Bribery an Favoritism by Auctioneers in Seale-Bi Auctions Roberto Burguet an Martin K. Perry Abstract We consier a moel of bribery in an asymmetric procurement auction. In return for a bribe from te isonest supplier, te auctioneer as te iscretion to allow tis supplier to revise is bi ownwar to matc te low bi of te onest supplier. Te isonest supplier can also win te contract outrigt witout paying a bribe by biing below te onest supplier. We investigate te effect of te bribe sare an te cost istributions on te biing functions, te allocative istortion, an te expecte price pai by te buyer. Te isonest supplier bis more aggressively to win te contract outrigt wen te auctioneer takes a larger bribe sare. Bribery an te implie rigt of first refusal introuce a new allocative istortion in favor of te isonest supplier. Finally, we use te power family of cost istributions to examine te expecte price pai by te buyer. Wen te isonest supplier as a more favorable cost istribution, tere exist bribe sares sufficiently large suc tat te expecte price pai by te buyer can actually ecline as a result of bribery. KEYWORDS: bribery, favoritism, auctions Roberto Burguet: Institute for Economic Analysis (CSIC), Campus UAB, Bellaterra, Barcelona, Spain 08193. Martin K. Perry: Department of Economics, Rutgers University, New Brunswick, New Jersey, USA 08901. Martin Perry woul like to acknowlege te financial support of te IAE (Institute for Economic Analysis, CSIC) an ICREA (Catalan Institute for Researc an Avance Stuies) locate in Barcelona, Spain. Roberto Burguet woul like to acknowlege financial support of FEDER (European Fun for Regional Development), te Spanis Ministry of Science an Tecnology, an CREA (Center of Reference in Economic Analysis, locate in Barcelona, Spain) an te Siney I. Simon Memorial Fun in te Department of Economics at Rutgers University. We also acknowlege comments from te participants of epartmental worksops an conferences were we ave presente tis researc, particularly Juan Jose Ganuza (University of Pompeu Fabra, Barcelona, Spain) an Ricar P. McLean (Rutgers University, New Brunswick, N.J.). Finally, we acknowlege te elpful questions an comments from te eitor an referees for te B.E. Journals in Teoretical Economics.
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions 1. Introuction In tis paper we analyze bribery as a monetary sie-payment by a supplier to an auctioneer in orer to alter te awar of a procurement contract in favor of te supplier. 1 Te auctioneer represents a buyer in te procurement of some goo. Te buyer coul be a government or a corporation, an te auctioneer woul be a procurement official or employee. Te awar of te contract in return for a payment coul take ifferent forms in ifferent contexts. In tis paper, we will analyze te excange between a corrupt auctioneer an a isonest supplier wereby te auctioneer allows te supplier to revise is bi, wen necessary to win te contract. Tat is, te auctioneer cannot awar te contract at a price above te lowest bi from te auction, but e can favor te isonest supplier by awaring im te contract at tat price. In effect, wen te isonest supplier oes not submit te lowest bi, e still as a rigt of first refusal to accept te contract at te lowest bi of te oter suppliers. We use te term favoritism to escribe te special case in wic te auctioneer oes not require a bribe from te isonest supplier in excange for awaring te contract uner tis rigt of first refusal. Tis specification of te auctioneer s iscretion in awaring te contract resembles some ocumente examples of bribery. For instance, Ingraam (2005) examines bribery in contracts aware by te New York City Scool Construction Autority from 1990-1997. Wen te bis were to be opene publicly to ientify te winner of te contract, te auctioneer save te bi from te bribing supplier to open last an ten submitte a new, winning bi for tis supplier. Since te bribing supplier coul witraw from te contract if te new winning bi unerestimate is costs, te auctioneer secretly create a rigt of first refusal uring te auction process. Lengwiler an Wolfstetter (2000) cite two major international construction projects (an airport in Berlin an a power station in Singapore) in wic bribes were pai to obtain te bis tat were submitte by te rivals. More generally, tis paper provies some insigts into favoritism wit or witout bribes. Government procurement officers are known to ave accepte bribes from suppliers an corporations are known to favor certain suppliers of various inputs. Te goal of tis paper is to examine te effects of tis form of bribery or favoritism on te allocation of contracts an prices. We first examine ow bribery or favoritism affects te biing strategies of suppliers competing in a 1 See Noonan (1984) for a istory of bribery. See Rose-Ackerman (1999) for a general survey of corruption an potential solutions wit specific examples of bribery to government officials for procurement contracts in Capter 3. Publise by Te Berkeley Electronic Press, 2007 1
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 procurement auction. We ten ientify te effects of bribery on te allocation of contracts an te expecte price pai by te buyer. In Section 2, we iscuss te literature most relate to our paper. Tis iscussion explains te key istinctions between te iffering moels an results in te literature. In oing so, we also explain ow our approac to bribery iffers from te existing literature. In Section 3, we present te moel were two suppliers compete in a firstprice auction (FPA) to win a procurement contract, but one supplier is favore by te auctioneer. Te supplier wo is favore will be calle te isonest supplier (DS), an te oter supplier calle te onest supplier (HS). Te auctioneer emans a sare of te post-auction surplus efine as te ifference between te low bi of te HS an te cost of te DS, wenever tis ifference is positive. We call tis a bribe sare. Bribery or favoritism alters te biing strategies of bot suppliers. Te biing strategy of te HS will account for te fact tat te DS will be favore by te auctioneer. Te biing strategy of te DS will account for bot te opportunity to bribe te auctioneer an te cost of oing so. In Section 4, we caracterize tese equilibrium biing strategies