Competing Auctions with Endogenous Quantities

Transcription

1 Competig Auctios with Edogeous Quatities Bey Moldovau, Aer Sela, Xiawe Shi December 6, 006 Abstract We study models where two sellers simultaeously decide o their discrete supply of a homogeous good. There is a ite, ot ecessarily large, umber of buyers who have uit demad ad privately ow valuatios. I the rst model, there is a cetralized maret place where a uiform auctio taes place. I the secod model, there are two distict auctio sites, each with oe seller, ad buyers decide where to bid. Our results shed some light o the coditios leadig to either the emergece of domiat maretplaces or to the coexistece of several competig sites. Usig the theory of potetial games, we show that i the oe auctio site model there is always a (almost symmetric) equilibrium i pure strategies. This equilibrium approximates the Courot outcome as the umber of buyers becomes large. I cotrast, if the distributio of buyers values has a icreasig failure rate, ad if the margial cost of productio is relatively low, there is o pure strategy equilibrium where both sellers mae positive pro ts i the competig sites model. We also idetify coditios uder which a equilibrium with a uique active site exists. Techically, we are able to deal with the ite ad discrete models by usig several results about order statistics developed by Richard Barlow ad Fra Proscha (1965, 1966, 1975). Moldovau: Departmet of Ecoomics, Uiversity of Bo, Leestr. 37, Bo, Germay; mold@uibo.de. Sela: Departmet of Ecoomics, Be Gurio Uiversity, Beer Sheva 84105, Israel; aersela@bgu.ac.il. Shi: Departmet of Ecoomics, Yale Uiversity, 8 Hillhouse Aveue, New Have, CT 0650, USA; xiawe.shi@yale.edu. 1

2 1 Itroductio We study models where two sellers simultaeously decide o their (discrete) supply of a homogeous good, ad where there is a ite, ot ecessarily large, umber of buyers who have uit demad ad privately ow valuatios. 1 I the rst model, the sellers brig their supply to a cetralized maret place where a uiform auctio taes place. Thus, by the itrisic rules of the mechaism, all sold uits commad the same price. I the secod model, there are two distict auctio sites, each with oe seller. Before observig their value, but after observig the respective supplied quatities at each site, buyers decide which auctio to atted (agai, each auctio is a uiform price oe). I priciple, each auctio may have its ow equilibrium price. Our results shed some light o the coditios leadig to either the emergece of oe domiat maretplace or the coexistece of several competig sites. I the oe auctio site model there is always a (almost symmetric) equilibrium i pure strategies. This equilibrium approximates the Courot outcome as the umber of buyers becomes large. I cotrast, if the distributio of buyers values has a icreasig failure rate (a coditio assumed i most of the literature, which implies that margial reveue is decreasig), ad if the margial cost of productio is relatively low, there is o pure strategy equilibrium where both sellers mae positive pro ts i the competig sites model. I other words, two distict auctio sites caot coexist i equilibrium uder these coditios. Coexistece becomes possible oly if the margial cost of productio is su cietly high. We also idetify coditios uder which a equilibrium with a uique active site exists. Aother goal of this paper is to revisit several classical scearios i moopoly ad oligopoly theory. These theories discard the stadard assumptio of competitive aalysis cocerig the large umber of producig rms (sellers), but they eep the assumptio that the umber of cosumers (buyers) is large. Our model has both a small umber of sellers with edogeous supply, ad a small umber of buyers. This framewor is appealig i may situatios, i particular i marets for iputs where both upstream ad dowstream marets are relatively cocetrated oligopolies. Techically, we are able to deal with the ite ad discrete models by usig several results about order statistics developed by Richard Barlow ad Fra Proscha (1965, 1966, 1975). 3 Oe way to uify the two mai models of the paper is i terms of buyers switchig cost. If the switchig cost is high, that is, oce buyers decide to place a bid i oe auctio they caot switch to the other, we obtai the model with two competig auctio sites. If the switchig cost is low, the a ascedig cloc auctio a la Demage-Gale-Sotomayor (this is a variat with moey o Gale-Shapley s 1 Most qualitative results ca be geeralized to settigs with more sellers. The parallels betwee a sigle-good auctio theory ad moopoly are well-ow - see Bulow ad Roberts (1989). Hase (1988) studies a model where sellers compete for the right to become the sole supplier to a uique buyer whose purchase depeds o the wiig bid. 3 For other applicatios of Barlow ad Proscha s results see Moldovau, Sela ad Shi (005) ad Hoppe, Moldovau ad Sela (006).

