Effi cient Entry in Competing Auctions

Transcription

1 Effi cient Entry in Competing Auction Jame Albrecht (Georgetown Univerity) Pieter A. Gautier (VU Amterdam) Suan Vroman (Georgetown Univerity) April 2014 Abtract In thi paper, we demontrate the effi ciency of eller entry in a model of competing auction in which we allow for both buyer and eller heterogeneity. Thi generalize exiting effi ciency reult in the competitive earch literature by imultaneouly allowing for nonrival (many-on-one) meeting and private information. We thank Andrzej Skrzypacz, our editor, and Jeremy Bulow for their helpful comment. We alo thank Xiaoming Cai, who made ubtantial contribution to the online appendix. 1

2 In thi paper, we conider the effi ciency of eller entry into a large market in which eller pot and commit to term of trade and buyer then allocate themelve acro eller. Buyer differ in their valuation for the good that i being old in thi market, and eller are heterogeneou with repect to their reervation value. In thi environment, a econd-price auction i a imple mechanim that allow the buyer who interact with a particular eller to compete for that eller good. The competing auction literature, e.g., McAfee (1993) and Peter and Severinov (1997), conider the characteritic of thee auction a the market get large, and it i now well undertood that competition lead each eller to pot an effi cient mechanim, e.g., a econd-price auction with reerve price equal to eller reervation value. The implication of a large market for eller entry are le well undertood in thi environment. When good are auctioned, buyer extract an information rent from eller. One might therefore expect that the equilibrium level of eller entry into the market would be below the level that a ocial planner would chooe. There i, however, a counterbalancing force, namely, that when a eller enter the market, that eller teal buyer from the eller who were already there and o potentially reduce the urplue aociated with thee eller. Our main reult i that in a large market when eller are free to chooe their preferred mechanim, thee two effect information rent veru buine tealing exactly offet each other, leading to the ocially effi cient level of eller entry. In addition, we how that the equilibrium and ocial planner allocation of buyer acro eller type coincide. The model we preent i one of competitive earch. Seller compete for buyer by poting and committing to elling mechanim, and buyer direct their earch baed on thee poted mechanim. Moen (1997) and Shimer (1996) demontrate the effi ciency of entry in a model of competitive earch that ha two pecial feature. Firt, meeting between buyer and eller are one-on-one, that i, each eller interact with at mot one buyer. Second, there i complete information in the ene that once a buyer and eller meet private information i not an iue. We generalize thi literature on effi cient entry in competitive earch in two direction. Firt, we allow for nonrival (many-on-one) meeting; that i, the fact that one or more buyer chooe to viit a eller doe not make 1

3 it more diffi cult for any other buyer to viit that eller. 1 Second, we allow for private information, which we model by auming that eller do not know how much the buyer at their auction value the good. 2 With a one-on-one meeting technology and complete information, the only relevant mechanim for elling a good i price poting. With manyon-one meeting and aymmetric information, econd-price auction rather than wage poting contitute an optimal mechanim. Thu, we have a model of competing auction. 1 Model We conider a market with a double continuum of buyer and eller. The meaure of buyer i B, and we normalize the meaure of eller that could potentially enter the market to one. In thi market, each buyer want to purchae one unit; each eller ha one unit of the good for ale. Every eller pot and commit to a elling mechanim, and each buyer, after oberving all poted mechanim, chooe one eller to interact with. The meeting technology i nonrival. Finally, a i tandard in the competitive earch literature, we aume that buyer cannot coordinate their viiting trategie. We aume that buyer valuation for the good are ditributed a X F (x), with correponding denity, f(x), and we normalize the range of X to [0, 1]. Buyer valuation are independently and identically ditributed and are private information; i.e., we are conidering a model of independent private value. In our baeline cae, we aume that buyer learn their valuation ex pot, that i, only after chooing a particular eller. 3 Seller are ex ante heterogeneou with reervation value ditributed a S G(), with correponding denity, g(). Again, we normalize the range of the random variable to [0, 1]. We interpret eller type a motivation the eller with high reervation value are relaxed" about elling their good while thoe with reervation value cloe to zero are deperate (or, a i often een in houing ad, motivated ). Finally, we decribe the 1 Nonrival i the terminology ued in the earch literature for uch meeting technologie. See Eeckhout and Kircher (2010). 2 Effi ciency of eller entry in competitive earch with private information wa hown by Albrecht and Jovanovic (1986) for one-on-one meeting uing a particular matching function. 3 We refer to our baeline model a the ex pot cae. We conider the ex ante cae in which buyer know their valuation before chooing which eller to viit in Section 3. 2

4 allocation of buyer acro the different eller type by a function θ(), which give the expected number of buyer per eller of type. We model the interaction of buyer and eller a a one-hot game with the following tage: 1. Each type eller decide whether or not to pay a cot A to enter the market. A type eller who doe not enter retain the reervation value. 2. Each eller who enter the market pot and commit to a mechanim to ell one unit of the good. 3. Each buyer, after oberving all poted mechanim, chooe a eller. 4. Given the allocation of buyer to eller, good are allocated and payment are made according to the poted mechanim. The payoff for a type eller conit of any payment received according to the poted mechanim plu if the good i retained. The payoff for a buyer with valuation x equal x if the good i purchaed and zero otherwie le any payment pecified by the poted mechanim. Definition 1 An equilibrium for thi game i a choice of a elling mechanim by each eller, an allocation of buyer acro eller, θ(), and a cutoff eller type,, uch that (i) each eller mechanim maximize the eller expected payoff given the mechanim poted by the other eller in the market and the rule that buyer follow to allocate themelve acro eller, (ii) each buyer chooe a probability ditribution over eller type that maximize the buyer expected payoff given the mechanim poted in the market and the allocation rule followed by the other buyer, and (iii) the payoff that eller expect on entering the market equal A +. In the independent private value environment that we conider, a econd-price auction with reerve price equal to eller reervation value i an effi cient mechanim. It guarantee 3

