), above

which defection is prevented by a  ̄rm with cost c, yielding

±

WC

(c; r) =

(r ¡ c)

1⁄4WC

S

(r) ¡ 1⁄4

NE

S

(r) + (r ¡ c)

: (2)

As 1⁄4WC

S

(r) and 1⁄4

NE

S

(r) are independent of a  ̄rm's current cost, it is straightforward to show that

±

WC

(c; r) is highest for a  ̄rm with the lowest cost. Therefore, ±

WC

(r)  ́ ±

WC

(c; r) is the critical

discount factor. It increases as the short-term gain from defection (r ¡ c) increases, and it increases

as the per-period loss from defection

¡

1⁄4WC

S

(r) ¡ 1⁄4

NE

S

(r)

¢

decreases.

Proposition 1 In a weak cartel using the symmetric bid rotation scheme with reserve price r, a

seller's ex ante expected pro ̄t from participating is

1⁄4

WC

S

(r) =

1 ¡ [1 ¡ F(r)]

N

NF(r)

Z r

c

F(c)dc:

Defection by a seller with the lowest possible cost can be deterred if and only if ±  ̧ ±

WC

(r), where

±

WC

(r)  ́

N (r ¡ c)

R r

c

F(c)

h

1¡[1¡F(r)]

N

F(r) ¡ N [1 ¡ F(c)]

N¡1

i

dc + N (r ¡ c)

:

±

WC

(r) may either increase or decrease as r increases.

3.2 Strong Cartels

Suppose that the sellers do not know each other's costs before prices are o®ered, but that they

are able to make side payments to each other. McAfee and McMillan [1992] show that in the

optimal scheme, the low-cost seller sets its price at the reserve price, pays each other seller an equal

15Note that the randomness in the assignment process creates for a  ̄rm incentives that are una®ected by whether

the  ̄rm has recently been the selected winner of another project. This greatly simpli ̄es the analysis by reducing the

number of constraints that must be satis ̄ed.

10
fraction of the cartel's surplus,

r¡T(c)

N¡1

, and receives T(c), where

T(c) = r ¡

R r

c

(r ¡ s)(N ¡ 1)[1 ¡ F(s)]

N¡1

f(s)ds

[1 ¡ F(c)]

N

:

Thus, a  ̄rm can receive a payment even if its cost exceeds the reserve price. The side payment

scheme can be implemented by the sellers' holding their own  ̄rst-price auction prior to the buyer's

 ̄rst-price auction, which Graham and Marshall [1987] refer to as a pre-auction knockout (PAKT).

I assume that the strong cartel attempts to collude in this fashion, with the seller winning the

PAKT making transfer payments to the losing sellers before the buyer's procurement auction.

Lemma 2 in the Appendix proves that a seller's expected pro ̄t from using the strong cartel

scheme is identical to its expected pro ̄t from using a rigid-price scheme in a setting in which

the sellers can credibly reveal their costs to one another but cannot make side payments. The

latter scheme allocates contracts e±ciently and avoids the asymmetric information that prevents

the cartel members from knowing which seller can most e±ciently ful ̄ll a particular contract.

Considering the setting in which costs can be credibly revealed does more than provide a baseline

for determining how asymmetric information across sellers a®ects their ability to collude. While the

assumption of credible cost revelation is challenged in the mechanism design approach, it has been

used in the literature on information sharing in oligopoly.

16 Moreover, this assumption may be

valid in certain settings, such as when transport costs are relatively high and colluding  ̄rms create

territorial restrictions. For example, in the Addyston Pipe conspiracy

17 and various conspiracies

in lumber contracting,

18

the designated winners were local  ̄rms and presumably were the low-cost

providers.

It is somewhat more complicated for a strong cartel to sustain collusion than it is for a weak

cartel, because with the strong cartel's side payment scheme there are more instances in which a

seller potentially has a pro ̄table deviation. As in a weak cartel, in a strong cartel a seller with

cost c that is designated to lose the buyer's auction must be deterred from slightly undercutting

the reserve price to beat the designated winner. That is, it must be the case that

μ

±

1 ¡ ±

¶

1⁄4

SC

S

(r)  ̧ (r ¡ c) +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r): (3)

16See Gal-Or [1986], Shapiro [1986], and Vives [1984].

17Addyston Pipe and Steel Company v. United States, 175 U.S. 211 (1899).

18See Baldwin, Marshall, and Richard [1997] for details.

11
Note that the side payment does not come into play in (3), because at the moment the losing seller

is deciding whether or not to deviate from the collusive arrangement, it already has received the

side payment.

The low-cost seller with cost c in the current period, who should win the PAKT, must be

deterred from the following two deviations. First, after the PAKT but before the buyer's auction,

the low-cost seller may elect not to make the agreed-upon side payments. As the PAKT reveals all

sellers' costs, the low-cost seller knows precisely how much pro ̄t it will earn in the buyer's auction,

assuming that the deviating seller will win by an arbitrarily small amount less than the secondlowest cost. Regardless of the low-cost seller's cost, the pro ̄t from this deviation is maximized

when all other sellers' costs exceed r. Thus, to prevent such a deviation it must be the case that

T(c) +

μ

±

1 ¡ ±

¶

1⁄4

SC

S

(r)  ̧ (r ¡ c) +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r): (4)

Second, the low-cost seller may elect to overstate its cost in the PAKT, receive a side payment, and

then slightly undercut the designated winner in the buyer's auction. To prevent such a deviation,

it must be the case that

T(c) +

μ

±

1 ¡ ±

¶

1⁄4

SC

S

(r)  ̧

E

£

r ¡ T(c(2))jc(1) = c

¤

N ¡ 1

+ (r ¡ c) +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r); (5)

where c(i) denotes the ith lowest order statistic from the cost distribution F(c).

