Procurement Auctions with Uncertainty in Corruption

Shinya Horie
March 2017
Discussion Paper No.1710

GRADUATE SCHOOL OF ECONOMI CS
KOBE UNIVERSITYROKKO, KOBE, JAPANProcurement Auctions with Uncertainty in Corruption
SHINYA HORIE1
Graduate School of Economics, Kobe University
2-1, Rokkodaicho, Nada-ku, Kobe, Hyogo, 657-8501, Japan

Abstract

This paper considers a situation in which a corrupt government official does not commit to using thecommon corruption scheme called right of first refusal in a procurement auction. Under the right of firstrefusal, the contractors (or bidders) participate in a sequential auction, and there is no inefficiency inproject allocation. However, in cases in which the scheme is not practiced, both contractors participate ina simultaneous auction, and the disadvantaged contractor bids more aggressively than the advantaged
contractor. I found that such uncertainty regarding the practice of corruption schemes can lead toinefficiency, even when the corruption scheme itself is not practiced.
Keywords: Procurement Auctions; Corruption, Right of first refusal
JEL classification: C72, D44, L14

1. Introduction
In this paper, I examine the impact of corruption on contractor bidding behavior, expected
price and allocation when the implementation of corruption schemes is uncertain. Corruption is amajor problem in both developed and developing countries. According to reports by the1

This paper is substantially based on my Ph.D. dissertation “Two Essays on Local Public Economics” submitted to

The Ohio State University. I thank Gene Mumy, Lexin Ye, Matt Lewis, Hajime Miyazaki, David Landsbergen, Jun
Iritani,and all of the participants of workshop at the Ohio State University for discussion regarding this paper and fortheir invaluable comments. All errors are mine. This paper is supported by the Ministry of Education, Culture,
Sports, Science and Technology in Japan (MEXT) Grant-in-Aid for Young Scientists (B) (grant number 26780139,
http://www.jsps.go.jp/english/e-grants/).

1

Department of Justice and Federal Investigation Bureau in the United States, there were 20,446
cases of stimulus fraud in the private sector between 1988 and 2007. Because corruption can
increase contract prices and result in an inefficient allocation of contracts, many researchers have
explored the impact of corruption on the outcomes of procurement auctions. However, thepossibility that the expected corruption schemes might not come to fruition has been ignored.
Practically speaking, it seems important to recognize that corruption schemes are not always
guaranteed to be practiced. Because regulatory authorities consider corruption to be a serious
problem, more thorough work is being done to uncover corruption, and the practice of corruption
schemes may depend on the judgments of corrupt government officials’ regarding how stringent
the oversight process is.
Corruption is a type of implicit contract between a briber and bribee, and one of corruption’s
features is that it is illegal. Because it is illegal, the implementation of the contract is kept secret.
Thus, other contractors are not aware of and are not involved in the corrupt scheme, though they
may be aware of the potential existence of a corrupt relationship between specific contractors
and government officials. For the same reason, corrupt contractors cannot take government
officials to court when corruption schemes are not implemented. In other words, the corrupt
government is not necessarily committing to following through with these schemes. The only
way for a corrupt contractor to make implicit illegal contracts credible is to control the timing of
payment at the close of auctions. However, such a bribery-payment scheme can only decrease,rather than eliminate, the risk of paying government officials to execute a plan on which they do
not follow through. How the use of corruption schemes might affect the prices of the procured
projects and the efficiency of project allocation remains unclear.
I consider the following situation. In a one-shot, two-stage game, the government procures a
project, and it has a collusive relationship with one of its contractors via a corruption scheme.
2Under the scheme, the government official can revise the bid submitted by the corrupt contractor(called the advantaged contractor in this paper) before the amount of the bids becomes public
information; thus, the advantaged contractor will be awarded the contract unless the bid
submitted by the opponent (called the disadvantaged contractor) is lower than advantaged
contractor’s costs. When the advantaged contractor wins the auction outright, the government
official makes changes that allow the advantaged contractor to derive a greater surplus from thecontract. The payment from the advantaged contractor to the government official is a portion of
the advantaged contractor’s surplus. However, such plans do not always come to fruition. The
existence of collusive relationships and of possible corruption schemes is common knowledge,
but whether such plans are actually executed is uncertain.
Using this scheme specification, I create a situation in which the advantaged contractor canbe almost automatically awarded the project without making his own strategic moves as long as
the corruption scheme is practiced. By adding the question of uncertainty to the problem ofcorruption schemes, I focus on the advantaged contractor, who joins the competition and chooses
his bid strategically. The results indicate that inefficiency in project allocation can occur when
the plan is not executed. The implications are as follows. Because the disadvantaged contractor
knows about the potential for corruption, he bids more aggressively than the corrupt contractor,
but it is uncertain that corrupt behavior will occur. As a result, the projects will be awarded to thecontractor whose costs are higher.
This paper continues as follows. In section 2, I briefly review the past literature, summarize
the major topics and findings in the field of corruption in procurement auctions, and discuss what
can be done to extend prior research. In section 3, I construct a model of corruption in a sealed
bid first-price procurement auction with an uncommitted government official. In section 4, I firstshow the bidding behavior of contractors and discuss the allocation of contracts in fair3

procurement auctions as a benchmark. Next, I model a corrupt procurement auction in which thegovernment official does not commit to the corruption scheme in order to characterize theequilibrium. However, because this is an asymmetric auction model without a closed-form
solution, I derive a numerical solution to the model and comparative statics in section 5. I alsopresent the characteristics of the bidding behavior of contractors and explain the potential
inefficiency of this game.