5 that the good i old whenever at leat one buyer ha a valuation above the eller reervation value and it alo guarantee that, if the good i old, it goe to the buyer with the highet valuation. Competition enure that eller pot effi cient mechanim. The intuition can be een in Levin and Smith (1994). They conider a ingle eller offering a econd-price auction with reerve price r who face a fixed number of potential buyer, each with a common outide option. Their reult, that endogenizing buyer entry drive r to the eller reervation value, even though there i only one eller in the market, i generated by the fixed outide option. Each buyer who participate in the auction mut get an expected payoff equal to thi outide option. It i therefore in the eller interet to pot an effi cient mechanim. Poting an ineffi cient mechanim cannot reduce the expected payoff per buyer that the eller ha to pay and therefore can only reduce the eller expected payoff. The ame reult hold in the competing auction environment (Albrecht, Gautier and Vroman 2012) with the difference that the outide option now equal to market utility, i.e., the payoff that a buyer can expect to get elewhere in the market i endogenou. 4 Levin and Smith (1994) alo how that it i not in the eller interet to pot an entry fee, and the ame reult can be derived in the competing auction framework uing the approach of Albrecht et al. (2012). The fact that buyer entry fee are zero and reerve price equal reervation value in a large market mean that not only i effi ciency enured in the competing auction equilibrium but alo that the eller expected payoff i limited to the expectation of the econd highet valuation among buyer participating in the auction. Thi reult i related to one derived in Gorbenko and Malenko (2011). They conider competition in ecuritie auction in which eller auction off the right to develop project for a combination of cah and a hare of the profit. Getting buyer to pledge a profit hare i a way for eller to recapture ome of the information rent aociated with buyer private information. Propoition 4 in Gorbenko and Malenko (2011) how that a the number of buyer and eller in the market get large, all-cah auction are poted in the competitive earch equilibrium. That i, a the market get large, 4 Of coure, thi effi ciency reult require a large market. In a market with a mall number of eller, the term of trade poted by any one eller affect market utility. 4

6 competition prevent eller from clawing back any part of the information rent. 5 2 Effi cient Seller Entry To dicu the effi ciency of eller entry, ome notation i ueful. Conider a type eller who pot a econd-price auction with reerve price r, which generate an expected buyer arrival rate of θ(r). Let Π(r, θ(r); ) denote thi eller expected payoff. Similarly, let V (r, θ(r)) denote the expected payoff for a buyer who participate in an auction with reerve price r. 6 Thi eller et a reerve price to maximize Π(r, θ(r); ) ubject to the contraint that V (r, θ(r)) = V, where V i market utility. Competition lead the eller to chooe r =. The urplu aociated with a econd-price auction poted by thi eller with reerve price equal to reervation value i the expected maximum of and the highet valuation drawn by buyer who interact with thi eller. Denoting thi urplu by Λ(, θ(); ), 7 we have Λ(, θ(); ) = Π(, θ(); ) + θ()v (, θ()). (1) The urplu aociated with thi auction equal the eller expected payoff plu the expected number of buyer participating in the auction, θ(), time the expected payoff per buyer. Becaue the mechanim i effi cient, no urplu i left on the table. Our proof that eller entry into thi market i effi cient relie on the fact that Π(, θ(); ) θ() V (, θ()) + θ() θ() = 0. (2) That i, if the expected number of buyer per type eller increae, the increae in thi eller expected payoff i jut offet by the expected decreae in payoff acro the eller 5 It i worth noting that the ame logic competition lead eller to pot effi cient mechanim applie with a rival meeting technology. However, the effi cient mechanim in thi cae doe include an entry fee. See Albrecht and Jovanovic (1986) and Faig and Jerez (2005). 6 The eller expected payoff depend on the reerve price poted, the expected arrival rate of buyer and on the eller type,, ince i retained if the good i not old. The buyer expected payoff only depend on the poted reerve price and the expected arrival rate of other buyer. 7 In Appendix A.1, we derive expreion for Π(, θ(); ), V (, θ()) and Λ(, θ(); ). 5

7 incumbent buyer. 8 The intuition for equation (2) can mot eaily be undertood in a ingle-eller etting. Conider a type eller poting a econd-price auction with reerve price. Suppoe n buyer initially participate in the auction. Now uppoe an (n+1) t buyer i added. Total urplu only increae if the new buyer win the auction. In thi cae, the buyer get hi or her valuation minu the maximum of and the highet valuation acro the original n buyer; that i, the marginal contribution to the urplu aociated with the auction equal the new buyer payoff. Thi mean, a equation (2) indicate, that the entry of a new buyer doe not change the um of the payoff to the eller and to the original n buyer. We can apply thi intuition to the large market etting. Suppoe a new type eller enter the market, poting a econd-price auction with reerve price. Thi eller can expect to attract θ() buyer away from eller who were already in the market. Since the market i large, the probability that two or more of thee buyer come from any one auction that wa already in the market i negligible. The new eller auction create a urplu equal to the highet valuation drawn by the buyer it attract (or if it attract no buyer drawing x ). Thi eller payoff equal the econd-highet valuation (or if the auction attract no buyer or only one buyer who draw x ). The difference between the urplu aociated with the auction and the eller payoff i the information rent. Now conider the buyer (if any) who are attracted to the new auction. If the new eller hadn t entered the market, each of thee buyer would have participated in ome other eller auction. The buyer who chooe the new auction decreae the urplue aociated with the auction they leave. In each of thee auction, there i a decreae in the expected eller payoff and an increae in the expected payoff acro the remaining 8 In Albrecht et al. (2012), we conider the problem of maximizing Π(r, θ(r); 0) ubject to V (r, θ(r)) = V uing the expreion for Π( ) and V ( ) derived in Peter and Severinov (1997). We verify equation (2) for the cae of = 0 in that paper by howing that thi equation i one of the firt-order condition for the eller contrained maximization problem. In Appendix A.1, we preent an alternative approach to verifying thi equation; namely, we ue the reult that competition lead eller to pot effi cient mechanim (r = ) to develop imple expreion for the term in equation (1). 6