By themselves, (4) and (5) look like they may cause substantial di±culty in calculating the

critical discount factor, relative to the calculation with a weak cartel. However, as T(c) > 0, then

(4) holds whenever (3) holds. Therefore, (4) is super°uous. Similarly, if (3) holds, then the only

way that (5) can fail to hold is if

T(c) <

E

£

r ¡ T(c(2))jc(1) = c

¤

N ¡ 1

:

However, because bidding truthfully is made incentive compatible in the optimal static mechanism,

a seller always prefers winning and making side payments over losing and receiving a side payment

when it should have won. Therefore, the preceding inequality cannot hold, and (5) holds whenever

(3) holds. Consequently, the only incentive constraint that requires checking for the strong cartel

12
is (3), which can be rearranged to yield

±

SC

(c; r) =

(r ¡ c)

1⁄4

SC

S

(r) ¡ 1⁄4

NE

S

(r) + (r ¡ c)

: (6)

(6) is quite simple and is analogous to the single incentive constraint that must be satis ̄ed for the

weak cartel to be able to sustain collusion. As was determined for the weak cartel, ±

SC

(c; r) is

highest for a seller with the lowest cost, c. Hence, the critical discount factor is ±

SC

(r)  ́ ±

SC

(c; r).

Proposition 2 In a strong cartel using the symmetric side payment scheme with reserve price r,

a seller's ex ante expected pro ̄t from participating is

1⁄4

SC

S

(r) =

r ¡ c

N

¡

1

N

Z r

c

[1 ¡ F(c)]

N dc:

Defection by a seller with the lowest possible cost can be deterred if and only if ±  ̧ ±

SC

(r), where

±

SC

(r)  ́

N (r ¡ c)

(N + 1)(r ¡ c) ¡

R r

c

[(N ¡ 1) F(s) + 1] [1 ¡ F(s)]

N¡1

ds

:

Collusion is more easily sustained as the reserve price increases. That is, ±

SC

(r) decreases as r

increases.

Note that expected pro ̄ts in the punishment regimes of the schemes used by the weak and strong

cartels are identical. Therefore, Propositions 1 and 2 indicate that the di®erence between the

critical discount factors for weak and strong cartels is caused by the di®erence in expected collusive

pro ̄ts. It can be shown that expected pro ̄ts are larger for the strong cartel. Hence, for a given

reserve price a strong cartel's critical discount factor is always lower than a weak cartel's. However,

the buyer's optimal reserve price may depend upon the type of cartel faced, which in turn will a®ect

the critical discount factor. The buyer's response to the potential for collusive behavior is the topic

discussed next.

3.3 The Buyer's Response

In the analysis so far, the reserve price has been taken as given. However, if the buyer has

discretion in setting the reserve price, then the buyer may be able to substantially a®ect the sellers'

ability to collude.

For any reserve price, the buyer's expected pro ̄t is identical when facing any two rigid-price

collusive schemes in which the seller selected to win sets its price at the reserve price, provided

13
that its cost is less than the reserve price. Thus, the optimal reserve price versus such collusive

price-setting is the same regardless of the collusive scheme.

Result 3 Facing the collusive scheme X in which a single seller o®ers the reserve price, the buyer

pays the reserve price if at least one seller's cost is below the reserve price. Thus, the buyer's

expected pro ̄t as a function of the reserve price, r, is

1⁄4

X

B (r) = (cB ¡ r)

3

1 ¡ [1 ¡ F(r)]

N

 ́

:

The buyer's optimal reserve price, r

X, solves

cB = r

X +

1 ¡

£

1 ¡ F(r

X)

¤N

N [1 ¡ F(rX)]

N¡1

f(rX)

;

provided that r

X · c. Otherwise, r

X = c and the reserve price is non-binding. If N  ̧ 2, then

the reserve price is always binding. Moreover, the optimal reserve price weakly decreases as N

increases.

When the N sellers collude by having the  ̄rm selected to win set its price at the reserve price,

the buyer essentially is facing a single seller whose costs are drawn from the distribution G(r) =

1 ¡ [1 ¡ F(r)]

N . G(r) is the distribution of the minimum of N draws from the distribution F(r).

Thus, Result 3's formula for the optimal reserve price in essence is identical to the one presented in

Result 2. Moreover, because the buyer's optimal reserve price versus the collusive scheme decreases

in the number of sellers, the buyer's optimal reserve price when facing such a collusive scheme is

less than the optimal reserve price when facing static Nash equilibrium price-setting.

19

While Result 3 determines the buyer's optimal reserve price when it knows it is facing a rigidpricing scheme by a strong or a weak cartel, that reserve price is not necessarily the one that would

be selected by a buyer concerned about collusion. Instead, the buyer may be able to deter collusion

by selecting a di®erent reserve price, which in turn a®ects the critical discount factor.

I focus on three reserve prices available to the buyer. First, the Nash reserve price, presented

in Result 2, is the reserve price that is optimal versus static Nash equilibrium price-setting. Second,

the naive reserve price, equal to the upper support of the cost distribution (c), is the reserve price

19This result follows because the optimal reserve price versus static Nash equilibrium price-setting is identical for

all N. Thus, the optimal reserve price versus collusive price-setting with N = 1 is the same as the optimal reserve

price versus static Nash equilibrium price-setting. The argument is completed by using the fact that the optimal

reserve price versus collusive price-setting decreases as N increases.

