Department of Eonomis and Institute for Poliy Analysis University oftoronto 15 St. George St. Toronto, Canada M5S 1A1 WORKIG PAPER UMBER UT-ECIPA-PETERS-97-2 Surplus Extration and Competition Mihael Peters Department of Eonomis University oftoronto 15 St. George St. Toronto, Canada M5S 1A1 First version June 1-1997 this version April 7, 1998 Copyright 1998by Mihael Peters Department of Eonomis University oftoronto ISS 829-499 Author's email: peters@hass.utoronto.a On-line Version: http://www.epas.utoronto.a:88/eipa/wpa.html
Abstrat A ompetitive eonomy is studied in whih sellers oer alternative diret mehanisms to buyers who have orrelated private information about their valuations. In ontrast to the monopoly ase where sellers harge entry fees and extrat all buyers surplus, it is shown that in the unique symmetri equilibrium with ompetition, sellers hold seond prie autions with reserve pries set equal to their ost. Most important, it is a best reply for sellers not to harge entry fees of the kind normally used to extrat surplus, even though it is feasible for them to do so. Journal of Eonomi Literature Classiation umbers: D82,D83,D44 I would like to thank a referee, Carolyn Pithik and seminar partiipants at Rohester, the University of orth Carolina, Washington University and Laval University for helpful omments. The author gratefully aknowledges the nanial support of the Soial Sienes and Humanities Researh Counil of Canada.
1. Introdution In a ompetitive environment, it seems natural that mehanism designers might onsider oering `softer' mehanisms that are less eient at extrating buyer surplus if they thought that this would attrat more buyers. This trade-o between surplus extration and partiipation is entral to the argument in Bulow and Klemperer [1] who show that in an independent information environment, sellers are better o using a simple seond prie aution without reserve prie than they are using an `optimal' negotiation mehanism if the `optimal' mehanism is sure to sare away one buyer. This result is true despite the fat that the deviation that sellers are fored to make from the optimal mehanism to keep the marginal buyer is `large' in any onventional sense. In the independent value ase disussed in [1], the ost of attrating buyers is modelled in a very speial way. The ost to the seller of attrating one more buyer is that instead of using an optimal reserve prie, he must use a reserve prie equal to his ost. This beomes more omplex in the ase where buyer valuations are orrelated, sine the optimal mehanism then extrats all the buyer surplus (MAfee and Reny [8] and Cremer and Malean [3]). Then, reverting to a simple seond prie aution with a zero reserve prie is extremely ostly to the seller and will not generally be warranted if all it aomplishes is to attrat one additional buyer. 1 This is unfortunate sine this trade-o between partiipation and surplus extration, at least as it is expressed by [1], suggests that it might pay sellers to use relatively simple mehanisms to attrat buyers. This in ontrast to the omplex belief dependent mehanisms predited by the theory of mehanism design. Indeed, [8] go so far as to suggest that the full surplus extration results for the orrelated value environment are evidene of the irrelevane of mehanism design sine the omplex random entry fees that they use to extrat the surplus have yet to be observed in pratise. The spei trade-o used in [1] may go too far, but the key idea remains. Even when valuations are orrelated, it is lear that if the deviation from optimality that is required to attrat one more buyer is small enough, sellers will make it. Obviously, this will mean that the stark preditions of the MAfee Reny theorem will have to be modied. The problem that remains is to gain some more 1 Reasonable restritions on the lass of feasible mehanisms (for example requiring that only the winning bidder in an aution be fored to pay the seller any money) an be used to resurret the theorem. 3
preise understanding of what the trade-o between surplus extration and buyer partiipation is. This paper analyzes the trade-o when the ontrat market is ompetitive. The result desribed for this environment is striking. Despite the fat that sellers an oer arbitrarily omplex (diret) mehanisms, in the unique symmetri equilibrium, sellers oer simple seond prie autions without reserve pries (i.e., with reserve pries equal to their osts). The notable thing about this result is that sellers do not use the ompliated entry fees of the kind desribed by [8] in equilibrium despite the fat that they ould do so, and despite the fat that buyers' valuations are orrelated. The results in the paper suggest that ompetitive ontrat markets have some interesting and useful properties. In terms of the trade-o between partiipation and surplus extration, [1] argue that seond prie autions with zero reserve pries are better than optimal mehanisms if reverting to them attrats just a single additional buyer. They explain why their result annot be true in the orrelated value environment. The ompetitive results given here show the sense in whih their intuition extends to the orrelated value ase. The equilibrium mehanism here is the zero reserve prie aution emphasized in [1], and as they suggested, omplex surplus extration tehniques are not used when sellers understand the surplus-partiipation trade-o.. As in the previous literature on ompetitive ontrat markets ([7, 11, 1]), the equilibrium mehanisms have a remarkable robustness property. They are invariant to the number of buyers and seller partiipating, and to buyers' and sellers' beliefs about the distribution of types of the other partiipants in the market. Prior papers on ompetition are restrited to situations in whih valuations are independent. In this sense, the results presented here extend this robustness property to markets where valuations are orrelated. Thus ompetition among sellers seems to be enough to explain why simple mehanisms are observed aross a spetrum of informational assumptions. Finally, the results presented here provide some vindiation for the methods and approah involved in the theory of mehanism design. The elegant results of MAfee and Reny presents a dilemma for mehanism design sine optimal mehanisms are omplex and highly sensitive to unobservable beliefs. Sine no ontrats have ever been observed that even remotely resemble 'optimal' ontrats, MAfee and Reny themselves onlude that their results argue for the irrelevane of mehanism design. 2 As will be seen below, the methods of mehanism design 2 It should be emphasized that it is not the fat that optimal ontrats extrat all the surplus 4