8 buyer, but thee two effect cancel, o the decreae in urplu aociated with each of thee auction equal the payoff the departing buyer could have expected there, namely, the market utility, V. Similarly, each of thee buyer increae the urplu aociated with the new eller auction. Each buyer who participate in the new eller auction increae that eller expected payoff but decreae the expected payoff for any other buyer who viit the new eller. Again, thee two effect are exactly offetting. That i, the increae in urplu at the new eller auction generated by adding one more buyer equal that buyer expected payoff at that auction. Since each of the buyer who participate in the new eller auction anticipate competing with θ() other buyer at that auction, each buyer expected payoff at the new auction i V (, θ()). In equilibrium, V (, θ()) = V, o the decreae in urplu at an auction that loe a buyer equal the increae in urplu that buyer add to the entrant auction. The decreae in urplu at auction that loe buyer i the buine-tealing effect, θ()v, and thi equal the um of the contribution thee buyer make to the new eller auction, θ()v (, θ()). From equation (1), we have Π(, θ(); ) = Λ(, θ(); ) θ()v (, θ()); that i, the expected payoff for a new entrant of type equal the increae in urplu generated by the auction net of the decreae in urplu caued by the departure of buyer from auction that were already in the market. Thu, the information-rent and buine-tealing effect are exactly offetting. We now conider the ocial planner problem. The ocial planner chooe a level of eller entry and an allocation of buyer acro eller type to maximize total urplu in the market net of entry and opportunity cot. That i, the ocial planner chooe a cutoff eller type,, all eller of type enter the market; all type > tay out and a buyer-eller ratio for each eller type, θ(), to maximize 0 Thi maximization i carried out ubject to (Λ(, θ(); ) (A + )) g()d. 0 θ()g()d = B; that i, the allocation of buyer acro eller type ha to atify an adding-up contraint. 7

9 Propoition 1 The ocial planner value of θ() and equal the free-entry equilibrium value. Proof. The Lagrangean for the ocial planner problem i ( ) (Λ(, θ(); ) (A + )) g()d + λ B θ()g()d 0 with firt-order condition 0 Λ(, θ( ); ) (A + ) λθ( ) = 0 (3) Λ(, θ(); ) λ θ() = 0 [0, ] (4) B Uing equation (1) and (2), equation (4) can be rewritten a 0 θ()g()d = 0 (5) V (, θ()) = λ [0, ]. (6) That i, the ocial planner want buyer to allocate themelve acro eller o that the value of viiting the variou eller type in the market are equalized. Thi ame condition hold in equilibrium ince buyer mut be indifferent acro eller type. Next, conider equation (3). Thi can be rewritten a Π(, θ( ); ) (A + ) + θ( ) (V (, θ( )) λ) = 0. From equation (6), we have V (, θ( )) = λ, o equation (3) require Π(, θ( ); ) = A +. That i, the marginal entrant hould be jut indifferent with repect to entering the market. Again, the ame condition hold in equilibrium with free entry. 9 3 The Ex Ante Cae In thi ection, we conider the cae in which buyer know their valuation ex ante, i.e., before chooing which eller to viit. We begin by decribing the equilibrium in competing 9 In our related working paper, Albrecht et al. (2013), we prove a imilar reult for the cae of two eller type. 8

10 auction for thi cae taking the ditribution of eller in the market a given. Then we dicu the effi ciency of eller entry and of the allocation of buyer acro eller type. The detail are given in Appendix A.2. Competition in elling mechanim in the ex ante cae wa firt conidered by McAfee (1993). Seller pot effi cient mechanim, a in the ex pot cae, but, interetingly, the reaoning i different. McAfee (1993) make the following argument to how that when eller all have reervation value 0, each pot a econd-price auction with a zero reerve price. 10 Suppoe all eller pot poitive reerve price, o buyer with low valuation are hut out of the market. Now conider the eller poting the lowet reerve price, ay r > 0. If thi eller deviate to r = 0, he or he capture the entire market between zero and r. What i more intereting i that the deviant eller doen t loe any buyer (in expectation) with valuation above r. Buyer with valuation above r don t care about competition from buyer with valuation below r. More preciely, a buyer with valuation x > r know that r i the lowet poible price irrepective of whether the deviant eller i poting r = r or r = 0 becaue all buyer with x r viit thi eller. Thu, buyer with valuation above r continue to allocate themelve acro all eller (including the deviant) exactly a they would have abent the deviation. Thi argument further implie that no eller (regardle of what reerve price are et by the other eller) want to et a poitive reerve price. Thu, competition drive the equilibrium reerve price to the common eller reervation value of zero in the ex ante cae. When eller are heterogeneou with repect to their reervation value, the equilibrium i more complicated. It i implet to tart with the intuition for the cae of two eller type. Suppoe there i a ma m 1 of eller with reervation value 1 and a ma m 2 of eller with reervation value 2 > 1. Competition lead the eller to pot reerve price equal to their reervation value. Buyer then ort themelve baed on their valuation. Buyer with valuation below 1 are hut out of the market. Similarly, it i pointle for a buyer with valuation x < 2 to viit a eller poting 2. Eentially the ame i true for buyer with valuation that are lightly higher than 2 thee buyer are alo better off viiting a eller poting 1, even 10 Our treatment follow Peter and Severinov (1997) and Peter (2013). 9

