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Procurement Auction with Corruption by Quality

Manipulation

Yangguang Huang∗, Jijun Xia†

April 15, 2015

Abstract

We studied an agency problem in procurement auction where the principal cares

both price and quality. The project that can be delivered at a range of quality level

chosen by the �rms. The quality evaluation job is delegated to the auctioneer and a

collusive auctioneer can manipulate �rm's quality score. The collusive �rm thus has

larger chance to win the contract and the principal's cannot procure at best combi-

nation of price and quality. We construct a model to show the equilibrium and prin-

cipal's optimal procurement mechanism with presence of this corruption by quality

manipulation. Depending on severeness of corruption, the principal pick a mechanism

inducing e�cient �rm to win by giving out more rent, or allow the collusive �rm wins

but procure at lower quality. The model allows us to compare principal's payo� and

e�ciency in two famous procurement auction format: scoring auction and minimum

quality under corruption.

1 Introduction

In procurement of some di�erential product or project, the buyer cares about bothprice and a number of other quality factors. Therefore, most procurement auction are con-ducted as a scoring auction, where the multidimensional objectives of the principal arere�ected in a scoring rule. By the law1 of tender in China, government and state-ownedenterprises must conduct their procurement via open auction at city government's public

∗University of Washington, Box 353330, Seattle, Washington, 98195. (email: [email protected])†Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, China, 200433. (email:

[email protected])1Law of the People's Republic of China on Tenders and Bids. This law entered into force on

Jan 1st, 2000. The original document has link on www.gov.cn. An English version can be found athttp://www.foreignercn.com/.

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resource trading center. The law also requires the auction shall base on a scoring rule con-taining three factors2

score = S(economic factor, technical factor, business factor)

The score function is usually based on linear weights and the �rm with highest score winsthe contract.

Another common way of procurement is called minimum quality. The principal an-nounce a minimum level of quality standard (and other requirement, like experience of thecompany). For all those who meets the standard, their bids are evaluated purely on pricefactor.

Both procurement format involve evaluation of quality factors. Due to the complexityof the project, the principal usually cannot fully observe quality and has to rely on expertsin the industry. Because of the complexity, delegation is necessary. Because of subjectiv-ity of quality evaluation, a natural problem of corruption rises: the procurement agent andsome �rm may collude, and manipulate the quality evaluation score. The result is the con-tract can be granted to less e�cient �rm at a high price and low quality. That's why wesee so many �jerry-built� projects and low quality but expensive government procurementcases. In this paper, we study this particular kind of collusion. To distinct from bid rig-ging problem among bidders, we the problem corruption by quality manipulation betweenthe auctioneer3 and �rm(s).A Motivating Story

The city government (principal, she) plans to build a bridge and cares about both theamount of spending and the quality of the bridge, for example, the number of lifespan,�rmness, material, maximum weight, travel capacity, beauty etc.. The city governmentrepresentative does not know the bridge building industry. In particular, she cannot eval-uate quality of bridge according to the submitted construction plan in the procurementauction. Therefore, she delegates the procurement auction task (or only quality evaluationpart) to an auctioneer (He) who knows the industry and can evaluate quality.

Suppose there are two �rms i = 1, 2 show up in the procurement auction. The auctionrule ask them to submit a price quality combination as their bids. If the auctioneer is hon-est, he will simply report the quality evaluation to the principal and the problem becomeshow to design the auction of two dimensional bid analyzed in Che (1993).

If the auctioneer is corruptible, then he will exert its ability of quality evaluation in

2Economic factor means price. In real practice, it is usually not just �the low the better�. Firm's theeconomic factor score is related to the engineer estimate and other �rm's submitted prices. Technicalfactor is an evaluation on the project proposal or construction plan concerning its technology, qualitystandard, follow-up service etc.. Business factor is an evaluation of the �rm itself, such as reputation,reliability, risk of bankruptcy.

3In reality, the procurement task is managed by a street level bureaucrat or representative within theprincipal's enterprise. Then it is usually delegated to a procurement agency specialized in that industry.Then the auction is conducted in some government supervised trading center. The score evaluation pro-cess is done a committee of experts. So there is several layer of agency between the buyer and the seller.We abstract all of them as one auctioneer.

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exchange for bribery. Suppose he has a collusion relation4 with �rm 1, then he can give ahigh score on the quality �rm 1's construction plan and thus increase its chance to win.Moreover, �rm 1 receives a higher compensation due to the reported high quality, whilethe actual quality delivered is lower. The rent is divided between the auctioneer and thecollusive bidder. This hierarchy agency problem is illustrated in Table 1.

Table 1: Hierarchy Agency in Procurement AuctionRole Main Action Information

Principal buyer set scoring rule price(She) (government, corporation) or minimum quality F (θ), c(q, θ)

Auctioneer procurement agency evaluate score price and quality(He) (specialist, committee of experts) F (θ), c(q, θ)

Firms sellers submit bids type θ(It) (producer, providers, contractor) price quality combination

Contributions

In this paper, we construct a model to analyze this corruption by quality manipulationproblem. This issue is much less studied comparing to bidding ring and bid revision cor-ruption. But we think it happens much more frequently in reality, especially in developingcountries. �Jerry-built� projects, reports of low quality but expensive government procure-ment, court cases of bribery in procurement are prominent phenomena everywhere. Rel-evant economic theoretical model, empirical analysis and corruption detection techniqueneeds to be developed to understand and restrict this corruption problem.

As the �rst step theoretical model, we make the following contribution and �ndings:1. In literature, when collusion relation is endogenous, the model is mostly focusing on

bribery competition and how it is di�erent than market competition. By allowingcollusion relation to be exogenous, we focus on the issue of corruption. The modelclearly de�ne and separate technological advantage and corruption advantage in thecompetition of procurement auction. We show that the equilibrium bids and out-come critically depends on the relative magnitude of these two advantages, both incomplete information and incomplete information case.

2. In literature of favoritism, usually the principal is passive and the auctioneer/agentis the mechanism designer. Hence the principal just passively su�ers a loss when cor-ruption is present. By allowing exogeneity of collusion relation, we take the principalas mechanism designer and treat auctionner as passive. We show that how the prin-cipal designs a optimal mechanism to �ght against corruption, in particular, makingtwo advantages o�set each other. We show that her payo� could be higher with cor-ruption than without in some cases.

4It may come from long run relation, favoritism or other reason outside of the model.

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3. With presence of corruption, the optimal mechanism in both scoring rule auctionand minimum quality have the following feature. When the size of quality manipu-lation is relaltive small, the principal may still design the mechanism so the e�cient�rm wins the contract. But when the size of quality manipulation is large, a mech-anism guarantee the e�cient �rm's winning will give out too much rent. So instead,the principal will choose to procure at lower quality level and allow which ever �rmis collusive to win.

4. Concerning the optimal mechanism, literature consistently predict that the principalshall use a mechanism that under-report her preference on quality. So the e�cient�rm is �Handicapped� and thus its rent is suppressed. In our model, we show that,to �ght against corruption, the principal may over-report her preference on quality.

5. In comparison of scoring auction and minimum quality mechanism, models in liter-ature predict that scoring auction always dominates. We study this issue with pres-ence of corruption and show that in some situation, minimum quality can be betterthan weighted scoring rule. Che (1993) and Asker and Cantillon (2008) show thatscoring auction strictly dominate price-only auction with minimum standards. How-ever, the empirical work by Tran (2009) show the opposite. They use a bribing com-pany's internal data and �nd that switching from scoring auction to minimum qual-ity reduce amount of bribery. In our model, we provide a little theoretical foundationthat why in some cases, minmum quality can dominate scoring auction and reduceroom of corruption.

6. We also construct an explicit model of incomplete information of collusion relation.We show that how mixed strategy equilibrium rises and the outcome become unsure.

1.1 Literature Review

Bidding ring/bid rigging - collusion among bidders

This is the most well-studied form of collusion in auction. It is a collusion among agroup of bidders. They form a cartel, reduce competition and try to win the contract athigher price (lower price if they are buyers). The theoretical works is covered in the text-book by Krishna (2009). Empirical works and collusion detection are studied by Porterand Zona (1993), Bajari and Ye (2003), Marmer (2014), Aryal and Gabrielli (2013).

Auctioneer-bidder corruption by bid revision

To distinct from collusion among bidders like bidding ring, we call the collusion be-tween auctioneer and bidder corruption. Another collusion problem in procurement is theprincipal-agent problem when the principal delegate the auction to an auctioneer. In lit-erature, starting by La�ont and Tirole (1991) and La�ont (1993), they call it favoritism,where they take the auctioneer as mechanism designer. The auctioneer is in favor of a �rmand design a unfair procurement rule, thus the principal's interest is jeopardized.

