Ariane Lambert Mogiliansky

Grigory Kosenok
email: gkosenok@nes.ru

Olivier Compte

Barry Ickes

Paul Milgrom

Eric Rasmusen

Larry Samuelson

Public markets tailored for the cartel -Favoritism in procurement auctions

Keywords: auction, collusion, favoritism, procurement auction, collusion, favoritism, procurement JEL: D44, D73, H57

In this paper, we investigate interaction between two …rms, which are engaged in a repeated procurement relationship modelled as a multiple criteria auction, and an auctioneer (a government employee) who has discretion in devising the selection criteria.

A …rst result is that, in a one-shot context, favoritism turns the asymmetric information (private cost) procurement auction into a symmetric information auction (in bribes) for a common value prize. In a repeated setting we show that favoritism substantially facilitates collusion. It increases the gains from collusion and contributes to solving basic implementation problems for a cartel of bidders that operates in a stochastically changing environment. A most simple allocation rule where …rms take turn in winning independently of stochastic government preferences and …rms'costs achieves full cartel e¢ ciency including price, production and design e¢ ciency. In each period the selection criteria is …ne-tailored to the in-turn winner: the "environment" adapts to the cartel. This result holds true when the expected punishment is a …xed cost. When the cost varies with the magnitude of the distortion of the selection criteria (compared with the true government's preferences), favoritism only partially shades the cartel from the environment. We thus …nd that favoritism generally facilitates collusion at a high cost for society. Our analysis suggests some anti-corruption measures that can be e¤ective to curb favoritism and collusion in public markets. It also shows that the rotation of o¢ cials is not one of them.

Introduction

Many cartels operate in a stochastically changing environment. In particular, this is the case of …rms involved in public procurement. The public demand for e.g., construction works typically depends on a number of factors that are di¢ cult to predict. They include social needs, elected representatives'political agenda, internal budget concerns etc... In addition, …rms'technology changes with time. Altogether this implies a signi…cant uncertainty about the pro…tability of future contracts. In face of such an uncertain environment, a cartel of …rms must devise a mechanism that while being responsive to changes does not open up for gaming opportunities. In this paper we claim that favoritism can contribute to solving key problems for a cartel of bidders that operate in a stochastically changing environment. A main motivation for the paper is the mounting body of evidence that collusion and corruption often go hand in hand in public procurement.

In France, practitioners and investigators in courts of accounts, competition authorities, and in the judiciary have long been aware of the close links between collusion and corruption in public procurement. [START_REF] Athey | Collusion and Price Rigidity[END_REF] According to one of the leading Parisian anti-corruption judges, there exists in France, almost not a single case of large stake collusion in public procurement without corruption. [START_REF] Burguet | Competitive Procurement with Corruption[END_REF] Beside empirical motivations, there are theoretical motivations for investigating the links between favoritism and collusion. In particular, a cartel typically faces a tension between the e¢ ciency goal and the need to provide …rms with incentives to reveal private information. A fair amount of attention has been given to the theoretical problems facing a cartel that operates within an imperfectly or privately observable environment. The general results of most relevance to our issue are due to Fudenberg, Levine and Maskin (1994). But their central e¢ ciency result has not always been applicable when approaching more concrete issues. Recently, Athey and al. (2004) show that it can simply be too costly [START_REF] Athey | Collusion and Price Rigidity[END_REF] The testimony of J. C. Mery provides suggestive evidence of those links (Le Monde, September 22 and 23, 2000). J. C. Mery, a City Hall o¢ cial, admitted that for ten years (1985-94) he organized and arbitrated collusion in the allocation of most construction and maintenance contracts for the Paris City Hall. In exchange, …rms were paying bribes used to …nance political parties.

A recent judgment in 'Les Yvelines'(Cour d'Appel de Versaille, January 2002) provides a vivid illustration as well. [START_REF] Burguet | Competitive Procurement with Corruption[END_REF] This judge from the Pole Financier was among other things involved in the investigation of corruption allegations in the procurement of a 4.3 billion euros market for the recontruction of Paris's lycees (see Le Monde April 23 2005). The case resulted in a series of convictions.

for a price cartel to provide the right incentives for …rms to reveal private information about shocks to costs so the optimal mechanism entails price rigidity. [START_REF] Jean | Corruption et Pratiques anti-concurrentielles: une première re ‡exion à partir d'une étude de cas[END_REF] Our analysis is concerned with a cartel of bidders that face both incomplete information about demand i.e., government preferences and asymmetric information about shocks to …rms'costs. Most closely related to our paper is Athey and Bagwell (2001). They study a price cartel that operates on a market with a given demand but under asymmetric information about costs. They show that cartel e¢ ciency (including production and price e¢ ciency) is achievable in a scheme where …rms are rewarded for truthfully reporting high costs by future market shares. Their e¢ ciency result relies on assumptions that secure the existence (with su¢ ciently high probability) of states where …rms have an identical cost structure. [START_REF] Celentani | Competition and Corruption in Procurement[END_REF] In those states utility can be transferred at no cost for the cartel. As a consequence future market shares can be used to provide incentives for revealing private information without relinquishing production or price e¢ ciency. Our main contribution is to show that, in an auction context, corruption can solve the cartel's information revelation problem in a situation characterized both by asymmetric information and stochastic government demand. Full cartel e¢ ciency including, production, price and design e¢ ciency (the contract is …ne-tailored to the cartel) is achievable in a very simple scheme relying on a non-contingent allocation rule so that …rms take turn to win in a predetermined manner. Favoritism e¤ectively shades the cartel from hazards in the environment.

The expected cost of corruption determines the extent of favoritism. This result is established for the case the expected punishment cost is independent of the magnitude of the distortion of government preferences. When the expected punishment varies with that magnitude, favoritism only partially shades the cartel from hazards in the environment. We …nd that favoritism generally exacerbates the social costs of collusion: the selected speci…cation is socially ine¢ ciency and the price paid by the government is higher than in the absence of favoritism.

We model the procurement procedure as a "…rst score auction". Two …rms characterized by a vector of cost parameters compete in scores with o¤ers that include a speci…cation of the project and a price. Public preferences are stochastic. The procedure is administered by an auctioneer who is a government employee. At the beginning of the period the auctioneer, privately observes a signal of public preferences. His duty is to devise and announce a scoring rule that re ‡ects the (current) public preferences. In the absence of favoritism, the procedure selects the socially e¢ cient speci…cation of the project.

The presence of asymmetric information between the government and its auctioneer implies that the auctioneer has some discretion when deciding over the scoring rule. We call favoritism the act of biasing the scoring rule in favor of one of the …rms. Corruption is modelled as an auction-like procedure that takes place before the o¢ cial auction. Firms compete in (menus of) corrupt "deals" including a bribe and a demanded scoring rule. We …nd that with favoritism the procedure selects a non-standard speci…cation of the project. The intuition is that the associated scoring rule induces minimal competition and thus maximal pro…t-if-win in the o¢ cial auction. In the one-shot setting favoritism turns the asymmetric information private cost procurement auction, into a symmetric information auction (in bribes) for a common value prize corresponding to in ‡uence over the design of the contract.

A main intuition here is that corruption works as a revelation mechanism: …rms truthfully reveal their private information to the corrupt agent who uses that information to maximize their rents because that maximizes the bribe he receives.

We then consider a situation where …rms meet repeatedly, each period on a new market (the auctioneers are short-run players). We show that favoritism fully solves the cartel's problems related to stochastic government preferences and privately observable costs. Provided each …rm is e¢ cient at producing some speci…cation of the project, the cartel can earn the maximal income in a scheme that selects the winner independently of the true preferences and of …rms'costs. The intuition is that a corrupt auctioneer has own incentives to …ne-tailor the scoring rule to the in-turn winner. Firms'main concern is to contain competition in bribes.

That is achieved by opting for a …xed in-turn allocation rule which makes any defection from the equilibrium strategies immediately observable.

In an extension we investigate a case where the expected punishment for favoritism is a function of the magnitude of the distortion between the announced scoring rule and the true preferences. We …nd that the central insights from the …xed punishment case carry over. In the stage game competition in bribes does not dissipate all the …rms'rents however. And in the repeated setting the cartel may face a problem due to imperfect public information. For high cost of punishment, the optimal scheme is contingent on the true government preferences which are never observed. The o¢ cial auction outcome is bounded away from full cartel e¢ ciency. For low cost of punishment the pre-determined in-turn allocation rule is optimal and full cartel e¢ ciency obtains.

The equilibrium allocation patterns emerging from the analysis is consistent with empirical …ndings. There exists ample evidence e.g., in developing countries of problems of maintenance of construction objects due to the non-standard design that was selected in the international procurement procedure (see Rose-Ackerman 1999). Evidence from corruption scandals in France also show that the tender winner is the most e¢ cient …rm and that its pro…ts often are larger than the average in the branch (30% contra 5%) as in the case with the court case concerned with the series of constructions contracts in Paris.

A central policy implication is that since collusion and corruption are linked they must investigated conjointly. A second implication of our analysis is that increasing the severeness of punishment can have a real impact on the extent of favoritism. On the other hand the much advocated anti-corruption policy aiming at reducing the time in any particular o¢ ce i.e., making procurement agents short-run players …nds no support in our analysis.