11 though there may be more competition from other buyer there. Let x be the lowet valuation that i conitent with indifference between viiting the two eller type. Buyer with x > x are alo indifferent between the two eller type and viit all eller with equal probability. Why do buyer with uffi ciently high valuation randomize acro all eller? To undertand thi, conider the buyer of type x. So long a buyer with higher valuation randomize their viit acro all eller, the buyer with valuation x i equally likely to be the high bidder in an auction with reerve price 2 a in an auction with reerve price 1. The threhold value x i then determined by equating the expected payoff in the two auction, conditional on being the high bidder. Next, conider a buyer with valuation x > x. If all buyer with even higher valuation randomize their viit acro all eller, then the buyer with valuation x i equally likely to be the high bidder in either eller type auction. Finally, given that expected payoff conditional on winning are the ame for the buyer with valuation x and given that buyer with valuation between x and x randomize their viit acro all eller, the expected payoff, conditional on winning, for the buyer with valuation x i the ame in an auction with reerve price 1 a one with reerve price 2. Continuing with two eller type, the next tep i to allow for free entry. That i, m 2 i endogenized in free-entry equilibrium. In Appendix A.2, we conider the two-eller type cae in detail. We characterize the equilibrium allocation of buyer acro the two eller type and the ditribution of eller type in the market in term of x and m 2, and we how that thee value are the ame a thoe that would be choen by a ocial planner. In hort, we how that the equilibrium level of eller entry and the allocation of buyer acro the two eller type are effi cient in competitive earch equilibrium. We then generalize to allow for an arbitrary, dicrete number of eller type. Finally, by taking the appropriate limit, we conider the model with a continuum of eller type. The equilibrium allocation of buyer acro eller, that i, the expected number of buyer chooing each eller type and the ditribution of buyer type acro eller, i decribed by a function x () that give the threhold valuation aociated with each eller type, and the marginal eller type i determined by free entry. The intuition for the effi cient allocation of buyer acro eller type and effi cient eller 10

12 entry in a model of competing auction when buyer know their valuation ex ante i not o different from the intuition given above in the ex pot cae. Conider again the model with two eller type. From the planner perpective, buyer with valuation below x contribute le to auction poted by eller of type 2 than they do to auction poted by the other eller type. In equilibrium, the contribution thee buyer would make to each auction type equal their expected payoff, o thee buyer have the correct incentive to direct their earch effi ciently. That i, the market ue the ex ante information effi ciently in the ene that low valuation buyer do not watefully viit high eller type. Similarly, buyer who find it worthwhile to earch both eller type hould do o in uch a way that the urplu they add by participating in an auction with reerve price 1 i the ame a what they add by participating in an auction with reerve price 2. Again, becaue thee buyer allocate themelve between the two auction type to equalize their expected payoff, equilibrium incentive give the ocially optimal allocation of buyer acro the two eller type. In term of eller entry, the marginal entrant ha reervation value 2. Only high-valuation buyer will conider viiting thi entrant. In equilibrium, the expected payoff for high-valuation buyer i the ame whether they viit eller poting 1 or 2. Thu, whichever auction type the marginal entrant draw buyer away from, the buinetealing effect i the ame. Exactly a in the ex pot cae, the expected contribution that a buyer make to the new entrant auction equal the expected lo in urplu that hi or her departure generate in the auction in which he or he would otherwie have participated. Once again, the information-rent and buine-tealing effect are exactly offetting, giving the correct incentive for entry. 4 Concluion The contrained effi ciency of competitive earch equilibrium i well undertood when meeting between buyer and eller take place on a one-on-one bai with complete information. However, in many ituation, e.g., in tandard auction etting, it i more appropriate to aume a nonrival meeting technology, i.e., many-on-one meeting. Thee are ituation in which buyer differ in term of how much they value the good that i 11

13 being offered for ale and in which thee valuation are private information. Our main contribution in thi paper i to how that the allocation of buyer acro eller type and the level of eller entry are effi cient in competitive earch equilibrium with nonrival meeting. To get effi cient entry, eller who could potentially enter the market need the correct incentive. The payoff expected by the marginal entrant hould equal hi or her expected contribution to market urplu. The expected contribution to market urplu from thi auction i the maximum of the eller reervation value,, and the expectation of the highet valuation drawn by buyer who participate in that auction, while the eller expected payoff i the expectation of the maximum of and the econd-highet valuation acro buyer in the auction. The difference between what the winning buyer expect to get and what the eller expect to receive i an information rent, and the exitence of thi information rent make it eem that there will not be enough eller entry. Thi, however, neglect the fact that the buyer who participate in the marginal entrant auction would have participated in ome other eller auction had the lat eller not entered the market. That i, the entry of the marginal eller create a buine-tealing externality. Our main contribution, therefore, can be undertood a howing that the information-rent and buine-tealing effect exactly offet each other in competitive earch equilibrium, thu generating the effi cient level of eller entry. 12