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In auction setting, auctioneer-bidder collusion is mostly modeled by bid/price revision.This form of collusion is sometimes called �magic-number�, because the auctioneer willchange the collusive �rm's bid in a way that it barely wins the contract. There are severalpaper explore this issue, like Lengwiler and Wolfstetter (2006), Burguet and Perry (2009)and Burguet and Perry (2014). There are empirical works like Büchner et al. (2008), Caiet al. (2013) and collusion detection methods by Ingraham (2005). Tian and Liu (2008)propose another way to model bid revision: they assume the collusive player can submittwo bids.

Auctioneer-bidder corruption by quality manipulation

This format of corruption is done by the auctioneer changing the collusive bidder'squality evaluation report. It �ts better into real procurement auction where quality is amajor concern and evaluating quality is both complex and subjective. It also explain phe-nomenon like jerry-built project, bribery in �fair� auction, low quality but expensive pro-curement etc..

To our knowledge, Burguet and Che (2004) and Celentani and Ganuza (2002) arethe only existing economic papers on this issue. Burguet and Che (2004) treat the collu-sion relationship as a endogenous result of bribery competition. Instead of only biddingfor contract, bidders also bid for collusive relationship with the auctioneer. Essentially, itadds another round of competition. At �rst glance, because the most e�cient �rm typi-cally has the largest advantage of acquiring collusion relationship from bribery. But theyshow that because the ine�cient �rm can use a mixed strategy on bribery competition andprice competition, so the e�cient �rm cannot guarantee its winning in every possibility. Soine�cient allocation arises.

Celentani and Ganuza (2002) treats the collusion relationship formation as a randommatching process. The procurement agency randomly select a �rm to o�er side contract,without knowledge of its type. Then dishonest �rm and other honest �rms compete to-gether in a procurement auction. Once a �rm become collusive, it surely wins the contract.Celentani and Ganuza (2002) then analyze how the principal designs an optimal mecha-nism to minimize her lost.

Collusion relation, exogenous or endogenous?

Burguet and Perry (2007) categorized the formation of collusion relation in two types.Type I corruption: collusion relation is endogenous. So any bidder can choose to bribe,

the auction turns to a bribery auction/competition. Usually, the presence of corrupted auc-tioneer has no e�ect on allocation of good because the bidder with highest value also havestrongest incentive to bribe. Bribery simply results in a transfer of rents from the buyerto the auctioneer. Most corruption literature take this setting, for example Burguet andChe (2004), Lengwiler and Wolfstetter (2006), Burguet and Perry (2009) and Burguet andPerry (2014).

Type I corruption: collusion relation is exogenous. The model speci�es only one sup-plier can bribe the auctioneer, but others cannot. In the actual analysis of auction, collu-sion relation is already �xed by side-contract. Most bidding ring literature take this set-ting, , for example Porter and Zona (1993), Bajari and Ye (2003), Marmer (2014), Aryal

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and Gabrielli (2013). In Krishna (2009) textbook, he discuss how bidding ring (side con-tract) can be supported by incentive within the ring.

In this paper, we take the second framework and assume there is one bidder has an ex-clusive collusive relationship with the auctioneer. They are the only pair among the agentsthat are able to collude. The model can be easily generalized to there is a group of bid-ders have collusive relationship with the auctioneer, and the result won't be signi�cantlydi�erent. This collusion relation may come from some long run relationship, auctioneer'sfavoritism (like prefer a domestic �rm) and a before-auction exclusive side-contract.

Multidimensional / scoring auction

To analyze quality manipulation in procurement auction, we must be studied a mul-tidimensional auction. The literature is well developed by theorist, like Rogerson (1990),Che (1993), Branco (1997), Asker and Cantillon (2008) and Asker and Cantillon (2010).We take the framework developed in Che (1993). Empirical works on multidimensional isstill rare. Lewis and Bajari (2011) explore a high way contract procurement data set of�A+B auction�, where bids are evaluated both by price and time of delivery.

2 Model

For the main model, to focus on the e�ect of corruption, we set up the model withtwo �rms under complete information on cost strucure and collusion relation. The agencystructure is described in Table 1.

2.1 Procurement by Scoring Rule

2.1.1 Scoring rule auction without corruption

Model setup

We start the model by a benchmark case of no corruption5. The principal seeks pro-curement of a project with di�erent level of quality q ∈ [0,∞). With quality q and price p,her payo� is linear

V (q, p) = q − p

The linearity of payo� function does not lose any generality (as long as we allow cost func-tion to be nonlinear) because quality is a subjective measure, so can always rescale q tomake it enter the payo� function linearly. The principal's tool to screen the �rms is by de-signing the scoring rule of the procurement auction. It is a function mapping quality andprice as a score S(q, p) : R2

+ → R. We focus on linear scoring rule

S(q, p) = αq − p, α ≥ 0.

If α = 0, the multi-dimensional auction reduced to a normal procurement auction withcost as each �rm's type. If α = 1, we say the scoring rule is truthful. In general, we expectα ∈ (0, 1). This is the prediction from Che (1993), Asker and Cantillon (2008) and Askerand Cantillon (2010), where they show that it is optimal for the principal to �handicap�

5Either the principal can observe quality, so there is no delegation, or the auctioneer is honest, so healways report the true observed quality.

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the e�cient �rm to achieve higher expected payo�. We also implicitly assume the principalhas the ability to commit to select the winning �rm by this scoring rule and use the reportof the auctioneer to evaluate each �rm's score.

Linearity argument: in reality... S(q, p) = w1q − w2pThere are two �rms characterized by their type as e�ciency parameters θ1 and θ2,

drawn from distribution Fθ. Without loss of generality, take θ1 < θ2 and refer �rm 1 ase�cient and �rm 2 as ine�cient. The cost function c(q, θ) satis�es the following followingthe assumptions6:

A1 - Standard properties: c(0, θ) = 0, cq > 0, cqq > 0, cqθ > 0, cqqθ > 0.A27 - Cost function is steep enough: cqqq > −c2

qq.A3 - The �inverse� of cost function also satis�ed Spence-Mirrlees single crossing prop-

erty: de�ne function q = f(C, θ) where C = c(q, θ), f satis�es fCθ > 0.Under complete information of cost structure, both �rms observes θ1, θ2. Facing the

scoring rule and given its type, each �rm chooses q and p simultaneously as its bid. Each�rm has payo� as

π(q, p) =

{p− c(q, θ), if wins the contract

0, otherwise

The auctioneer's role is passive. He reports submitted quality honestly to the principal.The following timeline describes the game of procurement by scoring rule:

t = 1, each �rm i has its type realized as θi and observe other �rm's type.t = 2, the principal choose a scoring rule by specifying a α and announce it.t = 3, each �rm submit a sealed bid as a price-quality combination (p, q).t = 4, the auctioneer evaluates each �rm's bid according to the scoring rule S(q, p) =

αq − p. The �rm with highest score wins the contract and it deliver the project at qualityq and will be compensated by price p according to its winning bid.

Equilibrium

Given the result in analysis of multidimensional auction, when scoring rule is addi-tively separate in quality and price, then the decision on quality and price can be sepa-rated. The following Lemma is a special case of Che (1993):

Lemma 1: Given S(q, p) = αq − p, it is a weakly dominant strategy for �rmwith type θ to choose quality as

q∗ = arg maxqαq − c(q, θ). (1)

q∗ satis�es �rst-order condition α = cq(q∗, θ) and q∗ is unique because the second-order

condition −cqq < 0 always hold under convex cost function. Treating α as parameter, wedenote the equilibrium quality as q1(α) = arg maxq αq − c(q, θ1) and q2(α) = arg maxq αq −c(q, θ2). They satisfy the following property:

Lemma 2: Equilibrium quality as a function of α has the following properties:

6Our assumptions hold for this parametric cost function: c1 = θ1qγ , c2 = θ2q

γ , γ > 2, θ2 > θ1.7For uniqueness of solution in theorem 2.

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(i) If the principal announces a higher quality weight α, then the optimal qual-ity increases, i.e. q′i(α) > 0.

(ii) The e�cient �rm picks higher quality under the same α and is more re-sponsive to change in α, i.e. q1(α) > q2(α) and q′1(α)− q′2(α) > 0.

(iii) qi(0) = 0.

(iv) De�ne each �rm's cost at producing equilibrium quality as c1(α) ≡ c(q1(α), θ1),c2(α) ≡ c(q2(α), θ2), c1(α) > c2(α).

Proof of Lemma 1:

Suppose �rm i submit (q, p) such that q 6= q∗, due to the monotonicity of scoring rule inboth quality and price, there always exist another bid (q∗, p′) such that αq∗ − p′ = αq − p.Because S(q∗, p′) = S(q, p), so both (q, p) and (q∗, p′) wins the contract with same proba-bility. Note p′ = αq∗ − αq + p, the payo� if the �rm wins the contract has the following re-lation

π(q∗, p′)− π(q, p) = p′ − c(q∗, θ)− p+ c(q, θ)

= αq∗ − αq + p− c(q∗, θ)− p+ c(q, θ)

= αq∗ − c(q∗, θ)− [αq − c(q, θ)] > 0,

by de�nition and uniqueness of q∗. So deviating from q∗ does not change winning probabil-ity but reduce �rm's payo�.Q.E.D.