14
that would be set if the buyer were unsophisticated, could not credibly impose a lower reserve price,

or simply had a large value relative to the sellers' costs.

20 Each of these reserve prices determines

a critical discount factor that the sellers' discount factor must exceed in order for collusion to be

sustainable.

Third, the minimal deterrent reserve price is the lowest reserve price for which the buyer's

expected pro ̄t versus static Nash equilibrium price-setting, using that reserve price, is equal to

the buyer's expected pro ̄t versus collusive price-setting, using the optimal reserve price versus

collusive price-setting (r

X from Result 3). As the buyer always can let the sellers collude and earn

1⁄4

X

B

(r

X), for X 2 fWC; SCg, then the buyer will never try so hard to induce static Nash equilibrium

price-setting that it actually earns less than that amount. The minimal deterrent reserve price

determines the critical deterrent discount factor, which is the lowest discount factor the sellers

can have in order to sustain collusion against a strategic buyer.

Because for any reserve price the buyer's expected pro ̄t versus static Nash equilibrium pricesetting exceeds its expected pro ̄t versus collusive price-setting, there always exists a set of discount

factors for which the buyer  ̄nds it pro ̄table to deter collusion through its selection of the reserve

price. The di®erences the buyer can create in the critical discount factor may not appear to be

substantial, say if the discount factors are distributed uniformly over the interval [0,1]. However, if

the true discount factors are distributed uniformly over [0.95,1], then increasing the critical discount

factor from, say, 0.98 to 0.991 decreases the probability that the sellers can collude by 55 percent.

21

The next section examines more carefully the extent of the buyer's ability to deter collusion through

its selection of the reserve price.

4 Simulation Results

The theoretical results in Section 3 analytically determine the minimal discount factor for which

collusion is sustainable, but they do not provide much insight into the actual value of the critical

discount factor, or how that value is a®ected by the number of sellers, their cost distributions, or

the level of the reserve price. These quantitative issues are important for several reasons. First,

concern about collusion via these mechanisms is warranted only if the critical discount factors

20

I am being somewhat imprecise. If the buyer's value exceeded the sellers' maximum cost, and the buyer could

not credibly impose a reserve price below its value, then the buyer would be even more susceptible to collusion than

the results I present will show. This is the case because, while colluding, the designated winning seller will o®er a

price equal to cB > c, but if punishment is entered into, then the highest price set will be c.

21There exists a 40 percent chance the sellers' discount factor is greater than 0.98, and an 18 percent chance it is

greater than 0.991. Hence, increasing the critical discount factor decreases the probability the sellers can collude by

(40 ¡ 18) =40 = 55 percent.

15
correspond to discount factors that  ̄rms might actually use to value future pro ̄ts. Second, it

is useful to know how the critical discount factor changes as a function of the primitives of the

strategic environment. For example, if there exist conditions under which collusion is more easily

sustained as the number of sellers increases, then under those conditions the buyer may wish to

restrict the number of sellers from which it accepts price o®ers. Similarly, if collusion is more easily

sustained as the variance of the sellers' costs falls, then there may exist incentives for the sellers to

adopt production technologies that reduce that variance.

Researchers have turned to computation to obtain both qualitative and quantitative insights

from models that are analytically cumbersome or intractable. For example, Quirmbach [1993]

provides computational evidence on the relationship between the incentives to perform R&D and

the extent of post-innovation collusive behavior. He  ̄nds that while the social welfare ranking

of Bertrand and Cournot competition can vary, both tend to outperform perfect collusion. Prior

to this  ̄nding, it was unclear whether the prospect of collusive pro ̄ts tended to lead to higher

welfare by promoting more intensive R&D. Haubrich [1994] provides computational evidence that

helps bridge the gap between the theory and the reality of executive compensation. He  ̄nds

that empirical regularities of compensation packages are consistent with reasonable risk aversion

parameters in standard principal-agent models, despite the widely held preexisting belief that such

models could not realistically explain observed compensation schemes.

This section complements Section 3's theoretical results by computing critical discount factors

for a varying number of sellers and for a wide range of speci ̄c functional forms of the sellers' cost

distributions. For the sake of space, I compute critical discount factors only when cB = c, for

which the Nash reserve price is binding. I evaluate the data by using a second-order polynomial

to approximate the nonlinear function that determines the critical discount factor. This technique

usefully summarizes how the critical discount factor is in°uenced by the number of  ̄rms (N) and

by the mean (1) and variance (3⁄4

2

) of the sellers' cost distribution, three parameters whose e®ect

one typically might wish to evaluate.

I compute the discount factor for each (N; 1; 3⁄4

2

) triplet in a grid of the three-dimensional

parameter space. The number of  ̄rms varies from 2 to 30. The cost distributions all are from the

family of Beta distributions, which has support [0; 1], includes the uniform distribution as a special

case, and varies smoothly in its strictly positive parameters p and q.

22 By varying the parameters,

the Beta distribution can take on a variety of shapes and can be made to look like many other

22Moving the support to [X; X + 1], while increasing cB by X, has no e®ect on the quantitative results to follow.

16
distributions. The mean and the variance uniquely determine the two Beta parameters, and hence

uniquely determine the distribution. One can show that in order for the Beta parameters to be

strictly positive, the mean and the variance must satisfy 1(1 ¡ 1) ¡ 3⁄4

2 > 0. Moreover, I further

restrict the mean and variance so that the Beta distribution is log-concave.

23 This condition is

required for the optimal mechanism without transfer payments to yield supracompetitive pro ̄ts.

24

The permissible set of means and variances is illustrated in Figure 1, with a grid size of 0.01.