an analyze ow te size of te bribe sare affects tis beavior. Altoug one migt conjecture tat bribery woul inuce te HS to bi more aggressively an te DS to bi less aggressively, we explain wy tis nee not be te case. In particular, wen te bribe sare pai to te auctioneer is large, te DS may bi more aggressively tan te HS, as well as more aggressively tan e woul ave in te absence of te opportunity to bribe te auctioneer. In Section 5 we first analyze te allocative effects of bribery. We compare te allocative istortion wit bribery to tat wic arises in an asymmetric FPA witout bribery an an optimal auction. Bribery favors allocation of te contract to te DS, weter e is ex ante stronger or weaker tan te HS in te sense of first orer stocastic ominance of te cost istribution. Tis contrasts wit bot te FPA an te optimal auction. Moreover, wit bribery, te allocative istortion is more pronounce for low cost realizations of te HS. Inee, te HS cooses is bi to compete against te cost of te DS. Tus, te allocative istortions will be solely etermine by te biing beavior of te DS. Uner mil conitions, te lower te cost of te HS, te iger te margin of is bi above is cost. In Section 6 we sow tat bribery may in fact result in a lower expecte price pai by te buyer, even toug te allocative istortions are not optimal an even toug te bribe pai to te auctioneer is a pure loss for te buyer. We illustrate tis result by analytically solving te moel for a convenient family of cost istributions. Te expecte price will be lower tan eiter te expecte price in a FPA or in an efficient auction (suc as a secon-price auction (SPA)) wen te bribe sare is large an te HS is ex ante weaker tan te DS. ttp://www.bepress.com/bejte/vol7/iss1/art23 2
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions Section 7 conclues te paper an briefly iscusses some oter interesting questions tat can be aresse wit tis moel. 2. Relate Literature Tere is a substantial an growing literature on bribery an favoritism in auctions. In tis section, we briefly summarize te major issues an finings of tat literature wic are most closely relate to our paper. 2 Several papers examine moels in wic any bier (or supplier) can bribe te auctioneer: Beck an Maer (1986) an Lien (1986), Lien (1990), Lengwiler an Wolfstetter (2000, 2005), Menezes an Monteiro (2001, 2006), Compte, Lambert-Mogiliansky, an Verier (2005), an Koc an Neilson (2005). Tese papers iffer in te specification of te bribe, but te common feature is tat te auctioneer for a procurement auction as iscretion to allow any supplier wit te lowest bi to receive a price equal to te secon lowest bi. Lengwiler an Wolfstetter (2005) escribe tis feature as type I corruption. Similarly, we will refer to tis feature as type I iscretion by te auctioneer. Te primary fining of all tese papers is tat te introuction of a corrupt auctioneer will moify a stanar auction el by a seller (or buyer) into a bribery auction el by te auctioneer. Te term bribery auction is use to escribe an auction in wic te presence of a corrupt auctioneer as no effect on te allocation of te goo (or te contract) an no effect on te expecte profits of te biers. In particular, te bribery auction remains efficient an te biers are inifferent to an auction wit or witout bribery. Tus, bribery simply results in a transfer of rents from te buyer to te auctioneer. In our paper tis transformation into a bribery auction oes not occur. Tere are two primary reasons for tis. First, te moel specifies tat only some suppliers can bribe te auctioneer, but tat te oter suppliers cannot. Secon, our moel assumes a ifferent specification for te auctioneer s iscretion. In particular, te auctioneer as te iscretion to awar te contract to a losing supplier at te price equal to te lowest bi by te winning supplier. Tus, te favore losing supplier will bribe te auctioneer to receive te contract wenever is cost is below te lowest bi of te oter suppliers. Lengwiler an Wolfstetter (2005) escribe tis feature as type II corruption, so we will similarly refer to it as type II iscretion by te auctioneer. In tis paper, we examine type II iscretion by te auctioneer. Tere are several reasons wy type II iscretion is interesting. First, type II iscretion oes 2 Tere is a relate literature in wic suppliers bribe a tir party wo provies an assessment of te quality in a multi-attribute procurement auction. For example, see Celentani an Ganuza (2002), an Burguet an Ce (2004). See also Laffont an Tirole (1991). Publise by Te Berkeley Electronic Press, 2007 3
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 not result in a bribery auction tat simply preserves efficiency wile transferring rents to te auctioneer. 3 Secon, we ave sown in Burguet an Perry (1999) tat a moel wit type II iscretion can be easily extene to inclue type I iscretion. Tir, favoritism by te auctioneer (or te buyer) towar one supplier is a natural special case of type II iscretion. Favoritism means tat te favore supplier wo loses te biing (or oes not bi) can still obtain te contract at a price equal to te lowest bi from te oter suppliers, but witout paying a bribe to te auctioneer (or te buyer). As suc, te favore supplier effectively as a rigt of first refusal on te contract at a price equal to te lowest bi from te oter suppliers, even if e never bis to win te contract. Several recent papers ave iscusse te implications of a rigt of first refusal in auctions. Tese papers inclue Arozamena an Weinscelbaum (2004), Porter an Soam (2005), an Bikcanani, Lippman, an Ryan (2005), Coi (2003) an Lee (2004). Applie to a procurement auction, tese papers examine te implications of te rigt of first refusal grante to one supplier by a buyer (or auctioneer) on te biing beavior of te oter suppliers an te resulting price pai by te buyer. 