2. Related Literature
Many of the related studies distinguish themselves from one another by using different
corruption schemes and by using different definitions of corrupt contractors. In many settings,
the corrupt government allows the corrupt contractor revise his or her bid after all bids are
submitted. This means that a sequential auction occurs in which the corrupt or advantaged
contractors bid after the honest contractors and corruption does not cause inefficiency, eventhough the expected prices are higher than in fair auctions (Jones and Menezes (1995), Menezes
and Monteiro (2003), Lengwiler and Wolfstetter (2004), Lee (2008), Burguet and Perry (2008)).
However, if there is a factor that drives honest or disadvantaged contractors to bid moreaggressively than corrupt contractors, then inefficiency occurs (Burguet and Perry (2008), Lee(2008)).
The studies most closely related to the present study are Arozamena and Weinschelbaum(2005) and Burguet and Perry (2008). Burguet and Perry (2008) employ a corrupt arrangement inwhich the government allows the corrupt contractor to revise the bid only when the contractor
fails to submit the lowest bid, whereas regular right of first refusal allows the contractor to revise
his bid even when he wins the auction outright. Because the government does not help thecorrupt contractor when he wins the auction outright, the contractor does not pay a bribe to the4government, which means that the corrupt contractor may enjoy a higher surplus than when heloses the auction during the first round but wins it by revising his bid. The essential factor is thesize of the payment for bribery relative to that of the total possible surplus. In this case, thecorrupt contractor has an incentive to join the competition from the beginning, which makes theauction game simultaneous and less inefficient.
Arozamena and Weinshelbaum (2005) use a strategy similar to mine in that at least one of
the contractors does not know that a corrupt relationship exists between the government and theopponent. The honest contractor assigns positive probabilities to both the possibility that
corruption will be practiced and the possibility that it will not be. This suspicion leads the honest
contractor to bid more aggressively than in a usual fair auction but less aggressively than when it
is certain that the corruption scheme will be executed. However, their setting does not allow thecorrupt contractor to face a simultaneous game. The only focus is on the bidding behavior ofhonest contractors, whereas my concern is the corrupt contractor’s behavior.

3. The Model
Information
Consider a situation in which a government official procures a public project. This is a
two-stage game with two contractors competing for the project. Each contractor is denoted aseither the advantaged contractor or the disadvantaged contractor by 𝐴 or 𝐷, respectively. (𝑖 ∈{𝐴, 𝐷}). The advantaged and disadvantaged contractors are defined by whether each has animplicit contract with the government. I describe the contract in the next subsection. Contractor
has a project completion cost of. Completion costis randomly distributed iid over theinterval [𝑐, 𝑐̅] . Let 𝐹(𝑐𝑖 ) be the probability cumulative function, and let 𝑓(𝑐𝑖 ) be thecorresponding density function. A contractor's (completion) cost is private information. I assume5

that 𝐹(⋅) is differentiable and that 𝑓(⋅) is continuously differentiable in [𝑐, 𝑐̅] . The probability
that the procurement auction is the target of a corruption scheme orchestrated by the government
official and the advantaged contractor is 𝑝 > 0. The probability is common knowledge, but thecorruption scheme is implemented only after the auction is closed.
The vNM utility function and the Corruption Scheme
The vNM utility functions in this game are represented as follows. Let 𝑏𝐴 be theadvantaged bidder's bid in the procurement auction. If the corruption scheme is not practiced(recall that the occurrence of collusion is revealed to the advantaged contractor only after hesubmits the bid) and if the advantaged contractor submits the lowest bid, then the contract isawarded to him. In such cases, the advantaged contractor's vNM utility is 𝑏𝐴 − 𝐶𝐴 . However, if
his bid is not the lowest, then the advantaged contractor's vNM utility is 0.
If the corruption scheme is not practiced and if advantaged contractor submits the lowest
bid, then the government official revises his bid based on the second lowest bid (i.e., thedisadvantaged contractor's bid, 𝑏𝐷 ) with the payment share 1 − 𝜆 (0 ≤ 𝜆 ≤ 1) and the contract
is awarded to the advantaged contractor. In this case, the advantaged contractor's vNM utility is𝜆(𝑏𝐷 − 𝑐𝐴 ).2 Here, it is assumed that the advantaged contractor shares cost-related information
with the government officials.
The vNM function for the advantaged contractor (𝑖 = 𝐴 ) is given by𝑏𝐴 − 𝑐𝐴
if 𝑏𝐷 > 𝑏𝐴
1
𝜋𝐴𝐹𝑎𝑖𝑟 (𝑐𝐴 , 𝑏𝐴 , 𝑏𝐷 ) = { (𝑏𝐴 − 𝑐𝐴 ) if 𝑏𝐷 = 𝑏𝐴
20
otherwise.

(1)

and

2

Here I assume that is exogenously determined. Usually, the contractor does not necessarily report to thegovernment official its true cost

. In some literature,

is determined in a bargaining game to maximize the joint

surplus of the advantaged bidder and the government officials. However, I do not focus on this aspect.

6

𝐶𝑜𝑟𝑟𝑢𝑝𝑡

𝜋𝐴(𝑐𝐴 , 𝑏𝐴 , 𝑏𝐷 ) = {𝜆(𝑏𝐷 − 𝑐𝐴 ) if 𝑏𝐴 ≥ 𝑐𝐷
0otherwise(2)

whereas that of the disadvantaged contractor (i = D ) is given by(𝑏𝐷 − 𝑐𝐷 )1
𝜋𝐷𝐹𝑎𝑖𝑟 (𝑐𝐷 , 𝑏𝐴 , 𝑏𝐷 ) = { (𝑏𝐷 − 𝑐𝐷 )20otherwise𝑏 − 𝑐𝐷
𝜋𝐷𝐶𝑜𝑟𝑟𝑢𝑝𝑡 (𝑐𝐷 , 𝑏𝐴 , 𝑏𝐷 ) = { 𝐷0

if 𝑐𝐴 > 𝑏𝐷
otherwise

if 𝑏𝐴 > 𝑏𝐷
if 𝑏𝐴 = 𝑏𝐷

(3)

and
(4)

Expected Profit
The expected profit functions are as follows. The advantaged contractor’s expected profit isgiven byΠ𝐴 (𝑏𝐴 , |𝑏𝐷 , 𝑐𝐴 , 𝑐𝐷 ) = 𝑝(𝑏𝐴 − 𝑐𝐴 )[1 − 𝐹(𝛽𝐷−1 (𝑏𝐴 ))]−1 (𝑏 )𝛽𝐷𝐴+(1 − 𝑝)𝜆 [∫𝑐̅(𝛽𝐷 (𝑥) − 𝑐𝐴 )𝑑𝑥 + ∫(5)−1 (𝑏 )𝛽𝐷𝐴𝑐𝐴

The first term starting with(𝛽𝐷 (𝑥) − 𝑐𝐴 )𝑑𝑥]

is the expected profit when the corruption scheme is not practiced.