14 Reference [1] Albrecht J., Gautier P.A. and S. Vroman (2013), Directed Search in the Houing Market, mimeo Georgetown Univerity. [2] Albrecht, J., P.A. Gautier and S. Vroman (2012), A Note on Peter and Severinov, Competition Among Seller Who Offer Auction Intead of Price, Journal of Economic Theory, 147, [3] Albrecht J. and B. Jovanovic (1986), The Effi ciency of Search Under Competition and Monopony, Journal of Political Economy, 94-6, [4] Eeckhout J. and P. Kircher (2010), Sorting veru Screening: Search Friction and Competing Mechanim, Journal of Economic Theory, 145, [5] Faig M. and B. Jerez (2005), A Theory of Commerce, Journal of Economic Theory, 122(1), [6] Gorbenko A.S. and A. Malenko (2011), Competition among Seller in Securitie Auction, American Economic Review 101(5), [7] Levin D. and J.L. Smith (1994), Equilibrium in Auction with Entry, American Economic Review, 84(3), [8] McAfee, R.P. (1993), Mechanim Deign by Competing Seller, Econometrica 61(6), [9] Moen, E. (1997), Competitive Search Equilibrium, Journal of Political Economy, 105, [10] Peter M. (2013), Competing Mechanim, in The Handbook of Market Deign, edited by N. Vulcan, A. E. Roth and Z. Neeman, Oxford Univerity Pre. [11] Peter, M. and S. Severinov (1997), Competition among Seller who Offer Auction Intead of Price, Journal of Economic Theory, 75, [12] Shimer, R. (1996), Eay in Search Theory, Ph.D. thei, MIT. 13

15 A Appendix A.1 Ex Pot Cae In thi appendix, we verify equation (2) and derive expreion for Λ(, θ(); ), Π(, θ(); ), and V (, θ()). A noted in the text, in Albrecht et al. (2012), we analyzed the problem max Π(r, θ(r); 0) ubject to V (r, θ(r)) = V r,θ(r) uing expreion for Π(r, θ(r); 0) and V (r, θ(r)) that were derived in Peter and Severinov (1997). Thi i the problem of a eller of type = 0 chooing a reerve price for a econdprice auction who take into account that the reerve price determine the expected arrival rate of buyer to the auction. In the proce of howing that r = 0, that i, that the optimal reerve price equal the eller reervation value, we derived Π(0, θ(0); 0) θ V (0, θ(0)) + θ(0) θ a one of the firt-order condition for the eller contrained maximization problem. The ame approach can be ued to confirm equation (2), for arbitrary. Π(, θ(); ) θ V (, θ()) + θ() θ Here we preent an alternative approach to verifying equation (2). We ue the reult that competition lead eller to pot effi cient auction; that i, eller pot reerve price equal to reervation value. Thi give equation (1) in the text, namely, Differentiating give = 0 = 0 Λ(, θ(); ) = Π(, θ(); ) + θ()v (, θ()). Λ(, θ(); ) θ = Π(, θ(); ) θ V (, θ()) + θ() + V (, θ()), θ and our proof of equation (2) conit of howing Λ(, θ(); ) θ = V (, θ()). (7) To do thi, we develop expreion for Λ(, θ(); ) and V (, θ()). We alo derive Π(, θ(); ) and verify that our expreion for eller and buyer expected payoff are equivalent to the one given in Peter and Severinov (1997). 14

16 We begin with an expreion for Λ(, θ(); ), the expected urplu aociated with the auction poted by eller, which equal the expected maximum of and the highet valuation greater than drawn by one of the buyer. Suppoe n buyer viit thi eller and draw valuation of or more. If n = 0, then Λ(, θ(); ) =. If n 1, then, conditional on n, the expected urplu i ( ) n F (x) F () 1 xd = 1 1 F () ( ) n F (x) F () dx, 1 F () that i, the expected maximum of the n draw that are greater than or equal to. The number of buyer viiting eller who draw valuation of or more i Poion with parameter θ()(1 F ()), o the unconditional expreion for expected urplu i θ()(1 F ()) Λ(, θ(); ) = e +e θ()(1 F ()) (θ()(1 F ())) n ( ( ) n F (x) F () 1 dx) n! n=1 1 F () = e θ()(1 F ()) + e ( θ()(1 F ()) e θ()(1 F ()) 1 ) e θ()(1 F ()) (θ()(f (x) F ())) n dx; i.e., n! Λ(, θ(); ) = 1 n=1 e θ()(1 F (x)) dx. (8) Next, conider V (, θ()), the expected payoff for a buyer who chooe a eller of type. Suppoe thi buyer draw a valuation x. Thi i the winning draw with probability e θ()(1 F (x)). Conditional on winning, the buyer payoff i the difference between x and the highet valuation drawn by any other buyer who viited thi eller or if no other buyer drew a valuation y. Now uppoe there were n other buyer who drew y [, x). Conditional on n, the winning buyer expected payoff i then x ( ) n F (y) F () x ( ) n F (y) F () x yd = dy F (x) F () F (x) F () Summing againt the probability ma function for n, the buyer expected payoff, conditional on winning with valuation x, i e θ()(f (x) F ()) (θ()(f (x) F ())) n n=0 n! x ( ) n F (y) F () x dy = e θ()(f (x) F (y)) dy. F (x) F () Multiplying by the probability of winning with valuation x at eller give V (, θ(); x) = x 15 e θ()(1 F (y)) dy. (9)