4 Te papers by Arozamena an Weinscelbaum (2004) an Porter an Soam (2005) introuce an auctioneer wo grants te rigt of first refusal instea of te buyer. However neiter paper as an explicit moel of te bribery payments to te auctioneer. As a result, te supplier wit te rigt of first refusal can be interprete as biing is cost, biing any price above is cost incluing is igest possible cost, or not biing at all. In contrast, our moel explicitly efines te bribery payments as a sare of te surplus generate by te rigt of first refusal. Te supplier wit tis rigt of first refusal still as an incentive to bi against te oter suppliers because e coul win te contract outrigt an avoi paying a bribe wen e as a low cost an submits te lowest bi. Tis will etermine te biing beavior of te supplier wit te rigt of first refusal, alter te bribery payments to te auctioneer, an affect te expecte price pai by te buyer in a variety of ways. 3 Lengwiler an Wolfstetter (2005) allow te auctioneer to coose between type I corruption an type II corruption. Using numerical examples, tey fin tat te equilibrium biing functions o not always result in efficient allocations. Tus, espite symmetry, te auction oes not egenerate into a bribery auction. Tese results suggest tat type II corruption introuces inefficiencies even wen type I corruption is present. 4 Bikcanani, Lippman, an Ryan (2005) examine a rigt of first refusal in a secon-price auction, wereas Coi (2003) examines a rigt of first refusal in a first-price auction. Wit private values, te gains of te favore buyer exactly offset te seller s loses in a secon-price auction, wereas te joint expecte profits of te seller an te favore buyer are iger in a first-price auction. ttp://www.bepress.com/bejte/vol7/iss1/art23 4
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions 3. Te Moel of Bribery Te buyer as a value v for a goo wit a fixe quantity an quality. Te buyer employs an auctioneer wo receives bis from suppliers an awars a contract for te buyer to purcase te goo from one of te suppliers using a seale-bi firstprice auction (FPA). In a fair auction witout bribery or favoritism, te contract woul be aware to te supplier wit te lowest bi at a price equal to tat bi. In contrast, we examine an auction in wic te auctioneer can provie a rigt of first refusal to one of te suppliers. If no bribe is require from tis supplier for te rigt of first refusal, ten we will refer to im as te favore supplier. But more generally, if tis supplier must pay a bribe for te rigt of first refusal, ten we will refer to im as te isonest supplier (DS). We assume tat tere is one oter supplier calle te onest supplier (HS). 5 We o not allow te auctioneer to consier bribes from bot suppliers. Moreover, we o not aress wic supplier is te DS an wic is te HS. 6 An important feature of our moel is tat te two suppliers are asymmetric in tat tey ave ifferent istributions for teir costs of proucing te goo. We assume tat eac supplier raws is cost of prouction c i, were i = (DS) or (HS), from a istribution G i (c) wit a common support [0,1], an a positive ensity g i (c) over tis support. Te cost c i is private information for eac supplier, but te istribution functions are common knowlege. For simplicity, we also assume tat te value of te buyer excees te igest possible cost realization (v > 1). Finally, we assume tat te costs of te suppliers are inepenently istribute. Tus, we will examine bribery an favoritism in an asymmetric inepenent private value (cost) FPA. Te suppliers simultaneously bi for te contract, knowing teir cost, knowing te cost istribution of te oter supplier, an also knowing te form in wic bribery or favoritism occurs. Te auctioneer runs a seale-bi FPA an must awar te contract at a price equal to te lowest bi. If te auctioneer as some iscretion in awaring te contract an/or setting te price pai by te buyer, e can extract a bribe from te isonest supplier. In tis paper, we will focus on a specification of te auctioneer s iscretion in wic e must set te price equal to te lowest bi, but tat e nee not awar te contract to te onest supplier wo makes te lowest 5 Te moel wit general cost istributions coul be caracterize in terms of multiple symmetric onest suppliers. However, te general insigts are fully illustrate wit one onest supplier. In Appenix 2 were Proposition 4 is prove, we iscuss te extension to multiple onest suppliers. 6 Tis question obviously requires tat te suppliers be asymmetric. Tis question as been partially aresse in a companion paper Burguet an Perry (2003, revise 2005) were we examine upfront payments irectly to te buyer (or similarly, upfront bribes to te auctioneer) in return for acquiring a rigt of first refusal (calle preference ) uring te subsequent auction. Publise by Te Berkeley Electronic Press, 2007 5
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 bi. If te bi of te DS is iger tan tat of te HS, te auctioneer can awar te contract to te DS at a price equal to te bi of te HS. 7 Even toug te DS submitte a iger bi, te DS as a rigt of first refusal at a price equal to te lower bi by te HS. Conversely, if te bi of te DS is below te bi of te HS, we assume tat te DS is simply aware te contract at a price equal to is bi. Let b an b, be te bis of te DS an te HS respectively. If b > b, te contract is aware to te DS at a price equal to is bi. However, if b > b, bribery or favoritism may occur. In particular, wen b > b > c, tere is surplus (b c ) wic can be ivie between te auctioneer an te DS. In tese cases, te auctioneer awars te contract to te DS at a price b, an te auctioneer receives a sare α [0,1] of te surplus (b c ). 8 We call α te bribe sare. Finally, if b > c > b, tere is no surplus tat te auctioneer an DS can sare, an te HS is aware te contract at a price equal to is bi. Te bribe sare α is etermine prior to te auction an tus is inepenent of te outcome of te auction. Te DS knows te value of α prior to submitting is bi. Te HS nee not know te specific value of te bribe sare α. Rater, all te HS nees to know is tat te DS as a rigt of first refusal at a price equal to tis bi an will tereby obtain te contract wenever b > c. Te bribe sare coul be interprete as te relative bargaining power between te auctioneer an te DS. It may ave arisen informally from past practice of te auctioneer. Also, it coul be etermine by giving te auctioneer a percentage of te stock in te subsiiary of te DS anling te contract. Te auctioneer woul ten receive bribery payments in te form of iviens from tat subsiiary after te contract is complete an te DS is pai by te buyer. In orer to calculate te bribery payments wen te bribe sare is strictly positive, te bi of te HS must be verifiable to te DS an te cost of te DS must be verifiable to te auctioneer. If te bis are submitte in writing, te auctioneer can verify te bi of te HS from te signe biing materials submitte by te HS. However, it may be more ifficult for te DS to verify is cost. Te cost of te DS migt be verifiable from te ex ante bi preparation ocuments of te DS. However, it seems more natural to assume tat te cost of 7 One justification for restricting te price to be equal or below te bi of te HS is tat te price may become known at te en of te auction. Te HS wo was not aware te contract coul complain to te buyer if is bi were below te resulting price. In government procurement, te HS may be able to sue te government. 