This paper contributes to a growing literature on corruption in auction. [START_REF] Compte | E¢ cacite et Transparence dans les Procédures de Spéci…cation sur les marchés Publics" Direction de la Prévision[END_REF] The auctioneer's abuse of discretion to devise the selection rule has been studied in Che and Burget (2004) in The paper is organized as follows. The model is described in section 2. Section 3 o¤ers an analysis of the one-stage game. In Section 4 we derive our central results. Section 5 proposes an extension to the case with varying punishment cost. Central assumptions are discussed in section 6 where we also suggest policy implications for procurement and control agencies.

The model

In each time period a project is allocated. A project allows for a multiplicity of speci…cations. A speci…cation is a vector q = (q 1 ; :::q k ) where q j represents the level of the j (quality) component. There are two …rms indexed i; i = 1; 2; which are characterized by their cost function

c i q; t i = k X j=1 t ij q 2 j 2
where t ij 2 ; ; j = 1; :::; k is …rm i 0 s cost parameter associated with quality component q j in period t. The vector of cost parameters t i = t i1 ; :::::: t ik is …rm i's private information. In each period there is a new draw of ( 1 ; 2 ) : For the sake of convenience we remove the realizations t i = ( ; ; :::; ); i = 1; 2 from the support and we assume that each …rm has a comparative advantage in at least one component. [START_REF] Compte | Corruption and Competition in Procurement[END_REF] The probability of the realizations left are proportional to the probabilities which we would have if parameters t ij are i.i.d. with prob( t ij = ) = across all i; j and t: 7 The government derives utility from the realization of a project in period t: W q t ; p t ; t = t 1 q t 1 + :::

+ t k q t k p t ; (1)
with t j 0; 8j = 1:::k;

k X j=1 t j = 1; (2)
where p t is the price paid to the …rm that delivers the project and t = t 1 ; :::; t k is a vector of parameters representing the true social preference in period t. A zero value for a component j; t j = 0 is understood as no social value of q j above a minimal level that de…nes a "basic good". The vector t is random with full support k 1 : The government does not know the true t : It hires an auctioneer who privately observes a signal of the true t at the beginning of each period. For simplicity we assume that the signal is fully informative. [START_REF] Fudenberg | The Folk Theorem with Imperfect Public Information[END_REF] The auction rule At the beginning of each period the auctioneer announces a selection criteria which is a function of both price p and quality q = (q 1 ; :::; q k ) : We consider a class of selection criteria [START_REF] Compte | Corruption and Competition in Procurement[END_REF] These restrictions basically imply that we disregard some special cases and focus on the main results.

Where that is relevant we comment on some implications of relaxing the restrictions. [START_REF] Che | Design Competition Through Multidimensional Auctions[END_REF] The precise characterization of the probability is rather complex but its details play no role for our results. [START_REF] Fudenberg | The Folk Theorem with Imperfect Public Information[END_REF] This is a not an assumption crucial to our results.

similar to the government's utility function:

S (q; b ) = s (q; b ) p = k X j=1 b j q j p; k X j=1 b j = 1;
where b is the vector of parameters announced by the auctioneer (see Timing below).

Throughout the paper we refer to b as the "scoring rule" which is a slight abuse of language since the score of an o¤er is determined by its price also according to the selection criterion. The …rms simultaneously submit in a sealed envelop an o¤er including a project speci…cation q t i and a price p i ; i = 1; 2: The contract is awarded to …rm i t whose o¤er maximizes (among submitted o¤ers) the announced selection criteria subject to a "reserve score" normalized to zero:

i t 2 arg max i=1;2 S q t i ; p t i ; b (3)
s:t : S q t i ; p t i ; b 0:

The winner is due to deliver the speci…cation q t i at price p t i . In case of tie in scores the project is awarded to the …rm whose "quality score"( i.e., s (q; b )) is highest. In case of tie in both price and quality the auctioneer randomizes. We refer to this procedure as a First Score Auction (FSA).

The …rm i's per-period pro…t-if-win is

t i = p t i c i q t i ; t i : (4)
Pro…t-if-lose is zero. We assume that when a …rm is indi¤erent between winning with zero pro…t and losing, it chooses (so as) to win. The game is in…nitely repeated with the same two …rms but with a di¤erent auctioneer in each period. The …rms discount future gains with a common factor . Their payo¤ for the whole game is the discounted sum of the per period payo¤s.

Corruption

The auctioneer is opportunistic. He accepts bribes in exchange for announcing a scoring rule i.e., some b . The auctioneer's utility is

U = w + b m;
where w is a wage that we normalize to 0; b is a bribe and m 0 is a term that captures the expected punishment cost including moral and other costs associated with distorting government preferences: [START_REF] Daniel | Collusive Bidder Behavior at Single-Object Second-Price and English Auctions[END_REF] In the basic model expected punishment is a …xed cost. This is consistent with e.g., the French legislation (Code Penal 432-14, 432-11). [START_REF] Steven | Deregulating Federal Procurement Nothing To Fear but Discretion Itself[END_REF] In an extension we consider the special case the expected punishment depends on the magnitude of the distortion of social preferences and where k = 2; so U = b m (b 1 1 ) 2 . Such a model can be relevant when the magnitude of the distortion signi…cantly a¤ects the probability of detection. We discuss these assumptions in section 6.

Corruption is modelled as a procedure whereby the …rms compete in corrupt "deals"

where a deal is an o¤er to pay a bribe in exchange for a speci…c scoring rule. The two …rms simultaneously and secretly submit a menu of deals M i = f( il ; b il ); l = 1; :::; n i g, where n i is (…nite and) freely chosen by …rm i:

The bribe is only paid by the o¢ cial auction's winner if the announced scoring rule corresponds to one he demanded. The assumption here is that the agent has some discretion to intervene after the submission of the o¢ cial o¤ers. He may invalidate some bids appealing to a formal default, or ask for a resubmission appealing to default in the tendering documents. [START_REF] La¤ont | Auction Design and Favoritism[END_REF] As a result the auction procedure does not result in any allocation under the current period.

Although this might in practice occur before the o¢ cial opening, for simplicity we refer to this action as "rejecting the auction's outcome". The key feature is that the agent can take an action that results in zero payo¤s for all. This makes the payment of bribes enforceable which is accounted for as an assumption. Another important consequence is that it also prevents free-riding in corruption which we will show in the analysis.

The stage game

The stage game is de…ned by the following Timing: step 0: Firms privately learn their cost parameters 1 and 2 ;

step 1 : The auctioneer learns ; the …rms submit each its menu of deals M 1 = f(b 1l ; 1l ) ; l = 1; :::; n 1 g and M 2 = f(b 2l ; 2l ) ; l = 1; :::; n 2 g) respectively;

step 2 : The auctioneer makes an announcement b ; b 2 k 1 ;

step 3 : The …rms simultaneously submit their o¤er (q i ; p i ) ; i = 1; 2; We …rst establish a result applying to the First Score Auction described by the Timing above when deleting step 1 from consideration and as we show later applying to any subgame starting from step 2. Throughout the paper we consider equilibria which are subgame perfect.

Lemma 1 The subgame perfect equilibrium o¤ ers of the FSA are characterized by speci…cation e¢ ciency: q i = arg max s (q; b ) c (q; i ).

All proofs are gathered in the appendix.

The result exploits separability between quality and price in the selection criteria. It can be shown that for any o¤er not including the e¢ cient values for the components, we can …nd another o¤er with the same score but that yields a higher expected pro…t.

The result in Lemma 1 greatly simpli…es the forthcoming analysis. Lemma 1 allows us to, at step 3, separate between …rms'o¤er of project speci…cation and their price bid. [START_REF] Lambert-Mogiliansky | Collusive Market-Sharing and Corruption in Procurement[END_REF] The equilibrium values of the components are the e¢ cient ones corresponding to the announcement

q i = arg max s (q; b ) c (q; i ) q ij ij ; c j = c j ij ; i = 1; 2; j = 1:::; k:
When the announcement corresponds to the true government preferences, Lemma 1 implies social e¢ ciency in the speci…cation of the project.

Favoritism

We now proceed to investigate the one-shot game described in Timing above.

Proposition 1 There exists a Perfect Bayes-Nash equilibrium such that 1 2 A similar result can be found in Che (1993). ii. The equilibrium o¤ ers are the competitive o¤ ers relative to the announced scoring rule.

A …rst result is that whatever the true government preferences, favoritism always entails an extreme (single-peaked) scoring rule b = b j = (0; :::; 1 j ; :::0) for some j: [START_REF] Mcafee | Bidding Rings[END_REF] The intuition for the single-peakness result is that the winner's pro…t is maximal when the scoring rule emphasizes a single component for which he has a comparative advantage. Alternatively, a selection rule including a single-peaked scoring rule induces the "weakest possible competitive pressure" among all FSA generated by any selection rule from the relevant class.

The interpretation of this result is that with favoritism the scoring rule tends to drive to a minimum the weight given to most components while emphasizing quite exclusively a component characterized by weak competition in production. [START_REF] Radner | An example of a Repeated Partnership Game with Discounting and with Uniformly Ine¢ cient Equilibria[END_REF] This means that the winning project has a speci…cation that tends to be "non-standard"in the sense of being unusual. We note that the true government preferences have minimal impact on the announced scoring rule. In case of ties in the corruption game, the auctioneer may choose the deal that is most congruent with the true preferences.