Subject to the parameter restrictions and the grid size, I compute the critical discount factor for

478 distributions. Given that the number of  ̄rms varies from 2 to 30, for each combination of the

two cartel schemes and the three reserve price strategies, I generate 13,862 critical discount factors

(478 distributions£29 numbers of  ̄rms).

Figure 1 Here

I use the least squares criterion to  ̄t the computed discount factors to the (mostly) second-order

polynomial equation

± = ®0 + ®1N + ®21 + ®33⁄4

2 + ®4N2 + ®5N1 + ®6N3⁄42 + ®712 + ®813⁄42 + ®93⁄44 + ®10N3 + ®11N4

:

I add the N3 and N4terms to the otherwise second-order polynomial because some of the generated data have a pattern requiring fourth-order terms. The results of the regressions for each

combination of the two cartel schemes and the three reserve price strategies are reported in Table

1. As is evident from the adjusted R2 values, the approximations provide a very good  ̄t of the

generated data. The importance of the second-order terms is evident from the reduction in the

adjusted R2 measures of on average about 0.245 points when only  ̄rst-order terms are used.

Table 1 Here

The worst  ̄ts predictably involves the deterrent reserve price strategy. With the naive reserve

price strategy, the reserve price does not change with N, 1, or 3⁄4

2

. With the Nash reserve price

strategy, the reserve price changes with 1 and 3⁄4

2

, but not with N. Because the critical discount

factor is a function of the reserve price and of N, 1, and 3⁄4

2

, there should exist greater variability

23One can show that the Beta distribution is log-concave provided that q  ̧ 1. The proof involves solving for the

conditions under which either the distribution itself is log-concave, or the density is log-concave (which implies the

distribution is log-concave).

24See Theorem 1 in McAfee and McMillan [1992].

17
in the critical discount factors with the Nash reserve price strategy than with the naive reserve

price strategy. With the deterrent reserve price strategy, the reserve price changes with N, 1, and

3⁄4

2

. Hence, the critical discount factors versus this strategy should exhibit the most variation, and

consequently the  ̄t should be worse than with the other two reserve price strategies.

The following three examples use the regression results in Table 1 to show how the critical discount factor is a®ected by changing the three parameters. Figure 2 illustrates the e®ect of changing

the mean of the sellers' cost distribution for a strong duopoly cartel whose cost distribution's variance equals the average level of the variance in the simulations (3⁄4

2 = 0:0382). The evaluated mean

costs are those from the parameter-grid that induce log-concavity of the cost distribution, given the

assumed variance. For each reserve price strategy in this example, collusion becomes more di±cult

to sustain as the distribution's mean increases. This e®ect is so pronounced that, against a buyer

using the deterrent reserve price strategy (panel (c)), collusion is extremely di±cult to sustain once

the mean cost exceeds 0.6. In fact, once the duopolists' expected costs exceed 0.5, their discount

factor must be at least 0.983 to possibly be able to sustain collusion versus the deterrent reserve

price strategy. In contrast, for the same range of expected costs the duopolists' discount factor

can be as low as 0.93 to sustain collusion versus the naive reserve price strategy.

Figure 2 Here

Figure 3 illustrates the e®ect of changing the variance of the sellers' cost distribution for a

strong duopoly cartel whose cost distribution's mean equals the average level of the mean in the

simulations (1 = 0:4221). The evaluated variances are those from the parameter-grid that induce

log-concavity of the cost distribution, given the assumed mean cost. For the naive and Nash reserve

price strategies, the actual data show a slight increase in the critical discount factor as the variance

increases, while the predictions show a slight decrease. For the deterrent reserve price strategy, the

predicted and actual data match closely and show a slight decrease in the critical discount factor as

the variance increases. The sellers' discount factor must always exceed 0.973 for a strong duopoly

cartel to sustain collusion against a strategic buyer using the deterrent reserve price strategy.

Figure 3 Here

Figures 4 illustrates the e®ect of changing the number of sellers in a strong cartel when the

cost distribution's mean and variance equal their average levels in the simulations. The predicted

and actual values match extremely closely, with one signi ̄cant di®erence for the Nash reserve price

18
strategy shown in panel (b). The actual data show the decrease in the critical discount factor as

the number of  ̄rms increases from two. Though the critical discount factor does increase in N

with three or more sellers, in this example collusion is more di±cult to sustain with two sellers

than it is with up to nine. This result is not uncommon in the data, though even the quartic

terms fail to capture it fully in the predictions. This failure exists even in panel (c), for which the

actual data illustrate a small but real decline in the critical discount factor as the number of sellers

increases from two to three.

Figure 4 Here

To examine the generality of the examples just presented, I use the regression results in Table

1 to provide a comprehensive numerical evaluation of comparative statics. The e®ect of increasing

the mean is predicted by the regression equation to be

d±

d1

= ®2 + ®5N + 2®71 + ®83⁄4

2

:

Similarly, the e®ect of increasing the variance is predicted to be

d±

d3⁄42

= ®3 + ®6N + ®81 + 2®93⁄4

2

:

These derivatives can be calculated for any (N; 1; 3⁄4

2

) triplet in the grid of feasible parameters.

General patterns in the signs of these derivatives provide evidence about the in°uence of changes

in the primitives on the ease of sustaining collusion.

Using the regression results in Table 1, the  ̄rst row of Table 2 reports the fraction of feasible

(N; 1; 3⁄4

2

) triplets for which the derivative of the critical discount factor, with respect to the mean

cost, is strictly positive. The second row similarly reports the fraction of strictly positive derivatives

with respect to the variance. Because one might be more curious about the derivatives solely in

more concentrated markets, rows 3 and 4 report about the same derivatives as rows 1 and 2, limited

to those (N; 1; 3⁄4

2

) triplets in the feasible parameter grid with ten or fewer sellers.