Then, the auction is a fair ordinary first-price sealed-bid procurement auction, and theadvantaged contractor wins the auction with a probability of 1 − 𝐹(𝛽𝐷−1 (𝑏𝐴 )). The second termis the expected profit when the planned corruption scheme is implemented. The first term insquare brackets is the expected profit when the advantaged contractor successfully submits thelowest bid, and the government official revises 𝑏𝐴 to the level of the disadvantaged contractor’s
bid 𝑏𝐷 so that the advantaged contractor has the greater surplus. The second term in square
brackets is the expected profit when the advantaged contractor fails to submit the lowest bid, andthe government official revises it to 𝑏𝐷 if 𝑏𝐷 ≥ 𝑐𝐴 . The disadvantaged contractor’s expected
profit is given byΠ𝐷 (𝑏𝐷 , |𝑏𝐴 , 𝑐𝐴 , 𝑐𝐷 ) = 𝑝(𝑏𝐷 − 𝑐𝐷 )[1 − 𝐹(𝛽𝐷−1 (𝑏𝐴 ))] + (1 − 𝑝)(𝑏𝐷 − 𝑐𝐷 )(1 − 𝐹(𝑏𝐷 ))7(6)

The first term is the expected profit when the scheme is not practiced, whereas the second term isthe profit when the scheme is practiced. In such a case, the disadvantaged contractor must matchthe advantaged contractor’s cost rather than his bid. Therefore, the probability of winning is
1-F(bD ) . Based on these settings, I obtain the Bayesian Nash equilibrium of this game.

4. Equilibrium Analysis
In this section, I determine the equilibrium in this game and provide the implications of
the factors at play. Because contractors have an identical cost distribution but exhibit different
utility functions, the strategies in equilibrium are expected to be asymmetrical. Because proving
the existence and uniqueness of the Bayesian Nash equilibrium is not the focus of this game, I
omit the proof here and simply note the conditions for the game.3 However, one can reference
the proof presented in Lebrun for more information (1995).
The corruption scheme is practiced with the probability one (p=0) and zero (p=1) Whenthe corruption scheme is practiced with the probability of one (p=0), the advantaged contractor
submits any bid 𝑏A ∈ ℝ+ because this value is changed by the government official to 𝑏𝐷 if 𝑏𝐷 ≥𝑐𝐴 , which means that disadvantaged contractor chooses his strategy as if he faced his opponent’s
hypothetical bid function𝛽𝐴∗ (𝑐𝐴 |𝑝 = 1) = 𝑐𝐴
Therefore, the disadvantaged contractor’s bid function in equilibrium is𝛽𝐷∗ (𝑐𝐷 |𝑝 = 1) = 𝑐𝐷

(7)

(8)

In this situation, efficient project allocation is achieved. The implication is that the disadvantaged
contractor faces the fiercest competition possible. Eventually, the government successfullycollects the information on both the advantaged and the disadvantaged contractor, although the
project price achieved is not always the lowest (i.e., 𝑝𝑟𝑖𝑐𝑒(A wins|𝑝 = 1) ≥ 𝑐𝐴 , and𝑝𝑟𝑖𝑐𝑒(D wins|𝑝 = 1) = 𝑐𝐷 ).
When the probability that the corruption scheme will be executed is zero (i.e., p=1, whichindicates a fair auction), the advantaged and disadvantaged contractor have symmetric bid
functions that satisfy the following first-order conditions.
3

The proof is available upon request.

8

1 − 𝐹 (𝛽𝑗−1 (𝑏𝑖 )) = (𝑏𝑖 − 𝑐𝑖 )𝑓 (𝛽𝑗−1 (𝑏𝑖 ))𝑑 𝛽𝑗−1 (𝑏𝑖 )
𝑑𝑏𝑖

(9)

Again, in this case, there is no inefficiency in project allocation. The probability that thecorruption scheme will be practiced is p (0<p<1) Given, the advantaged contractor’s

expected profit maximization problem is given by𝑐̅̅−1(𝛽𝐷 − 𝑐𝐴 )𝑑𝐹(𝑐)max 𝑝(𝑏𝐴 − 𝑐𝐴 )[1 − 𝐹(𝛽𝐷 (𝑏𝐴 )] + (1 − 𝑝)𝜆 ∫𝑏𝐴−1 (𝑏 )𝛽𝐷𝐴

(10)The disadvantaged contractor’s expected profit maximization problem is given by
max 𝑝(𝑏𝐷 − 𝑐𝐷 )[1 − 𝐹(𝛽𝐷−1 (𝑏𝐴 )] + (1 − 𝑝)(𝑏𝐷 − 𝑐𝐷 )(1 − 𝐹(𝑏𝐷 ))
𝑏𝐷

(11)

Maximization problems (6) and (7) imply the following first-order conditions.
1 − 𝐹(𝛽𝐷−1 (𝑏𝐴 )) = (𝑏𝐴 − 𝑐𝐴 )𝑓(𝛽𝐷−1 (𝑏𝐴 ))1 − 𝐹(𝜙𝐷 (𝑏𝐴 )) 𝑑𝛽𝐷−1 (𝑏𝐴 )𝑑𝑏𝐴(𝑏𝐴 − 𝜙𝐴 (𝑏𝐴 ))

(12)

and
𝑝[1 − 𝐹(𝛽𝐴−1 (𝑏𝐷 ))] + (1 − 𝑝)(1 − 𝐹(𝑏𝐷 ))= 𝑝 [(𝑏𝐷 − 𝑐𝐷 )𝑓(𝛽𝐴−1 (𝑏𝐷 ))𝑑 𝛽𝐴−1 (𝑏𝐷 )] + (1 − 𝑝)[(𝑏𝐷 − 𝑐𝐷 )𝑓(𝑏𝐷 )]
𝑑𝑏𝐷

(13)