17 Thi i the expected payoff for a buyer who viit eller conditional on drawing valuation x, where x. Finally, the unconditional expected payoff for a buyer who viit eller i V (, θ()) = (1 F ()) = x f(x) V (, θ(); x) 1 F () dx e θ()(1 F (y)) dyf(x)dx = (1 F (x))e θ()(1 F (x)) dx, (10) where the lat equality follow by integration by part (u = x e θ()(1 F (y)) dy, v = (1 F (x))). With = 0, equation (10) i the expreion for expected buyer payoff derived in Peter and Severinov (1997). Differentiating equation (8) and uing equation (10) give equation (7). Thi complete the derivation of equation (2). For completene, we alo derive Π(, θ(); ). Conider a eller of type who pot reerve price equal to, and uppoe thi eller attract n buyer who draw valuation of or more. The eller get a payoff of if n = 0 or 1; if n 2, the eller expected payoff equal the expected value of the econd highet valuation acro the buyer, i.e., E[Y n 1 ]. The ditribution function of Y n 1 i ( ) n F (y) F () G n 1 (y) = + n 1 F () ( F (y) F () 1 F () ) n 1 ( ) 1 F (y) 1 F () for y 1. Thu, conditional on n 2, the eller expected payoff i ( ) n F (y) F () 1 ( ) (F (y) F ()) n 1 (1 F (y)) ydg n 1 (y) = 1 dy n dy. 1 F () (1 F ()) n Summing againt the probability ma function for n, Π(, θ(); ) = e θ()(1 F ()) (1 + θ()(1 F ())) + e θ()(1 F ()) (θ()(1 F ()) n Π(, θ(); ) = 1 e θ()(1 F ()) (θ()(f (y) F ()) n dy n! n=2 e θ()(1 F ()) (θ()(f (y) F ())) n 1 (1 F (y))θ() ; i.e., (n 1)! e θ()(1 F (x)) dx n=2 n=2 θ()(1 F (x))e θ()(1 F (x)) dx. (11) Uing equation (8), (10) and (11), it i traightforward to check equation (1). Finally, uing 1 e θ()(1 F (x)) dx = θ() 16 xe θ()(1 F (x)) f(x)dx, n!

18 the eller expected payoff can be rewritten a ( Π(, θ(); ) = θ() x 1 F (x) ) e θ()(1 F (x)) f(x)dx. f(x) With = 0, thi i the expreion for expected eller payoff that i given in Peter and Severinov (1997). A.2 Ex Ante Cae In thi appendix, we give the detail and prove the effi ciency of free-entry equilibriuim for the ex ante cae, i.e., the cae in which buyer draw their valuation before deciding which eller to viit. A dicued in the text, competition lead eller to pot effi cient mechanim in the ex ante cae jut a it doe in the ex pot cae; in particular, a eller with reervation value pot a econd-price auction with reerve price. However, with a ditribution of eller type in the market, the fact that buyer learn their valuation ex ante complicate the analyi. The extra complication arie becaue the ditribution of buyer will vary acro eller type. The urplu aociated with an auction poted by a type eller hould therefore be written a Λ(, θ(), F (x; ); ); that i, the urplu depend on the poted reerve price,, on the expected number of buyer attracted by that reerve price, θ(), and on the ditribution of valuation acro the buyer viiting eller of type, F (x; ), a well a on the eller type,. Similarly, the expected payoff for eller i Π(, θ(), F (x; ); ), and the expected payoff for a buyer with valuation x who viit a eller poting reerve price i V (, θ(), F (x; ); x). Applying the approach ued in Appendix 1 give Λ(, θ(), F (x; ); ) = 1 Π(, θ(), F (x; ); ) = θ() V (, θ(), F (x; ); x) = x e θ()(1 F (x;)) dx (12) ( ) x e θ()(1 F (x;)) f(x; )dx (13) 1 F (x; ) f(x; ) e θ()(1 F (y;)) dy (14) For eae of notation, however, we ue Λ(), Π() and V (; x), repectively. We alo implify the notation by normalizing the meaure of buyer, B, to one. We ue the following approach to characterize the equilibrium and the ocial planner olution in the ex ante cae. Firt, we conider the cae of two eller type with a ma m 1 of eller of type 1 and a ma m 2 of potential eller of type 2, where 0 1 < 2 < 1. Second, we extend the analyi to the cae of N eller type with mae m 1,..., m N 17

19 of eller type 0 1 <... < N conidering the appropriate limit. < 1. Finally, we move to a continuum of eller by A.2.1 Two Seller Type We have argued in the text of the paper that equilibrium i characterized by a cutoff, x, and a meaure, m 2, of type 2 eller uch that V ( 1 ; x ) = V ( 2 ; x ) (15) Π( 2 ) = A + 2. (16) Note that x i the lowet value of x atifying equation (15); in particular, V ( 1 ; x) > V ( 2 ; x) for all x [ 1, x ) while V ( 1 ; x) = V ( 2 ; x) for all x [x, 1]. The correponding ocial planner problem i to chooe a cutoff, x, and a meaure, m 2, to maximize m 1 (Λ( 1 ) (A + 1 )) + m 2 (Λ( 2 ) (A + 1 )). Conider the partial derivative of the ocial planner maximand with repect to x; in particular, conider an increae in the cutoff from x to x + dx. The key to undertanding thi derivative i to recognize that the only agent who change their behavior are buyer with valuation in the interval [ x, x + dx). Buyer with valuation x [0, x) randomized over eller of type 1 before the change; they continue to do o after the change. Similarly, buyer with valuation x [ x + dx, 1] randomized over all eller before the change; they continue to do o afterward. Buyer with valuation x [ x, x+dx) randomized over all eller before the increae in the cutoff; after the increae, thee buyer randomize over eller of type 1. Thu, there are ome buyer with valuation in [ x, x + dx) who would have participated in an auction run by a eller of type 2 before the change but intead participate in an auction run by m 2 a eller of type 1 after the change. To be more precie, approximately f( x)dx m 1 + m 2 buyer are expected to witch eller type. When a buyer witche from an auction with reerve price 2 to one with reerve price 1, there i an increae in urplu aociated with the auction poted by the type 1 eller but a decreae in urplu aociated with the auction poted by the type 2 eller. The ocial planner want thee two effect to offet each other. Were thi not the cae, e.g., if moving buyer with valuation cloe to x from type 2 eller to type 1 eller increaed the urplu aociated with auction poted by type 1 eller more than it decreaed the urplu aociated with auction poted by type 2 eller, then the ocial planner hould increae the cutoff value. 18