Table 2 Here

The results in Table 2 suggest that collusion typically becomes more di±cult to sustain as the

mean cost increases, particularly in more concentrated markets. While this result potentially is

due partly to the change in the reserve price as 1 changes, the fractions reported for the naive

19
reserve price, which is unchanged as 1 changes, indicate that the result is largely due to the

increase in the mean cost causing a decrease in the di®erence between the per-period cooperative

and noncooperative payo®s. With the exception of a weak cartel facing the naive reserve price

strategy, the results in Table 2 provide even stronger evidence that collusion typically becomes easier

to sustain as the variance of the sellers' costs increases. This result is caused by increasing the

di®erence between the per-period cooperative and noncooperative payo®s at 3⁄4

2

increases. With no

a priori reasoning about how the di®erence between the per-period cooperative and noncooperative

payo®s changes as the mean or the variance change, the results in Table 2 provide useful insights

into the determinants of sellers' ability to collude.

Examining the level of the critical discount factor is just as valuable as is examining the preceding comparative statics, because the critical discount factors that emerge from the simulations

are quite high. For example, the average critical discount factors for duopoly facing the deterrent

reserve price strategy are approximately 0.966 for a strong cartel and 0.979 for a weak cartel. In

contrast, the critical discount factor for successful collusion in a homogeneous product Bertrand

duopoly is

1

2 when  ̄rms have constant identical marginal costs and split the market each period. If

the Bertrand duopolists alternate production, which is more comparable to the procurement setting

examined here, then the critical discount factor is

2

3

. While the di®erence between the Bertrand

games and the procurement setting is striking, it is not clear whether the critical discount factors

for strong and weak cartels are high from a practical perspective.

Relating the discount factors to plausible real-world values helps one to evaluate the likelihood

of tacit collusion using the two proposed schemes. By de ̄nition, the one period discount factor

has a one to one relationship to the one period discount rate, R, which is given by ± =

1

1+R

: If a

 ̄rm's discount rate were known, then one could determine the  ̄rm's discount factor. This discount

factor then could be compared to the critical discount factors derived in the simulations.

One measure of a  ̄rm's discount rate is its opportunity cost of capital. Another measure of

a  ̄rm's discount rate is its \hurdle" rate, which is the minimum annual return on investment

necessary for an investment opportunity to be undertaken. While many factors are considered

by  ̄rms selecting investment opportunities, it seems reasonable to use a  ̄rm's hurdle rate as its

annual discount rate.

If the contracts considered in the repeated game model are not solicited annually, then one

must adjust the hurdle rate to calculate the correct discount factor. If contracts are o®ered every t

years, where t years constitute one period from the repeated game perspective, and R is the hurdle

20
rate, then the one period discount factor is ± =

1

(1+R)

t

. If contracts are o®ered more than once per

year, then t is less than one.

25

Table 3 presents the per-period discount factors associated with four hurdle rates over six

di®erent time periods, as measured by the number of contracts o®ered per year. For hurdle rates

of 15 percent and higher, the resulting discount factors are below those reported in Table 4 as

being necessary for successful collusion by a weak cartel versus a sophisticated buyer, even with a

small number of sellers. With a 10 percent hurdle rate, the same conclusion holds if six or fewer

contracts are o®ered per year. The critical discount factors in Table 4 illustrate that a strong

cartel also faces di±culty when the buyer uses the deterrent reserve price strategy. Hence, in many

instances sellers with costs drawn from the \average" Beta distribution examined in Table 4 could

not successfully collude against a sophisticated buyer by using the optimal mechanism with or

without side payments. In contrast, if the buyer cannot credibly impose a reserve price, then

collusion can be sustained by strong and weak cartels for several values of the hurdle rate, as shown

by the critical discount factors in Table 4 associated with the naive reserve price strategy.

Table 3 Here

Table 4 Here

Even if the sellers' discount factor is such that they can collude, the buyer can decrease the

sellers' discount factor by decreasing the frequency with which contracts are let. As Tables 3 and

4 show, a sophisticated buyer can eliminate the possibility of collusion by a strong cartel with a

15 percent hurdle rate by o®ering contracts every two months rather than every month. Note that

the change in frequency does not change the critical discount factor, but does change the sellers'

true discount factor.

Finally, the buyer may prefer to reduce the number of sellers from which it accepts price o®ers.

As the computational results revealed, in some instances collusion may be easier to sustain with

more  ̄rms. Of course, the desire to eliminate collusion must be balanced against the expected

pro ̄t in both cases.

5 Conclusion

25One could envision a procurement setting in which there are multiple buyers within an industry, which in many

respects would appear to be equivalent to the single buyer setting with a further increase in the number of contract

o®erings. One potential di±culty is that individual buyers may have incentives to game the system by raising their

reserve price without inducing collusion, because the sellers are unable to collude against the remaining buyers who

chose their reserve prices for deterrent purposes.

21
This paper uses tools from repeated game theory to examine the ability of sellers in repeated  ̄rstprice procurement auctions to sustain two collusive bidding schemes that are optimal from a static

mechanism design perspective. Speci ̄cally, for di®erent numbers of  ̄rms, di®erent reserve price

strategies, and a large number of cost distributions, I numerically determine the value that  ̄rms

must place on future pro ̄ts such that abiding by the collusive scheme is more pro ̄table than is

defecting from it and inciting retaliation. Computing the critical discount factors necessary for

collusion to be sustainable illuminates the analytically complex e®ects of the various parameters

of the strategic environment. Moreover, relating the computed discount factors to plausible realworld discount factors helps one to assess the practical relevance of the static mechanism design

approach to collusion in auctions.