Quite remarkably we …nd that …rms'asymmetric information is a minor concern in our context. The intuition is that a corrupt auctioneer has incentives to use that information to devise a scoring rule that maximizes the winning …rm's rents as it also maximizes his bribe. [START_REF]Corruption in Government. Cause, Consequences and Reform[END_REF] Therefore …rms have an incentive to reveal their cost. Yet, the agent's favor is costly and there may exist incentives for …rms to free-ride. Indeed with some probability the two …rms have low cost on the same component. Therefore there is an incentive to let the other …rm demand a scoring rule in exchange for a bribe and undercut its o¢ cial bid to win. Such deviation deprives the auctioneer from the bribe. He has therefore incentives to prevent it which he can do by credibly threatening to reject the auction's result. In the appendix we show that in equilibrium there is no free-riding so …rms' cost structure is fully revealed to the auctioneer in the submitted menus of deals. The …rms infer all relevant information about each other's cost structure from the announced scoring rule. The equilibrium o¤ers are the (unique) competitive Nash equilibrium o¤ers of the symmetric information FSA de…ned by the announced scoring rule (for details see the appendix). The o¤ered speci…cations are e¢ cient relative to b and the equilibrium price is determined by the second score.

Competition for favors drives up the bribe to b = ( )

2

( m) which is the pro…t that yields with a scoring rule that is most favorable to the winner. Since this pro…t is the same for the two …rms, the auctioneer captures the totality of the winning …rm's rents. 16 In the remaining of the paper we assume that m < ( )

2
so the stage game is characterized by favoritism. 17 Note that this feature i.e., that the equilibrium yields zero payo¤ to both …rms makes it a particularly good candidate for serving as a threat equilibrium in the repeated game that we investigate next.

We see that, in e¤ect, favoritism turns the asymmetric information (private cost) auction, into a symmetric information common value auction (in bribes) for a prize. The prize is in ‡uence over the design of the selection rule which has a common value corresponding to the gain when winning the o¢ cial auction with a maximally favorable selection criteria. This gain is common knowledge and identical for both …rms. 18 The social cost of favoritism is twofold.

First, a socially ine¢ cient project speci…cation is selected. Second, the price paid by the government is higher than in the absence of favoritism. The bias in project speci…cation due to favoritism minimizes competition between …rms. The equilibrium depicted in Proposition 1 will serve a threat point in the collusive schemes we study next.

Collusion and Favoritism: A Strategic Complementarity

We now proceed to investigate a situation when the two …rms interact repeatedly. In each period they meet on a public market administered by a new auctioneer, e.g., di¤erent local governments. In each period there is a new draw of t 1 ; t 2 ; t : We are interested in collusion between the two …rms under the assumption that transfers between them are precluded.

Information assumptions: At the end of each period the submitted contract o¤ers are publicly observed by the two …rms and the active auctioneer. The corrupt deal o¤ers remain private information to the involved parties. The true value of is never revealed. Each auctioneer is appointed for one period only and there is no communication between auctioneers from di¤erent periods.

We consider a repetition of the game described in Timing (Section 3). Proposition 2 constitutes the central result of this paper. Proposition 2 i. There exists 1 < 1 and m such that for 1 and m m full cartel e¢ ciency is achievable in a Perfect Bayes-Nash equilibrium of the repeated game.

ii. In the o¢ cial auction …rms take turn in winning independently of government preferences and …rms' costs.

iii. The equilibrium scoring rule is single-peaked (i.e., b = (0; :::; 1; :::0)) and the winning …rm i pays a bribe b i = m:

Full cartel e¢ ciency is de…ned for the o¢ cial auction as follows. i. In each period the winner is (one of) the most e¢ cient …rms relative to the announced selection criteria (productive e¢ ciency); ii. The price paid to the winner is the highest price the government is willing to pay (price e¢ ciency); iii. The selection criteria that applies yields the highest gains to the winning …rm from among all possible selection criteria (design e¢ ciency). Note that the third part of our criteria goes beyond the standard de…nition of cartel e¢ ciency. Proposition 2 establishes that with favoritism full cartel e¢ ciency is achievable in spite of incomplete and asymmetric information. 19 , 20 The cartel needs not adapt to the "environment" i.e., to the current cost structure or to the current government preferences. Instead the environment 1 9 Notice that the …rm that wins …rst has a higher expected discounted payo¤. We can equate the two …rms' discounted payo¤s by randomly designing the …rst winner. 2 0 We here wish to remark that the results in proposition 2 hold true even when …rms have an identical cost structure (a case that we excluded in the model). However full cartel e¢ ciency can obviously not obtain if one of the …rm has high cost on all components (but the other has not).

adapts to the cartel: in each period the auctioneer …ne-tailors the scoring rule to the in-turn winner. The optimal allocation rule is extremely simple: …rms take turn for winning in a noncontingent manner. A main concern for the cartel is to contain competition in bribes which can be very costly as we learned from Proposition 1. Proposition 2 shows that competition for favors is eliminated when opting for a non-contingent in-turn allocation rule. At the corruption stage both …rms o¤er a menu of deals each with a single-peaked scoring rule as in Proposition 1. The out-of-turn …rm o¤ers a zero bribe while the in-turn winner o¤er a bribe that just covers the expected punishment cost m: The out-of-turn …rm may deviate and (unobserved) bribe the auctioneer to announce a scoring rule favorable to itself. This is immediately detected however -the pre-determined in-turn rule is violated -and punished by reverting to the equilibrium of proposition 1 which yields zero payo¤ to both …rms from next period on: This explains why the bribe can be kept to a minimum of m: In the o¢ cial auction the out-of-turn …rm submits an o¤er that scores at most zero. Since contract o¤ers become public information any defection at that stage is detected after the o¢ cial opening and punished similarly.

We thus see that favoritism facilitates collusion in several ways. The gains from collusion are higher than with an honest auctioneer: the scoring rule is …ne-tailored to maximize the winner's pro…t. While the threat payo¤s are lower than in the absence of corruption because competition in bribes dissipates the rents. Most importantly we …nd that favoritism solves key problems for a repeated cartel in a stochastic environment. The auctioneer's selfinterested determination of the scoring rule e¤ectively shades the cartel from ‡uctuations in the pro…tability of projects due to stochastic government preferences and changing costs.

The environment "adapts" to the cartel and ex-post e¢ ciency i.e., e¢ ciency relative to the announced scoring rule, is secured. But this comes at a cost, the bribe.

Finally we note that in the equilibrium of the one-shot game the agent is limited to choosing a scoring rule that minimizes competition i.e., a scoring rule emphasizing a (single) component such that …rms have di¤erentiated technologies (costs). In the repeated setting with collusion, the agent needs not bother about cost di¤erentials. So, in a sense, collusion makes favoritism "easier": the agent's equilibrium choice set is larger.

The impact of the expected punishment cost

In this section we discuss the impact of an increase in the expected cost of punishment on players' incentives, payo¤s and strategies. In the appendix we show that defection from the equilibrium strategies supporting our result in Proposition 2 can be deterred i¤

(1 2 ) 1 2 m 1 2 :
Recalling that all our analysis is performed for m < ( )

2
= N E we know that 1 2 m > 0 and so 1 < 1: A …rst interesting point is that @ 1 @m > 0 which means that the larger the cost of punishment, the more patient …rms must be to be able to sustain collusion with favoritism. This suggests that anti-corruption policy may have a real impact on competition in procurement. But as favoritism becomes more expensive (m grows), there may exist equilibria without favoritism but with collusion that Pareto dominate the equilibrium of proposition 2. In order to investigate that issue we must spend some time on "pure collusion".

Pure collusion in our context means that …rms do not compete with each other but there is no manipulation of the selection criteria which always re ‡ects the true government preferences. We do not go into the details of the pure collusion case. Instead, we shall rely heavily on Athey and Bagwell (2001) who provide a thorough analysis of similar case. We only focus on a few features of relevance to our discussion.

Under pure collusion there is no manipulation of the selection criteria. As a consequence the selection of the winner must be contingent on t 1 ; t 2 ; t i.e., on …rms'cost parameters and on current government preferences in order to secure the cartel's production e¢ ciency goal. Clearly, for any realization of t 1 ; t 2 ; t , the gain from winning the contract is largest for the …rm whose cost structure is most congruent with current government preferences.

Assume …rst that the agent is honest. Under symmetric information about costs First Best can be achieved in a simple mechanism that designates the winner as the …rm with the cost structure most congruent with current government preferences. The designated winner submits a bid that scores exactly 0 and the other …rm a bid that scores less than zero. Any deviation from those strategies is observable and triggers a reversal to a competitive Nash equilibrium. 21 Let E F Bc denote …rms expected First Best collusive payo¤. Appealing to our result in Proposition 1(i) (that establishes the optimality of the single-peaked scoring rule) we know that F Bc < 1 2 where the left-hand-side is the pure collusion pro…t-if-win and the right-hand-side of the inequality is the pro…t-if-win (gross of the bribe) under favoritism (see Proposition 2).

Under asymmetric information things are much more tricky, production e¢ ciency requires a mechanism that induces …rms to truthfully reveal their private information. Athey and Bagwell (2001) show that for some combination of parameters, the cartel's …rst-best is achievable provided there exist states where utility can be transferred at no cost for the cartel.