20 Conider the reallocation of a buyer with valuation x [ x, x + dx) from a type 2 eller to a type 1.eller. The increae in urplu at the auction poted by the type 1 eller i the um of three component: (i) the increae in the eller expected payoff, (ii) the decreae in the expected payoff of any incumbent buyer, and (iii) the expected payoff, V ( 1 ; x), of the buyer who witched eller type. A we have argued in the text, the firt two term are exactly offetting; thu, the increae in urplu aociated with an auction poted by a type 1 eller that gained a buyer of type x equal V ( 1 ; x). By the ame argument, the decreae in urplu aociated with an auction poted by a type 2 eller that lot a buyer of type x equal V ( 2 ; x). Now let dx 0, o x x. Satifying the firt-order condition of the ocial planner problem with repect to the cutoff value require V ( 1 ; x) = V ( 2 ; x). The cutoff x i the lowet value of x atifying thi equation. Equation (15) thu implie x = x ; that i, the equilibrium and ocial planner cutoff coincide. Next conider the partial derivative of the ocial planner maximand with repect to m 2. We can write the firt-order condition a m 1 Λ( 1 ) m 2 + m 2 Λ( 2 ) m 2 + Λ( 2 ) (A + 2 ) = 0. (17) Since the ocial planner want eller to pot effi cient mechanim, we have Λ( 2 ) = Π( 2 ) + θ( 2 )V ( 2 ), where f(x) V ( 2 ) = V ( 2 ; x) dx (18) x 1 F (x ) i a horthand notation for the expected payoff per buyer viiting a type 2 eller. Equation (17) implie equation (16) if m 1 Λ( 1 ) m 2 + m 2 Λ( 2 ) m 2 = θ( 2 )V ( 2 ). (19) We now verify equation (19). Suppoe a type 2 eller enter the market. In expectation, thi eller take θ( 2 ) buyer away from incumbent eller; thu, m 1 + Λ( 1 ) m 2 Λ( 2 ) m 2 i the buine-tealing effect aociated with the entrant. Any buyer who attempt to purchae the good from the new entrant ha a valuation of x or more, m 2 and thee buyer randomize their viit acro both eller type. If a buyer with valuation x move from an incumbent eller to the new entrant, the lo in urplu at the incumbent 19

21 eller auction i V ( 2 ; x), and ince V ( 1 ; x) = V ( 2 ; x) for all x x, i.e., high-valuation buyer are indifferent between the two eller type, thi lo i the ame irrepective of the type of the incumbent eller. The valuation x i a draw from the truncated denity, f(x) 1 F (x ) ; thu, the expected lo to the incumbent eller i V ( 2), a given in equation (18). Multiplying thi by the expected number of buyer who viit the new entrant give equation (19). A.2.2 N Seller Type Suppoe there are N eller type, 0 1 < < N < 1, with repective meaure m 1,..., m N, where we conider the entry deciion of the marginal eller type, N. equilibrim, there will exit N 1 threhold x ( 2 ),..., x ( N ) uch that buyer type in [x ( k ), x ( k+1 )] randomize among eller 1,..., k. Buyer of type x < 1 do not participate in the market; that i, x ( 1 ) = 1. Equilibrium i characterized by the cutoff, x ( 1 ), x ( 2 ),...x ( N ), and a meaure of eller of type N uch that 0 = V ( 1 ; x ( 1 )) V ( 1 ; x ( 2 )) = V ( 2 ; x ( 2 )) (20) V ( 1 ; x ( 3 )) = V ( 2 ; x ( 3 )) = V ( 3 ; x ( 3 ))... V ( 1 ; x ( k )) = V ( 2 ; x ( k )) =... = V ( k ; x ( k )) (21)... V ( 1 ; x ( N )) = V ( 2 ; x ( N )) =... = V ( N ; x ( N )) Π( N ) = A + N. (22) The correponding ocial planner problem i to chooe cutoff, x( 1 ), x( 2 ),..., x( N ), and a meaure, m N, of type N eller to maximize N m k (Λ( k ) (A + k )). k=1 The proof that x( k ) = x ( k ) for k = 1,..., N i eentially the ame a the one given for the cae of two eller type. Firt, it i obviou that x ( 1 ) = x( 1 ) = 1. Buyer with x 1 have no incentive to participate in the market nor doe the ocial planner want them to do o. Second, given any collection of cutoff for eller of type 3 and above and given any level of entry by type N eller, the choice of x( 2 ) doe not affect N k=3 m k (Λ( k ) (A + k )). The 20 In