The computational results reveal that the buyer's choice of the reserve price has a large impact

on the sellers' ability to collude. If the buyer is sophisticated in its choice of the reserve price, then

collusion tends be sustainable only for extremely high discount factors that correspond to what

appear to be unreasonably low hurdle rates within the  ̄rm. Successful collusion requires the sellers

to place such high value on future pro ̄ts because the short-term gain from cheating on the collusive

agreement is large relative to the per-period loss of collusive pro ̄ts. Thus, sellers must value those

foregone pro ̄ts highly in order to resist their temptation to cheat. The necessity of such extremely

high discount factors suggests it may be unlikely that tacit collusion using the previously identi ̄ed

bidding schemes can be supported as a subgame perfect equilibrium of the repeated game. This

conclusion is consistent with the assertion by Graham and Marshall [1987] that cartels in  ̄rst-price

auctions in the United States are unstable. Firms in such auctions may attempt to collude, only

to  ̄nd that their arrangement cannot be sustained.

The results also provide strong evidence that collusion is more di±cult to sustain when the

sellers' costs have a high mean or a low variance. Surprisingly, collusion in highly concentrated

markets might be more easily sustained as the number of  ̄rms increases, which is not a feature

of traditional oligopoly models. The likelihood of collusion initially increases in the number of

sellers if the di®erence between the expected per-period collusive and punishment pro ̄ts increases

as the number of colluding sellers increases, because increasing that di®erence decreases the critical

discount factor. The buyer may be able to use this phenomenon to eliminate collusion by reducing

the number of sellers from which it accepts price o®ers. As a complementary strategy, the buyer

also can deter collusion through the more familiar method of increasing the duration of contracts,

which a®ects sellers' per-period discounting.

22
The strong conclusion that the probability of rigid-price collusion is low versus strategic buyers

in an independent private values environment raises the issue of how the many reported instances

of collusion in auction and procurement markets were supported. Relaxing three assumptions used

in the present analysis generates three possible explanations of how collusion might more readily

be sustained. First, buyers may not be very sophisticated in their reserve price selection, they may

not be able to credibly commit to the reserve prices necessary to severely limit collusion, or they

may have large values relative to the sellers' costs. If so, then the results for naive buyers apply,

and they indicated that collusion was more easily sustained.

Second, collusion might be more pro ̄table in common value environments, and it is possible

that the reported instances of collusion tended to occur in procurement settings in which common

value elements were more important than private value elements. Collusion in common value

environments generates better information about the contract's true value and permits sellers to

avoid the winner's curse. If the value of collusion in common value environments exceeds its value

in private value environments, then it may be easier to sustain.

Third, the two schemes I examine may be less e®ective than other collusive schemes one could

design. In particular, colluding  ̄rms might coordinate on a price less than the reserve price, in

order to make deviation less attractive. However, coordinating on a lower price reduces the value

of collusion, which reduces the per-period loss from defection and makes collusion more di±cult to

sustain. Moreover, if sellers coordinate on a price less than, but a®ected by, the reserve price, then

one must account for the possibility that the buyer will select its reserve price di®erently. Clearly,

determining more e®ective means of colluding in procurement markets is a worthwhile area for

future research.

Appendix

Lemma 1 Consider a bid rotation scheme in which each of N  ̄rms is equally likely to be selected

to win the contract at price r. If a  ̄rm is selected but its cost exceeds r, then one of the remaining

 ̄rms is randomly selected to win the contract at price r: This  ̄rm can also decline, and so on. A

 ̄rm's ex ante expected pro ̄t using this scheme is

1⁄4

BR

S

(r) =

1 ¡ [1 ¡ F(r)]

N

NF(r)

Z r

c

F(c)dc:

Proof of Lemma 1: Suppose that a  ̄rm has cost less than r and is selected to win, either initially

or subsequent to other  ̄rms being selected but having costs exceeding r. The  ̄rm's expected pro ̄t,

23
conditional on having cost less than r, is

R r

c

(r ¡ c) f(c)dc

F(r)

=

R r

c

F(c)dc

F(r)

;

where the term on the right hand side following from integrating the left hand side by parts. The

probability the  ̄rm's cost is less than r is F(r), and the probability the  ̄rm is selected from the

N  ̄rms is

1

N

+

1

N

[1 ¡ F(r)] +

1

N

[1 ¡ F(r)]

2 + ¢ ¢ ¢ +

1

N

[1 ¡ F(r)]

N¡1 =

1 ¡ [1 ¡ F(r)]

N

NF(r)

:

Therefore, the  ̄rm's ex ante expected pro ̄t is

1⁄4

BR

S

(r) =

1 ¡ [1 ¡ F(r)]

N

NF(r)

£ F(r) £

R r

c

(r ¡ c) f(c)dc

F(r)

=

1 ¡ [1 ¡ F(r)]

N

NF(r)

Z r

c

F(c)dc;

which is the desired result. ¥

Proof of Proposition 1: The determination of ex ante expected pro ̄t is made in McAfee and

McMillan [1992]. The following incentive constraint must be satis ̄ed to prevent defection by a

 ̄rm with cost c that is not selected to win:

μ

±

1 ¡ ±

¶

1⁄4

WC

S

(r)  ̧ r ¡ c +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r):

The smallest ± for which this is satis ̄ed is

±

WC

(r)  ́

N (r ¡ c)