8 We assume tat any compensation to te auctioneer from te buyer is inepenent of te auctioneer s actions in awaring te contract, an tat tere are no punisments for accepting bribes. We are not attempting to moel te agency problem between te buyer an te auctioneer. On tis, see Krueger (1974), Rose-Ackerman (1975, 1978), Rasmusen an Ramseyer (1994), an Mookerjee an Png (1995). ttp://www.bepress.com/bejte/vol7/iss1/art23 6
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions te DS is verifiable from te ex post accounting recors of expenitures by te DS. It soul be note tat wen te bribe sare is zero, te costs of te DS nee not be verifiable. 9 4. Equilibrium Conitions for te Biing Functions In tis section, we caracterize te equilibria in a seale-bi FPA for te basic moel of bribery assuming general istribution functions for te costs of te two suppliers. Bribery will generally alter te equilibrium biing functions of bot suppliers. However, we fin tat bribery oes not make te equilibrium biing functions uniformly more or less aggressive for eiter supplier. In general, it is ifficult to solve for te equilibrium biing functions in an asymmetric FPA. Te asymmetric FPA becomes more tractable wit bribery. In particular, te game is ominance solvable. Once we exclue bis by te DS below is cost an exclue te rejection of contracts by te DS at a price above is cost, te HS as a ominant strategy against wic te DS can ten coose is biing strategy. Inee, te HS is effectively biing against cost of te DS. As a result, te biing strategy of te HS is inepenent of bot te biing strategy of te DS an te value of te bribe sare α. Te HS calculates is biing strategy b (c) by solving te following problem: (1) max Π [ b; c] = ( b c) [ 1 G ( b) ]. Te first-orer conition of tis problem is (2) [ G ( b) ] b 1 ( b c) g ( b) = 0. 10 Te best response of te DS against te biing strategy b (c) of te HS is obtaine by solving te following problem: (3) maxb Π ( b c) [ b; c, b ( ) ] = 1 1 b ( b) [ 1 G ( b ( b)) ] + (1 α) ( b ( x) c) G ( x). 1 b ( c) 9 If te DS as no creible way to inform te auctioneer about its true cost, ten bargaining woul occur uner asymmetric information. In tis case, efficient bargaining is possible only in te special case in wic α = 0 (favoritism). See Myerson an Sattertwaite (1983). 10 Note tat tis biing function is equivalent to te best take-it-or-leave-it offer tat a supplier wit cost c can make to a buyer wit a ranom valuation in te interval [0,1] given by te istribution function G. Publise by Te Berkeley Electronic Press, 2007 7
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 Te first term is te expecte profit wen te DS wins te auction outrigt witout bribery or favoritism. Te secon term is te expecte profit wen te DS loses te auction (b > b ), but bribes te auctioneer because is costs are below te bi of te HS (b > c). In tis case, te DS retains te sare (1 α) of te surplus (b - c). Te first-orer conition for tis problem is 1 1 b ( b) (4) [ 1 G ( b ( b)) ] α ( b c) g ( b ( b)) = 0. b In orer to ensure tat te first orer conitions (2) an (4) are sufficient to efine te equilibrium biing functions of te HS, b (c), an te DS, (c), we nee some conitions on te istributions G an G. 1 b Lemma: Assume G i, for i=,, is twice ifferentiable an wit a ecreasing 1 G ( ) inverse azar rate i x 1 G ( x). Also, assume tat J g i ( x) (x) = x - is convex. g ( x) Ten an equilibrium of te FPA wit bribery is ( b (c), b (c) ), were b (c) is implicitly efine by (2), b (c) = max { b ) (0, b (c) }, an b (c) is te solution to (4). Te function J (x) is equivalent to wat is known as te "virtual valuation" in te literature on auction teory. Te proof of te lemma is containe in te Appenix 1. 11 Te effects of te bribe sare α on te biing beavior of bot suppliers are easy to caracterize. Proposition 1: Te biing function of te onest supplier, b (c) is inepenent of te bribe sare α. However, te biing function of te isonest supplier, b (c), sifts ownwar as te bribe sare increases. Proof: Te first result is obvious from inspection of te first-orer conition (2) efining b (c). Te secon result follows from ifferentiating te first-orer conition (4) wit respect to te bi b of te isonest supplier an te bribe sare α : 11 Tis lemma an te proof are ue to Ricar P. McLean at Rutgers University. ttp://www.bepress.com/bejte/vol7/iss1/art23 8
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions b 1( b) b ( c) ( b c) g ( b 1( b)) = b α 2π / b2 < 0, 2 2 were π / b is obtaine by ifferentiating te left-an sie of (4) wit respect to b. Tis erivative is negative for interior solutions of (3). QED Te DS woul never bi below te minimum bi of te HS. Oterwise, te DS coul raise is bi witout reucing te probability of winning te auction outrigt. Tus, if te solution to (4) is less tan b (0), te DS simply bis b (0). At te igest cost realization, bot suppliers bi teir cost: b (1) = b (1) = 1. Wen α = 0, te DS bis unity for all cost realizations: b (c) = 1. Tus, wit favoritism, te DS oes not effectively bi for te contract, but is aware te contract wenever is cost is below te bi of te HS. On te oter an, wen α > 0, te DS will bi to win te auction outrigt in orer to avoi paying te bribe: b (c) < 1 for all c < 1. It is easy to sow tat te DS prefers a smaller bribe sare. 12 We can now examine te effect of bribery on te biing strategies of te suppliers. Proposition 1 oes not fully answer tis question because no value of te bribe sare correspons to a fair auction witout bribery or favoritism. Even in te case for wic te DS pays te full