Equation (12) indicates a marginal condition under a fair sealed-bid first-price procurement
auction, which implies that the advantaged contractor bids as if he were participating in a fairprocurement auction as long as there is a strictly positive probability that the corruption scheme
is practiced. Remember that if p=0 (i.e., the corruption scheme is always implemented), theadvantaged contractor’s optimal choice of bid is any real number that is greater than his costs
because the advantaged contractor wins the auction if the disadvantaged contractor’s bid ishigher than the advantaged contractor’s costs. In such a case, consequently, the disadvantaged
contractor needs to behave as if she were facing a sequential auction in which the advantaged
contractor would submit his bid after the disadvantaged contractor submitted her bid. However,once the probability of a fair auction becomes strictly positive (p>0), the advantaged contractor
9begins to consider the possibility that the scheme will not be implemented, and the current
procurement auction becomes a fair sealed-bid first-price auction in which the advantaged
contractor needs to bid simultaneously with the disadvantaged contractor. The portion of theadvantaged contractor's payments to the government official has no impact on the advantaged
contractor’s marginal decision.
By adding two boundary conditions that are common in competitive procurement
auctions, I derive the following necessary conditions of the Bayesian Nash equilibrium for thisgame.
Proposition 1. Given the assumptions (A1) and (A2), a pair of pure strategies (β*A , β*D ) in theBayesian Nash equilibrium for this game satisfies the following conditions.
1. ∀𝑖 ∈ {𝐴, 𝐷}, 𝜙𝑖 (𝑐̅) = 𝑐̅
2. ∀𝑖 ∈ {𝐴, 𝐷}, ∃𝑏 ∈ ℝ such that 𝜙𝑖 (𝑏) = c
3. The functions 𝜙𝐴 (⋅) ≡ 𝛽𝐴−1 and 𝜙𝐷 (⋅) ≡ 𝛽𝐷−1 are the solutions of the system of ordinary
differential equations of𝜙𝐷′ (𝑏𝐴 ) =𝜙𝐴′ (𝑏𝐷 ) =1 − 𝐹(𝜙𝐷 (𝑏𝐴 ))(𝑏𝐴 − 𝜙𝐴 (𝑏𝐴 ))(1 − 𝑝)[1 − 𝐹(𝑏𝐷 ) − (𝑏𝐷 − 𝜙𝐷 (𝑏𝐷 ))𝑓(𝑏𝐷 )] + 𝑝[1 − 𝐹(𝜙𝐴 (𝑏𝐷 ))]𝑝(𝑏𝐷 − 𝜙𝐷 (𝑏𝐷 ))𝑓(𝜙𝐴 (𝑏𝐷 ))

(14)

(15)

That corruption schemes become more likely to be practiced indicates that thedisadvantaged contractor has a higher chance of competing with the advantaged contractor interms of cost, as that figure is lower than the advantaged contractor’s bid. To match his
opponent’s cost, the disadvantaged contractor shifts his bidding schedule downward, and hislowest bid

decreases to the lowest cost . However, the advantaged contractor considers theother side of the situation. If it is more likely that the government official will implement the10corruption plan, then the advantaged contractor will experience softer price competition, and thus,
he shifts the bidding schedule upward. At the same time, however, the advantaged contractor
knows that it is still possible that the plan will not be implemented and that he may therefore face
more aggressive bidding behavior on the part of the disadvantaged contractor under a fair auction.
As a result, the advantaged bidder needs to decrease the cost of his lowest bid

to his lowest

possible cost . Assuming these conditions and using a numerical approach developed by Bajari(2001), I conduct the computations in the next section.

5. Computations
Methodology
Because I cannot derive closed-form inverse bid functions 𝜙𝐴 (𝑏𝐴 ) and 𝜙𝐷 (𝑏𝐷 ) , I
numerically derive them using the computation algorithm developed by Bajari (2001). Bajari’s(2001) algorithm allows us to approximate inverse bid functions using a flexible functional form
(e.g., a high-order polynomial) and find a set of coefficients that approximately satisfy the firstorder conditions and the boundary conditions.
First, define the distribution of costs to be truncated normal with support [c, c]. . The
probability density function for bidder 𝑖 ∈ {𝐴, 𝐷} is as follows.
1−(𝑐𝑖 −𝜇)2exp[] ∀𝑐𝑖 ∈ [𝑐, 𝑐]𝑓(𝑐𝑖 ) = {2𝜋𝜎 22𝜎20otherwise.

(16)

Next, I assume that the form of the inverse bid function for a contractor 𝑖 ∈ {𝐴, 𝐷} takes the
fourth order polynomial:
4

𝜙𝐴 (𝑏𝐴 ; 𝑏, 𝛾𝐴 ) = 𝑏 + ∑ 𝛾𝐴,𝑘 (𝑏𝐴 − 𝑏)
𝑘=0

11

𝑘

(17)

4

𝜙𝐷 (𝑏𝐷 ; 𝑏, 𝛾𝐷 ) = 𝑏 + ∑ 𝛾𝐷,𝑘 (𝑏𝐷 − 𝑏)

𝑘

(18)

𝑘=0

4

4

where 𝛾𝐴 ≡ {𝛾𝐴,𝑘 }𝑘=0 and 𝛾𝐷 ≡ {𝛾𝐷,𝑘 }𝑘=0 . In equilibrium, the inverse bid functions 𝜙𝐴 (𝑏𝐴 ) and𝜙𝐷 (𝑏𝐷 ) satisfy:
1−

1−𝑓(𝜙𝐷 (𝑏𝐴 ))𝜙𝐷′ (𝑏𝐷 )(𝑏𝐴 − 𝜙𝐴 (𝑏𝐴 ))=0
1 − 𝐹(𝜙𝐷 (𝑏𝐷 ))𝑓(𝜙𝐴 (𝑏𝐷 ))𝜙𝐴′ (𝑏𝐷 )(𝑏𝐷 − 𝜙𝐷 (𝑏𝐷 ))(1 − 𝑝){(1 − 𝐹(𝑏𝐷 )) − (𝑏𝐷 − 𝜙𝐷 (𝑏𝐷 ))𝑓(𝑏𝐷 )} + 𝑝𝐹(𝜙𝐴 (𝑏𝐷 ))

(19)

= 0.