A mechanism similar to theirs could be devised. A full characterization of such a mechanism is beyond the scope of this section. We just note that a key feature of the mechanism is to exploit states where …rms are equally e¢ cient to secure incentives for truth-telling. 22 In our context such states do also exist i.e., when …rms are equally e¢ cient relative to government preferences: 23 Just as in Athey and Bagwell, cartel e¢ ciency is demanding on parameters so it is likely that the optimal pure collusion mechanism is plagued by some ine¢ ciency so E c < E F Bc where E c is the expected pro…t from pure collusion: 24 But that is not crucial to our argument.

We now return to the case with a corruptible agent. Suppose that m = m t so m changes with time. Assume that in period t; 1 2 m t < E c where E c ( E F Bc ) is the expected payo¤ of pure collusion. Then both (the in-turn and the out-of-turn) …rms would prefer to refrain from bribery if they could collect the expected pure collusion payo¤. In the mechanism of proposition 2 that would not happen however. A violation of the in-turn rule leads to a reversal to the zero payo¤ Nash equilibrium of the stage game. But …rms could make their strategies contingent on m: Namely for m t 1 2 E c ; they play the equilibrium of proposition 2 but for m t > 1 2 E c they play an optimal pure collusion equilibrium.

Note importantly that the presence of a corruptible agent relaxes the (o¤-schedule) incentive constraint compared to the case when the agent is honest. 25 This is because the threat payo¤s 2 2 In our context truth-revelation could be achieved by conditioning …rms's probability to be selected as the winner on their cost announcement.

2 3 Assume that government only values the two …rst components and that …rms have the same cost for those, then they are equally e¢ cient. 2 4 When truth revelation is not achievable costlessly, it may be achieved at some cost in productive e¢ ciency or price e¢ ciency. 2 5 This terminology is borrowed from Athey and Bagwell (2001). The o¤-schedule incentive constraint re ‡ects …rms' incentives to take observable defection steps. As opposed to the in-schedule incentive constraint that are related to incentives to take actions designed for their type. Defection in that respect is not observable.

are the ones corresponding to the Nash equilibrium of the one-stage game with favoritism which yields zero payo¤s.

The discussion above shows that the cost of favoritism to society depends critically on the expected cost of punishment m. For m < 1 2 E c …rms achieve the highest payo¤ in an equilibrium with favoritism. The social cost of favoritism includes a distortion of the selection criteria. The winning project is not the social e¢ cient one and it is paid at the high collusive price. In that equilibrium the higher the cost of punishment the more patient …rms need to be to be able to sustain collusion in equilibrium. For m

1 2
E c the social cost due to the presence of corruptible agent may be limited to facilitating collusion by making it sustainable for lower level of the discount factor (because the "threat payo¤" is then zero).

There is no social cost due to design ine¢ ciency (but we may have production ine¢ ciency i.e., when the First Best is not feasible) and the government pays a lower (collusive) price than in the case of favoritism. Finally we note that further increase in m (up to ( )

2

) has no impact on players'incentives.

Extensions

In this section we extend the analysis by considering the case when the expected punishment for favoritism depends on the magnitude of the distortion of social preferences. We do that in a simpler setting with k = 2 so 1 = and 2 = (1 ) and is uniformly distributed on The time-line of events in the stage game is as follows:

step 0: Firms learn privately their cost parameters 1 and 2 : iii. The contract o¤ ers are the competitive equilibrium o¤ ers relative to the announced scoring rule.

A …rst important result is that the equilibrium scoring rule is single-peaked as in the …xed punishment case. In the appendix we prove this as a lemma. The intuition is that in the corruption game …rms compete in the auctioneer's utility levels. This utility is separable in bribe and expected punishment cost. We show that for any deal with = 2 f0; 1g that achieves a given utility to the auctioneer there exists a deal with 2 f0; 1g that achieves the same utility level but yields a higher expected pro…t for the …rms. We also note that, as in Proposition 1, the o¢ cial auction o¤ers are the (e¢ cient) competitive equilibrium o¤ers relative to the announced scoring rule.

In contrast with earlier results competition for favors does not dissipate all …rms'rents.

The intuition is that contingent punishment costs introduces an asymmetry between …rms:

the …rm whose demanded scoring rule is closer to the true preferences has more bribing power than the other …rm. Firms'incomplete information about the true preferences therefore induces a continuity of the probability to win in the submitted bribe. As a result competition for favor is mitigated.

The results in proposition 3 apply for m ( )

2 : Since ( ) 2
is the competitive pro…t-if-win associated with the most favorable scoring rule, this range covers most interesting cases. 27 Over that range of value for m the pro…t-if-win is simply i = m:

Summing up, with an expected punishment cost that is a function of distortion, the scoring rule always induces a "non-standard" project. What government expenditure concerns there is no advantage in the more sophisticated punishment rule. Finally, because it mitigates competition in favors, some of the rents stays with the …rms. We conclude that in the stage game, the sophisticated punishment scheme o¤ers no advantage from the point of view of social e¢ ciency.

We now consider a repeated version of the game described above. As in the case with kcomponents, the two …rms meet in each period with a new (short-run) auctioneer. At the end of each period, the submitted contracts o¤ers are publicly observed by the two …rms and the active auctioneer. The corrupt deal o¤ers remain private information to the involved parties.

The true value of is never revealed. There is no communication between auctioneers from di¤erent periods.

Proposition 4 i. For 2 2 (0; 1), there exists a Public Perfect Equilibrium equilibrium of the repeated game with collusion in contract o¤ ers and in corruption deals.

ii. For m small a simple pre-determined in-turn allocation rule is optimal while for m large any optimal collusive scheme entails a contingent allocation rule.

A …rst important remark is that collusion in contract o¤ers and in bribes is achievable in a simple pre-determined in-turn scheme at b = m. The reasoning is similar to that in proposition 2. However, for m relatively large, the simple scheme implies a signi…cant loss in revenue for the cartel. This is because in such a scheme the bribe always covers the punishment cost associated with the maximal distortion of the scoring rule relative to the true one. The bribe cost can be reduced in a contingent scheme but that may not always be worthwhile because of imperfect public information which induces new ine¢ ciencies.

We …rst note that once the winner has been designated, collusion in the o¢ cial auction is sustainable relying on a standard folk theorem argument. This is because o¤ers become public information with the o¢ cial opening of the envelops. As in earlier Propositions, in the scheme 2 7 In an earlier version we investigated the whole solution. For m > 4 of Proposition 4, the announcement of the agent results from competition in corruption deals and determines the winner (as the …rm whose cost structure is most congruent with the announced scoring rule). A main issue for the cartel is therefore to sustain collusion in corrupt deals in order to contain competition in bribes and make an e¢ cient use of stochastic government preferences as an allocation rule. The problem is that …rms do not know the true scoring rule and do not observe the submitted bribe deals. They only observe the announced scoring rule which is an imperfect public signal of …rms'action in the corruption game. Therefore …rms must sometime be "punished" even when complying (this result is a similar to results in Green and Porter (1986) and Radner, Myerson and Maskin (1986)). In the appendix we provide an example showing that collusion is sustainable in a Public Perfect Equilibrium (PPE) with b = 1 4 m: A PPE is a pro…le of public strategies that, beginning any date t and given any public history till time t; forms a Nash equilibrium. 28 Deterrence from defection at the bribing stage is achieved by the threat of competition in the o¢ cial auction.

In case a …rm wins twice in a row, it is "punished" by the other …rm which then submits an o¤er that scores more than zero. This reduces the cartel's revenue (we have price ine¢ ciency).

In our example, we have an equilibrium with a contingent scheme that is bounded away from full cartel e¢ ciency but that dominates the …xed in-turn rule scheme for m not too small. We did not aim at characterizing an optimal contingent mechanism but it can be shown that any contingent mechanism is bounded away from full cartel e¢ ciency. This may seem to con ‡ict with the results in Fundenberg, Levine and Maskin (1994). A main reason is that our model does not satisfy their property of pairwise identi…ability. But most importantly e¢ ciency fails because the cartel only has a limited set of instruments to achieve con ‡icting goals. When applying the general theory to real interaction situations Athey and Bagwell (2001) hit upon that problem. In their model e¢ ciency is achievable but only under some con…guration of the parameters (and with direct communication between …rms). The situation is similar here with a tension between information revelation, design e¢ ciency and production or price e¢ ciency. In the present context with only two components and with government preferences uniformly distributed over [0; 1] ; there is no state where …rms are identically e¢ cient (and thus utility cannot be transferred at no cost for the cartel). As a consequence to achieve information revelation (or collusion in bribes) one must relinquish price e¢ ciency (see appendix). In an earlier version we considered a case where there existed states where …rms were identical. Even then full cartel e¢ ciency including, price, production and design e¢ ciency could not obtained because we have a trade-o¤ between design e¢ ciency and bribe minimization.

The main insight from Proposition 4 is that even with a variable expected punishment cost favoritism facilitates collusion. First, for m not too large, favoritism increases the collusive gain and it always reduces the threat payo¤s. Second, for m not too large the simple …xed in-turn rule is optimal. For larger m and unlike in the case with a …xed punishment cost, favoritism here does not fully shade …rms from future hazards in preferences and costs. Yet, matters are simpli…ed for …rms. With favoritism the pro…t-if-win is fully known by force of single-peakness and depends minimally on the environment. As a result favoritism allows for a reasonably simple contingent collusive scheme to sustain collusion.