surplus as a bribe to te auctioneer (α = 1) an tus bis aggressively to win te contract outrigt wit te low bi, te HS still bis against te cost of te DS, an not against te bi of te DS. Consier te first-orer conitions (2) an (4) for te symmetric case G = G. Te first term of eac conition is te probability of winning te contract outrigt for any given bi. Tis is te incentive to raise te bi because te marginal increase in expecte profit from a iger bi is te probability of winning te contract outrigt. In oter wors, iger bis increase te profit on contracts tat te supplier wins outrigt in te auction. For a given cost realization common to bot suppliers, te first term in (2) for te HS is always less tan te corresponing term in (4) for te DS. Tis follows immeiately from symmetry 1 (G = G ) an te fact tat b ( b) < b. Te secon term in te two first-orer conitions is te isincentive to raise te bi. Higer bis increase te probability of losing te auction an losing te corresponing profit on tose contracts. Te secon term for te DS is multiplie by te bribe sare α because te DS loses 12 For any given cost realization c an bi b, te profit function (3) of te DS is ecreasing in α. As a result, te envelope teorem implies tat a lower bribe sare will increase te expecte profits of te DS. Publise by Te Berkeley Electronic Press, 2007 9
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 only te fraction α of te ifference between is bi an is cost on te contracts for wic is bi excees te bi of te HS. Tus, wen α < 1, te DS as a smaller isincentive to raise is bi, inucing im to bi less aggressively. Tese two forces suggest tat te DS bis less aggressively tan te HS. However, tere is an aitional factor in te secon term of te first-orer 1 conition of te DS: b ( b) / b. Tis factor is te slope of te inverse biing function of te HS. Wen te DS increases is bi marginally above b, tis factor is efine as te aitional cost realizations of te HS for wic te HS submitte a bi lower tan b. For tese contracts wic are no longer won outrigt, te DS losses te fraction α of te surplus wen e obtains tese contracts by a bribe. Tis factor may well be greater tan unity, 13 in wic case te DS experiences a greater probability of losing te auction outrigt wen e increases is bi marginally. Tus, tis factor increases te isincentive of te DS to raise is bi. Tis tir force works against te two previous forces, an is te reason wy tere is no general result tat te DS bis less aggressively tan te HS. We can also compare te biing function of te HS to te biing function of a supplier in a symmetric FPA witout bribery. Te first-orer conition for te biing function of it supplier witout bribery is 1 j (6) [ 1 G ( b ( b)) ] ( b c) g ( b ( b)) = 0, j were j enotes te oter supplier. Comparing tis conition to te first-orer conition (2) for te HS, we see tat te same traeoffs apply. Te first term representing te incentive to raise te bi woul be larger witout bribery. However, te secon term representing te isincentive to raise te bi can also 1 be larger because te factor b ( b) / b may be greater tan 1. Tus, it is not clear in general weter favoritism towar te DS will cause te HS to bi more aggressively tan e woul in a FPA witout bribery. Even if te DS an HS bot a te same cost istribution, tere is no one-sie result on weter te HS woul bi more or less aggressively tan in a FPA witout bribery. For tis symmetric case, Arozamena an Weinscelbaum (2004) an Porter an Soam (2004) ave sown tat convexity (or concavity) of te inverse azar rate of te cost istribution is te conition tat woul etermine weter one or more onest suppliers woul bi more (or less) aggressively against a remaining supplier wo a a rigt of first refusal at te j 1 j b 1 j b ( b) 13 Tis factor will be greater tan unity wenever te margin between bi an cost of te HS, b (c) c, is ecreasing in c. For example, tis will be true wen g (c) is non-ecreasing in c. It will also be true for te family of cost istributions efine in Section 6. ttp://www.bepress.com/bejte/vol7/iss1/art23 10
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions lowest bi of te onest suppliers. 14 In our asymmetric case in wic te HS an DS ave ifferent cost istributions, tis conition is not sufficient to etermine te biing beavior of te HS. In Burguet an Perry (1999), we examine te moel wit bot type I an type II iscretion by te auctioneer. Wit te aition of type I iscretion, te price pai by te buyer is always equal te bi of te HS. Given te form of te bribe, te resulting biing beavior of te DS is solely esigne to reuce te bribery payments to te auctioneer. If te DS must pay te same bribe sare wen type I iscretion is exercise, α (b - b ), te biing function of te DS woul ten be te same as te solution to (4) for α = 1. Tere are also two oter simple cases. If te DS must pay a bribe sare for type I iscretion, but not for type II iscretion, te DS woul bi is igest possible cost realization, b (c) = 1, te same as te solution to (4) for α = 0. As suc, te DS woul only obtain te contract wit te exercise of type II iscretion by te auctioneer. Conversely, if te DS must pay a bribe sare for type II iscretion, but not for type I iscretion, te DS woul bi is cost (or lower) an only obtain te contract wit te exercise of type I iscretion by te auctioneer. 5. Te Effect of Bribery on te Allocation of Contracts Wit bribery, te allocation of contracts is inepenent of te biing beavior by te DS, an te bribe sare. As suc, te allocation is etermine solely by te biing function of te HS. Te biing function of te HS epens on te istribution of costs of te DS, G (see equation (2)). A cange in G tat causes te HS to bi more aggressively reuces te expecte allocative istortion for any given value of c. Wen te ifference between te bi an te cost of te HS is smaller, it is less likely tat te cost of te DS c lies between te two. In fact, if te cange in te istribution of te isonest supplier lowers te inverse azar rate, te HS unambiguously bis more aggressively for any cost realization. Tus, by inspection of equation (2), we obtain te following proposition. Proposition 2: Given te cost realization of te isonest supplier, te allocative istortion from bribery is larger for cost istributions of te isonest supplier wit a lower inverse azar rate [ 1 G ( c) ]. Te allocative istortion is inepenent of te bribe sare α. g ( c) 14 In Section 6, we use te power family of cost istributions to illustrate te effects of bribery on te expecte price pai by te buyer. Tis family of cost istributions as te special property tat te inverse azar rate is linear in te cost. Tus, if te DS an HS ave te same cost istribution witin tis family, te HS woul bi te same in a FPA wit or witout bribery. Publise by Te Berkeley Electronic Press, 2007 11