work very well in ompetitive ontrat markets, and are entral to the proof of the main result. Despite this, the equilibrium ontrat is an aution. Thus even the most ardent believer in 'diret mehanisms' an be satised with the apparently ad ho restrition to autions that is made in muh of the theoretial literature. A ompetitive ontrat market is one where sellers take the payo that they need to oer buyers to attrat them to be xed. This idea 3 is natural in the setting examined here sine pries are speied as part of the ontrat. It has a number of advantages. First, though sellers need to know the number of potential buyers, they do not need to know preisely who these buyers are. This is harateristi of markets like real estate where sellers have good information about aggregate demand, but little information about the identities or tastes of individual buyers. Seondly, sellers do not need to know exatly who their ompetitors are, or even to know what pries their ompetitors have oered (though it is important that they believe that buyers know these pries). This is important sine ompetition may be relevant even in autions or sales that appear to be isolated. For example a privately held ompany might try to sell o a ontrolling interest in the ompany to some group of potential investors. The sale appears unique, but the potential investors have many alternative investment opportunities that the seller only poorly understands. Many of the alternative opportunities will be atively ompeting for the investors' money. The trade-o between surplus extration and partiipation is aptured in the following way. Eah seller observes (or formulates a onjeture about) the market payo. The seller then believes that he an ahieve any partiipation-surplus ombination having the property that every buyer type who the seller expets to partiipate reeives at least the market payo. The trade-o then behaves muh like a standard demand urve whih an be interpreted as a desription of the quantity that the seller an expet to sell at dierent pries. Tougher mehanisms make it less likely that buyers will partiipate in the same way that higher pries are expeted to redue demand. The properties of this simple tradeo are preisely what makes it possible to exploit the standard tehniques of the theory of mehanism design to get preise preditions about the mehanisms that sellers will use. The results of the paper follow from one key property - that sellers' best replies to any market payo must involve an 'eient' mehanism that awards that reates the problem, it is the fat that optimal ontrats are suh omplex objets (see below) that reates problems. 3 This idea is due to Gale [5] and is now widely used in labor eonomis, for example [9]. 5
the ommodity to the buyer with the highest valuation among all the buyers who visit him. Ironially, the MAfee-Reny argument is ruial to this. To see how, observe rst that the seller's prots are equal to the expeted gain to trade generated by his mehanism, less the expeted surplus buyers get by partiipating in it. In the ompetitive environment, the expeted surplus that buyers get by partiipating in the seller's mehanism must be the same as what they an get elsewhere on the market. ow itiswell know that an eient mehanism will maximize the expeted gains from trade provided partiipation is exogenously xed. So onsider a seller who is using a soft (ineient mehanism). When he deviates to an eient mehanism he ould ompensate buyers (provided valuations are orrelated) by implementing a MAfee Reny partiipation fee that generates an expeted payment (onditional on the buyer's type) exatly equal to the dierene between the expeted surplus before and after the deviation assuming onstant partiipation. The deviation along with this new partiipation fee prevents buyers' surplus from hanging, and therefore maintains partiipation. The expeted surplus omponent of the seller's payo inreases, while the surplus that he oers buyers does not hange - so the deviation is protable. It is then immediate that there will be an equilibrium in whih sellers use seond prie autions with zero entry fees. Suppose that all but one seller is oering a seond prie aution with zero entry fee. Then when they visit the remaining seller, buyers market payo is the payo that they get from partiipating in a seond prie aution with no entry fee. The remaining seller believes that there is nothing he an about this, so he will simply hoose a mehanism to maximize expeted gains from trade. Sine the seond prie aution without entry fees will aomplish this latter task and give buyers the market payo, there is no inentive for the remaining seller to deviate from this strategy. Perhaps the more remarkable fat is that among symmetri equilibria, this outome is unique. It is diult to give a good intuitive explanation for this result, however, the assertion that an equilibrium is symmetri along with the result that sellers must use eient mehanisms in equilibrium ompletely determines the distribution of buyer types that eah seller faes (as is shown below). The seller an hange this distribution by modifying his mehanism if he likes, so to verify uniqueness it is suient to show there is a unique market payo funtion for whih the distribution implied by symmetry and eieny of mehanisms is atually a best reply to this market payo. The ontribution here is not that sellers will be unable to extrat all the surplus 6