Our conclusion is that the central insight from proposition 2 i.e., that favoritism facilitates collusion in face of demand uncertainty (incomplete information about the true government preferences) and privately observed shocks to costs, partly carries over to the case when the punishment cost that varies with the magnitude of the distortion. Favoritism relaxes the incentive constraints by increasing the collusive gain and decreasing the threat payo¤. With a variable punishment cost, favoritism may not always bring about larger gains in terms of a simpli…ed scheme compared with no corruption collusion.

Discussion

The main insights of the analysis can be summarized as follows:

Favoritism facilitates collusion because -It induces the revelation of …rms'private information as that information is used by the corrupt auctioneer to maximize the winner's rent; -It shades …rms from ‡uctuations in government preferences. The selected contract speci…cation re ‡ects the cartel's interests instead of social preferences; Favoritism exacerbates the cost of collusion for society. The contract speci…cation is socially ine¢ cient and the price is higher than with collusion alone. Anti-corruption policy can have a real impact on …rms'behavior.

The analysis thus reveals that favoritism fundamentally perverts the auction mechanism

both what concerns the use of …rms'private information (about their costs) and that of the agent's private information about government preferences.

A central intermediary result is that the equilibrium scoring rule is extreme i.e., "singlepeaked". In the one-shot setting this allows to minimize competition between …rms. As a result the selected project tends to be non-standard in the sense that the winning …rm is alone to be e¢ cient at its production. In the repeated setting competition is less of an issue because of collusion. As a result the winning …rm's rents can be maximize for a larger range of project speci…cations each of which responding to a single-peaked scoring rule. Most procurement codes include provisions that preclude the use of non-standard (a fortiori …rm speci…c) speci…cations and that encourage generic technical speci…cation. Interestingly, even for the simplest objects such as print paper one may not be able to de…ne a unique standard (see Compte and Lambert-Mogiliansky (2000)). When dealing with complex procurement projects, it is simply not realistic to expect being able to de…ne a unique generic speci…cation.

Choices have to be made either by settling for a technical solution or in a scoring rule.

Often it is mistakenly believed that a …rst price auction of a technically speci…ed object precludes favoritism. In Compte and Lambert-Mogiliansky (2000), it is demonstrated that such procedure can be even more vulnerable to favoritism. A technical speci…cation can bias competition at a larger cost for the government than a scoring rule. Generally, the use of a scoring rule (that weights technical components or performance measures) increases competition and thereby reduces the stake of favoritism. Our analysis applies within the spectrum of discretion consistent with typical anti-favoritism provisions. It says that within that spectrum, favoritism results in the selection of a project speci…cation that maximizes the winner's rent. It also says that collusion relaxes a constraint on equilibrium scoring rule (i.e., it needs not minimize competition), which presumably makes favoritism more di¢ cult to detect.

Single-peakness as the solution to rent maximization, obtains from the conjunction of a series of assumptions most of them are standard or reasonable. Two assumptions deserve some comments: separability in costs between components and separability in bribes and punishment cost. There is a natural way to reinterpret the single-peakness result when relaxing the assumption of separability in costs. If we have complementarities in costs, one should group components that are complementary in production into a composite component that is given full weight in a proper manner. Clearly, a more involved cost structure would entail more complex computation of the demanded scoring rule(s) and a more involved operation to compute the scoring rule that maximizes the winner's rent (used in the stage game). A conjecture is that the menu of deal o¤ers is su¢ ciently rich a message language to allow for quite sophisticated information to be revealed so the auctioneer can minimize competition as in the basic model. With (ex-ante) symmetric …rms the prize i.e., winning the contract with minimal competition is the same for both …rms in which case most of the results carry over. Some additional analysis may be required if we want to relax the assumption about separability in bribes and expected punishment. In particular to investigate the case when the auctioneer is not willing to take a bribe so high that it covers the expected cost of all distortions. However, evidence suggests (see footnote 8 and policy implications below) that the expected cost is rather low in which case the problems related to bribe cap would not arise.

Our conjecture is thus that the main insights of the analysis do not depends on the …ne details of the model but capture central features of the reality of favoritism in procurement as revealed by empirical evidence. First there exists numerous anecdotal evidence e.g., from developing countries. In one case an Africa country set its telephone speci…cation to require equipment that could survive in frigid climate. Only one telephone company from Scandinavia could satisfy this obviously worthless speci…cation (Rose-Ackerman (1999), p.64). Similarly problems of maintenance of construction objects are often due to the non-standard project speci…cation that was selected by the international procurement procedure. Second, the allocation pattern emerging from the analysis: a pre-determined in-turn rule that allocates the contract to the most e¢ cient …rm while generating large pro…ts is very close to the patterns observed in Paris Hall case mentioned in the Introduction. Interestingly, people have argued that the fact that the contract were allocated to the most e¢ cient …rm was an indication that there was no collusion. The present analysis shows that it is su¢ cient that each …rm has a comparative advantage in some component for this outcome to obtain in a collusive equilibrium with favoritism.

22

A central message of the analysis is that the risks of collusion and favoritism are linked and must be addressed simultaneously. Yet, the investigation of collusion is often the jurisdiction of Competition Authorities while that of corruption is the jurisdiction of criminal courts. A …rst recommendation is to develop cooperation to overcome this institutional separation so as to improve e¢ ciency in the prosecution of cases that involve both favoritism (corruption) and collusion.

The analysis con…rms earlier results (see e.g., La¤ont and Tirole (1993) and Che and Burget (2004)) that discretion to devise the scoring rule is subject to capture by …rms. This seems to suggest that one should eliminate the agent's discretion i.e., let the agent administer a …rst price auction. But that would be highly naive a conclusion. Indeed scoring rules are used to give ‡exibility in design which generally increases competitive pressure. In a pure …rst price auction the object has to be fully de…ned by the technical speci…cations. Compte and Lambert-Mogiliansky (2000) show that the decisions related to the technical speci…cation is even more sensitive to capture than those related to the scoring rule because they are linked with higher rents. On the other hand if a …rst price auction is associated with a standardization of the technical speci…cation, the agent's discretion can be truly reduced.

Our result gives support to a policy that reduces the agent's discretion to devise the scoring rule but only when the technical speci…cation can be standardized. When standardization is too costly (or not feasible), the auctioneer's decision should be subjected to close scrutiny. This recommendation is in line with Steven Kelman (1994) 29 who argues in favor of ‡exibility in association with increased accountability of procurement o¢ cials. Concretely this means for instance an obligation to motivate their decisions in writing. Another type of measures recognizes that …rms often have a superior information about each other than the government has. They can be in a position to recognize when a scoring rule is …ne-tailored to some other …rm. A recommendation would be to consider devising a mechanism to reveal this information e.g., by performing an anonymous consultation prior the o¢ cial submission.

Our results suggest that there is a real role to play for anti-corruption policy (controls and punishments) to play. From proposition 2 we know that the minimal discount factor 1 increases with m from 1 (m = 0) :62: An anti-corruption policy aiming at induc-ing a positive m has the immediate e¤ect of precluding the most impatient …rms. As we noted in section 4.1, further increasing in m over some threshold may even induce …rms to refrain from favoritism altogether so the cost to society is then limited to a high collusive price. The risk of favoritism disappears when the expected punishment cost is so high that it exceeds the stage game competitive payo¤.

The potential e¢ ciency of the repressive tools contrasts with the current legislation in the European Union that makes it very di¢ cult to convict for favoritism. A central reason for this is that favoritism is di¢ cult to prove. Indeed, generally any selection criteria would favor some …rm(s) at the expense of others. "Deciding to build a swimming pool rather than a stadium is good for …rms that have a comparative advantage in building swimming pools." The problem is thus to compare between selection criteria that favor di¤erent …rms. The honest auctioneer picks up the one that is congruent with public preferences while the corrupt selects another one. But public preferences are seldom so well-de…ned that congruence can be measured in a way that is non-controversial (which also suggests that a …xed punishment cost model maybe the relevant one). Generally, detecting and proving the occurrence of favoritism is di¢ cult.

An implication of the analysis is that attention should be paid to a careful study of allocation patterns over time. Unfortunately courts tend to focus on bribery and few cases of favoritism are brought to court. We thus suggest that sophisticated economic expertise be given more power in cases where there is a suspicion of favoritism. Indeed, while this is the rule in cases of standard collusion, economist expertise appears to be seldom requested in cases involving corruption and favoritism.

Finally, in our analysis we have assumed that the agents were short-run players. The idea is that the …rms are rather specialized and meet on public markets that are set up in di¤erent jurisdictions and therefore administrated by di¤erent agents. The demand for e.g., public sport facilities is not recurrent in any single jurisdiction. But we could also interpret our results in the context of …rms who meets on public markets organized by the same administration but with o¢ cials who are often moved from one position to another. Such a policy is often advocated to prevent corruption which is presumed to be easier to sustain within the frame of long-running relationships. Our result shows that this presumption is not warranted here. On the contrary the short-run character of the agents permits …rms to earn all the rents from collusion. So our results suggest that a high turnover of o¢ cials certainly does not make favoritism more di¢ cult.