Proof of Lemma 2:

(i) qi(α) satis�es equation f(α, q) = α − cq(q, θi) = 0. By implicit function theorem and

cqq > 0, q′i(α) = −∂f∂α∂f∂q

= 1cqq(q,θi)

> 0.

(ii) Because cqθ > 0, ∂q∗

∂θ= −

∂f∂θ∂f∂q

= − cqθcqq

< 0. By assumption θ1 < θ2, so q1(α) > q2(α).

Because cqqθ > 0, cqq(q, θ1) < cqq(q, θ2) and

q′1(α)− q′2(α) =1

cqq(q, θ1)− 1

cqq(q, θ2)> 0.

(iii) Because when α = 0, q∗ = arg maxq 0 × q − c(q, θ) = arg maxq {−c(q, θ)}. The mono-tonic increasing function c(q, θ) take minimum at q = 0, so qi(0) = 0.(iv) Recall assumption A1 and A3, cqθ > 0 and fCθ > 0 are the Spence-Mirrlees singlecrossing property.∀α > 0, cq(q, θ) = α. Let C = c1(α), q2(C) = f(C, θ2). fCθ > 0 ⇒ c2(α) < c(q2(C), θ2) =C = c1(α). Hence, c1(α) > c2(α).Q.E.D.

After pinning down quality choice, �rms choose their prices in a way of Bertrand com-petition and the �rm 1 can always wins by slightly out-bidding �rm 2 in its score.

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Theorem 1: . The equilibrium prices are{p1(α) = αq1(α)− αq2(α) + c2(α) = c1(α) + ∆S(α)

p2(α) = c2(α),

where ∆S(α) ≡ αq1(α)− c1(α)− αq2(α) + c2(α) represents �rm 1's rent and wecall it technological advantage (under scoring rule auction). In the equilibrium,∆S(α) ≥ 0, ∆′S(α) > 0 and ∆′′S(α) > 0.

Proof of Theorem 1:

Let ε be some small positive number. Under Bertrand competition, �rm 2 chooses price asits cost p2 = c2(α) in equilibrium. Firm 1 chooses p1(α) = αq1(α) − αq2(α) + c2(α) − ε tomatch �rm 2's score and slightly over-bids it

S1 = αq1(α)− p1(α)

= αq1(α)− [αq1(α)− αq2(α) + c2(α)− ε]= αq2(α) + c2(α) + ε

= αq2(α) + p2(α) + ε > S2

Given p1(α) = αq1(α) − αq2(α) + c2(α) − ε, �rm 2 cannot decrease its price below itsmarginal cost and don't have incentive to increase its price. Ignoring ε for conciseness,p2(α) = c2(α) and p1(α) = αq1(α)− αq2(α) + c2(α) are equilibrium prices.Because in the equilibrium, �rm 1 earns a positive rent, ∆S(α) = p1(α) − c1(α) > 0 . By�rst-order condition in Lemma 1,

∆′S(α) = q1(α) + αq′1(α)− cq(q, θ1)q′1(α)︸ ︷︷ ︸=0

−q2(α)−αq′2(α) + cq(q, θ2)q′2(α)︸ ︷︷ ︸=0

= q1(α)− q2(α) + q′1(α) [α− cq(q, θ1)]︸ ︷︷ ︸=0

−q′2(α) [α− cq(q, θ2)]︸ ︷︷ ︸=0

= q1(α)− q2(α) > 0

∆′′S(α) = q′1(α)− q′2(α) > 0

Q.E.D.We can use the following �gure to illustrate this equilibrium. In quality-price space, q

and αq represents principal's bene�t and �rst term of the score of quality q respectively.Firm's quality choice depends on slope of scoring rule and its cost function. the maximumscore �rm 2 can get is S2 = αq2(α) − c2(α), while �rm 1 matches �rm 2's score to beat�rm 2 by score S1. They are both illustrated by the red part. Hence, in the equilibrium,�rm 1's rent being its technological advantage is represented by ∆S(α) and principal's pay-o� VH(α). When principal is the mechanism designer, he use α to make trade-o� betweenreaching a desired quality level and �rm 1's rent.

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Figure 1: Illustration of Scoring Rule Auction without Corruption

Principal's payo� and optimal scoring rule

With honest auctioneer and quality weight α, at the equilibrium{q1(α), p1(α), q2(α), p2(α)},�rm 1 always wins and the principal's payo� is

VH(α) = q1(α)− p1(α)

= q1(α)− c1(α)−∆S(α) (2)

= (1− α)q1(α) + αq2(α)− c2(α)

Theorem 2: There exist a unique quality weight αH characterizes the optimal(linear) scoring rule which maximizes VH(α) and 0 < αH < 1, i.e. the e�cient�rm is �handicapped8�.

8A similar result is shown in Burguet and Che (2004) section 5.

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Proof of Theorem 2:

αH satis�es the following �rst-order condition:

V ′H(α) = (1− α)q′1(α)− q1(α) + αq′2(α) + q2(α)− cq(q, θ2)q′2(α)

= (1− α)q′1(α)− q1(α) + q2(α) + q′2(α) [α− cq(q, θ2)]︸ ︷︷ ︸=0

= (1− α)q′1(α)− q1(α) + q2(α) = 0,

By Lemma 2 and cqq > 0,

V ′H(0) =1

cqq(q, θ1)− q1(0)︸ ︷︷ ︸

=0

+ q2(0)︸ ︷︷ ︸=0

> 0,

V ′H(1) = 0− q1(1) + q2(1) < 0.

VH is continuous in α, therefore, there exists at lease one αH ∈ (0, 1) satis�es V ′H(α) = 0.So the solution to V ′H(α) = 0 lies within (0, 1). By Lemma 2 (ii), q′1(α)−q′2(α) > 0, q′′i (α) =d( 1cqq(q(α),θi)

)

dα= − cqqq

c2qq× q′i(α) = − cqqq

c2qq× 1

cqq= − cqqq

c3qq. The second-order condition of concavity

is

V ′′H(α) = (1− α)q′′1(α)− q′1(α)− [q′1(α)− q′2(α)]︸ ︷︷ ︸>0

.

< (1− α)q′′1(α)− q′1(α)

= −(1− α)cqqqc3qq

− 1

cqq< 0

⇔ cqqq > −c2qq

1− α

Therefore, a su�cient condition for the second-order condition to hold on α ∈ [0, 1] iscqqq > −c2

qq.Q.E.D.

2.1.2 Scoring rule auction with corruption

Model setup

As we describe in the motivating story, because of the principal lacks the ability toevaluate quality at the stage of determining the procurement winner, so she has to del-egate the job of score evaluation to the auctioneer. The auctioneer is given some discre-tion/manipulation power on quality evaluation. Following Burguet and Che (2004), we

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assume that he can raise9 the collusive �rm's quality score by m. Burguet and Che (2004)show that, if there is no monitoring or the monitoring is not related to m, then the auc-tioneer will always exert full manipulation power. So the result is the collusive �rm's re-ported q on the bid will be changed to q +m.

This form of quality manipulation can re�ect a wide class of corruption:(i) Direct quality manipulation by the auctioneer (Burguet and Che (2004)).(ii) The quality evaluation process is fair, but the quality requirement design by the

auctioneer is biased, so that it is in favor of the collusive �rm's technological factor (Laf-font and Tirole (1991)).

(iii) Both the quality evaluation process and quality requirement are fair. But as thecollusive �rm wins a fair auction, afterward, the auctioneer allows the �rm to deliver aquality lower than the one written on the bid (Celentani and Ganuza (2002)).

We restrict quality manipulation by a �xed number for two reasons. First, the princi-pal receives the project will learn the quality ex post. Due to the complexity of the project,some discrepancy between reality and the evaluation report in procurement is reasonableand can be caused by unintended mistake of the auctioneer. This allowance is of courselimited and the auctioneer don't want to trigger investigation by manipulating quality toomuch. Second, the auctioneer is typically liable if the project went wrong, for example, thebridge collapses. So he does not want to announce the quality too high and the excessiveuse of low quality project reveals problem in the future.

Concerning collusion relation, we assume the principal knows that the auctioneer ismatched with �rm 1 (the e�cient �rm) with probability p and �rm 2 with probability1 − p, but not exactly who he matched. We also assume complete information of collusionrelation among the auctioneer and both �rms: if a �rm knows it has the collusion relation,it knows the other �rm don't, vice versa. This assumption is relaxed in extension sectionon incomplete information and many �rms.