when there is ompetition (or more generally when partiipation is endogenous). That muh is obvious. The ontribution here is that ompetition generates equilibrium mehanisms that are muh simpler than optimal mehanisms. Ironially, the key to the simpliity of equilibrium mehanisms is preisely the fat that sellers an imagine oering arbitrarily ompliated mehanisms, speially random entry fees of the kind desribed by MAfee and Reny ([8]). On the equilibrium path mehanisms will be very simple, but o the equilibrium path they need not be. Thus the tehniques of mehanism design are ritial to understanding the result. On the other hand, sine sellers' best reply to the market is to oer simple seond prie autions, the paper also justies the extensive analysis of autions in environments where they are known to be non-optimal mehanisms. 2. The Model 2.1. Primitives There is a nite number J of sellers, eah of whom owns a single unit of a homogenous and indivisible ommodity. Seller j has a ost represented by a real number j. It will be assumed that the sellers' osts are all ommon knowledge though the model ould be extended to allow this to be private information. There are buyers partiipating in the market. Buyers all have private information haraterized by their valuation. The valuation of buyer i is x i 2 [; 1]. The ex post payos that buyers and sellers get depend only on their own information and both are assumed risk neutral. Thus a buyer of type x who trades with a seller with ost at a prie p gets surplus x, p. In the same situation the seller gets p,. 2.2. Prior Beliefs Let F denote the joint distribution funtion for buyer valuations. Two assumptions are made about F. Assumption 2.1 There is a random variable y distributed aording to G on the interval [; 1] suh that x 1 ;:::x are identially and independently distributed onditional on y. This assumption is used primarily to larify the role that the buyers' hoie strategy plays in determining beliefs. It appears that this assumption ould be 7
weakened, but the extension is not trivial. Formally, it implies that there is a onditional distribution ~ F (jy) suh that F (x 1 ;:::x )= Y i=1 ~F (x i jy) dg (y) It is assumed heneforth that F has a ontinuous and stritly positive density denoted by f (x 1 ;:::;x ). Assumption 2.2 For eah x 1 and any probability measure on the Borel sets of [; 1] satisfying (fx 1 g) 6= 1 f(x 2 ;:::x jx 1 )6= f(x 2 ;:::x jx)d (x) Assumption 2.2 is the neessary and suient ondition for (approximate) surplus extration given by [8]. In this paper, it's signiane is that it implies that for any ontinuous funtion (x) on[;1], and any ">, there are nitely many funtions y i (x 2 ;:::;x ) for i =1;:::I suh that sup (x), min i y i (x 2 ;:::x )df (x 2 ;:::x jx) <" uniformly in x. This is the method that MAfee and Reny use to extrat buyers' surplus. In their approah, (x) is the payo that a buyer of type x reeives by partiipating in the seller's mehanism and the z i are a series of paymentshedules from whih the buyer is allowed to hoose. It is not hard to onstrut funtions that satisfy both assumptions 1 and 2. For example, suppose that " i for i =1;::: and y are independently and uniformly distributed on [; 1] and let x i = " i y. Then F (x i jy) =Prf~x x i j~y= yg= Pr f~" ~y x i j~y = yg =Prf~"x i =yg = ( x i =y if x i y 1 otherwise and G (yjx) =Prf~yyj~x=xg=Prf~yyj"y=xg=Prf"x=yg = ( 1, x=y if x y otherwise 8
Then using the densities dened by these funtions, we have ~f (xjy) dg (yjx 1 ) Y f(x;:::xjx )j = x =x i=2!,1! 1 x = dy max(x;x ) y y 2 It is straightforward to show that f (x;:::xjx ) <f(x;:::xjx) whenever x 6= x (for x < xthe expression is linear in x, while straightforward dierentiation veries that the expression is stritly dereasing as x inreases above x). This implies that if u is not a point mass at x, f (x;:::xjx)> f(x;:::xjx )d (x ) Unfortunately, MAfee and Reny's theorem annot be applied diretly to the ompetitive environment beause beliefs in the ompetitive environment are not absolutely ontinuous (with respet to Lebesque measure) and beause beliefs are endogenous and depend on the mehanism that the seller oers. 2.3. The market proess The market game 4 that determines all trades now proeeds. At the beginning of the game, eah seller oers a diret mehanism to buyers. Mehanisms are desribed in more detail below. They speify how the seller will determine a prie and trading partner (if there is one) among the buyers who visit him. The outome will depend on messages that the buyers send to the sellers. For example, the seller ould promise to hold an aution with a xed reserve prie. Alternatively he ould simply oer his unit of output at a xed prie One the buyers see the mehanisms that are being oered by sellers, they selet one and only one seller as a potential trading partner. After buyers selet sellers, they ommuniate with the sellers they have hosen as speied by the seller's announed mehanism. Trades and payments our, then the game ends. Eah seller's problem is similar to any stati mehanism design problem exept for the fat that buyers have many alternative sellers to whom they might turn, and sellers reognize this 5. 4 It is important to note that despite the fat that the trading proess is desribed as a game, the solution onept that is employed is not ash. The solution uses ompetitive ideas that resemble the prie taking assumptions of market theory. 5 It is not diult to onvert this problem to a dynami one in whih buyers and sellers who 9
2.4. Mehanisms It is assumed here, as in the rest of the literature 6, that sellers oer buyers anonymous diret mehanisms : T!fR[; 1]g +1 where T [; 1] [fx ; g and x ; is a message that tells the seller that the buyer deided not to partiipate. These mehanisms have the property that eahvetor of messages determines a sequene of transfers and trading probabilities fp i ;q i g, one for eah buyer i =1;:::. These mehanisms should be inentive ompatible and satisfy a number of obvious restritions (like the transfer assigned to a buyer who signals that he is not partiipating must be zero). Sine these restritions will be evident in ontext, disussion of them will be suppressed and referene will be made to the set, of feasible (though not neessarily inentive ompatible) anonymous diret mehanisms. Armed with this notation, it is possible to dene the buyers' seletion strategies i :[;1], J! n 2 R J+1 : j ; P J j= j =1 o.inwords, for eahvaluation x and eah array of mehanisms that are oered by the sellers the seller hooses the probability j i with whih he will hoose to partiipate in the mehanism oered by seller j. The notation i is interpreted as the probability with whih the buyer hooses to not to partiipate in any mehanism at all. In what follows attention will be foussed on symmetri ontinuation equilibria in whih buyers all adopt the same mixed strategy. This seletion aptures the buyers' inability to