22 ame argument that wa ued to characterize x( 2 ) in the two-eller cae then implie that the ocial planner et x( 2 ) o that V ( 1 ; x( 2 )) = V ( 2 ; x( 2 )). Comparing thi with equation (20) give x( 2 ) = x ( 2 ). The final tep in the argument ue induction. Suppoe x( i ) = x ( i ) for i = 1,..., k 1, and take { x( k+1 ),..., x( N ), m N } a given. The choice of x( k ) ha no effect on N m i (Λ( i ) (A + i )). Since i=k+1 { x( 1 ), x( 2 ),..., x( k 1 )} are aumed to have been et optimally, the ocial planner i indifferent between aigning the buyer with valuation x( k ) to eller k veru aigning that buyer to any eller with a lower reervation value. That i, V ( 1 ; x( k )) = V ( 2 ; x( k )) =... = V ( k ; x( k )); thu, by comparion with equation (21), x( k ) = x ( k ). By induction, the equilibrium and ocial planner cutoff value coincide for i = 1,..., N. Finally, in order that m N = m N, it mut be that N k=1 m k Λ( k ) m M = θ( N )V ( N ); that i, the buine-tealing effect aociated with the entry of a type N eller ha to equal the expected number of buyer drawn away by the entry of the marginal eller time the lo in urplu per buyer who leave an incumbent auction. The argument for why thi equation hold i exactly the ame a in the cae with two eller type. A.2.3 Continuum of Seller Type In the model with N eller type, for each eller type k, there i a correponding buyer type x ( k ) who i indifferent between viiting a eller of type k veru any eller poting a lower reerve price. The function x ( k ) i defined on a dicrete et of point, { 1,..., N }. To move to a continuum of eller type, we let the ditance between eller type k+1 and k go to zero and derive a differential equation that give a continuou function x () a the limit of the N-eller cae. The purpoe of thi ubection i to derive thi equation. Since the continuum-of-eller-type olution i the limit of the dicrete eller type cae, our effi ciency reult carry over to the continuum. A in the ex pot cae, we normalize the total meaure of potential eller to one, and we denote the ditribution of reervation value acro thee eller by G(). We begin with a dicrete ditribution over eller type. Let 1 = 0, 2 =,..., k+1 = k +, and 21

23 let m 1 = g( 1 ), m 2 = g( 2 ), etc. We denote the arrival rate of buyer to type k eller by θ( k ) and the ditribution of valuation among buyer viiting type k eller by F (x; k ). Lemma 1 x ( k ) 1 θ( k ) = θ( k+1 ) + k j=1 m f(x)dx for k = 1,..., N 1 (23) j x ( k ) 1 x θ( k )F (x; k ) = k j=1 m f(z)dz for x ( k ) x x ( k+1 ). (24) j x ( k ) Proof. Buyer with valuation x x ( k+1 ) randomize acro all eller of type k+1 or below. Thu, a type k eller can expect a many buyer of thi type a can a type k+1 eller. In addition, a type k eller attract ome additional buyer, namely, thoe with valuation x [x ( k ), x ( k+1 )). Buyer with valuation in thi range randomize over eller of type k and below, and there i a ma k j=1 m j of uch eller. Thi give equation (23). To undertand equation (24), note that (i) the meaure ( of the buyer with ) valuation between x ( k ) and x < x m k x ( k+1 ) viiting type k eller i m m f(z)dz x ( k ) k while (ii) the meaure of type k eller can be written a m k θ( k ). Since F (x; k ) = 0 for x x ( k ), it follow that ( F (x; k ) = m k m m k m k θ( k ) ) x x ( k ) f(z)dz Multiplying both ide by θ( k ) give equation (24). for x ( k ) x x ( k+1 ). The cutoff valuation x ( k+1 ), i.e., the lowet buyer type who i indifferent between viitinga type k eller veru a type k+1 eller, i defined by V ( k ; x ( k+1 )) = V ( k+1 ; x ( k+1 )). Uing equation (14) give x ( k+1 ) k e θ( k)(1 F (x; k )) dx = x ( k+1 ) k+1 e θ( k+1)(1 F (x; k+1)) dx. (25) Note that F (x; k ) = 0 for x < x ( k ), and imilarly F (x; k+1 ) = 0 for x < x ( k+1 ). Equation (25) can thu be rewritten a e θ( k) (x ( k ) k ) + x ( k ) k + x ( k+1 ) x ( k ) x ( k+1 ) x ( k ) e θ( k)(1 F (x; k )) dx = e θ( k+1) (x ( k+1 ) k+1 ) or e θ( k)f (x; k ) dx = e θ( k) θ( k+1 ) (x ( k+1 ) k+1 ) 22

24 Uing e θ( k)f (x; k ) 1 + θ( k )F (x; k ) and e θ( k) θ( k+1 ) 1 + θ( k ) θ( k+1 ) equation (25) can be further rewritten a x ( k+1 ) k + x ( k+1 ) x ( k ) θ( k )F (x; k )dx = (1 + θ( k ) θ( k+1 )) (x ( k+1 ) k+1 ). (26) We ue the notation k+1 =, k =, x ( k+1 ) = x (), and x ( k ) = x () x () and note that G() = k j=1 m j. Then uing equation (23) and (24), we can rewrite equation (26) a x x x f(z)dz () + + x () x () x () x () G() dx = (x () ) 1 + x () x () x () f(x)dx G() The left-hand ide of equation (27) ued equation (24); the right-hand ide ued equation x x f(z)dz (23). The term dx on the left-hand ide of thi equation i x () x () x () x () G() o( x ()). On the right-hand ide, x () x () x () Equation (27) therefore reduce to f(x)dx = F (x ()) F (x () x ()) f(x ()) x (). (27). x () = G() (x () ) f(x). (28) Together with the initial condition, x (0) = 0, equation (28) determine the function x (). 23