The following timeline describes the game of procurement by scoring rule (S):t = 1, �rm types and become common knowledge among �rms. Collusion relation

realized and become common knowledge among �rms and the auctioneer.t = 2, the principal choose a scoring rule by specifying a α and announce it.t = 3, each �rm submit a sealed bid as a price-quality combination (p, q).t = 4, the auctioneer raises the collusive �rm's quality by m, then evaluate each bids

by scoring rule S(q, p) = αq − p. So the collusive �rm submitting (q, p) bid receives a scoreα(q +m)− p.

Equilibrium

We assume the manipulation power m satis�es m ≤ q2(α) − c2(α), which means thatthe su�cient condition that the principal want the project despite corruption.

Theorem 3: In a scoring rule auction with corruption, �rms still choose equi-librium following (1). The equilibrium prices, outcome and principal's payo�s

9

We assume the auctioneer does not use manipulation power to extort the other �rm, �if you don't payme a bribe, I will reduce your quality score�. When both bribery and extortion exists, the problem becomecomplicated, see Fahad Jacques paper.

12

outcome price principal's payo�Firm 1 �rm 1 wins p1(α) = c1(α) + ∆S(α) + αm q1(α)− p1(α)collusive p2(α) = c2(α)[p]

�rm 1 wins p1(α) = c1(α) + ∆S(α)− αm q1(α)− p1(α)Firm 2 αm ≤ ∆S(α) p2(α) = c2(α)collusive[1− p] �rm 2 wins p1(α) = c1(α) q2(α)− p2(α)

αm > ∆S(α) p2(α) = c2(α)−∆S(α) + αm

are

Note that αm represents the extra score gain by collusive �rm and we call it corruptionadvantage (under scoring rule auction). When the e�cient �rm is collusive, it wins thecontract for sure. When the ine�cient �rm is collusive, the outcome depends on the rela-tive magnitude of technological advantage ∆S and αm.

13

Proof of Theorem 3:

(1) Equilibrium qualitySuppose �rm i is collusive and (q, p) such that q 6= q∗ de�ned by (1), there always existanother bid (q∗, p′) such that αq∗ − p′ + αm = αq − p + αm. Because S(q∗, p′) = S(q, p),so both (q, p) and (q∗, p′) wins the contract with same probability. Following the samemethod in the proof of Lemma 1, can show that π(q∗, p′) − π(q, p) > 0, so q∗ is optimal for�rm i. For the non-collusive �rm, the proof is similar.

(2) If �rm 1 is corruptedLet ε be some small positive number. Under Bertrand competition, �rm 2 chooses price asits cost p2 = c2(α) in equilibrium. Firm 1 chooses p1(α) = c1(α)+∆S(α)+αm−ε to match�rm 2's score and slightly over-bids it

S1 = αq1(α)− p1(α) + αm

= αq1(α) + αm− [c1(α) + ∆S(α) + αm]

= αq1(α) + αm− [αq1(α)− αq2(α) + c2(α) + αm− ε]= αq2(α) + c2(α) + ε

= αq2(α) + p2(α) + ε > S2.

Principal's payo� in this case is q1(α)− p1(α) = q1(α)− c1(α)−∆(α)− αm.

(3) If �rm 2 is corruptedWhen αm ≤ ∆S(α), technological advantage dominates corruption advantage. Firm 2 stillhas no chance to win. It chooses p2 = c2(α) and received raised score S2 = αq2(α)−p2(α)+αm. Firm 1 chooses p1(α) = c1(α) + ∆S(α) − αm − ε to match �rm 2's score and slightlyover-bids it

S1 − S2 = αq1(α)− p1(α) + ε− [αq2(α)− p2(α) + αm]

= αq1(α)− c1(α)−∆S(α) + αm+ ε− αq2(α) + c2(α)− αm= αq1(α)− c1(α)− αq2(α) + c2(α)−∆S(α) + ε

= ε > 0.

Principal's payo� is q1(α)− p1(α) = q1(α)− c1(α)−∆S(α) + αm.When αm > ∆S(α), corruption advantage dominates technological advantage. Firm 1 nowhas no chance to win. It chooses p1 = c1(α) and received score S1 = αq1(α) − p1(α). Firm2 chooses p2(α) = c2(α)−∆S(α) + αm− ε to match �rm 1's score and slightly over-bids it

S2 − S1 = αq2(α)− p2(α) + αm− [αq1(α)− p1(α)]

= αq2(α)− c2(α) + ∆S(α)− αm+ ε+ αm− αq1(α) + c1(α)

= αq2(α)− αq1(α) + c1(α)− c2(α) + ∆S(α) + ε

= ε > 0

Principal's payo� is q2(α)− p2(α) = q1(α)− c2(α) + ∆S(α)− αm.Q.E.D.

14

The principal's payo� at the equilibrium is computed by considering two cases, re-ferred as outcome A and B hereafter. Outcome A: When technological advantage domi-nates corruption advantage, i.e. αm ≤ ∆S(α). Firm 1 wins for sure regardless collusionrelation. By Theorem 3 and (2), we can write

VA(α) = p [q1(α)− c1(α)−∆S(α)− αm] + (1− p) [q1(α)− c1(α)−∆S(α) + αm]

= q1(α)− c1(α)−∆S(α)− (2p− 1)αm

= VH(α)− (2p− 1)αm (3)

Outcome B: When corruption advantage dominates technology advantage, i.e. αm >∆S(α). The collusion relation determines which �rm wins the contract,

VB(α) = p [q1(α)− c1(α)−∆S(α)− αm] + (1− p) [q2(α)− c2(α) + ∆S(α)− αm]

= p [q1(α)− c1(α)−∆S(α)] + (1− p) [q2(α)− c2(α) + ∆S(α)]− αm= pVH(α) + (1− p) [q2(α)− c2(α) + ∆S(α)]− αm. (4)

Given a �xed manipulation power m, the occurrence of case A or B is determined by therelative magnitude of αm and ∆S(α), which in turn depends on the quality weight choiceof the principal. ,. There exists a threshold quality weight α̃ satis�es the following condi-tion:

Lemma 3: De�ne α̃ as the positive10 solution of equation αm = ∆S(α).

(i) De�ne k(α) ≡ ∆S(α)α

. If m < supα∈[0,∞) k(α), then there exists a uniqueα̃ > 0 such that α̃m = ∆S(α̃).

(ii) For α < α̃, αm > ∆S(α); for α ≥ α̃, αm ≤ ∆S(α).

(iii) α̃ is monotonically increasing in m.

Note that when m > supα∈[0,∞) k(α), the manipulation power is so large that the corrup-tion advantage always dominates. In this case, we can treat α̃ =∞.

Proof of Lemma 3:

(i) By Lemma 2 (iv), k′(α) = 1α2 [∆′S(α)−∆S(α)] = 1

α2 [c1(α)− c2(α)] > 0, k(α) is increas-ing. If m < supα∈[0,∞) k(α), by continuity of k(α), there exists a unique α̃ > 0 such thatm = k(α̃)⇔ α̃m = ∆S(α̃).(ii) Following monotonicity of k(α), obviously, for α < α̃, k(α) < m ⇔ αm > ∆S(α); forα ≥ α̃, k(α) ≥ m⇔ αm ≤ ∆S(α).(iii) By identity m− k(α̃) = 0

α̃′(m) = − 1

−k′(α)> 0.

Q.E.D.

10α = 0 always makes αm = ∆S(α), but the principal does not want o induce zero quality

15

With Lemma 3, principal's payo� function V (α) depends on parameter m and herchoice of α,

V (α) =

{VA(α) = VH(α)− (2p− 1)αm , if α ≥ α̃⇔ αm ≤ ∆S(α)

VB(α) = pVH(α) + (1− p) [q2(α)− c2(α) + ∆S(α)]− αm , if α < α̃⇔ αm > ∆S(α).

(5)Given a m, the principal can induce the occurrence of outcome A or B. Because techno-logical advantage is a convex function of α and corruption advantage is linear of α, if theprincipal chooses a large quality weight, �rm 1's technological advantage dominates �rm2's corruption advantage, thus �rm 1 is guaranteed as winner. If the principal chooses asmall quality weight, �rm 2 dominates �rm 1 when it is corrupted.

Optimal scoring rule

Lemma 4: Recall (3) and (8), VA and VB have the following properties

(i) VA(·) has a unique maximum αA.

(ii) VB(·) is not generally concave. Because VB is continuous, we can still de�neits maximum on the compact interval αB ∈ arg maxα∈[0,α̃] VB(α).

(iii) For p > 12, αA < αH and αA decreases in m; for p < 1

2, αA > αH and

increases in m.

(iv) ∀α ∈ [0, α̃), VA(α) > VB(α).

16

Proof of Lemma 4:

(i) The second-order condition for maximization is satis�ed because

V ′A(α) = V ′H(α)− (2p− 1)m = 0

V ′′A(α) = V ′′H(α) < 0.