predit eah others' partiipations deisions. This inability to oordinate is an important aspet of all deentralized mathing models. This symmetry assumption ensures that the distribution of types faed by any partiular seller will satisfy the exhangeability requirements of Assumption 2.2. Exhangeability is a reasonable defense of anonymity in a monopoly problem. In a ompetitive environment, anonymity is a strong assumption. It prevents a seller from piking out a partiular buyer and promising to treat him well if he partiipates in the seller's mehanism. The solution onept used in this paper does not require that sellers be able to predit individual buyer strategies. In this sense, it is onsistent with an environment in whih sellers have little idea about who the buyers are and how toontat them (though sellers ertainly need to know something about the number of potential buyers). In suh anenvironment, fail to trade right away, an try to do so again in the following period. This extension is straightforward provided it is reasonable to assume that buyers and sellers believe that their urrent ations have no impat on future payos (whih is reasonable in the large game environment that we study here). 6 For example MAfee [7], Peters [?] orpeters and Severinov [11]. 1
a restrition to anonymous mehanisms is reasonable. The symmetry and anonymity assumptions together will ensure that it is suf- ient to onsider the problem from the point of view of buyer 1. 2.5. Seller's Beliefs Let j () denote the joint distribution of types that seller j expets to fae. The proess by whih buyers hoose among the various sellers imposes some important struture on seller's beliefs. This struture is used in the proof of the main theorem below. By assumption, buyer valuations are independently and identially distributed onditional on some outside variable, y. Thus onditional on y and the seletion strategy used by all buyers inluding buyer 1, the probability that buyer 1 either has a valuation below x or hooses not to partiipate in the mehanism is given by 1, R 1 x j(s) f(sjy)ds. ~ 7 Thus any equilibrium belief j () should have the property that there is a hoie strategy j suh that j (x 1 ;:::x n )= Y 1, j (s) d F ~ (sjy) i=1 x i dg (y) (2.1) Denition 2.1. A belief funtion j that satises (2.1) will be said to be admissible. One property of (2.1) that bears mention is that j (;:::) = Y i=1 1, j (s) d ~ F (sjy) dg (y) > whenever j (x) < 1 on a set of non-zero measure. This means that the distribution j will typially have atoms at zero. This simply reets the fat that the seller should believe that a lot of potential buyers just won't turn up at all with high probability. One nal denition is required for the argument below. Assuming that j is dierentiable, the density funtion is given for every vetor (x 1 ;:::x ) 2 [; 1] by z j (x 1 ;:::x )= Y i=1 j (x i ) ~ f (x i jy) dg (y) 7 Here we suppress the fat that j depends on the array of mehanisms on oer to ease the notation. 11
From this desription, it is apparent that a buyer of type x i partiipates in seller j's mehanism with positive probability if j (x i )> or equivalently if This suggests the following denition z j (x i ;x i; :::x i )> Denition 2.2. A buyer of type x i is expeted to partiipate in seller j's mehanism if (x i ;x i; :::x i ) is in the support of j. Alternatively, symmetry in the hoie strategy and the fat that the x i are i.i.d onditional on y guarantees that all the marginal distributions of j are the same. Hene it is legitimate to say that x i is expeted to partiipate in seller j's mehanism if x i is in the support of the marginal distribution for j. 2.6. Payos Let j =(p j ;q j ) denote seller j's mehanism. Fix a ommon seletion strategy for buyers. This implies a distribution of buyer types at seller j given by some xed j. The expeted payo that buyer 1 gets by hoosing to partiipate in the mehanism j depends on his valuation x 1, and his beliefs about the other buyers, given by j (x 2 ;:::x jx 1 ). Say that a pair ( j ; j )isinentive ompatible if j is inentive ompatible when eah buyer who partiipates in seller j s mehanism has beliefs j (jx). The payo that buyers expet from the inentive ompatible pair ( j ; j ) is given by j (x 1 ; j ; j )=E x2 ;:::x jx 1 fq 1j (x 1 ;x 2 ;:::x )x,p 1j (x 1 ;x 2 ;:::x )g (2.2) where the expetation is taken using the onditional distribution assoiated with j. Taking j to be the xed belief by seller j about the distribution of types that he faes, the surplus that seller j enjoys from the inentive ompatible mehanism j is equal to j ( j ; j )= E x fp 1j (x 1 ;:::x ),q 1j (x 1 ;:::x ) j g = fe x f[x 1, j ]q 1j (x 1 ;:::x )g,b j (x 1 ;:::x )g 12
where = fe x f[x 1, j ] q 1j (x 1 ;:::x ), j (x 1 ; j ; j )g (2.3) B j (x 1 ;:::x ) q 1j (x 1 ;:::x )x 1,p(x 1 ;:::x ) is the expeted payo of a buyer of type x 1 given the valuations of the other buyers. 3. Equilibrium Mehanisms When a monopoly seller raises prie he weighs the gain he gets by reduing the surplus of buyers who ontinue to purhase from him at the higher prie, against the lost prots from buyers who deide that they no longer wish to trade with him at the higher prie. When there is ompetition the gain is mitigated beause the existene of alternative suppliers limits the seller's ability to extrat the surplus of buyers with higher types. In equilibrium, buyers in large markets an always nd some other seller who is willing to oer the good at the original prie. Thus if the seller wants to trade with buyers he must oer them the market prie, or equivalently he must oer them the same payo that they an get elsewhere in the market. This leads to the following denition: Denition 3.1. A ompetitive equilibrium in mehanisms is a payo funtion (), an array ofmehanisms f 1 ;:::; Jg and a symmetri buyer strategy = ( ;:::; J) suh that 1. for eah j and eah (x 1 ;:::x ) j (x 1 ;:::x )= Y i=1 1, x i j (s) f (sjy) ds dg (y) 2. for eah j =1;:::J, j ; j is an inentive ompatible pair and j j ; j j ( j ; j ) for any inentive ompatible pair ( j ; j ) satisfying j (x; j ; j ) (x) for eah type x who is expeted under j to partiipate in seller j's mehanism. 3. for all x 2 [; 1] h i (x) = max ;j x; j j=1;j ; j 13