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 Asymmetric FPAs generate allocative istortions because a supplier wo bis more aggressively will be aware te contract in some situations were is costs are iger tan tat of te rival wo bis less aggressively. As a consequence, te allocative istortion in an asymmetric FPA witout bribery occurs by awaring te contract to te weaker supplier in cases were e as a iger cost. 15 In contrast, an asymmetric FPA wit bribery allocates te contract to te DS wenever te cost of te DS is below te bi of te HS. Since te HS bis above is cost, te FPA wit bribery istorts te allocation by awaring te contract to te DS in cases were e as a iger cost. However, tis istortion occurs irrespective of weter te DS is stronger or weaker tan te HS. Wen te HS is stronger, a FPA wit or witout bribery favors te DS by allocating te contract to im in some cases wen e as a iger cost. However, unlike a FPA witout bribery, te allocative istortion in a FPA wit bribery oes not vanis as te costs of te suppliers approac te lowest possible realization. As in a FPA, an optimal mecanism favors allocating te contract to te weaker supplier. 16 Ten, once again, a FPA wit bribery istorts te allocation in te opposite irection from te optimal mecanism wen DS is te stronger supplier. Moreover, even wen te DS is te weaker supplier, te istortion wit bribery is very ifferent from te optimal istortion. Inee, in orer to reuce informational rents wit te smallest negative impact on efficiency, an optimal mecanism typically istorts te allocation in favor of te weaker DS more at te iger cost realizations for te HS, wereas te istortion isappears as te cost approaces te lowest possible realization. In a precise sense, bribery istorts te allocation in te opposite irection from te optimal allocation mecanism. Inee, if te cost istribution of te DS is caracterize by a monotone inverse azar rate, i.e., if [ 1 G ( c) ] is ecreasing, ten te margin ( c) c is g ( c) ecreasing in c. Tis follows immeiately from (2). Tat is, b ( c) c attains its igest value at c = 0, an ten ecreases to zero as te cost c increases to one. Terefore, wen te cost of te HS is close to 1, te HS almost always wins te contract wenever it is efficient for im to win. However, wen te cost of te HS is close to 0, te DS obtains te contract frequently wen is cost is iger tan te cost of te HS: b ( c ) > c > c. b 15 See Lebrun (1999) an Waerer (1999) for a more general examination of tis issue. 16 See Myerson (1981), an McAfee an McMillan (1989). Rotkopf, Harsta, an Fu (2003) iscuss a moel in wic a particular form of type I corruption can be use by a buyer to subsiize a weaker supplier wit iger costs. Tey fin tat te buyer can benefit by increasing te price pai to te weaker supplier above is winning bi. ttp://www.bepress.com/bejte/vol7/iss1/art23 12
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions Tis iscussion emonstrates tat bribery is not an alternative way of introucing istortions in te allocation of te contracts tat will reuce te rents of suppliers an reuces te expecte price pai by te buyer in line wit optimal mecanisms. 17 Tis notwitstaning, te next section sows tat bribery nee not result in a iger expecte price pai by te buyer. 6. Te Effect of Bribery on te Expecte Price Pai by te Buyer In tis section, we examine te effect of bribery on te expecte price. One migt expect tat te expecte price pai by te buyer woul increase wit bribery. First, bribery inuces non-optimal allocative istortions. Secon, te bribery payment to te auctioneer is a tax on te procurement transaction. One migt also expect tese problems to be more acute wen te bribe sare pai to te auctioneer is iger. For several reasons, we will sow tat tese results nee not obtain. First, bribery may inuce te HS to bi more aggressively. Secon, from Proposition 1, te DS bis more aggressively wen te bribe sare is iger. In fact, we can sow tat Proposition 3: Te expecte price pai by te buyer is lower wen te bribe sare α is larger. Proof: Notice tat te bi of te HS is inepenent of α. On te oter an, from b ( ) our previous lemma, we foun tat c < 0 for c < 1. Since te price pai by α te buyer is simply te min{ b, b}, te expecte price is lower wen α is larger for any realization of te cost c. QED. Now, if we again consier equations (2) an (4), we notice tat for α = 1, (4) is te same first-orer conition tat DS solves wen competing in a FPA witout bribery. Equation (2), on te oter an, is te biing function of a supplier competing against a virtual competitor tat bis its cost. We know tat te HS nee not bi uniformly more aggressively tan in a FPA witout bribery. Even if te HS oes bi more aggressively, te DS nee not bi uniformly more aggressively. However, bot te HS an DS coul bi more aggressively over a 17 Wen suppliers ave uniform istributions wit ifferent lower bouns on teir omains, Lee (2004) as sown tat awaring a rigt of first refusal (α=0) to te weaker supplier, introuces precisely te type of istortions tat an optimal auction introuces, an may result in a lower price for te buyer as well. Publise by Te Berkeley Electronic Press, 2007 13