So there exists a unique maximum αA.(ii)

V ′B(α) = pV ′H(α)− (1− p) [q′2(α)− c′2(α) + ∆′(α)]−mV ′′B(α) = p

[(1− cq(q, θ1)) q′′1(α)− cqq(q, θ1)(q′1(α))2

]+(1− p)

[(1− cq(q, θ2)) q′′2(α)− cqq(q, θ2)(q′2(α))2

]−(1− 2p) [q′1(α)− q′2(α)]︸ ︷︷ ︸

>0

The sign of V ′′B(α) is not deterministic. A numerical example is shown in Appendix B.(iii) For p > 1

2, (2p − 1)m > 0, V ′A(α) < V ′H(α). Because V ′A(αA) = 0, V ′H(αH) = 0 and

both V ′A(α) and V ′H(α) is are decreasing function, hence αA < αH . For p <12, similarly, we

can show that αA > αH .By implicit function theorem and identity V ′A(αA) = V ′H(αA)− (2p− 1)m = 0

dαAdm

=2p− 1

V ′′H(α)

{< 0, if p > 1

2

> 0, if p < 12

.

(iv)

VA(α)− VB(α)

= VH(α)− (2p− 1)αm− pVH(α)− (1− p) [q2(α)− c2(α) + ∆S(α)] + αm

= (1− p) [VH(α)− q2(α) + c2(α)−∆S(α) + 2αm]

= (1− p)

q1(α)− c1(α)− [q2(α)− c2(α)]︸ ︷︷ ︸>0 by ∆S(α)>0

+2 [αm−∆S(α)]︸ ︷︷ ︸>0 for α<α̃

> 0.

Q.E.D.Intuitively, when e�cient �rm is more likely to be corrupted, choose a low α to reduce

both technological advantage and corruption advantage. When e�cient �rm is less likelyto be corrupted, choose a high α to possibly use �rm 2's corruption advantage o�setting�rm 1's technological advantage.

Theorem 4: De�ne m̄S as the solution11 of α̃(m) = αA(m). The optimalchoice of quality weight α∗ in scoring rule auction is described below:

11α̃ and αA are de�ned in Lemma 3 and Lemma 5. More rigorously, de�ne m̄S as m̄S ={m0 such that α̃(m0) = αA(m0)

∞ if αA(m) > α̃(m), ∀m.

17

(i) m ≤ m̄S, α∗ = max{α̃, αA} [diagram (a)].

(ii) If m > m̄S, there exists a unique α̂ > αA such that VA(α̂) = VB(αB).

If α̃ ≤ α̂, α∗ = α̃ [diagram (b)].

If α̃ > α̂, α∗ = αB [diagram (c)].

(iii) The relative magnitude of α̃ and α̂ depends on parameter m and p.12

Figure 2: Principal's Payo�s under Scoring Rule

Proof of Theorem 4:

(1) If m ≤ m̄S, α̃ ≤ αA, the maximum of VA can be achieved. In Lemma 4, ∀α ≤α̃, VB(α) ≤ VA(α) < VA(αA). So α∗ = αA.If m > m̄S, α̃ > αA, the maximum of VA can be achieved. Inducing outcome B yields theprincipal at most VB(αB). VB(αB) < VA(αB) ≤ VA(αA). By concavity of VA, VA is decreas-ing on [αA,∞), so there exists a unique α̂ > αA such that VA(α̂) = VB(αB). The optimalα∗ now depends on the relative magnitude of α̃ and α̂.If α̃ ≤ α̂, VA(α̃) ≥ VA(α̂) = VB(αB), then α∗ = α̃.If α̃ > α̂, ∀α ∈ [α̃,∞), VA(α) < VA(α̃) < VA(α̂) = VB(αB), the optimal solution is α∗ = αB.(2) Discussion on α̃ and α̂De�ne m̂S as α̃(m) = α̂(m).Q.E.D.

**We haven't been able to characterize the set of parameter that makes α̃ > α̂, so theprincipal induces outcome B.

Theorem 4 tell us the following intuitive results. When corruption is present, thereare two e�ects: when �rm 1 is corrupted, the principal su�ers a loss; but when �rm 2 iscorrupted, �rm 1's rent is suppressed and the principal bene�ts from it.

If manipulation power is relative small, the second e�ect dominates, the principal choosesa high α to induce outcome A, where �rm 1 always win. In particular, when p < 1

2and m

is small, �rm 1 wins the contract with less rent and the principal's payo� is higher thanno corruption case. In the case that p < 1

2and αA > αH , the optimal quality weight can

exceed 1 . So the principal over-report his quality preference.

12In Appendix B, we show that there is cases when α̃ > α̂ and thus the principal induces outcome B.

18

However, when manipulation power is large, the �rst e�ect dominates and allowing anoncollusive �rm 1 win will require a very high α. Doing so, the rent given up is largerthan the loss from allowing corrupted �rm 2 win. The principal will rather set a low α andinduce outcome B, where the collusive �rm will win.

2.2 Procurement Auction by Minimum Quality

Besides scoring rule, the other focus format of procurement auction is imposing a min-imum quality requirement. Each participating �rms needs to meet a �xed quality level tobe a valid bidder in the auction. Once a �rm passes the level, then the competition is fullydepend on price. Similar quality manipulation problem may arise because the collusive�rm can enter by a lower quality than other �rms.

In the literature, scoring auction and minimum quality are compared without corrup-tion and the prediction is that scoring auction always dominates minimum quality. Weshow that with quality manipulation corruption, this result still holds in outcome A, butmay not hold if the principal induces outcome B.

2.2.1 Minimum quality auction without corruption

Model setup

The principal sets up a minimum quality requirement q. Firms need to meet this qualityq ≥ q. The following timeline describes the game of procurement by minimum quality:

t = 1, �rm types realized and become common knowledge among �rms.t = 2, the principal choose a minimum quality q and announce it.t = 3, both �rms submit their bid (q, p) and the quality is checked by the auctioneert = 4, among �rms that meet minimum quality q, the �rm with lowest price wins the

contract.If the principal commits on only considering price besides minimum quality require-

ment, there no bonus to for �rms to make a higher quality than q. So if a �rm decides tobid, it will pick quality just meet the requirement, i.e. q = q. Therefore, the principal isjust picking procurement quality q as her strategy space. Assume that q is not too large,so the ine�cient �rm 2 is not �shut down�: it makes pro�t given upon winning. The coststructure is assume to be the same as previous models.

Equilibrium and optimal minimum quality

When there is an honest auctioneer, provided the principal sets q, assume both �rms chooseto enter the auction. Denote their costs as c1(q) ≡ c(q, θ1) and c2(q) ≡ c(q, θ2). Becausec1(q) < c2(q), �rm 1 wins for sure and equilibrium prices are

p1(q) = c2(q), p2(q) = c2(q).

The proof is similar to Bertrand competition model and is trivial. Firm 1 earns a rent∆M(q) ≡ c1(q) − c2(q): the technological advantage under minimum quality auction. Prin-

19

outcome price principal's payo�Firm 1 �rm 1 wins p1(q) = c2(q) q −m− c2(q)collusive p2(q) = c2(q)[p]

�rm 1 wins p1(q) = c2(q −m) q − c2(q −m)Firm 2 c1(q) ≤ c2(q −m) p2(q) = c2(q −m)collusive[1− p] �rm 2 wins p1(q) = c1(q) q −m− c1(q)

c1(q) > c2(q −m) p2(q) = c1(q)

cipal's payo� is thenUH(q) = q − c2(q) (6)

The optimal minimum quality qH satis�es �rst-order condition U ′H(q) = 1− c′2(q) = 0. It isunique maximum because U ′′H(q) = −c′′2(q) = −cqq(q, θ2) < 0.

2.2.2 Minimum quality auction with corruption

Model setup

Keep the same assumption on cost structure, collusion relation and information on them.The game of procurement by minimum quality under corruption is described as:

t = 1, �rm types and become common knowledge among �rms. Collusion relationrealized and become common knowledge among �rms and the auctioneer.

t = 2, the principal choose a minimum quality q and announce it.t = 3, both �rms submit their bid (q, p) and the quality is checked by the auctioneer.

The auctioneer raises the quality evaluation of the collusive �rm so it can pass the mini-mum quality by only producing at q −m.

t = 4, among �rms that meet minimum quality q, the �rm with lowest price wins thecontract.

Similar to the case without corruption, the principal is essentially choosing its procure-ment quality q and she knows if the collusive �rm wins, she will only receive q −m.