A Proof of Lemma 1

For any announcement b ; the e¢ cient speci…cation for …rm i is de…ned: q i = arg max s (q; b ) c i (q; i ). We claim that in the equilibrium of the FSA both …rms o¤er the e¢ cient speci…cation corresponding to their cost structure. Assume that this was not the case i.e., that, in equilibrium, …rm i o¤ers (b q;b p) with b q 6 = q i : We …rst note that under asymmetric information prob f winj ((b q;b p))g > 0.

To show this we order the …rms'type according to the number of components for which they have high cost (so each such type corresponds to a group of "elementary types") 0 <; :::

; < k 1 ;
where k 1 is the highest cost type (recall the fully inef-…cient types have been deleted from the distribution). Let S ( ) = max q s (q; b ) c i (q; ) ; in our model we have that prob

n winj S k 1 o > 0:
If the o¤er (q;p) for some j < k 1 was such that prob f winj (q;p)g < prob

n winj S k 1
o ; it could not be an equilibrium o¤er since the score in decreasing in cost i.e., there would exist another o¤er that would yield higher expected pro…t.

We now show that o¤er (q i ;p 0 ) with p 0 = b p + s (q i ; b ) s (b q; b ) dominates (b q;b p) : Note that S (q i ;p 0 ) = S (b q;b p) so in particular prob f winj (b q;b p)g = prob f winj (q i ;p 0 )g : Now the expected pro…t from submitting (q i ;p 0 ) is i q i ;p 0 ; i = p 0 c i (q i ; i ) prob winj q i ;p 0 = [b p c i (b q; i ) + s (q ; b ) c i (q i ; i ) (s (b q; b ) c i (b q; i ))] prob f winj (b q;b p)g > [b p c i (b q; i )] prob f winj (b q;b p)g = i (b q;b p; i )g:

The last inequality holds because s (q ; b ) c i (q i ; i ) > s (b q; b ) c i (b q; i ) :

The argument applying to the symmetric information case which we also use below is even simpler. Consider the case when …rm 1 has a cost structure that is more congruent with the announced scoring rule than …rm 2. Firm 1 is sure to win when submitting the second highest score (corresponding to …rm 2's e¢ cient speci…cation associated with a price bid equal to its cost) because the tie breaking rule favors quality. Suppose …rm 2 submits an o¤er that does not include the e¢ cient speci…cation and …rm 1 matches that score. Then …rm 2 could switch to an o¤er that includes the e¢ cient speci…cation to achieve a higher score and win. Suppose now that …rm 1 matches …rm 2's score with an o¤er that does not include the e¢ cient speci…cation. Appealing to the argument above (setting the winning probability equal to 1), we see that it cannot be optimal. Firm 1 could earn a higher pro…t with an o¤er that scores the same but includes the e¢ cient speci…cation. Similar reasoning applies when …rms are identically e¢ cient. Hence, in equilibrium …rms submit o¤ers that include the e¢ cient speci…cation. QED

B Proof of proposition 1

We consider the following strategies for the players:

Firms:

At step 1 submit a menu of deal o¤er j ; b with a deal for each component j; ji = with a single-peaked scoring rule i.e., j = (0; :::; 1 j ; :::; 0). The same bribe is o¤ered in each one of the deals belonging to the o¤ered menu. Both …rms o¤er b = ( ) 2 :

At step 3, when the announced scoring rule is single-peaked, …rms submit the competitive equilibrium o¤ers under the assumption that they are anti-symmetric in cost (for b = j if ji = then j i = ). When the scoring rule is not single-peaked, the (non-deviating) …rm believes the other has a cost structure fully congruent with the announced scoring rule (i.e., that the …rm has low cost on all emphasized components) and submits the corresponding competitive o¤er.

The auctioneer:

At step 2, the auctioneer selects from among the submitted corrupt deals a deal that includes the highest bribe provided the bribe covers the cost m. From among the highest bribe deals he selects a one associated with a scoring rule only demanded by one …rm if there is any. If there are none, he randomizes among the highest bribe deals. If there are several deals demanded by one …rm only he randomizes among those. He announces the associated scoring rule.

At step 5, the auctioneer maintains the outcome of the auction if the winner is one who demanded the favor (and by assumption will pay for it) or if no favors were demanded.

Otherwise he rejects the outcome.

Note that the players's action at Step 4 are not speci…ed. This is because they are fully determined by the rules and assumptions of the game.

We below show that the strategies described above form a Perfect Bayes-Nash equilibrium of the game. …rm 1. In equilibrium …rms submit deal on all their low cost components with the same bribe. By assumption each …rm has a comparative advantage on at least one component and we know that the auctioneer chooses to announce a scoring rule demanded by one …rm only. Firms therefore correctly infer from b = j that they are anti-symmetric in cost with respect to j so in particular when a …rm has high cost (…rm 2), it infers that its opponent has low cost. We note that these are precisely the beliefs that we assumed when deriving the competitive equilibrium above. The only distinction is that they apply to b = j : Firm 2's best response is to to bid p 2 de…ned in ( 5) for b = j . If 2 wins, it pays no bribe. Firm 1 bids p 1 as de…ned in (6) which secures win and precisely covers the bribe cost b = ( )

2

:

In an (out-of-equilibrium) subgame where ^ 6 = j the non-deviating …rm believes that the opponent has low cost on all emphasized components, so it is optimal to submit the Nash equilibrium o¤ers with the price bid given by (5). By construction this is optimal.

At step 2, the auctioneer chooses a deal among the submitted menus (M 1 ; M 2 ) with

M i = f( il ; b il )g n i k l=1 :
The auctioneer expects …rms to ask for scoring rules that emphasize components in which they have low cost

. Since U = b m, he selects a deal (^ ; b k ) such that b k 2 max fb 1l ; b 2l g n i k
l=1 and b k m. If no such deal has been submitted the auctioneer announces the true . From among acceptable highest bribe deals (b k m) he is indi¤erent between those with a scoring rule demanded by one or by the two …rms. So it is optimal to choose a deal that emphasizes a component demanded by one …rm only. By assumption there are at least two such deals at equilibrium.

At step 1, we know from step 2 that the auctioneer selects a deal associated with the highest bribe and among those deal with a scoring rule demanded by one …rm only. We show that (i) …rms demand single-peaked scoring rules, (ii) they submit deals for each low cost component, (iii) they bid the same bribe b in each deal.

i. The scoring rule only determines the pro…t-if-win which for arbitrary b is 1 = P n j c j 2 ( )

2

; since P n j c j = 1; 1 is maximized with any j 1 = (0; :::; 1 j ; :::; 0); 1j = and 2j = . Hence, …rms demand a single peaked-scoring rule.

ii. The …rms don't know each other's cost, but they assume the other …rm acts according to the equilibrium strategy i.e., submits deals on each of its low cost component. Consider …rm 1's incentives to submit a deal with j 1 and some b and 1j = : It knows that the agent only chooses a deal with a highest bribe so the bribe must be b : But we show below that b corresponds to the rents of a low cost …rm winning over a high cost …rm so if the …rm was to win with such a deal on a high cost component it would earn a negative payo¤. Assume that 1j = : When …rm 1 submits a deal, it wins if …rm 2 has high cost (and thus 1 is alone to submit on that component). If …rm 1 does not submit a deal with j 1 ; it can still win if …rm 2 has low cost (and 2 is alone to submit a deal on that component) by slightly undercutting 2's o¤er in the o¢ cial auction. But …rm 1 knows that in that case the auctioneer will reject the auction's outcome because the winner will not pay the bribe which results in zero payo¤s for the …rm so free-riding does not pay. Now for any given M 2(1) the probability that the auctioneer …nds a deal with a scoring rule demanded by only …rm 1 only increases with the number of submitted deals by …rm 1. So it is optimal to submit a deal on each low cost component.

iii. A …rm's pro…t-if-win with any of the j it demands is equal to ( ) We show that full cartel e¢ ciency can be achieved in an equilibrium supported by a Trigger strategy with a punishment phase corresponding to the play of the equilibrium of proposition 1. The cooperative phase is characterized by the following:

Firms'strategy

At step 1 the in-turn-…rm (referred to with subscript in) submits a menu M in = b ; j with in;j = ; b = m. The out-of-turn …rm (referred to with subscript out) submits M out = 0; j for some out;j ; b = 0:

At step 3 for any b t the in-turn …rm submits an o¤er that scores less than zero. The out-of-turn …rm bids to score strictly less than zero.

The auctioneer's strategy:

At step 2 the auctioneer selects from among the submitted corrupt deals a one associated with the highest bribe. If that bribe covers the costs, he announces the associated scoring rule.

At step 5 the auctioneer maintains the outcome if the winner is the one who demanded the favor (and by assumption will pay for it) otherwise he rejects the outcome.

Let H t 1 = H denote a public history of the game when it is in a cooperative phase i.e., in all t 0 = 1; :::; t 1 the outcome is characterized by the …rm wining in alternation i.e., every second period.

The trigger strategy entails that in any subgame following H t 1 6 = H ; the …rms move to (stay in) the punishment phase. Since it is a Nash equilibrium, conforming is by construction a best response for all players.

We now consider a subgame following H t 1 = H to show that cooperating according to the strategies de…ned above is optimal: We proceed by backward induction.

At step 5 the short-run auctioneer has the same incentives as in the stage game, see proof of proposition 1.