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 sufficient range of costs suc tat te expecte price pai by te buyer woul ecline. In orer to examine tis possibility, we efine te one-parameter power family of istribution functions G(c;t) = 1 [1 c] t over te support [0,1] were c is te cost an t > 0 is a parameter wic can vary between te suppliers. 18 Te corresponing ensity function is g(c;t) = t [1 c] t 1. For iger t, G(c;t) is iger an [ 1 G ( c) ] is lower. Te supplier wit a iger t is stronger in te g ( c) sense of bot first-orer stocastic ominance an azar rate ominance. Let t efine te cost istribution of te HS an t efine te cost istribution of te DS, an enote G (c) = G(c;t ) an G (c) = G(c;t ). Te equilibrium biing function of te HS from (2) as te following linear form: 1 t (7) b ( c) = + c. 1+ t 1+ t Te equilibrium biing function of te DS from (4) is sligtly more complicate. It takes te following form: 1 α t (8) b ( c; α) = + c for c> c, 1+ α t 1+ α t 1 = 1 + t, for c c, α t t were c = max{0, }. Figure 1 epicts representative biing α t (1 + t ) functions of te suppliers for te two expressions of c in (8). If te two suppliers are symmetric witin tis power family of cost istributions (t = t = t ), ten te HS bis just as e woul in a FPA witout bribery. 19 Te DS bis uniformly less aggressively tan te HS for all bribe sares α < 1. However, if te auctioneer extracts all te surplus from te DS (α = 1), ten bot suppliers bi just as tey 18 Waerer an Perry (2003) sow tat tis istribution function is not as restrictive as it migt seem. Distributions of tis form follow irectly from natural properties, particularly a property corresponing to constant returns to scale. 19 Tis fining follows from tat fact tat te inverse azar rate is linear for te power family of cost istributions. See Arozamena an Weinscelbaum (2004) an Porter an Soam (2004). ttp://www.bepress.com/bejte/vol7/iss1/art23 14
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions woul in a symmetric FPA witout bribery. In Appenix 2, we generalize tese biing functions to te case of multiple symmetric onest suppliers. b(c;0) = 1 1 b(c;1/9) α b(c) b(c;1/3) b(c;1) c' Figure 1: Biing Functions t = 1 an t = 3 α = {0, 1/9, 1/3, 1} 1 c We can now examine te expecte price wit tis family of istributions. We first compare te expecte price in te FPA wit bribery to te expecte price tat woul arise wit an efficient auction, like te secon-price auction (SPA). 20 Let Ep( α ; t, t ) be te expecte price in te FPA wit bribery, an Ep ( t, t e ) be te expecte price in an efficient auction. See Appenix 2 for te expressions of tese expecte prices. Te following proposition provies comparisons of tese two expecte prices. Proposition 4: (i) Wen t > t, te expecte price in a first-price auction wit bribery can be below te expecte price in a secon-price auction. In particular, 20 Witout bribery, te allocation of te contract virtually etermines te total surplus tat can be ivie between te buyer an te suppliers, an ow it is ivie. Tis is an implication of te Revenue Equivalence Teorem. See Myerson (1981). Tus, since a SPA allocates te contract to te lowest cost supplier, te expecte price in a SPA provies te natural reference point for te ivision of total surplus in any efficient auction. Publise by Te Berkeley Electronic Press, 2007 15
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 tere exists a set of bribe sares (α,1] suc tat for any α in tis set, Ep t, t ) > Ep( α; t, t ). e ( (ii) Wen t t, te expecte price in a first-price auction wit bribery is above te expecte price in a secon-price auction. In particular, Ep e t, t ) Ep( α; t, t ) for all α 1, an Ep e ( ( t, t) Ep( α; t, t < ) for t t or α < 1. Te proof of Proposition 4 is containe in Appenix 2. Appenix 2 also provies a generalization of Proposition 4 to te case of multiple symmetric onest suppliers. Proposition 4 states tat, for tis power family of cost istributions, te istortions tat bribery introuces (iscusse in Section 4) elp reuce te price pai by te buyer wen te DS is te stronger supplier. As we iscusse before, te optimal rent-reucing istortion woul favor allocation of te contract to te weaker supplier in orer to reuce te information rents of te stronger supplier. However, allowing te opportunity for te stronger supplier to bribe te auctioneer can also acieve te ultimate goal of a lower expecte price. Tis may seem paraoxical. Te explanation is tat, wen te auctioneer appropriates part of te rents in te form of bribe, te buyer soul also be concerne tat tese rents will cause an increase in te expecte price. As a result, te auctionteoretic evice of ecomposing te expecte price into te sum of te expecte cost an te information rents is less elpful in analyzing te effect of istortions on expecte prices in auctions wit bribery. Instea we soul consier te expecte price irectly as te winning bi. By oing so, we realize tat te only avantage of bribery for te buyer is tat te HS faces fiercer competition in tat e must beat te cost of te DS instea of te "bi" of tis DS. 21 Consier te case α = 1. If te DS is te weaker supplier, bribery can only ave a small impact on te bi of te stronger HS. Te reason is tat te "bi" of te weaker DS is very close to is cost. Tus, competing against te cost of te DS instea of is "bi" provies very little incentive for te stronger HS to bi more aggressively. Te situation is reverse if te DS is te stronger supplier. In an (efficient) auction tere is a large gap between te cost an te "bi" of te stronger DS. Tus, te competition face by te weaker HS becomes muc touger wen e as to beat te cost of te DS instea of te "bi" of te DS. Tis provies a stronger incentive for te weaker HS to bi more aggressively. 21 Since we are iscussing te comparison to an efficient auction ere, "bi" soul be unerstoo as te expecte price conitional on winning. ttp://www.bepress.com/bejte/vol7/iss1/art23 16
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions As in Section 4, Proposition 4 illustrates te irect effect of bribery on ow aggressive te DS bis. But tere is also a "slope effect". Te cange in te slope of te biing function of te HS as less intuitive effects on te biing function of te DS. Te importance of tese effects is clearer wen we consier te effects of bribery on a FPA. Proposition 4 only compares te asymmetric FPA wit bribery to an efficient auction wit no allocative istortions. However, it is also interesting to compare te expecte prices in an FPA wit an witout bribery. Let Ep 1( t, t ) be te expecte price pai by te buyer in a FPA witout bribery. Tis comparison is straigt-forwar for te symmetric case. Wen α < 1, Ep( α ; t, t) > Ep ( 1; t, t) = Ep 1( t, t) = Ep 2 ( t, t). For te asymmetric case, te comparison is more ifficult because te asymmetric FPA witout bribery oes not ave analytic solutions for te biing functions even wit tis power family