Equilibrium

Theorem 5: Given q, with corruption, the non-collusive �rm chooses quality qand collusive �rm chooses quality q −m. The equilibrium prices, outcome andprincipal's payo�s are

The result is similar to scoring rule auction. When the e�cient �rm 1 is collusive, it winsthe contract for sure. When the ine�cient �rm 2 is collusive, the outcome depends on therelative magnitude of �rm 1's technological advantage and �rm 2's corruption advantage.The corruption advantage is cost saved by the collusive �rm 2: c2(q) − c2(q − m). Hence,the relative magnitude of them depends on c1(q) and c2(q −m).

20

Proof of Theorem 5:

The proof is straight forward. When �rm 1 is corrupted, �rm 2 bid at its cost c2(q), �rm 1can match and slightly over-bid it by p1 = c2(q)− ε. The principal thus receives a distortedquality compare to no collusion case: q −m− p2(q) = q −m− c2(q),.When �rm 2 is corrupted, the outcome is determined by relative magnitude of c1(q) andc2(q − m). When c1(q) ≤ c2(q − m), technological advantage still dominates, �rm 1 winsby slightly over-bidding �rm 2 at its cost c2(q − m) − ε. The principal receives the non-distorted quality q produced by �rm 1 at a lower cost, q − c2(q − m). On the other hand,when c1(q) > c2(q−m), corruption advantage now dominates technological advantage, �rm2 wins for sure by slightly over-bidding �rm 2 at its cost c1(q) − ε. The principal receivesthe project from �rm 2 with distorted quality. The payo� is then q −m− c1(q).Q.E.D.The principal's payo� at the equilibrium, similar to scoring rule, can be divided into

two outcomes:(A) When technological advantage dominates corruption advantage, i.e. c1(q) ≤ c2(q −

m). Firm 1 wins for sure regardless collusion relation. By Theorem 5 and (6)

UA(q) = p [q −m− c2(q)] + (1− p) [q − c2(q −m)]

= q − pc2(q) + (1− p)c2(q −m)− pm= q − c2(q) + (1− p)c2(q)− (1− p)c2(q −m)− pm= UH(q) + (1− p) [c2(q)− c2(q −m)]− pm (7)

(B) When corruption advantage dominates technology advantage, i.e. c1(q) > c2(q − m).The collusion relation determines which �rm wins the contract,

UB(q) = p [q −m− c2(q)] + (1− p) [q −m− c1(q)]

= q − pc2(q)− (1− p)c1(q)−m= q − c2(q) + (1− p)c2(q)− (1− p)c1(q)−m= UH(q) + (1− p) [c2(q)− c1(q)]−m (8)

Given a �xed manipulation power m, the occurrence of outcome A or B can be determinedby the relative magnitude of c1(q) and c2(q −m), which in turn depends on the minimumquality choice of the principal. There exists a threshold minimum quality q̃ satis�es thefollowing condition:

Lemma 5: For any q > 0, because c2(q) > c1(q) > 0 and c′2(q) > c′1(q) > 0,there exists a unique t > 0, such that c2(q − t) = c1(q). We de�ne this t as afunction of q.

(i) If m < supq∈[0,∞) t(q), there exists a unique q̃ > 0 such that c1(q̃) = c2(q̃ −m).

(ii) For q ≥ q̃, c1(q) ≤ c2(q −m); for q < q̃, c1(q) > c2(q −m).

21

Figure 3: Cost and Threshold Qualities

(iii) q̃ increases in m. If m = 0, q̃ = 0.

(iv) For q ≥ q̃, c′2(q −m) > c′1(q).

Note that, similar to scoring rule auction, when m > supq∈[0,∞) t(q), the manipulationpower is so large that the corruption advantage always dominates. In this case, we cantreat q̃ =∞.

22

Proof of Lemma 5:

(i) By assumption A3, given that c2(q − t) = c1(q) = C, c′2(q − t) − c′1(q) > 0. So t(q) ismonotonically increasing because

t′(q) =c′2(q − t)− c′1(q)

c′2(q − t)> 0.

By the continuity of t(q), if m < supq∈[0,∞) t(q), then there exists a unique q̃ > 0 such thatt(q̃) = m, and thus c1(q̃) = c2(q̃ −m).(ii) Because t(q) is increasing, for q ≥ q̃, t(q) ≥ t(q̃) = m, c2(q −m) = c2(q − t(q̃)) ≥ c2(q −t(q)) = c1(q); for q < q̃, t(q) < t(q̃), c2(q −m) = c2(q − t(q̃)) < c2(q − t(q)) = c1(q).(iii) By assumption on cost function, at a �xed cost level C = c2(q − m) = c1(q) andc′2(q) > c′1(q). So c′2(q − m) − c′1(q) = c′2(q − t(q)) − c′1(q) > c′2(q) − c′1(q) > 0. Becauseq̃ satis�es identity c1(q)− c2(q −m) = 0,

dq̃

dm=

c′2(q −m)

c′2(q −m)− c′1(q)> 0.

When m = 0, c1(q̃) = c2(q̃) has a unique solution at q̃ = 0.(iv)Q.E.D.

A fuzzy proof:De�ne ψ(q) = c2(q −m)− c1(q)When q = m, limq→m ψ(q) = c2(0) − c1(m) < 0; when q → ∞, limq→∞ ψ(q) = c2(∞) −c1(∞) > 0.ψ′(q) = c′2(q −m)− c′1(q)When q = m, limq→m ψ

′(q) = c′2(0) − c′1(m) < 0; when q → ∞, limq→∞ ψ′(q) = c′2(∞) −

c′1(∞) > 0.

With Lemma 5, principal's payo� function U(q) depends on value of parameter m andher choice of q

U(q) =

{UA(q) = UH(q) + (1− p) [c2(q)− c2(q −m)]− pm , if q ≥ q̃ ⇔ c1(q) ≤ c2(q −m)

UB(q) = UH(q) + (1− p) [c2(q)− c1(q)]−m , if q < q̃ ⇔ c1(q) > c2(q −m).

(9)Intuitively, due to the convexity of cost function, the gap between c1(q) and c2(q) increasesas the minimum quality rises. So when q is large, �rm 1's technological advantage domi-nates �rm 2's corruption advantage, thus guarantee its as winning �rm. When q is small,�rm 2 dominates �rm 1 when it is corrupted. So given13 a m > 0, the principal can inducethe occurrence of outcome A or B by choosing q above or below q̃. Given a higher manip-

13

When there is no corruption (m = 0), U(q) take UA(q) on its whole support and the principal cannot(and won't) induce outcome B.

23

ulation power, q̃ is higher, which means the principal needs to set up a higher q to makesure �rm 1 wins. The optimal choice of q depends on parameter m and p.

Optimal minimum quality

By adjusting q, the principal controls the outcome leading to UA or UB. We �rst discusssome basic property of UA and UB.

Lemma 6 : These two payo� function have the following property:

(i) UA(q) has a unique maximum qA . UB(q) has a unique maximum qB and qBdoes not depend on m.

(ii) qA > qH and qB > qH .

(iii) ∀q ∈ [0, q̃], UA(q) > UB(q).

24

Proof of Lemma 6:

(i) By Lemma 6 (ii), for q < q̃, c2(q −m)− c1(q) < 0

UB(q)− UA(q) = (1− p)

c2(q −m)− c1(q)︸ ︷︷ ︸<0

−m

< 0.

(ii) For UA, the second order condition satis�es because

U ′A(q) = 1− pc′2(q)− (1− p)c′2(q −m)

U ′′A(q) = −p c′′2(q)︸ ︷︷ ︸>0

−(1− p) c′′2(q −m)︸ ︷︷ ︸>0

< 0.

So there exists a unique maximum qA.For UB, the second order condition satis�es because

U ′B(q) = 1− pc′2(q)− (1− p)c′1(q)

U ′′B(q) = −p c′′2(q)︸ ︷︷ ︸>0

−(1− p) c′′1(q)︸ ︷︷ ︸>0

< 0.

So there exists a unique maximum qB. Because the �rst-order condition U′B(q) = 0 does

not involve m, so qB does not depends on m.(iii)

U ′A(q) = U ′H(q) + (1− p) [c′2(q)− c′2(q −m)]︸ ︷︷ ︸>0

= 0,

U ′B(q) = U ′H(q) + (1− p) [c′2(q)− c′1(q)]︸ ︷︷ ︸>0

= 0

Because qH satis�es U ′H(q) = 0, hence qA > qH and qB > qH .Q.E.D.

(iv) ∀q ≥ q̃, there exists a unique q̂ such that for q ≤ q̂, UA(q) ≥ UB(q); q > q̂for UA(q) < UB(q).

(v) q̂ increases in m. For any m, q̂ > q̃.