(20)

Now, define the following functions 𝐺𝐴 (𝑏𝐴 ; 𝑏, 𝛾𝐴 , 𝛾𝐷 ) and 𝐺𝐷 (𝑏𝐷 ; 𝑏, 𝛾𝐴 , 𝛾𝐷 ):𝐺𝐴 (𝑏𝐴 ; 𝑏, 𝛾𝐴 , 𝛾𝐷 ) ≡ 1 −𝑓 (𝜙𝐷 (𝑏𝐴 ; 𝑏, 𝛾𝐷 )) 𝜙𝐷′ (𝑏𝐴 ; 𝑏, 𝛾𝐷 )(𝑏𝐴 − 𝜙𝐴 (𝑏𝐴 ; 𝑏, 𝛾𝐴 ))1 − 𝐹(𝜙𝐷 (𝑏𝐴 ; 𝑏, 𝛾𝐷 ))(21)

𝐺𝐷 (𝑏𝐷 ; 𝑏, 𝛾𝐴 , 𝛾𝐷 )≡ 1−

𝑓 (𝜙𝐴 (𝑏𝐷 ; 𝑏, 𝛾𝐴 )) 𝜙𝐴′ (𝑏𝐷 ; 𝑏, 𝛾𝐴 )(𝑏𝐷 − 𝜙𝐷 𝑏𝐷 ; 𝑏, 𝛾𝐴 ))(22)(1 − 𝑝) {(1 − 𝐹(𝑏𝐷 )) − (𝑏𝐷 − 𝜙𝐷 (𝑏𝐷 ; 𝑏, 𝛾𝐴 )) 𝑓(𝑏𝐷 )} + 𝑝𝐹(𝜙𝐴 (𝑏𝐷 ; 𝑏, 𝛾𝐴 ))These functions correspond to the left-hand side of equations (18) and (19), where the inverse
bid functions are replaced with approximated inverse bid functions. I evaluate the first-orderconditions on a grid of points that are uniformly spaced on the compact interval. Letting𝑛𝑔𝑟𝑖𝑑 denote the number of points in the grid, I define the m-th point in the grid 𝑏𝑔𝑟𝑖𝑑𝑚 , asfollows
𝑏𝑔𝑟𝑖𝑑𝑚 (𝑏) = 𝑏 +𝑚(𝑐 − 𝑏)𝑛𝑔𝑟𝑖𝑑 + 1

12

(23)

I bind the vectors of coefficients 𝛾𝐴 , and 𝛾𝐷 and the lowest possible bid , thereby allowing theapproximated bid functions to satisfy the first-order conditions at the grid points and theboundary conditions simultaneously. To do so, I define the following function 𝐻(𝑏, 𝛾𝐴 , 𝛾𝐷 ).
𝐻(𝑏, 𝛾𝐴 , 𝛾𝐷 )
𝑛𝑔𝑟𝑖𝑑
2

2

≡ ∑ (𝐺𝐴 (𝑏𝑔𝑟𝑖𝑑𝑘 (𝑏); 𝑏, 𝛾𝐴 , 𝛾𝐷 )) + (𝐺𝐷 (𝑏𝑔𝑟𝑖𝑑𝑘 (𝑏); 𝑏, 𝛾𝐴 , 𝛾𝐷 ))
𝑘=0

(24)
2

2

2

+ (𝑐 − 𝜙𝐴 (𝑏, 𝑏, 𝛾𝐴 )) + (𝑐 − 𝜙𝐷 (𝑏, 𝑏, 𝛾𝐷 )) + (𝑐 − 𝜙𝐴 (𝑐, 𝑏, 𝛾𝐴 ))2+ (𝑐 − 𝜙𝐷 (𝑐, 𝑏, 𝛾𝐷 )) .
If the function 𝐻(𝑏, 𝛾𝐴 , 𝛾𝐷 ) is equal to zero, then the first order conditions are satisfied at the grid
points and the boundary conditions are also satisfied. Since it is hard to find such 𝛾𝐴 , 𝛾𝐷 , and 𝑏, I
find their values by minimizing the value of 𝐻(𝑏, 𝛾𝐴 , 𝛾𝐷 ):min 𝐻(𝑏, 𝛾𝐴 , 𝛾𝐷 ) subject to 𝑐 ≤ 𝑏 ≤ 𝑐𝑏,𝛾𝐴 ,𝛾𝐷(25)

To find the solutions, we can use a standard non-linear least square algorithm
Characteristics of the Equilibrium
I compute the inverse bidding functions using the algorithm introduced above. I let thecost be distributed over [𝑐, 𝑐] = [0,50] and the mean, and I set the standard deviation of theprobability distribution to (𝜇, 𝜎) = (25,10). Furthermore, I set the probability that the corruption
scheme is not implemented at p=0.5. The results are reported in Figure 1. As shown, the bidfunction for both advantaged and disadvantaged contractors is increasing in their costs. Figure 1
also reports that both contractors bid at the same price when their technology is at the highest
level (i.e., when their costs are the lowest) or the lowest level (i.e., when their costs are thehighest). Except in these two cases, the disadvantaged contractor always bids more aggressively
than the advantaged contractor.

13

The disadvantaged contractor has a chance to win the contract only when his bid
profitably matches the advantaged contractor's cost, which is lower than advantaged contractor'sbid. However, the advantaged contractor does not have an incentive to bid lower than the
disadvantaged contractor. Suppose that the advantaged contractor bids more aggressively thanthe disadvantaged contractor. In such cases, he can always profitably deviate from the strategy
without changing the probability of winning both an auction with a fixed outcome and an auction
without a fixed outcome. Therefore, the disadvantaged contractor always bids lower than theadvantaged contractor for the given cost.
Proposition 2: When the probability that the corruption scheme will be implemented increases,
1. The advantaged and disadvantaged contractors bid more aggressively than when they arein a fair auction (p=1) and less aggressively than when they are in a perfectly corrupt
auction (p=0).
2. The disadvantaged contractor bids more aggressively when there is a strictly positive
probability that the corruption scheme will be implemented (0<p<1).
Efficiency in Project Allocation
Figure 2 and Table 1 provide insight into how the possibility of corruption and an
uncommitted government official can cause the contract to be allocated efficiently or
inefficiently. I compare possible combinations of ex-post costs and bids for advantaged anddisadvantaged contractors. In a procurement auction with an uncommitted government official,
there exist two potential inefficiencies with respect to the costs associated with the advantaged
contractor.
Type I inefficiency: If the advantaged contractor has a lower cost than the disadvantaged
contractor, then the contract may be allocated inefficiently when the corruption scheme ispracticed.
14Type II inefficiency: If the disadvantaged contractor has a lower cost than theadvantaged contractor, then the contract may be allocated inefficiently when thecorruption scheme is practiced.