R r

c

F(c)

h

1¡[1¡F(r)]

N

F(r) ¡ N [1 ¡ F(c)]

N¡1

i

dc + N (r ¡ c)

;

which is the desired result. ¥

Proof of Proposition 2: A seller's interim expected pro ̄t is

1⁄4

SC

S

(cjr) =

8

<

:

[T(c) ¡ c] [1 ¡ F(c)]

N¡1 +

h

1 ¡ [1 ¡ F(c)]

N¡1

i R c

c

h

r¡T(s)

N¡1

i h

(N¡1)[1¡F(s)]

N¡1

1¡[1¡F(c)]

N¡1

i

f(s)ds if c · r

h

1 ¡ [1 ¡ F(r)]

N¡1

i R r

c

h

r¡T(s)

N¡1

i h

(N¡1)[1¡F(s)]

N¡1

1¡[1¡F(r)]

N¡1

i

f(s)ds if c > r

Through tedious integration of the interim expected pro ̄t, one can show that the seller's ex ante

expected pro ̄t is

Z c

c

1⁄4

SC

S

(cjr)f(c)dc =

r ¡ c

N

¡

1

N

Z r

c

[1 ¡ F(c)]

N dc:

24
As shown in the text, only the following incentive constraint must be satis ̄ed to prevent defection

by a  ̄rm with cost c that is not the selected winner :

μ

±

1 ¡ ±

¶

1⁄4

SC

S

(r)  ̧ r ¡ c +

μ

±

1 ¡ ±

¶

1⁄4

NE

S

(r):

The smallest ± for which this is satis ̄ed is

±

SC

(r)  ́

N (r ¡ c)

(N + 1)(r ¡ c) ¡

R r

c

[(N ¡ 1) F(c) + 1] [1 ¡ F(c)]

N¡1

dc

:

Finally, it is straightforward to show that the derivative of ±

SC

(r) with respect to r is negative. ¥

Lemma 2 Consider a setting in which the sellers can credibly reveal their costs to one another,

and in which the low-cost seller is designated to win by setting a price equal to the reserve price r.

There are no side payments, so  ̄rms designated to lose earn nothing. A seller's ex ante expected

pro ̄t from participating in the credible revelation scheme is

1⁄4

CR

S

(r) =

r ¡ c

N

¡

1

N

Z r

c

[1 ¡ F(c)]

N dc:

Proof of Lemma 2: A seller's interim expected pro ̄t is

1⁄4

CR

S

(cjr) = (r ¡ c)[1 ¡ F(c)]

N¡1

:

The seller's ex ante expected pro ̄t is

Z r

c

1⁄4

CR

S

(cjr)f(c)dc =

r ¡ c

N

¡

1

N

Z r

c

[1 ¡ F(c)]

N dc;

which is the desired result. ¥

References

[1] Athey, S., Bagwell, K., and Sanchirico, C. \Collusion and Price Rigidity" unpublished

manuscript (2000).

[2] Baldwin, L., Marshall, R., and Richard, J., \Bidder Collusion an Forest Service Timber Sales."

Journal of Political Economy, Vol. 105, No. 4 (1997), pp. 657-699.

25
[3] Comanor, W., and Schankerman, M., \Identical Bids and Cartel Behavior." Bell Journal of

Economics, Vol. 7 (1976), pp. 281-286.

[4] Cramton, P., and Palfrey, T., \Cartel Enforcement with Uncertainty About Costs." International Economic Review, Vol. 31, No. 1 (1990), pp. 17-47.

[5] Gal-Or, E. \Information Transmission-Cournot and Bertrand Equilibria." Review of Economic

Studies, Vol. 53 (1986), pp. 85-92.

[6] Graham, D., and Marshall, R., \Collusive Bidder Behavior at Single-Object Second-Price and

English Auctions." Journal of Political Economy, Vol. 95, No. 6 (1987), pp. 1217-1239.

[7] Haubrich, J., \Risk Aversion, Performance Pay, and the Principal-Agent Problem." Journal

of Political Economy, Vol. 102, No. 2 (1994), pp. 258-276.

[8] McAfee, R.P., and McMillan, J. \Bidding Rings." American Economic Review, Vol. 82, No. 3

(1992), pp. 579-599.

[9] McCutcheon, B., \Do Meetings in Smoke-Filled Rooms Facilitate Collusion?" Journal of

Political Economy, Vol. 105, No. 2 (1997), pp. 330-350.

[10] Porter, R., and Zona, J.D., \Detection of Bid Rigging in Procurement Auctions." Journal of

Political Economy, Vol. 101, No. 3 (1993), pp. 518-538.

[11] Quirmbach, H., \R&D: Competition, Risk, and Performance." RAND Journal of Economics,

Vol. 24, No. 2 (1993), pp. 157-197.

[12] Scherer, F.M., \Focal Point Pricing and Conscious Parallelism." Antitrust Bulletin, Summer

(1967), pp. 495-503.

[13] Scherer, F.M., and Ross, D. Industrial Market Structure and Economic Performance. Boston:

Houghton Mi2in (1990).

[14] Shapiro, C. \Exchange of Cost Information in Oligopoly." Review of Economic Studies, Vol.

53 (1986), pp. 433-446.

[15] Vives, X. \Duopoly Information Equilibrium: Cournot and Bertrand." Journal of Economic

Theory, Vol. 34 (1984), pp.71-94.