ote that the denition of j along with ondition (3) ensures that the buyers' ommon strategy is a ontinuation equilibrium strategy for the subgame in whih the buyers hoose among the sellers. The market payo is the ontinuation equilibrium payo assoiated with this. A seller who onsiders deviating takes the market payo to be xed in alulating his best reply aording to (2). The market payo plays a role analogous to a market prie whih the sellers believe isbeyond their ontrol. Hene the title 'ompetitive equilibrium in mehanisms". As this market payo adjusts to 'lear the market', sellers oer mehanisms to maximize prots against this xed market payo. 4. Charaterization To begin, the main result of MAfee and Reny is adapted to the model presented in this paper. The proof follows the argument in their paper losely, but needs to be adapted for the atoms that the distributions j have when any of their arguments are zero. Theorem 4.1. Suppose that the joint distribution F of types satises Assumption 2.2. Let j be any mehanism for rm j and let j () be any admissible onjeture. Let j (; j ; j ) be dened by (2.2) and let () be any ontinuous funtion whose domain and range are both [; 1]. Then for any ">there exist a nite set of ontinuous fee (subsidy) shedules fs k g k=1;:::k depending only on x,1 = fx 2 ;:::x g suh that min E x,1 jxs (x,1 ), ( j (x; j ; j ), (x)) k for all x in the support of the marginal distribution for j. Proof. See the Appendix <" Remark 1. The idea is that a seller who wishes to ensure partiipation in his aution an add these fees on to his existing mehanism to aet partiipation. The fees are random and the buyer who agrees to them does not know what the atual fees will be at the time that he deides to partiipate. So for example, if the buyer hooses to partiipate and latter nds that he is the only buyer who has done so, he will pay a fee (or possibly reeive a subsidy) equal to s k (;:::) where s k is the shedule that he hose. 14
Remark 2. The payo funtion () in the statement of the theorem is intended to be the market payo. However, when the theorem is stated in this more general way it is possible to imagine as the market payo plus some small positive onstant. Then entry fees that get the buyer within " of the market payo plus are sure to give the buyer at least as muh as the market payo. This devie is used below to get around the fat that the MAfee Reny theorem only gives approximate surplus extration. 5. Autions The result that follows is readily derived from Theorem 4.1 and (2.3). It illustrates the manner in whih the surplus extration result an be used in the ompetitive setting. Lemma 5.1. Suppose that the joint distribution of valuations satises Assumptions 2.2 and 2.2. Then in any equilibrium q j1 (x 1 ;:::x )= for j -almost all (x 1 ;:::x ). 8 >< >: 1 if x 1 > max k6=1 x k if 9k : x k >x 1 2[;1] otherwise (5.1) Proof. In any equilibrium j (x; j ; j ) (x)8x : z j (x;:::x) >. Using assumption 2.2, seller j spayo an be written fq(x 1 ;:::x )x 1, j, j (x 1 ; j ; j )gd ~ j (x 1 jy):::d ~ j (x jy) dg (y) (5.2) Suppose that seller j replaes the mehanism fq j () ;p j ()g with the seond prie aution where q a is given by (5.1) and ( if q (x1 ;:::x )= p a (x 1 ;:::x )= max k6=1 x k otherwise Let a (x) be the payo assoiated with the aution assuming that the seller believes that the probability distribution j () ofvaluations for buyers will be the same for this aution as it was for his original mehanism (whih means that 15
j is admissable). Let (x) = a (x), j (x), where > is small. By Theorem 4.1, for any " >, there exists a nite set of entry fees fs k ()g k=1;k depending only on (x 2 ;:::x ) suh that min E x,1 jxs k (x 2 ;:::x ),(x) k <" for all x in the support of the marginal distribution for j. If the seller's onjeture j is true, then the buyer's payo in the seond prie aution augmented by the hoie of a random entry fee is given by a (x), mine x,1 jxs k (x 2 ;:::x ) a (x),(x)," k = j (x)+," j (x) The last inequality follows from the fat that " an be hosen to be arbitrarily small. Sine this augmented seond prie aution yields every buyer type who selets it with positive probability a payo that is no smaller than the market payo, the seller's prots under this new mehanism are n x 1 ~,1 j (x 1 jy), j, j (x 1 ) o d ~ j (x 1 jy) dg (y), (5.3) Sine an be hosen arbitrarily small, and sine q a stritly inreases the integrand in (5.2) for eah (x 1 ;:::x ) suh that q diers from q a, this will exeed the value given by (5.2) if q diers from q a on a set that has non-zero probability under j. This theorem says roughly that for any vetor of valuations that the seller might observe with positive probability, the seller's mehanism must award trade to the buyer with the highest valuation, as in an aution. The reason is simply that the entry fees allow the seller to extrat all the additional surplus that is reated by this hange in his mehanism. Analytially this is advantageous beause it makes it possible to use (5.3) to represent seller's prots. The main haraterization theorem, however, is given by the following. Theorem 5.2. In any ompetitive equilibrium in mehanisms seller j's payo is given by n x 1,1 j o (x 1 jy), j, (x 1 ) d j (x 1 jy) dg (y) 16 n () ; 1 ;::: j ; 1 ;::: j o,
Furthermore, the pair j ; j maximizes seller j's prots given relative to the set of all diret mehanisms if and only if there does not exist a buyer strategy suh that and n n x 1 ^,1 j x 1 ^ (xjy) =1,,1 j x (s)f(sjy)ds (5.4) (x 1 jy), j, (x 1 ) o d ^j (x 1 jy) (x 1 jy), j, (x 1 ) o d j (x 1 jy) dg (y) > dg (y) (5.5) Proof. The Proof is similar to the proof of the previous Lemma and is relegated to the appendix This theorem simplies the problem of haraterizing equilibrium. Suppose that rms ompete in `levels of demand' j instead of in mehanisms. A series of demands f ;::: J g that maximize n [x1, ],1 j (x 1 jy), (x 1 ) o d j (x 1 jy)dg (y) (5.6) for eah seller will, aording to the `if' part of Theorem 5.2, oinide with a series of seond prie autions with entry fees that maximize sellers prots relative to the entire lass of diret mehanisms. Aording to the `only if' part of the Theorem, a series of mehanisms and assoiated beliefs n j ; jo that all onstitute best replies to the market payo must all maximize (5.6). Thus the equilibrium mehanism problem an be redued to the problem of haraterizing an equilibrium in whih rms ompete in `levels of demand' ( j ). A symmetri ompetitive equilibrium is simply a ompetitive equilibrium where all sellers oer the same mehanism. The haraterization result given above is used to prove the following result. Theorem 5.3. Suppose that all sellers have a ommon ost. Then there exists a unique symmetri ompetitive equilibrium in mehanisms in whih all sellers oer seond prie autions with reserve pries equal to and zero entry fees. Proof. Begin with existene. Dene the seletion rule j (x) =1=J for all x, and j (x) = otherwise. This seletion rule generates the beliefs ( 1, 1 [1, F (xjy)] (xjy) = if J x 1, 1 [1, J F (jy)] otherwise (5.7) 17