At step 3 whatever b t , the in-turn …rm expects the out-of-turn …rm to bid less that zero.

The maximal payo¤ c = 1 2 yields when the in-turn …rm o¤ers the e¢ cient speci…cation and a price so its o¤er scores just zero. So the proposed strategy is optimal. The out-of-turn …rm may deviate. The most pro…table deviation occurs when the announced scoring rule is single-peaked and the out-of-turn …rm also has low cost on the emphasized component and submits p = 1 "; " > 0: Its gain is d = 1 2 ": However the in-turn rule is violated and from the next period on the …rms revert to the zero payo¤ competitive equilibrium of proposition 1. So the out-of-turn …rm complies with the collusive strategy whenever

IC : 1 2 1 2 m 1 2 " (8)
which is satis…ed for 1 ; 1 2 (0; 1) with @ 1 @m > 0: At step 2 since the auctioneer is a short-run player, the argument developed in the proof of proposition 1 carries over. A distinction is that the auctioneer may announce a scoring demanded by both. This is because there is no competition in the o¢ cial auction, the outof-turn …rm submits an o¤er that scores less than zero. A step 1 we start with a Lemma Lemma 2 In a symmetric equilibrium …rms always demand the "cartel e¢ cient" scoring rule contingent on their cost structure i.e., 1 = 1 and correspondingly 2 = 0:

For …rm 1, the "cartel e¢ cient" scoring rule is de…ned 1 = arg max b 1 ( )

2 (2b 1 1) m 1 ) 2 :
It is the scoring rule that maximizes the cartel's payo¤ given that there is a cost associated with deviations from the true scoring rule. We know that the auctioneer selects the …rm whose deal maximizes U (b i ; i ) = b i m ( i

In a symmetric equilibrium the auctioneer never distorts more than by :5 so the highest cost for distortion is 1 4 m. Firms bid the bribe in (9) which is a best response to the other …rm. Hence, for ( ) In this proof we consider two types of collusion, characterize the condition for their sustainability and compare them in terms of cartel e¢ ciency.

"In-turn rule" collusion:

This type of collusion is similar to the one in Proposition 2. The strategies are the same as the ones described in the proof of proposition 2 when putting k = 2: Any deviation from those strategies triggers the play of the Nash equilibrium of proposition 3 from the next period on.

As usual we investigate the game by backward induction and focus on incentives to comply in a period following a history of compliance play.

  the context of a single auction. The present article is most closely related to Compte, Lambert-Mogiliansky and Verdier (2005) and Lambert-Mogiliansky and Sonin (2006). Both articles are concerned with links between corruption and collusion. They address a cartel's enforcement problem in a one-shot setting and focus on the impact of the auctioneer's abuse of discretion to let …rms readjust their bid. In Compte et al. the auctioneer sells an illegal opportunity to resubmit, which is shown to permit sustaining collusion in a single object auction. In Lambert-Mogiliansky and Sonin, the auctioneer abuses a legal right to let all …rms simultaneously readjust their o¤er in the context of a multiple-object auction. As a consequence collusive market-sharing becomes sustainable. The contribution of the present paper is to demonstrate corruption's role with respect to another central problem of a cartel: how to achieve (cartel) e¢ ciency in a stochastically changing environment.

step 4 :step 5 :

 45 The auctioneer publicly opens the envelops and he selects the …rm whose o¤er maximizes the selection criteria corresponding to the announced scoring rule; The auctioneer decides to con…rm or reject the outcome of the auction. If he con…rms the outcome, the winner i pays a bribe b if and only if ( b ; b) 2 M i : Otherwise no bribe is paid. If he rejects the outcome no transaction takes place.

i. For m ( ) 2 the 2 for all j: When m > ( ) 2 there

 222 equilibrium scoring rule is b = (0; :::; 1 j ; :::0) for some j ; 1j 6 = 2j and b 1j = b 2j = ( ) is no favoritism.

[0; 1 ] 2 :

 12 : The auctioneer's utility is U = b m (b ) Note that since, by assumption, each …rm has a comparative advantage in at least one component (see the model description), the k = 2 case boils down to symmetric information. 26

step 1 :step 2 :step 3 :step 4 :step 5 : 5 ( ) 2 Proposition 3

 12345523 The auctioneer learns ; the …rms submit their corruption deals f(b 1l ; 1l )gand f(b 2l ; 2l ) ; g ; l = 1; ::n i ; I = 1; 2; The auctioneer makes an announcement b ; b 2 [0; 1] ; The …rms submit their contract o¤ers (q i ; p i ) ; The auctioneer publicly opens the envelops and he selects the …rm whose o¤er maximizes the selection criteria corresponding to the announced scoring rule; The auctioneer decides to con…rm or reject the outcome. If he con…rms the winner i pays a bribe b if and only if ( b ; b) 2 M i : Otherwise no bribe is paid. If he rejects no transaction takes place. Proposition 3 characterizes symmetric Perfect Bayes-Nash equilibria of the stage game described above. We show that for m 4 Any symmetric Perfect Bayes-Nash equilibrium is characterized by i. The equilibrium scoring rule is b (m; ) = equilibrium bribe o¤ er is b 1 = b 2 = b (m) = ( )

5 D 2

 52 the equilibrium bribe is equal to the expected punishment cost and favoritism occurs less often. As the cost grows larger, it becomes fully prohibitive.

2 :

 2 The corruption game boils down to a symmetric information common value auction. By a standard argument, …rms submit the common value b 1 = b 2 = ( ) 2 for all component with low cost:QED C Proof of Proposition 2

At step 1

 1 the …rms submit their menu of deals. Since the auctioneer only cares about the bribe the argument of proposition 1 carry over and …rms always propose deals with single-peaked scoring rules. The in-turn …rm expects the out-of-turn …rm to o¤er b = 0: It is su¢ cient to o¤er b = m to cover the auctioneer's cost so he announces one of the in-turn …rm's preferred scoring rule. The out-of-turn …rm can defect and o¤er b = m + " associated with a menu including a most preferred scoring rule out . It knows that the auctioneer would respond by announcing that b = out : But such a deviation brings at most a pro…t by one …rm only: By assumption and given the …rms'equilibrium strategies, we know that there is at least one such. The auctioneer announces b i if i b i = b i. If no bribe deal can secure win in the o¢ cial auction or if b b i < m b i 2 , the auctioneer announces the true alpha.

) 2 :b 1 # 2 .Hence 1 = 1 1 E 1 (b 1 ; b 2 ) = 2 b 1 !:

 212111111 Suppose by contradiction that an equilibrium o¤er is (b 1 ; 1 ) with 1 6 = 1 : We now construct o¤er (b 0 1 ; 1 ) with b0 1 = b 1 m(1=2 1 ) 2 + m(1=2 1 ) 2 : By construction prob U b 0 1 ; 1 > U (b 2 ; 2 ) = prob (U (b 1 ; 1 ) > U (b 2 ; 2 )) = 1=2: prob (U (b 1 ; 1 ) > U (b 2 ; 2 ))where the inequality holds because m < ( ) and similarly for …rm 2: 2 = 0.We now consider the determination of b 1 and b 2 : The expected pro…t of …rm prob fU (b 1 ; 1) > U (b 2 ;Taking the derivative with respect to b 1 an interior solution satis…esIn a symmetric equilibrium we obtainb = b 1 = b 2 = 2 m;

2 ;

 2 the investigated strategies described above including the deal o¤ers (b ; 1) and (b ; 0) form a Bayes-Nash equilibrium of the FSA with favoritism: QED: E Proof of proposition 4

See alsoGreen and Porter (1984) for the analysis of a price cartel on a market with a demand subject to shocks.

Even then cartel e¢ ciency is only achievable under some con…guration of the parameters.

See for instance La¤ont J-J. and J. Tirole (1991), Celentani, M. and J. Ganunza, (2002), Che and Burget (2004), Compte, Lambert-Mogiliansky and Verdier (2005).

The government can engage a procedure to …nd out its true preferences and punish the auctioneer if he distorted them in his announcement.1 0 The punishment for favoritism is independent of the social-economic loss it induced.1 1 As argued in Lambert-Mogiliansky and Sonin (2006) empirical evidence shows that there is ample opportunities for the auctioneer to intervene in the procedure so as to e¤ectively relaunch it and delay its outcome.

[START_REF] Jean | Corruption et Pratiques anti-concurrentielles: une première re ‡exion à partir d'une étude de cas[END_REF] We show in section 5 below (Extensions) that this result is robust to other speci…cation of the punishment costs.1 4 Strictly speaking the interpretation of zero weight as a minimal level is equivalent to assuming that …rms are (more or less identical) in the production of a "basic good" while they di¤er in the production of speci…cations of the project in excess of the requirement de…ning the "basic good".1 5 An honest auctioneer uses information about costs to minimize …rms' rents. If he knew the …rms' costs, he would simply give the contract to the most e¢ cient …rm and pay it its actual cost i.e., he would leave no rents to the winner.