There are four cases that generates either type I or type II efficiency.
̂𝑨 )Case A: 𝒄 ≤ 𝒄̂𝑫 ≤ 𝜷−𝟏
𝑫 (𝒄Suppose that advantaged contractor has the cost of 𝑐𝐴 = 𝑐̂𝐴 and disadvantaged
contractor’s cost 𝑐𝐷 = 𝑐̂𝐷 is in 𝑐 ≤ 𝑐̂𝐷 ≤ 𝛽𝐷−1 (𝑐̂𝐴 ) . Their bids are 𝑏𝐴 = 𝛽𝐴 (𝑐̂𝐴 ) , and 𝑏𝐷 =𝛽𝐷 (𝑐̂𝐷 ) respectively. If the corruption scheme is practiced, disadvantaged bidder wins thecontract since 𝛽𝐴 (𝑐̂𝐴 ) >𝛽𝐷 (𝑐̂𝐷 ) and if the scheme is not practiced, disadvantaged contractor wins
the contract again since 𝛽𝐴 (𝑐̂𝐴 ) >𝛽𝐷 (𝑐̂𝐷 ). Since 𝑐̂𝐴 >𝑐̂𝐷 , the allocation is always efficient.
̂ 𝑨 ) ≤ 𝒄̂𝑫 ≤ 𝑐̂𝐴Case B: 𝜷∗−𝟏
𝑫 (𝒄Suppose that advantaged contractor has the cost of 𝑐𝐴 = 𝑐̂𝐴 and disadvantaged
contractor's cost 𝑐𝐷 = 𝑐̂𝐷 in 𝛽𝐷∗−1 (𝑐̂𝐴 ) ≤ 𝑐̂𝐷 ≤ 𝑐̂𝐴 . Their bids are 𝑏𝐴 = 𝛽 ∗𝐴 (𝑐̂𝐴 ) , and 𝑏𝐷 =𝛽 ∗ 𝐷 (𝑐̂𝐷 ) respectively. If the scheme is practiced, advantaged contractor wins since 𝛽 ∗ 𝐷 (𝑐̂𝐷 ) >𝑐̂𝐴 If the corruption scheme does not occur, disadvantaged contractor wins the contract because𝛽 ∗ 𝐷 (𝑐̂𝐷 ) > 𝛽 ∗𝐴 (𝑐̂ _𝐴) . Since 𝑐̂𝐴 > 𝑐̂𝐷 , the contract allocation is inefficient if the corruption
scheme is not practiced, but inefficient if the scheme is practiced. (Type II inefficiency)∗̂ 𝑨 ))
Case C: 𝒄̂𝑨 ≤ 𝒄̂𝑫 ≤ 𝜷∗−𝟏
𝑫 (𝜷𝑨 (𝒄Suppose that advantaged contractor has the cost of 𝑐𝐴 = 𝑐̂𝑨 and disadvantaged
contractor's cost 𝑐𝐷 = 𝑐̂𝐷 is in 𝑐̂𝐴 ≤ 𝑐̂𝐷 ≤ 𝛽𝐷∗−1 (𝛽𝐴∗ (𝑐̂𝐴 )). . Their bids are 𝑏𝐴 = 𝛽 ∗𝐴 (𝑐̂𝐴 ) , and𝑏𝐷 = 𝛽 ∗ 𝐷 (𝑐̂𝐷 ) respectively. If the corruption scheme is practiced, advantaged contractor wins
since 𝛽𝐷∗ (𝑐̂𝐷 ) > 𝑐̂𝐴 . If the corruption scheme is not practiced, disadvantaged contractor wins thecontract because 𝛽𝐴∗ (𝑐̂𝐴 ) > 𝛽𝐷∗ (𝑐̂𝐷 ). Because 𝑐̂𝐷 > 𝑐̂𝐴 the contract allocation is efficient if thecorruption scheme is practiced, but inefficient if the corruption scheme is not practiced.
15∗̂ 𝑨 )) < 𝒄̂𝐷Case D: 𝜷∗−𝟏
𝑫 (𝜷𝑨 (𝒄Suppose that advantaged contractor has the cost of 𝑐𝐴 = 𝑐̂𝑨 and disadvantaged
contractor's cost 𝑐𝐷 = 𝑐̂𝐷 is in 𝛽𝐷∗−1 (𝛽𝐴∗ (𝑐̂𝐴 )) < 𝑐̂𝐷 . Their bids are 𝑏𝐴 = 𝛽 ∗𝐴 (𝑐̂𝐴 ) , and 𝑏𝐷 =𝛽 ∗ 𝐷 (𝑐̂𝐷 ) respectively. If the corruption scheme occurs, advantaged contractor wins because
𝛽𝐷 (𝑐̂𝐷 ) > 𝑐̂𝐴 . If the corruption scheme does not occur, advantaged contractor wins the contract
because 𝛽 ∗ 𝐷 (𝑐̂𝐷 )𝛽𝐴∗ (𝑐̂𝐴 ). Because 𝑐̂𝐷 > 𝑐̂𝐴 , the contract allocation is always efficient. (Type I
inefficiency)
In Type I inefficiency, inefficient contract allocation occurs not due to corruption but due
to the cautionary behaviors of the disadvantaged contractor, and this is the key result of thisstudy. Because the existence of a corruption scheme in this context is not something that theparticipants in the auction are actively aware of, the contractors may bid more aggressively or
less aggressively depending on whether individual contractors have an illegal contract withgovernment officials. Because the contractors are preparing for possible corruption, inefficient
allocation can occur even when there is no artificial price manipulation by the government. TypeI inefficiency can occur because of the disadvantaged contractor's behavior, which is predicatedon his knowledge that a corruption scheme may be at work.
Type II inefficiency occurs in an auction that includes the right of first refusal. If thecontractor with the right of refusal has the lower-level technology ex-post, then corruption canallow the contractor to bid high enough to have the opportunity to observe the opponent's bid anddetermine whether the opponent will be awarded the contract or not. Because the disadvantaged
contractor will bid less aggressively than when he knows with certainty that a corruption scheme
is in place, the advantaged contractor has more room to win the contract. Type II inefficiency
occurs because the disadvantaged contractor takes cautionary measures, even when corruption
does not occur.
16