The distribution funtion () = R (jy) dg (y) is admissible by onstrution and therefore satises ondition (1) in the denition of a ompetitive equilibrium in mehanisms. ext dene the funtion (xjy) suh that (xjy) =ifxand (xjy) = (xjy),1 otherwise. Finally let (x) = R (xjy)dg (yjx). With dened this way, wenow show that R (xjy) dg (y) maximizes (5.6). Rewriting (5.6) using the denition of the onditional expetation gives n o [x, ],1 (xjy), (xjy) d (xjy)dg (y) (5.8) Consider the problem of maximizing this by hoosing some funtion (xjy) onstrained only to be non-negative. From the fat that (xjy) = for x<and the fat that,1 (xjy) for all x, it is apparent that a neessary ondition for maximization of the expression (5.8) in is that d (xjy) = for all x<. ote that (xjy) has this property by onstrution for all y. Sine (xjy) is dierentiable by onstrution, (5.8) an be integrated by parts to get 1,, (1), h (xjy), (xjy) (xjy) i dxdg (y) (5.9) Sine (xjy) is onstrained only to be non-negative, the funtion an be maximized by seleting the funtion (xjy) that minimizes the integral pointwise. The rst order neessary ondition for this is,1 (xjy) = (xjy) =,1 (xjy). Sine the integrand is onvex in for any xed x, this neessary ondition is also suient. It follows that (xjy) minimizes the integrand in the above expression point-wise in x and y for x when the buyer's payo is given by.thus must maximize (5.6). It remains to alulate the entry fee that generates the payo for buyers when beliefs are. This an be aomplished by alulating the payo to partiipating in the aution then nding the dierene between this payo and the market payo. Let (x; jy) be the expeted payo earned in a seond prie aution with reserve prie by a buyer of type x onditional on y. Sine the aution awards the good to the buyer with the highest valuation uniformly in y, and sine buyer valuations are independently distributed ondition on y, the argument in [12] gives (x; jy) = x,1 (sjy)ds = 18 x (sjy) ds = (xjy)
where the seond equality is by onstrution and the third is by denition. Thus the payo from the seond prie aution is equal to the market payo and the desired entry fee is uniformly. Sine the payo that the buyers get with every seller is () in the symmetri equilibrium, ondition (3) in the denition of equilibrium is also satised. Thus the market payo along with the array onsisting of J seond prie autions is a ompetitive equilibrium in mehanisms. To show uniqueness, suppose that there is a seond equilibrium in whih the market payo is (x) with assoiated onditional payos (xjy). By Theorem 5.2 we an assume that sellers oer seond prie autions with reserve pries equal to ^ in this equilibrium. Thus sellers' prots are given by n o [x, ],1 (xjy), (xjy) d (xjy) dg (y) ^ From this expression and Theorem 5.2, we an take^=without loss of generality. 8 Integrating this by parts gives [x, ] (xjy) 1, 1 (sjy) ds dg (y),, (xjy) (xjy)j 1, (1, ), = (1jy), (jy) (jy), (sjy) d (sjy) dg (y) (sjy) ds dg (y) (sjy) d (sjy) dg (y) Sine (jy) = for all y, by the denition of the reserve prie, this expression will be maximized whenever the expression (sjy) ds, is minimized. This expression an be written ( 1, (s) f (sjy) ds dx, 1, x (sjy) d (sjy) x dg (y) ) (s) f (sjy) ds d (xjy) dg (y) 8 It is lear from this expression that the seller would not want a reserve prie below ^.Ifhe wants a reserve prie ^ >he an take d (xjy) =forx2[; ^]. 19
In any symmetri equilibrium sellers oer the same mehanism and share the same beliefs about the distribution of types who will partiipate. Thus the onditional beliefs j (xjy) and k (xjy) for any two rms must be the same, or 1, x j (s) f (sjy) ds = 1, x k (s) f (sjy) ds Sine this must be true for all x, wemust have R x x ( j (s), k (s)) f (sjy) ds on every interval [x; x ]. This implies that j (x) = k (x) almost everywhere, or by the summing up restrition, j (x) =1=J for x. This implies that in any symmetri equilibrium, beliefs must be given by as dened by (5.7). Theorem 5.2 implies that if is an equilibrium payo, then there is no admissible belief that yields the seller a higher payo than that assoiated with.thus the buyer strategy (x) = 1 should satisfy the neessary onditions for J unonstrained optimization. That is, for all w, ( w,1 w ), 1, (s) f (sjy) ds f (wjy) dx + f (wjy) d (xjy) dg (y) = x or using the fat that (s) = 1=J in a symmetri equilibrium, 8 < w ", : 1, 1, F (xjy) #,1 9 w = dx + d (xjy) f J ; (wjy) dg (y) = h Using the fat that 1, gives z i 1,F (xjy),1 J = (xjy) from the rst part of the proof, (xjy) dx + z d (xjy) f (zjy) dg (y) = Using the fat that (jy) = for all y gives f (zjy), (zjy)g f (zjy) dg (y) = whih gives (z) = (z) for all z. This ontradition gives the result. The existene of an equilibrium without entry fees is perhaps not so surprising - in a large market sellers need to onform. This argument an be readily generalized to the ase where sellers have dierent osts so that the equilibrium is not 2