(iv) For q ≥ q̃, UB(q)−UA(q) = (1− p)

c2(q −m)− c1(q)︸ ︷︷ ︸≥0

−m

, cannot determine the sign

without discussing m. By Lemma 6 (ii) and (iv), for q ≥ q̃, c2(q − m) − c1(q) ≤ 0, c′2(q −m)− c′1(q) > 0. So there exists a unique q̂ such that c2(q−m)− c1(q) ≤ m for q ≤ q̂; c2(q−m)− c1(q) > 0, for q > q̂, which in turn gives the result.(v) By identity c2(q −m)− c1(q)−m = 0

dq̂

dm=

c′2(q −m) + 1

c′2(q −m)− c′1(q)> 0

It is easy to see that when m = 0, q̂ > 0, q̃ = 0. For any m, dq̂dm

> dq̃dm

> 0, so q̂ > q̃.25

The last property implies that the optimal quality choice rises when corruption arises.While in Lemma 5, the optimal quality weight could rise or fall depending on p, the princi-pal's belief of the probability that the e�cient �rm is collusive. With the result in Lemma5 and 6, we reach the following theorem on optimal minimum quality choice.

Theorem 6: De�ne m̄M as the solution14 of q̃(m) = qB. The optimal choice ofminimum quality q∗ in minimum quality auction is describe below:

(i) If m ≤ m̄M , q∗ = max{q̃, qA} [diagram (a)].

(ii) If m > m̄M , there exists a unique q̂ > qA such that UA(q̂) = UB(qB).

If q̃ ≤ q̂, q∗ = max{q̃, qA} [diagram (b)].

If q̃ > q̂, q∗ = qB [diagram (c)].

(iii) The relative magnitude of q̃ and q̂ depends on parameter m and p.15

Figure 4: Principal's Payo�s under Minimum Quality

14q̃ is de�ned in Lemma 6.15In Appendix B, we show that there is cases when q̃ > q̂ and thus the principal induces outcome B.

26

Proof of Theorem 6

(1) qB is �xed and does not depend on m. q̃ rises in m.If m ≤ m̄M , q̃ ≤ qB, by Lemma 5 UB(q̃) < UA(q̃) ≤ UA(qA). The principal will choose qsuch that �rm 1 always win. If q̃ ≤ qA, then can reach the maximum of UA by picking qA.If q̃ > qA, UA is decreasing on [q̃,∞), hence pick q̃. In summary, q∗ = max{q̃, qA}. (Figure)If m > m̄M , q̃ > qB, inducing outcome B can at most get the principal UB(qB) < UA(qB) ≤UA(qA). By concavity of UA, there exists a unique q̂ > qA such that UA(q̂) = UB(qB). q∗

depends on the relative magnitude of q̃ and q̂.If q̃ < q̂, then inducing outcome A yields higher payo�. For the similar argument above,q∗ = max{q̃, qA}. (Figure )If q̃ > q̂, because UA is decreasing, ∀q > q̃, UA is decreasing. So inducing outcome B yieldshigher payo�, q∗ = qB. (Figure )(2) Discussion on relative magnitude of q̃, q̂

By identity UA(q̂(m)) − UB(qB) = 0, dq̃dm

=c′2(q−m)

c′2(q−m)−c′1(q)> 0. Recall from the proof of

Lemma 6, dq̂dm

= 1(1−p)c′2(q−m)+p

> 0. Both q̃ and q̂ are monotonically increasing.

Q.E.D.Theorem 6 has the same structure and implication as theorem 4. If manipulation power

is relative small, the second e�ect dominates, the principal chooses a high q to induce out-come A, where �rm 1 always win. However, when manipulation power is large,allowing anoncollusive �rm 1 win will require a very high q. Doing so, the rent given up is too largethat the principal would rather pick a low q to induce outcome B.

2.3 Scoring rule or minimum quality?

2.3.1 Comparison of principal's payo�

The comparison is based on the expected payo� received by the principal under opti-mal mechanism in scoring rule and minimum quality. When the auctioneer is honest andthere is no corruption, we can show that the principal receive higher expected payo� byusing scoring rule auction than using minimum quality. The basic intuition is, under scor-ing rule, the principal can better suppress the technology advantage of the e�cient �rm.

Theorem 716: Given the same manipulation power m, cost structure and col-lusion relation structure, the principal's payo� from the procurement auction ishigher when she uses optimal scoring rule than optimal minimum quality.

16There is a short proof. When α = 1, then q2(α = 1) = arg maxq q − c2(q) = qH , then VH(α = 1) =q2(α = 1) − c2(q2(α = 1)) = UH(qH). Because αH < 1, VH(αH) > VH(α = 1) = UH(qH), scoring ruledominates.

27

Proof of Theorem 7:

Recall (2) and (6), with honest auctioneer, under scoring rule

VH(αH) = (1− αH)q1(αH) + αq2(αH)− c2(αH).

Under minimum qualityUH(qH) = qH − c2(qH).

To compare these two mechanism, we transfer scoring auction payo�s V from function ofα to a function of q, the actual procurement quality. If �rm 1 wins, his quality is optimalunder weight α is q = q1(α). Because q1(α) in monotonically increasing, can de�ne the

inverse function as α(q) = q−11 (q). By Lemma 2 Note dα(q)

dq> 0 and q2(α) < q1(α), De�ne

q2(α(q)) ≡ q2(q) and we have q2(q) < q.By de�nition of q2(α), it satis�es 2's FOC

α(q) = c′2(q2(q))

Then dα(q)dq

= c′′2(q2(q))dq2(q)dq

, dq2(q)dq

> 0.Principal's payo�

VH(α(q))d= VH(q) = q − c1(q)−∆(α(q))

UH(q) = q − c2(q) = q − c1(q)− [c2(q)− c1(q)]

2nd order Taylor expansion of c2(q) at q2(q)

c2(q) = c2(q2(q)) + c′2(q2(q)) [q − q2(q)] +1

2c′′2(q2(q̌)) [q − q2(q)]2

where q̌ is between q and q2(q).

⇒ c2(q)− c2(q2(q)) = c′2(q2(q)) [q − q2(q)] +1

2c′′2(q2(q̌)) [q − q2(q)]2

= α(q) [q − q2(q)] +1

2c′′2(q2(q̌)) [q − q2(q)]2︸ ︷︷ ︸

+

Technology advantage of �rm 1 under scoring rule and minimum quality

∆S(α(q))d= ∆S(q) = α(q)[q − q2(q)] + c2(q2(q))− c1(q)

∆M(q) = c2(q)− c1(q)

∆S(q)−∆M(q) = α(q)[q − q2(q)] + c2(q2(q))− c2(q)

= −1

2c′′2(q2(q̌)) [q − q2(q)]2 < 0

So, for any induced quality q > 0, we have the relation of technology advantage as∆S(q) < ∆M(q), so VH(q) > UH(q).Q.E.D.

28

Theorem 8: Principal's payo� is higher when she uses optimal scoring rulethan optimal minimum quality when outcome A (�rm 1 wins) occurs.

**(Theorem 8 is not �nished because the occurrence of outcome B haven't beencharacterized by parameter value)

Case 1, �rm 1 is corruptedFollowing the notation above (**)

V (q) = q − α(q)(q − q2(q))− c2(q2(q))− αm

U(q) = q − c2(q)−m

V (q)− U(q) = −α(q)(q − q2(q))− c2(q2(q))− αm+ c2(q) +m

= −α(q)(q − q2(q)) + c2(q)− c2(q2(q))︸ ︷︷ ︸+

+(1− α)m > 0

Hence V (q) > U(q). So when �rm 1 is corrupted, scoring rule always dominates mini-mum quality.

Case 2, �rm 2 is corrupted, but m is small, �rm 1 wins in equilibrium.

m ≤ min{m̂S, m̂M}

V (q) = q − α(q)(q − q2(q))− c2(q2(q)) + αm

UH(q) = q − c2(q −m)

V (q)− U(q) = −α(q)(q − q2(q)) + c2(q −m)− c2(q2(q)) + αmd= g(q,m)

∂g(q,m)

∂q= −α′(q)[q − q2(q)]− α(q) + α(q)q′2(q) + c′2(q −m)− c′2(q2(q))q′2(q) +mα′(q)

= −α′(q)[q − q2(q)−m]− α(q) + c′2(q −m) + [α(q)− c′2(q2(q))]︸ ︷︷ ︸=0

q′2(q)

= −α′(q)[q − q2(q)−m]− α(q) + c′2(q −m)

De�ne q0 as when q0 −m = q2(q0).

When q = q0, c′2(q −m) = c′2(q2(q)) = α(q), then both g(q,m) = 0 and ∂g(q,m)

∂q= 0.

∂2g(q,m)

∂q2= −α′′(q)[q − q2(q)−m] + α′(q)q′2(q) + c′′2(q −m)

????