6. Comparative Statistics
Let me return to Figure 3 and compare the two sets of curves in Figure 1. These are thebid functions for advantaged and disadvantaged contractors when the probability of corruption is
p=0.05 and p=0.2. An increase in p means that the possibility of corruption is more credible.
The intercepts of the curves, are the bids corresponding to the lowest possible cost 𝑐. When the
chances that the corruption scheme will be implemented increase (i.e., when p=0.2 changes top=0.05), the lowest bids increase from 𝑏(p=0.05) to 𝑏(p=0.2). Recall that the bid functions
increase with cost and that they are continuous. Also remember that both of the contractors’ bids
correspond to the highest cost. Thus, both of the contractors will bid more aggressively as thepossibility that the scheme will be implemented increases. Furthermore, based on Proposition 1,
the magnitude that the advantaged contractor lowers his bid for a given cost is less than theamount by which the disadvantaged contractor lowers her bid.
Expected Price
Expected price of contract in equilibrium is given by
𝐸[𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑐𝑜𝑠𝑡] =−1 (𝑐̂ )𝛽𝐷𝑐

𝛽𝐷 (𝑐) 𝑓(𝑐)𝑑𝑐 + (1 − 𝑝) ∫∫ {𝑝 ∫0−1 (𝑐̂ )𝛽𝐷

0𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐

0

𝑐̂

𝑐̂

𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐 + (1 − 𝑝) ∫+𝑝∫

−1 (𝑐̂ )𝛽𝐷−1 (𝑐̂ )𝛽𝐷−1 (𝛽̂𝛽𝐷
𝐴 (𝑐̂ ))−1 (𝛽̂𝛽𝐷
𝐴 (𝑐̂ ))+𝑝∫𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐 + (1 − 𝑝) ∫

𝑐̂

𝑐̂
𝑐

+𝑝∫

−1 (𝛽 (𝑐̂ ))𝛽𝐷𝐴𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐

𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐𝑐

𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐 + (1 − 𝑝) ∫−1 (𝛽 (𝑐̂ ))
𝛽𝐷
𝐴

17

𝛽𝐷 (𝑐)𝑓(𝑐)𝑑𝑐 } 𝑑𝑐̂(26)

Figure 4 shows the how the relationship between the probability of that the corruption scheme is
not practiced (the case of fair auction) p and the expected price of the contract. The expected costis decreasing in the probability of fair auction. This is because that as the probability of thepractice of corruption scheme increase, the prices corresponding to the possible prices become
higher when the corruption scheme is practiced with higher probabilities than the case with lower
probabilities, although the entire curve of bid function of lower probability lie in below the ones
with lower probabilities.
Winning Probabilities
The advantaged and disadvantaged contractors’ winning probabilities are given by
Pr(𝐴 𝑤𝑖𝑛𝑠. ; 𝑐𝐴 = 𝑐̂𝐴 . 𝛽𝐴∗ , 𝛽𝐷∗ )𝑐̅

(1 − 𝑝) + 𝑝 ∫−1 (𝛽 (𝑐̂ ))
𝛽𝐷
𝐴 𝐴

=

𝑓(𝑐)𝑑𝑐if 𝑐̂𝐴 < 𝑏−1𝛽𝐷(𝛽𝐴 (𝑐̂𝐴 )){(1 − 𝑝) ∫𝑐̅

𝑓(𝑐)𝑑𝑐 + ∫−1 (𝑐̂ )𝛽𝐷𝐴−1 (𝛽 (𝑐̂ ))
𝛽𝐷
𝐴 𝐴

𝑐̅

Pr(𝐷 𝑤𝑖𝑛𝑠. ; 𝑐𝐷 = 𝑐̂𝐷 . 𝛽𝐴∗ , 𝛽𝐷∗ ) = 𝑝 ∫(27)𝑓(𝑐)𝑑𝑐 if 𝑐̂𝐴 ≥ 𝑏𝑐̅

𝑓(𝑐)𝑑𝑐 + (1 − 𝑝) ∫−1 (𝛽 (𝑐̂ ))𝛽𝐴
𝐷 𝐷

𝑓(𝑐)𝑑𝑐(28)

𝛽𝐷 (𝑐̂𝐷 )The first term of equation (27) of the advantaged contractor’s probability of winning comes fromthe cost domain in which the advantaged contractor wins because his costs are lower than thecommon lowest bid among contractors. In this domain, the advantaged contractor is definitely
awarded the contract regardless of his bid and the disadvantaged contractor’s bid if thecorruption scheme is practiced because 𝑏𝐴 = 𝑏𝐷 > 𝑐̂𝐴 ≥ 𝑐.
Figures 4 and 5 show the relationship between each contractor’s probability of winningand the probability of a fair procurement auction. Whereas the advantaged contractor’s
probability of winning has a negative relationship with the probability of a fair auction, thedisadvantaged contractor’s probability of winning has a positive relationship with the probability18

of a fair auction. This is again because

𝑏𝐴 = 𝑏𝐷 ≥ 𝑐. As long as there is a strictly positive

probability that the corruption scheme will be implemented, the disadvantaged contractor runs
the risk of losing the contract even when he has the lower cost 𝑐𝐷 = 𝑐.
Proposition 3:
1.

Expected prices decrease with the probability that the corruption scheme will be executed.

2.