symmetri. Uniqueness on the other hand, is more diult to prove. The argument involves two ideas. First, from Lemma 5.1, attention an be restrited to situations in whih sellers oer 'eient' mehanisms. In a symmetri equilibrium these mehanisms are the same so that buyers will hoose every seller with the same probability. Sine the seller will need to understand this in equilibrium, this determines the seller's beliefs in suh an equilibrium. Theorem 5.2 then insures that the beliefs that this generates will have to be optimal given the market payo and this ultimately ties down the market payo funtion. This argument makes use of the symmetry assumption. It is important to note, when interpreting this result, that the entry fees are not being bid away in the way that prots are in a Bertrand equilibrium. Sellers who oer seond prie autions with reserve prie equal to their osts still earn positive prots. The trade-o that eah seller faes at the margin is onventional. To raise the probability of being hosen by buyers of type x the seller needs to oer exatly the market payo (x). This is exatly oset by the inrease in surplus enjoyed by the seller at the margin when the other sellers are oering seond prie autions. The seller ould attrat ertain buyer types with probability 1. The drawbak is that when the seller is attrating a partiular buyer type with very high probability,hemust still pay the market payo to attrat another buyer of that type. Sine the new buyer will fae a lot of ompetition, it is unlikely the seller will be able to reoup this payo by trading with the new buyer. 6. Conlusions and Problems The interpretation oered by [8] for their surplus extration result is a negative one. Sine the ompliated entry fees that are required to implement the optimal mehanism are essentially never observed in pratie, the theory of optimal mehanism design is irrelevant. This paper shows that the ompliated entry fees that the theory predits disappear in the presene of ompetition among the sellers. The fees are essentially ompeted away, leaving seond prie autions with reserve pries set equal to sellers's opportunity osts. This predition is muh loser to the autions that are observed in pratie. The result leaves open a number of issues. The assumption that sellers behave ompetitively seems defensible on it's own merits. The informational requirements of a full game theoreti treatment seem inappropriate to many ompetitive environments. A ompetitive model in whih sellers have little spei information about other sellers and about their potential ustomers, seems well suited 21
to markets like the real estate market, or to nanial markets. onetheless, the relationship between the ompetitive solution and the full game theoreti one is of some interest. It has been shown that the impat that sellers have on market payos shrinks as the eonomy gets large ([1]) when valuations are independent and sellers are required to use diret mehanisms. I do not know if similar results are true for the ase where valuations are orrelated. Furthermore the existing mathematial results are not suient to onlude that ompetitive onverge to exat equilibria, or that ompetitive equilibria will be approximate equilibria when the number of traders is large even in the ase of independent valuations, though it would seem plausible that both onjetures ought to be true under reasonable onditions. It seems likely that when the number of buyers and sellers is small enough, there will still be room for entry fees in equilibrium. The study of non-ompetitive environments where there are small numbers of sellers is ompliated by two major problems. First, it is known that when sellers are restrited to ompetition in autions and valuations are independent, pure strategy equilibria typially do not exist when the number of sellers is small ([7]). This is partly due to the inherent omplexity of the two stage interation in whih sellers have to antiipate the impat that their oers have on buyers' ontinuation equilibrium behavior.. Seondly, the appropriate set of mehanisms for sellers to use in the small numbers problem is not the set of 'diret' mehanisms as has been desribed in this paper ([4]). Sellers an learn about the mehanisms that have been oered by their opponents by asking buyers to report this information to them. Foring sellers to use simple diret mehanisms in whih buyers report their willingness to pay is therefore restritive. Though it is possible to desribe the appropriate set of mehanisms, not enough is urrently known about this set to make qualitative preditions about what equilibrium mehanisms look like. 7. Appendix 7.1. Proof of Theorem 4.1 Proof. The belief j is admissible so that there is a buyer strategy j suh that j (x 1 ;:::x n )= Y 1, j (s) d F ~ (sjy) dg (y) i=1 x i 22
The theorem is vauously satised when j (x) = Lebesque almost everywhere, hene we an assume without loss of generality that j (x) > on a set of stritly positive Lebesque measure. Seondly, note that (again by the admissibility of j ) that eah term in the expansion of the onditional distribution Y j (x 1 ;:::;x n jy)= 1, j (s) d F ~ (sjy) x i i=1 an be deomposed into a mass point at and a distribution that is absolutely ontinuous with respet to Lebesque measure elsewhere. Dene ~ j (jy) = 1, j (s)d F(sjy) ~ and let ~z j (xjy) denote the (Radon-ikodym) derivative of this onditional distribution. This is almost everywhere equal to j (x) f (xjy). Finally, observe that inentive ompatibility guarantees that j (x; j ; j )is ontinuous. Sine the payo funtion is ontinuous as part of the hypothesis of the theorem, ( j (x; j ; j ), (x 1 )) (x 1 ) is ontinuous and hene in C [; 1]. Following [8], dene r ( j )= y2c[; 1] : 9K; fs k g k=1;:::k (8x 2 [; 1]) y (x) = min k E x,1 jxs k (x,1 ) The expetation in this expression is given by = E x,1 jxs (x,1 )= s(x 2 ;:::x )d j (x 2 ;:::x jx) s(x 2 ;:::x )d~ j (x 2 jy):::d~ j (x jy)dg (yjx) = ~j (jy),1 s (;:::) dg (yjx)+,1 X i=1, 1 i! ~ j (jy),i,1 fs(x 2 :::x i+1 ; ;:::)g ~z j (x 2 jy) :::~z j (x i+1 jy) dx 2 :::dx i dg (yjx) The assertion in the Theorem amounts to showing that lies in the losure of r ( j ). MAfee and Reny [8] provide restritions on j suh that the losure of 23