29

Case 3, �rm 2 is corrupted, m is large, �rm 2 win in equilibrium.

m ≥ max{m̂S, m̂M}

Induce a q to let �rm 2 wins.

q = q2(α)

α(q) = q−12 (q)

q1(q) > q

V (q)d= V (α(q)) = q − c2(q) + ∆(q)− αm

= q − c2(q) + αq1(q)− αq − c1(q1(q)) + c2(q)− αm= q − αq + αq1(q)− c1(q1(q))− αm

U(q) = q −m− c1(q)

V (q)− U(q) = αq1(q)− αq − c1(q1(q)) + c1(q)− αm+m

= α[q1(q)− q] + [c1(q)− c1(q1(q))] + (1− α)m

Also compare e�ciency and probability of corruption happening?Compare the possibility that corruption advantage dominates technology advantagem̄S and m̄M

αm = α(q1 − q2) + (c1(q1)− c2(q2))c1(q) = c2(q −m)

2.3.2 �Optimal� corruption level

When p < 12, that is the e�cient �rm has lower probability to win, the principal's payo� is

maximized at some m > 0. In both scoring rule and minimum quality, we can show that

∂(V (αA(m))

∂m

∣∣∣∣m=0

= · · · > 0.

∂(U(qA(m))

∂m

∣∣∣∣m=0

= · · · > 0.

Intuitively,without corruption, �rm 1 can reap all its technological advantage as rent. Butwith a positive amount of corruption, the principal will design the mechanism so that

because corruption of the other �rm has To the principal, �rm 2 serve as the role ofphony bidder.

arg maxm V (m), m∗ > 0.A positive amount of bribery can increase principal's payo�s.When m is too large, m ≥ m̂S, then there is e�ciency loss. Firm 2 wins in outcome B.

30

2.3.3 Other comparison

**Not �nishedComparison of e�ciency

Compare the probability of the e�cient �rm wins the contract.

Comparison of frequency of corruption

Compare the probability that the collusive �rm wins the contract (Outcome B happens)Besides principal's payo�, the e�ciency and probability of corruption happening is also

of interest.(1− p)(1− F (m))Compare the possibility of having �rm 2 wins.That is compare m̂S and m̂M , when m is above these threshold, VB(αB) and UB(qB) is

induced and �rm 2 has positive probability to win the contract.If m̂S > m̂M , it means minimum quality, more likely �rm 2 can win.Very hard to compare

3 Extensions

**Not �nished

3.1 Incomplete information on Collusion Relation

A desirable model of collusion shall consider the case that both the cost structure andcollusion relation are private information. However, setting up an economic model withboth cost and collusion relation as private information is very di�cult. In the literature,the explicit assumption on incomplete information of collusion relation is usually avoid.Bajari and Ye (2003) assume collusion relation is common knowledge (the identity of �rmsinvolve in the bidding ring is known to all bidders), Aryal and Gabrielli (2013) and Tianet. al. (2008) assume bidders outside the bidding ring is completely unaware of the ring,so they follow their equilibrium strategy without collusion. Other literature (?) assumebidders outside the ring believe there is collusion with a �xed probability.

In one of the extension, we analyze two cases under incomplete information. In the�rst case, we give an explicit structure of incomplete information on collusion relation andkeep cost as common knowledge. In the second case, we allow incomplete information oncost while keeping non-collusive bidders follow their equilibrium strategy without collusion.

In the main model, we assume complete information on both collusion relation andcost structure. Modeling will be very di�cult if we allow two of them become incompleteinformation. So as two extension, we allow collusion relation to be incomplete information�rst in this section. Then in next section, we study the problem of incomplete informationon cost.

It is not trivial to spell out the information structure of each agent's knowledge on thecollusion relation. In modeling bidding ring, to our knowledge, there is no paper explic-

31

outcome price state

�rm 1 wins

p1(α) = c1(α) + ∆S − αm �rm 1 collusiveαm ≤ p1(α) = c1(α) + ∆S(α) + αm �rm 1 not collusive(1− h)∆S(α) p2(α) = c2(α) �rm 2 collusive

p2(α) = c2(α) �rm 2 not collusive�rm 1 wins p1(α) = c1(α) + ∆S(α) + αm �rm 1 collusive

αm > ? p1(α) ∼ fS(p1) �rm 1 not collusive(1− h)∆S(α) ? p2(α) ∼ gS(p2) �rm 2 collusive

�rm 1 wins p2(α) = c2(α) �rm 2 not collusive

itly models the incomplete information structure of collusion relation. Literature take twopaths circumvent the problem. The �rst one is assuming common knowledge on collusionrelation, like Krishna (2009) and Bajari and Ye (2003). The other is assuming that collu-sion is completely unaware to non-collusive bidders, like Aryal and Gabrielli (2013) andTian et al. (2008). Taking a step forward, we explicitly model the information on collusionrelation.

Model setup

The procurement auction game is the same as the counterpart in section 2. There aretwo �rms. Firm 1 is e�cient and �rm 2 is ine�cient. Cost structure are common knowl-edge among �rms. Assume that there is at most one collusive �rm. The auctioneer andthe collusive �rm of course knows their relation, but the principal and non-collusive �rmsdo not know it for sure. Let (1, 0), (0, 1) and (0, 0) denote three state in state space of col-lusion relation, the information structure is described in the following table

state indicator probability of being collusive(t1, t2) principal's belief �rm 1's belief �rm 2's belief

Firm 1 is collusive (1, 0) p1 1 p11−p2

Firm 2 is collusive (0, 1) p2p2

1−p1 1

No collusion (0, 0) 1− p1 − p2 h ≡ 1−p1−p21−p1

1−p1−p21−p2

We restrict p1 + p2 ∈ (0, 1). When p1 + p2 = 0 and p1 + p2 = 1, the model reduced tothe without corruption and with corruption counterparts in section 2.

Equilibrium

If �rm 1 is non-collusive, its belief on the probability of �rm 2 also non-collusive is de-noted as h ≡ 1−p1−p2

1−p1 . Given m, we have the following theorems of the equilibrium of pro-curement auction:

Theorem 9: Under scoring rule S(q, p) = αq − p, equilibrium quality followsqi = arg maxq αq − c(q, θi), i = 1, 2. The equilibrium outcome and prices are

where fS(p1) is some density function with support [c1(α) + h∆S(α), c1(α) +∆S(α)] and gS(p2) is some density function with support [c2(α)−(1−h)∆S(α)+αm, c2(α) + αm].

32

outcome price state

�rm 1 wins

p1(q) = c2(q) �rm 1 collusivec2(q −m) ≥ p1(q) = c2(q −m) �rm 1 not collusivec1(q) + h∆M(q) p2(q) = c2(q −m) �rm 2 collusive

p2(q) = c2(q) �rm 2 not collusive�rm 1 wins p1(q) = c2(q) �rm 1 collusive

c2(q −m) < ? p1(q) ∼ fM(p1) �rm 1 not collusivec1(q) + h∆M(q) ? p2(q) ∼ gM(p2) �rm 2 collusive

�rm 1 wins p2(q) = c2(q) �rm 2 not collusive

Proof of Theorem 9:Q.E.D.

Comments

Theorem 10: Under minimum quality q, equilibrium quality follows qi =arg maxq αq − c(q, θi), i = 1, 2. The equilibrium outcome and prices are

where fM and gM are some density functions with property that

fM(p1) =

{1− 1

p1−c2(q−m)if (1− h)c1(q) + hc2(q) ≤ p1 < c2(q)

1 if p1 = c2(q),

gM(p2) =

{1 + h− 1

p2−c1(q)if (1− h)c1(q) + hc2(q) ≤ p1 < c2(q)

1 if p1 = c2(q).

Proof of Theorem 10:Q.E.D.

CommentsMixed strategyCompare to complete informationComparing scoring rule and minimum quality

3.2 Incomplete information on cost structure

3.3 Many Firms

Go back to the assumption with complete information on both cost and corruption rela-tion

2 �rms case to n �rms caseGeneral prediction: attracting more non-collusive �rm is bene�cial to the principal

33

3.4 Discussion on auctioneer and collusion

3.4.1 Discussion on the auctioneer

Auctioneer's incentive to introduce a �not going to win� collusive �rm 2.What if auctioneer is making scoring rule.

3.4.2 Endogenous manipulation power and bribery

Bribery payment y ≤ πC − πH . y(m)

3.5 Empirical work

4 Conclusion

Empirical work followingExplain exclusion phenomenon. The entry threshold of large procurement auction is

usually set high. Nearly all procurement auction require bidders have certain certi�catesor reaching certain standards. Typically, this kind of document is long and complex. Asa result, there won't be many bidders in a procurement auction. It seems the auctioneerdon't want to maximize entry and competition - as auction theory suggested. The pro-curement law usually requires a minimum number of bidders. (In China, this minimumnumber is 3 for public sector projects)

Appendix

Appendix A - Proofs

Appendix B - Numerical Example

Setup

Cost function c(q, θ) = θqγ. Take e�ciency parameter θ2 = θ > θ1 = 1.c1 = qγ, c2 = θqγ, γ > 2Quality choice under scoring rule auction

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