The advantaged contractor’s probability of winning increases and the disadvantaged
contractor’s probability of winning decreases with the probability that the corruption scheme
will be executed. Moreover, the disadvantaged contractor’s probability of winning does notequal one when her cost is the lowest possible.
We can interpret our findings by comparing bid function curves. When the advantaged

contractor considers that the scheme is always practiced (p=0), he submits a bid any value, since
he knows that it is going to be revised by the government. However, once the contractors
perceive that the probability of a fair auction is strictly positive (p>0), the advantaged contractor
faces the possibility that his bid will become public information without being revised by thegovernment officials. Therefore, the advantaged contractor has to submit a meaningful bid that isgreater than his cost 𝑐𝐴 , but because there is still a positive probability that the corruption scheme
will be implemented, he bids less aggressively than the disadvantaged bidder. In contrast, for the
disadvantaged contractor, p>0 indicates weaker price competition than p=0. Thus, thedisadvantaged contractor bids less aggressively than when p=0. However, because he knows that
there is a strictly positive probability that the scheme will be implemented (1-p>0), thedisadvantaged contractor bids more aggressively than the advantaged contractor. As a result, thebid function curves of both contractors are located above the 45-degree line, with thedisadvantaged contractor’s curve located below the advantaged contractor’s bid function curve.

19

As the probability that the corruption scheme will be implemented increases, bothcontractors’ bidding curves shift downwards, as demonstrated by the vertical differences
between the advantaged and disadvantaged contractor bid function curves ceteris paribus. This
movement of bid functions can be interpreted as follows. Let me restate my findings by adding
the positive probability that the (promised) corruption scheme will not be executed.
1. When the procurement auction is fair (p=1) or perfectly corrupt (p=0), project allocation is
not inefficient.
2. When the probability that the corruption scheme is not practiced becomes strictly positive(), two possible types of inefficiencies emerge. One is achieved when the scheme ispracticed, whereas the other is achieved when the scheme is not practiced.
3. As the probability that the corruption scheme will be implemented increases, the verticaldifferences between the advantaged and disadvantaged contractor bid function curves
increases, indicating that project allocation inefficiency can occur within a wider range ofcosts.

Keeping in mind that regulatory agencies detect and punish corruption because it creates
unfair situations, the results presented here become quite interesting. When the government isperfectly corrupt and everybody in the economy knows so, there is no inefficient allocation, andprojects are delivered with the lowest cost, although the convention itself is unfair. If a
government prohibits bribes from contractors to the government, whether such plans are actually
executed may depend on the detection capabilities of the regulatory authorities. However, if thedetection level is low, economic performance could be very low, meaning that inefficiency andhigher project costs will more likely occur, even in an auction without corrupt arrangements.
This provides a rationale for very stringent investigation procedures by third parties such as the20

FBI, which will presumably decrease to very close to zero the probability that corruption
schemes will be carried out. This means that efficient and reasonable project costs become more
likely.

7. Conclusion
Although a great deal of literature has investigated the impact of variations in corruption
schemes on the prices of contracts and the efficiency of contract allocation, these studies havenot taken into account cases in which the probability that corruption schemes will not bepracticed is strictly positive. In this paper, I introduced the idea of a government official who
does not commit to implementing a corruption scheme in the context of sealed-bid, first-priceprocurement auction with identical type distribution and different utility functions. In such asituation, the advantaged contractor bids more aggressively than in a corrupt auction but lessaggressively than in a fair auction. Alternatively, the disadvantaged contractor bids lessaggressively than in a corrupt auction but more aggressively than in a fair auction. The
implications of these different behaviors can be interpreted with respect to cautionary measuresthat can be taken in both situations. I have shown that because of differences in the certainty that
corrupt plans will actually be executed; there can be two types of inefficiency in project
allocation.

21

Figure 1: Advantaged (upper curve) and Disadvantaged (lower curve) Contractors’ Bid
Functions22bA , bD

 A* ĉ A 

ĉ A
b


c


 D*1 (cˆ A )ĉ A  D*1  A* (cˆ A ) 

Figure 2: Efficiency in Project Allocation

23cc A , cDFigure 3: Comparative Statics in p

24

32.5

32

31.5

31

30.5

30
0

0.2

0.4

0.6

Figure 4: Expected Cost of the Project(X: probability of fair auctions (p), Y: expected cost)

25

0.8

1.2
1
0.8

p=0.2
p=0.4

0.6

p=0.6
0.4

p=0.8

0.2

0
0

10

20

30

40

50

Figure 5: Advantaged Contractor’s Probability of Winning(X: Cost for Advantaged Contractor, Y: Probability of Winning)

26

1
0.9
0.8
0.7
0.6

p=0.2

0.5

p=0.4

0.4

p=0.6

0.3

p=0.8

0.2
0.1

0
0

10

20

30

40

50

Figure 6: Disadvantaged Contractors Probability of Winning(X: Cost for Disadvantaged Contractor, Y: Probability of Winning)27

For given 𝑐̂𝐴 ∈ [𝑐, 𝑐̅]Case A. 𝑐 ≤ 𝑐𝐷 ≤ 𝛽𝐷−1 (𝑐̂𝐴 ) < 𝑐̂𝐴Contractor with 𝑐𝐷 always wins and the allocation isefficient.

Case B. 𝛽𝐷−1 (𝑐̂𝐴 ) < 𝑐𝐷 < 𝑐̂𝐴(1) If price manipulation occurs: Contractor with𝑐̂𝐴 wins the allocation is inefficient.

(Type I inefficiency)(2) If price manipulation does not occur: Contractor
with 𝑐𝐷 wins the allocation is efficient

Case C. 𝑐̂𝐴 < 𝑐𝐷 < 𝛽𝐷−1 (𝛽𝐴 (𝑐̂𝐴 ))(Type II inefficiency)(1) If price manipulation occurs: Contractor with 𝑐̂𝐴
wins the allocation is efficient.

(2) If price manipulation does not occur: Contractor
with 𝑐𝐷 wins the allocation is inefficient

Case D. 𝛽𝐷−1 (𝛽𝐴 (𝑐̂𝐴 )) ≤ 𝑐𝐷 ≤ 𝑐̅Contractor with 𝑐̂𝐴 always wins and the allocation isefficient.

Table 1: Efficiency in Allocation28

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