26
Table 1

Regression Results by Cartel Type and Reserve Price Strategy for

δ α α α μ α σ α α μ α σ α μ α μσ α σ α α =+ + + + + + + + + + + 01 2 3

2

4

2

5 6

2

7

2

8

2

9

4

10

3

11

4

N NNN N N

Weak Cartel

Naive RP

Weak Cartel

Nash RP

Weak Cartel

Deterrent RP

Strong Cartel

Naive RP

Strong Cartel

Nash RP

Strong Cartel

Deterrent RP

Intercept 0.69379149 0.86431262 0.91699998 0.68188789 0.84017995 0.89139666

Number of

Firms

0.03704046 0.00248617 0.00267516 0.03874445 -0.00017688 0.00342790

Mean 0.21043461 0.29331887 0.26129209 0.17315251 0.25770755 0.29180795

Variance 0.24967026 -0.40937132 -0.51774748 -0.16285821 -0.42003522 -0.57022403

Number of

Firms2

-0.00250669 0.00036730 -0.00000426 -0.00253982 0.00085835 0.00003237

Number of

Firms*Mean

-0.00792688 -0.00569608 -0.00300236 -0.00711913 -0.00671399 -0.00378106

Number of

Firms*Variance

-0.01335749 -0.00272237 -0.00146551 0.00159331 -0.00248975 -0.00237113

Mean2 0.00772701 -0.15093804 -0.19797418 0.02759866 -0.07577537 -0.20118931

Mean*Variance 0.33454753 0.38034936 0.46419345 -0.18393295 0.03053952 0.47785869

Variance2 -1.12080580 0.67125324 1.45184525 0.84536447 2.30518151 1.67026818

Number of

Firms3

0.00008605 -0.00002024 -0.00000176 0.00008394 -0.00004206 -0.00000427

Number of

Firms4

-0.00000109 0.00000032 0.00000004 -0.00000103 0.00000063 0.00000008

Data Points 13,862 13,862 13,862 13,862 13,862 13,862

Adjusted R2 0.94958884 0.90662891 0.78330459 0.96408683 0.93118992 0.87697972

Adjusted R2

(1st order only)

0.67843531 0.68311733 0.48413183 0.71895946 0.75560037 0.61968881
Table 2

Percentage of Feasible (N,μ,σ2) Triplets with Strictly Positive Derivatives

Weak Cartel

Naive RP

Weak Cartel

Nash RP

Weak Cartel

Deterrent RP

Strong Cartel

Naive RP

Strong Cartel

Nash RP

Strong Cartel

Deterrent RP

( ) d

d N δ

μ ≤ 30 94.75% 86.57% 76.32% 86.62% 90.92% 80.41%

( ) d

d N δσ2 ≤ 30 71.98% 0% 0% 0% 0.02% 0%

( ) d

d N δμ ≤ 10 100% 99.33% 85.61% 100% 100% 90.66%

( ) d

d N δσ2 ≤ 10 100% 0% 0% 0% 0.07% 0%

Table 3

Per-Period Discount Factors Associated with Various Hurdle Rates and Time Horizons

Hurdle Rate 12 contracts/year 6 contracts/year 4 contracts/year 3 contracts/year 2 contracts/year 1 contract/year

5% 0.996 0.992 0.988 0.984 0.976 0.952

10% 0.992 0.984 0.976 0.969 0.953 0.909

15% 0.988 0.977 0.966 0.954 0.933 0.870

20% 0.985 0.970 0.955 0.941 0.913 0.833

Table 4

Critical Discount Factors for Weak and Strong Cartels 

Versus Naive and Deterrent Reserve Price Strategies

(μ and σ2 at average levels from parameter set)

Number of Firms Weak Cartel

Naive RP

Weak Cartel

Deterrent RP

Strong Cartel

Naive RP

Strong Cartel

Deterrent RP

2 0.849492 0.990717 0.810981 0.977497

3 0.877683 0.990965 0.837246 0.977051

4 0.899341 0.991818 0.861076 0.978607

5 0.914829 0.992628 0.879408 0.980327

6 0.926262 0.993319 0.893588 0.981893
Figure 1

Feasible and Log-Concave Beta Parameters

(Grid size = 0.01)

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mean Cost

Cost Variance

Log-Concave Beta Parameters Feasible Beta Parameters
Figure 2

Critical Discount Factors for Strong Duopoly Cartel with σ2

 at Average Level

(a) Naïve RP

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mean Cost

Critical Discount Factor

Actual Values Predicted Values

(b) Nash RP

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mean Cost

Critical Discount Factor

Actual Values Predicted Values

(c) Deterrent RP

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mean Cost

Critical Discount Factor

Actual Values Predicted Values
Figure 3

Critical Discount Factors for Strong Duopoly Cartel with μ at Average Level

(a) Naive RP

0.75

0.80

0.85

0.90

0.95

1.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Cost Variance

Critical Discount Factor

Actual Values Predicted Values

(b) Nash RP

0.75

0.80

0.85

0.90

0.95

1.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Cost Variance

Critical Discount Factor

Actual Values Predicted Values

(c) Deterrent RP

0.75

0.80

0.85

0.90

0.95

1.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Cost Variance

Critical Discount Factor

Actual Values Predicted Values
Figure 4

Critical Discount Factors for Strong Cartel with μ and σ2

 at Average Levels

(a) Naive RP

0.75

0.80

0.85

0.90

0.95

1.00

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Firms

Critical Discount Factor

Actual Values Predicted Values

(b) Nash RP

0.75

0.80

0.85

0.90

0.95

1.00

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Firms

Critical Discount Factor

Actual Values Predicted Values

(c) Deterrent RP

0.75

0.80

0.85

0.90

0.95

1.00

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Firms

Critical Discount Factor

Actual Values Predicted Values