r ( j )=C[;1]. There are two problems to resolve before applying their theorem: j has atoms when any of it's arguments are zero, and j depends indiretly on an endogenous variable. The argument in this proof simply provides a slightly modied version of the part of their argument that depends on these things. Dene U ("; ;x )= fu2c[; 1] : (u (x) 8t 2 [; 1]) ; (u (x ) ");(u(x)18x:jx,x j >)g It is straightforward to show that r ( j ) satises the restritions (2.7) through (2.1) of [8] Theorem 1 (pp 43). 9 This same theorem shows that if U ("; ;t) has a non-empty intersetion with the losure of r ( j ) for all " >, >, and t 2 [; 1] then the losure of r ( j ) will be equal to C [; 1]. If theorem 4.1 is false, then this annot be true, and so there must be some ",, and x suh that the intersetion of U (" ; ;x ) and the losure of r ( j ) is empty. Dene R ( j )= n y2c[; 1] : (9s)(8x2[; 1]) y (x) =E x,1 jxs(x,1 ) o It is apparent from the denition that R ( j ) is a linear subspae of C [; 1] that is entirely ontained in r ( j ). Thus R ( j ) ontains no interior point in ommon with U (" ; ;x ) whih is readily seen to be onvex, and to have a non-empty interior ([8, p42,ftnote 7]). Thus by the separating hyperplane theorem[6, Theorem 1, page 133], there exists an additive set funtion whose total variation is nite and a onstant having the property that y (t) d (t) =8y2R( j ) (7.1) and y (t) d (t) < (7.2) for all y in the interior of U (" ; ;x ). From (7.2) and the denition of U (" ; ;x ), the funtion () is non-zero.. Sine R ( j ) is in fat a linear subspae, =, otherwise for >1 y (t) d (t) 6= 9 The only dierene between the r () here and the one in their paper is the atoms in the distribution used to take expetations. It is easy to see that the properties (2.7) through (2.1) follow from the denition of r () and not from any properties of the distribution used to dene it. 24
Thus by denition s(x 2 ;:::x )d j (x 2 ;:::x jx)d (x) = for all s () 2 C [; 1],1. Using the denition of j gives ~j (jy),1 s (;:::) g (yjx) dyd (x)+,1 X i=1, 1 i! ~ j (jy),i,1 fs(x 2 :::x i+1 ; ;:::)g ~z j (x 2 jy) :::~z j (x i+1 jy) dx 2 :::dx i g(yjx)dyd (x) = ow using Fubini's theorem this implies s (;:::) ~j (jy),1 g (yjx) dyd (x)+ X i=1, 1 i! fs(x 2 :::x i+1 ; ;:::)g ~j (jy),i,1 ~z j (x 2 jy) :::~z j (x i+1jy) g (yjx) dyd (x) dx 2 :::dx i = This must be true for all s () 2 C [; 1],1, so that ~j (jy),1 g (yjx) dyd (x) = and ~j (jy),i,1 ~z j (x 2 jy) :::~z j (x i+1jy) g (yjx) dyd (x) = for all i =1;:::,1. Let + and, be the Jordan Hahn deomposition [2, Theorem 5-1G,p53] of. Sine +,, =, this implies ~j (jy),1 g (yjx) dyd + (x) = ~j (jy),1 g (yjx) dyd, (x) 25
and ~j (jy),i,1 ~z j (x 2 jy) :::~z j (x i+1jy) g (yjx) dyd + (x) = ~j (jy),i,1 ~z j (x 2 jy) :::~z j (x i+1jy) g (yjx) dyd, (x) (7.3) for i =1;:::,1. Summing these equalities over i gives = d (x 2 ;:::x jx)d + (x),1 X i=1,1 X i=1 iy ~ j (jy),i,1 ~z j (x k jy) g (yjx) dyd + (x) k=2 iy ~ j (jy),i,1 ~z j (x k jy) g (yjx) dyd, (x) = k=2 Integrating both sides over (x 2 ;:::x ) yields d + (x) = d(x 2 ;:::x jx)d, (x) (7.4) d, (x) (7.5) Sine has bounded variation, it is nite [2, Theorem 51B,p 51], hene both + and, are nite. Thus it is readily veried that if satises 7.1 and 7.2, then so does for any >. Hene the integrals in (7.5) an both be taken to be equal to 1 without loss of generality, implying that both + and, an be interpreted as probability measures. It is next shown that neither + nor, an have all of their mass onentrated at some x. For suppose the ontrary that + is a point mass at x. By the Jordan Hahn deomposition there is a set A [; 1] suh that + (A) =1and, (A )=1. Thus if + is a point mass at x,, annot be (and onversely). So set i =, 1 in ondition (7.3) to get Y ~z j (x k ) g (yjx) dyd + (x) = k=2 Y k=2 Y k=2 j (x k ) Y k=2 ~z j (x k ) g Y j (x k ) k=2 Y k=2 yjx dy = ~f (x k jy) g yjx dy 6= ~f (x k jy) g (yjx) dyd, (x) = 26
Y k=2 ~z j (x k ) g (yjx) dyd, (x) where the seond equality follows from the fat that j is admissible and the inequality follows from the fat that F satises Assumption 2.2 and the assumption that j (x) is stritly positive on a set of positive Lebesque measure. This inequality violates ondition (7.3) so we onlude that + annot have unit mass at x. The same argument shows that, annot be a point mass at x. Finally, there exists a set B A, (where + (A) = 1 as in the previous paragraph) suh that + (B) > and x =2 B (for if this were false, + would be a point mass at x ). ow use the set B and follow the argument in[8, p46] to verify the existene of a funtion y 2 U (" ; ;x ) suh that (7.2) is violated, a ontradition. 7.2. Proof of Theorem 5.2 Proof. The haraterization of the seller's payo and the `if' part of the seond assertion follow from Lemma 5.1 and the fat that the seller's onjeture j must be admissible (i.e. must satisfy (2.1) in equilibrium). To show the `only if' part, suppose to the ontrary that there is a strategy rule satisfying (5.4) and (5.5). We will show that by appropriately hoosing (or modifying) the entry fee, the seller an ensure that the probability distribution indued by is admissible. The argument mimis the one given in Lemma 5.1. Let ^ j be the distribution indued by. Let a j (x) denote the payo that a buyer of type x gets by partiipating in the seller's mehanism when the distribution of types is ^ j. Dene (x) =j(x), a (x),where >is small. Sine the prots assoiated with are given by n x 1 ^,1 j (x 1 jy), j, (x 1 ) o d ^ j (x 1 jy) dg (y) = n x 1^,1 j (x 1 jy), j, (x 1 ) o (x 1 )f(x 1 jy)dx 1 dg (y) the inequality implied by the ontrary hypothesis annot be true unless (x) > on a set of non-zero (Lebesque) measure. Then by Theorem 4.1, for any " >, there exists a nite sequene of entry fees fs k ()g k=1;k depending only on (x 2 ;:::x ) suh that min k E x,1 jxs k (x 2 ;:::x ),(x) 27 <"
for all x for whih (x) >, where the expetation is taken using the distribution ^ j. If the seller's onjeture ^j is true, then the buyer's payo in the mehanism augmented by this random entry fee is given by a (x), mine x,1 jxs k (x 2 ;:::x ) a (x),(x)," k = (x)+," (x) The last inequality follows from the fat that " an be hosen to be arbitrarily small. This veries that the onjeture ^j is admissible. As an be taken arbitrarily small, the seller's prots are arbitrarily lose to n n n x 1 ^,1 j x 1 ~,1 j x 1 ~,1 j (x 1 jy), j, (x 1 ) o d ^ j (x 1 jy) dg (y) > o (x 1 jy), j, (x 1 ) d j ~ (x 1 jy) dg (y) (x 1 jy), j, j (x 1 ) o d ~ j (x 1 jy) dg (y) The latter inequality follows from the fat that j (x) (x). This ontradits ondition (2) in the denition of the ompetitive equilibrium in mehanisms. Referenes [1] J. Bulow and P. Klemperer, Autions vs. negotiations, Amerian Eonomi Review 86 (1996), 18{194. [2] C. Burrill, Measure Integration and Probability, MGraw Hill (1972). [3] J. Cremer and R. MLean, Full extration of the surplus in bayesian and dominant strategy autions, Eonomtria 56 (1988), 345{361. [4] L. Epstein and M. Peters, A revelation priniple for ompeting mehanisms, University oftoronto, (1997). [5] D. Gale, A walrasian theory of markets with adverse seletion, Rev. Eon. Stud. 59 (1992), 229{255. [6] D. Luenberger, Optimization by Vetor Spae Methods, John Wiley and Sons (1969). 28