[START_REF] Compte | Corruption and Competition in Procurement[END_REF] We wish to remark that if …rms have an identical cost structure, favoritism has no value. If one of the …rms is ine¢ cient on all components but not the other, we would have favoritism but no competition so the bribe would just cover the punishment cost m. 1 7 When reviewing court cases, it appears quite clear that the cost of favoritism is very low. The only instances of conviction for favoritism in France pertain to cases where the auctioneer explicitly required a …rm speci…c technology. (Cour des Graces

).1 8 We consider a symmetric case but the logic would be the same if we allowed for some asymmetry in the cost structure. All that a …rm needs to know is the other …rm's value of winning the contract under the most favorable circumstances i.e., with a selection rule that gives full weight to a component such that the …rm has the largest comparative advantage in its production.

[START_REF] Athey | Collusion and Price Rigidity[END_REF] We do not characterize the equilibrium of the stage game without favoritism. By lemma 1, we know that the equilibrium speci…cation is socialy e¢ cient. The …rms compete in price given their beliefs about the other's cost structure.

[START_REF] Compte | Corruption and Competition in Procurement[END_REF] We learned from proposition 1 that asymmetric information is not a big concern in our setting. Therefore, the k = 2 provides a satisfying model to focus on the new features due to a variable expected punishment cost.

[START_REF] Fudenberg | The Folk Theorem with Imperfect Public Information[END_REF] A strategy for player i is public if, at each time t; the strategy depends only on the public history and not on i 0 s private history.

[START_REF] Daniel | Collusive Bidder Behavior at Single-Object Second-Price and English Auctions[END_REF] Harvard professor Steven Kelman was the director of the O¢ ce of Federal Procuremnt Policy under 1993-1997.

Athey S. and K. Bagwell (2001) " Optimal Collusion with Private Information" Rand Journal of Economics 32 (3) 428-465.

For that purpose it is useful to …rst derive the competitive o¤ers that form the Nash equilibrium of the First Score Auction described by step 3 and 4 with no bribes with the following out-of-equilibrium beliefs. The non-deviating …rm (2) plays competitively as if the other …rm has the cost structure which is most congruent with the announced scoring rule i.e., low cost on all emphasized components). The deviating …rm (1) plays as if the other …rm had the least congruent cost structure i.e., high cost on all emphasized components. Although these beliefs are not consistent they generate the largest possible gain from deviation with respect to the demanded scoring rule at the corruption deal submission stage. Those beliefs are designed so as to make the deviation most attractive.

We know from lemma 1 that …rms choose q i = arg max q s (q; b ) c (q; i ) : Suppose b emphasizes n component n < k; …rm 1 expects …rm 2 to make an o¤er including q 2 = q ij ( i ) so its quality score is P j j j = P j 2 j ; for j = 1; :::n: By a standard argument, …rm 2 bids the lowest price that just secures non-negative pro…t

Firm 1's best response is to bid the highest price that secures win:

The deviation pro…t (gross of bribe) is

We call the bids (q 1 ; p 1 ) ; (q 2 ; p 2 ) de…ned in Lemma 1 and the formulas in ( 5) and ( 6) the out-of-equilibrium competitive bids.

We now proceed to investigate the whole game by backward induction.

At step 5, the auctioneer chooses whether to reject or maintain the outcome. If the winner is not a …rm that demanded the scoring rule the auctioneer expects no bribe. Rejecting the auction outcome earns him zero as well so it is optimal to do so. When the winner is a …rm that demanded the favor, if he rejects he earns 0 while if he maintains the outcome he collects the bribe. So maintaining the outcome is optimal. At Step 3 …rms make their o¤ers.

By lemma 1 we know that any o¤er includes the cost e¢ cient speci…cation. Consider …rst the subgame which corresponds to single-peaked b = j for some j = 1; :::; k and say it favors of 1 2 m ". Since we know that such a win triggers a punishment phase defection is not pro…table under (8) since there the gain from defection is larger. Hence for satisfying (8) the proposed strategies do form a Bayes-Nash equilibrium of the repeated game. The cartel's gain is maximized. In each period, the scoring rule is the most favorable to the winner, the price is given by the reserve score and the bribe is the lowest possible.QED

D Proof of Proposition 3

Firms'strategy:

At step 1 submit a menu of deal o¤ers j ; b with a deal for each component j; ji = ; j = 1; 2; with 1 = (1; 0) and 2 = (0; 1). If the menu contains two deals, the same bribe b (m) de…ned below is o¤ered in both.

At step 3 …rms submit the competitive equilibrium o¤ers under the assumption that they are anti-symmetric in cost.

The auctioneer:

At step 2 the auctioneer selects from among the submitted corrupt deals a deal that includes the highest bribe provided the bribe covers the cost m ( b ) 2 . From among the highest bribe deals he selects a one associated with a scoring rule only demanded by one …rm if any. If there are several such deals he randomizes. He announces the associated scoring rule.

At step 5 the auctioneer maintains the outcome of the auction if the winner is a …rm who demanded the favor (and by assumption will pay for it). Otherwise he rejects the outcome.

We below show that the strategies described above form a symmetric Perfect Bayes-Nash equilibrium with favoritism. We develop the proof in terms of …rm 1 which has its advantage in the production of q 1 : Firm 2 is symmetric with advantage in component 2. We proceed by backward induction.

The reasoning for step 5 is the same as in proposition 1 and 2. At Step 3, the reasoning here is identical to the one in the proof of proposition 1 for k = 2; 1 = and 2 = (1 ) :

A step 2 the auctioneer's utility function is b i m ( i ) 2 : So it is optimal to choose a deal among the submitted ones as

In case of ties, he is indi¤erent and may just as well select a deal demanded At step 5 the same argument as in the proof of proposition 3 applies since our agent is short-run.

At

Step 3 the designated winner, say …rm 1, has no incentive to deviate while the other …rm might undercut the o¤er of …rm 1 and get at most 1 2 in the current period and a continuation payo¤ of 1 2 ) ( )

m; the incentive constraint writes

where the rhs is the compliance payo¤ and the lhs is the deviation payo¤. We show below that deterring deviation at step 1 is more demanding so we postpone the derivation of the limit on the discount factor.

At step 2 there is no incentive to deviate for the agent for reason similar to those in proposition 3. At step 1 …rm 2 might make a secret bribe bid of b 2 (0; 2m] and demand ^ = 0. The agent will grant …rm 2 the favor of choosing its demanded scoring rule when 2 [0; b=(2m)]:

In order for b = 0 to be an equilibrium strategy, the above expression needs to be maximized at b = 0. Since the expression is strictly concave in b we get the following restriction for the discount factor

The expression in the bracket is negative and increasing in m: By assumption we have

( )

but to simplify the calculation, we check for m = ( )

the constraint is more restrictive then. Thus a conservative formulation of the constraint on the discount factor yields where

For = 2:2; = 0:8:

Contingent rule

We use the notation: w 1(2) (Y ES) and w 1(2) (N O) to denote …rm 1(2) continuation after a period when it won respectively when it lost. We focus on a smaller set of strategies including b 2 1 4 m; 5 4 m so in particular we only consider a defection that secures win. To out-compete …rm 1 when the true scoring rule is most favorable to 1 e.g. = 1, …rm 2 must o¤er [START_REF] Compte | E¢ cacite et Transparence dans les Procédures de Spéci…cation sur les marchés Publics" Direction de la Prévision[END_REF] 4 m: Restricting the set of possible deviations is not crucial to the result but it simpli…es the presentation. Let H t 1 = H denote a history of the game where in all periods t 0 ; t 0 = 0; ::t 1; we have b t 0 2 f0; 1g and S p t 0 ; q t 0 = 0.

We propose the following strategies for the players: i. If H t 1 6 = H ; the …rms and the auctioneer play the equilibrium strategies depicted in proposition 3.

ii. If H t 1 = H ; the …rms'strategy is At step 0 …rm 1 (2) learn its cost structure:

At step 1 …rm 1(2) submits a menu of deals one for each component where cost is low demanding a single-peaked scoring rule and o¤ering the same bribe b = 1 4 m: At step 3 -If b t is single-peaked, the …rm with the congruent cost structure submits an o¤er including the corresponding e¢ cient speci…cation such that the o¤er scores zero. Firm 2 with the non congruent cost structure submits an o¤er that scores at most zero.

-If the announced scoring rule is not single peaked, the winner is designated by a random rule characterized below. The designated winner submits and o¤er that scores zero and the other …rm submits an o¤er that scores slightly less than zero. The auctioneer's strategy At step 2, the auctioneer selects the corruption deal among that maximize his utility provided the bribe covers expected costs and he announces the corresponding scoring rule.

In case of ties he randomizes.

At step 5 if the scoring rule is single peaked but the winner is not a …rm that demanded that scoring rule he rejects the auction's result. Otherwise he maintains it.

We below show that these strategies form a Perfect Public Equilibrium of the repeated game with the stage game as described in section 5. Collusive bidding at step 3 is sustainable relying on an argument similar to the one developed in Proposition 2 when setting k = 2: The non-favored (say 2) …rm's incentives to comply with the collusive strategy is satis…ed for > 2 where 2 is de…ned by the following equality (1 )

ne the lowest compliance payo¤ is set equal to the defection payo¤.

Consider the following strategy that satis…es the constraint. The full punishment is taken in the next-following period and "the clock is reset" i.e., the next following payo¤s in t + 2 are determined as if b t = 0. Note that when (12) holds incentives to comply in a period following an announcement of = 1 also are satis…ed. This is because the gain from defection are lower then. Hence, for 1; the contingent scheme yields a higher expected payo¤. We note that the condition m