of cost istributions. Neverteless, using numerical computations, we can sow tat bribery may increase or ecrease te expecte price pai by te buyer in a FPA wen α is close to 1. Wen t = 4 an t = 1, te expecte price pai by te buyer is.4943 in a FPA witout bribery. (See Marsall, Meurer, Ricar, an Stromquist,1994.) Alternatively, wit bribery an α = 1, te expecte price is.4958. Tus, bribery increases te expecte price pai by te buyer, even toug α = 1. However, wen t = 3 an t = 2, te expecte prices in a FPA witout an wit bribery are.4125 an.4122, respectively. Tus, bribery reuces te expecte price for tis case. Notice tat bot cases are in te region t > t. Tus, even toug te stronger supplier is te DS in bot cases, te effect of bribery on te expecte price can ave eiter sign. We can furter illustrate te comparison using te Bicomp 2 program of Li an Riley (1999) to compute te asymmetric FPA witout bribery. Figure 2 illustrates te expecte price wit an witout bribery for values of te capacity parameters (t,t ) wit α = 1. Publise by Te Berkeley Electronic Press, 2007 17
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 8 t 6 Ep(α=1) > Ep 1 4 Ep(α=1) < Ep 1 2 * (3,2) 3 (4,1) * 4 Ep(α=1) > Ep 1 6 8 t Figure 2: Expecte Prices First-Price Auction wit (α = 1) an witout Bribery Te expecte prices are equal along te iagonal (a property special to te power family of cost istributions), an te expecte price wit bribery is always greater wen t < t. However, wen t > t, te expecte price wit bribery is lower wen t is only somewat larger tan t. Te expecte price wit bribery is again iger in te lower rigt region were t is substantially greater tan t. Te previous examples from Marsall, Meurer, Ricar, an Stromquist(1994) are marke by asterisks in Figure 2. 7. Conclusion We ave examine te effects of bribery on te beavior of suppliers an te outcome of a FPA. In te particular form of bribery tat we ave consiere, one supplier bribes te auctioneer in orer to revise is bi ownwar wen tis is necessary to win te contract an profitable. We ave sown tat tis form of bribery nee not make te isonest supplier bi less aggressively or te onest supplier bi more aggressively. Bribery istorts te allocation of te contract. In cases were te isonest supplier is te ex ante stronger supplier, bribery reverses te istortion tat woul exists in a FPA witout bribery. Even wen te isonest supplier is ex ante weaker, bribery alters te allocation of te contract in funamental ways. In particular, an inefficient allocation of te contract to te ttp://www.bepress.com/bejte/vol7/iss1/art23 18
Burguet an Perry: Bribery an Favoritism by Auctioneers in Seale-Bi Auctions weaker isonest supplier occurs wit ig probability even wen te cost of te stronger onest supplier is very low. For similar reasons, tese properties of te allocation in te FPA wit bribery contrast sarply wit te allocation inuce by optimal mecanisms. Te effects of bribery on te expecte price pai by te buyer are subtle. Wen te auctioneer appropriates a large fraction of te surplus, te price for te buyer may in fact be lower tan in te absence of bribery. Tis in fact occurs wen te stronger supplier is te isonest supplier wit te opportunity to bribe te auctioneer. Tere are oter interesting questions tat coul be aresse wit variants of tis moel. We ave use some to analyze te effect of bribery on te incentives to invest, an te resulting implications for trae (Burguet an Perry (2006)), an te incentives for te sale of preference by a buyer in te form of a rigt of first refusal (Burguet an Perry (2003, revise 2005)). Despite its simplicity, te moel igligts subtle but intuitive consequences of auctioneer iscretion in a wie range of contracting scenarios. Appenix 1: Secon-Orer Conitions Monotonicity of te inverse azar rate implies (inverse) monotonicity of J (b). Te first orer conition (2) can be written as: g ( b) [ c J ( b)] = 0. Te erivative of te left-an sie is g' ( b) [ c J ( b)] g ( b) J ' ( b), wic, using te first-orer conition, sows tat te secon-orer conitions are satisfie uner our assumptions. Tus, (2) efines an interior solution of (1). Let b ( c) = J 1 ( b) as implicitly efine in (2). Note tat since J (b) is increasing an convex, b (c) is increasing an concave. Now (4) can be written as α( b c) 1 G ( J ( b)) g ( J ( b)) = 0. b' ( J ( b)) g ( J ( b)) As before, te secon-orer conitions are satisfie if te term in brackets is increasing in b. Since J (b) is increasing, it suffices to sow tat α( b ( x) c) 1 G ( x) b' ( x) g ( x) Publise by Te Berkeley Electronic Press, 2007 19
Te B.E. Journal of Teoretical Economics, Vol. 7 [2007], Iss. 1 (Contributions), Art. 23 is increasing in x. Te secon term is ecreasing, an te first term is increasing since b (c) is concave. Tus, secon-orer conitions are satisfie. Te Lemma follows immeiately. Appenix 2: Proof of Proposition 4 Using te biing functions from Section 6, we can calculate te expecte price pai by te buyer in te FPA wit bribery: t 1+ t 1+ t + t t 1+ t α t (A1) Ep( α; t, t ) = (1 + t )(1 + t ) (1 + t )(1 + t + t ) t t 1+ α (A2) for α t (Case 1), t t t 1 t t 1+ α t Ep( α, t, t ) = + (1 + t ) (1 + t )(1 + t + t ) t t 1+ α for α t (Case 2). t Also, we can calculate te expecte price in a SPA: (A3) ( 1+ t + t t t Ep2 t, t ) =. (1 + t )(1 + t ) (1 + t )(1 + t )(1 + t + t ) Proof of Proposition (4): (i) Wen t > t, expression (A1) for te expecte price applies over te entire range of te bribe sare of α 1. Tis expecte price is continuous an eclines as α increases over tis range. At α = 0, tis expecte price is clearly greater tan te expecte price uner a SPA from (A3). However, at α = 1, te conition tat t > t implies tat tis expecte price is lower tan te expecte price uner a SPA. Tus, tere exists an α suc tat for all α >α, bribery woul result in a lower expecte price. Te value of α can be easily efine as t 1/(1+ t ) 1 Q t t α =, were Q =. t 1 Q 1 t 1 t + + Note tat α nee not be close to unity. If t = 1, ten α =.89 wen t = 2,α =.75 wen t = 3, α =.67 wen t = 4. It is also easy to sow tat α is ecreasing in t. (ii) Wen t < t, expression (A1) for te expecte price applies only for α t /t. But uner te conition tat t < t, tis expecte price is greater tat te expecte ttp://www.bepress.com/bejte/vol7/iss1/art23 20