3 deferred acceptace algorithm) where buyers shift demad to the cheaper uit, leads to a e ciet outcome (coditioal o supplied quatities). This outcome is equivalet to the oe of the uiform price auctio i the sigle site model - i particular, all uits are sold at a sigle maret price. Viewed i this perspective, the competig auctios model is oe where, i the rst (quatity-settig) stage, the sellers compete for maret share, i.e., for the buyers that will become loced-i at the secod stage (see Klemperer, 1987 for a early aalysis of the forces at play i such models). By icreasig quatity, a seller attracts more buyers. Loc-is appear i may settigs ad have a variety of reasos such as trasactio costs associated with operatig via several techologically di eret platforms/clearig houses, ucertaity about the quality/quatity o ered at other maretplaces, loyalty cotracts with operators, ex-ate maret speci c ivestmets, ad various forms of etwor exteralities. O the other had, a model with a cetralized auctio site ad with edogeous supply is better suited for the study of some moder marets where all trasactios are executed at maret price, i.e., order drive periodic auctios i acial marets, or marets for iputs such as electricity or gas. 4 olie auctio sites such as ebay there are ofte several simultaeous auctios for idetical commodities (e.g., CPU s). Sajid et al. (004) empirically study such parallel auctios ad d that a sigi cat proportio of bidders are ideed active across the competig auctios: they repeatedly place bids i the auctios with the lowest stadig bid. Moreover, prices teded to be uiform across auctios. I cotrast to our paper, most of the papers i the relatively thi literature o competig auctios cosider models with several sellers edowed with a sigle uit of a homogeous good, ad with several buyers with uit demad who decide which auctio to atted. Thus, total demad ad supply are xed exogeously, whereas total supply is edogeous i our models. Moreover, most of these papers use some id of large maret assumptio for their mai results (i.e., idividual agets igore the e ect of their ow actios o prices) while such a assumptio is ot eeded here. It is ot always clear whether such assumptios are cosistet with a limit obtaied i small marets of icreased size. Peters ad Severiov (006) cosider several sellers who set reserve prices at their ow auctios sites ad the coduct secod price auctios. A large umber of buyers with uit demad ca move amog auctio sites. The mai cosequece of this freedom (i.e., absece of switchig costs) is that there exists a e ciet equilibrium (coditioal o reserve prices). This is cosistet with the isight obtaied from the models that mimic features of the Gale-Shapley deferred acceptace algorithm (such as our model of competig sellers at a uique auctio site) ad should be cotrasted with earlier models such as McAfee (1993) 5 or Peters ad Severiov (1997) where buyers ca place oly oe bid, thus 4 Legwiler (1999), Bac ad Zeder( 001) ad McAdams (006) aalyze edogeous supply i moopolistic models where the seller adjusts supply after seeig the realizatio of demad, ad where bidders demad several uits. LiCalzi ad Pava (006) allow the moopolist to ex-ate commit to a supply fuctio. This is i the spirit of Klemperer ad Meyer (1989) who study supply-fuctio equilibria i oligopoly with demad ucertaity. 5 He cosiders several sellers o erig auctios at xed sites (each seller at aother site). His aalysis assumes that the At 3

4 leadig to a ie ciecy stemmig from the coordiatio problem. The role of a uique maretplace is also the theme of Moreo ad Ubeda (006) who itroduce a explicit elemet of price formatio i the traditioal Courot oligopoly story (while eepig the assumptio that there is a large umber of buyers). Their startig poit is the classical paper by Kreps ad Scheima (1983) who study a two stage game, with capacity choice i the rst stage, ad Bertrad competitio i the secod stage. Kreps ad Scheima show that the uique equilibrium coicides with the Courot outcome. Moreo ad Ubeda (006) loo at the two stage model where rms rst choose capacity ad the set reservatio prices at which they are willig to sell their etire capacity. The esuig supply fuctio is used to clear the maret. The mai di erece to the Kreps ad Scheima aalysis is that all supply is sold at maret price whereas, i Kreps ad Scheima s model each rm charges its ow price. This allows them to avoid several di culties appearig i the Kreps-Scheima model. 6 If we itroduce reserve prices i our model of competig sellers at oe auctio site, ad if we let the umber of buyers go to i ity (which is the stadard assumptio i the literature) it ca be show that, if quatity is optimally adjusted to the umber of buyers, the margial gai from a reserve price goes to zero. 7 The limit outcome is the the classical Courot oe - correspodig to the mai result of Moreo ad Ubeda (006). A iterestig strad that allows for both a small umber of buyers ad a small umber of sellers is the literature o the so-called strategic maret games, followig the pioeerig wor by Shapley ad Shubi (1975). I these models both buyers ad sellers bid quatities, ad trasactios are cleared at a price equal to the ratio of demad to supply (thus, commodities are allocated i proportio to bids). Almost the etire literature ideti es the umber of tradig sites (or posts ) with the umber of traded commodities. I other words, there is a uique tradig site for each traded commodity where all demad ad supply of that commodity is cleared. A otable exceptio is Koutsougeras (003) who allows for two separate tradig sites for the same good. I his model traders ca sed quatity bids to either oe, or to both sites. He shows via a example that the Law of Oe Price eed ot hold i equilibrium. Although di eret tradig prices ope up arbitrage opportuities for agets who ca shift their demad/supply (which seems icosistet with the equilibrium idea), these opportuities ecoomy is large, ad therefore it igores the e ects of idividual buyers o prices. I particular, it igores the fact that a chage i oe seller s mechaism a ects the distributio of buyers at his ow ad at other sites. 6 The K-S result critically depeds o a particular ratioig rule used whe capacity is bidig. Davidso ad Deecere (1986) show that the Courot outcome caot be the equilibrium of the two stage game for ay other ratioig rule. Moreover, for a wide rage of capacity choices, the equilibrium at the price competitio stage is i mixed strategies, leadig to ex-post regret. 7 This result is based o a aalogous observatio for a moopolistic rm that optimally adjusts quatity i respose to a varyig umber of buyers. Sice a icrease (decrease) i quatity causes a decrease (icrease) i price, a bidig reserve price becomes super uous as a istrumet to cotrol price. This should be cotrasted to what happes if competig sellers have oly oe uit to sell (Burguet ad Saovics, 1999) 4

5 disappear whe idividual agets (who have, relatively speaig, a large impact if the umber of traders is small) try to tae advatage of them. Such pheomea, which also appear i our model, icely illustrate the cautio that eeds to be employed whe dealig with models with a small umber of traders. I particular, maret structure may matter a lot: i our model, eve mere equilibrium existece is a ected by it. Our model of competig auctio sites with two possibly distict prices is related to that of Elliso, Fudeberg ad Mobius (004). These authors study a situatio where both sellers ad buyers (each with uit demad/supply) simultaeously choose oe of two auctio sites. 8 Thus, both buyers ad sellers are loced-i at their respective sites, ad both total supply ad demad i the maret are xed. Prices are determied by uiform-price auctios at each site (ad thus by the ratio of buyers to sellers at each locatio). 9 Their elegat aalysis igores the iteger costraits, ad some results hold oly for marets with large umbers of traders. 10 Elliso et al. focus o the coditios leadig to the existece of equilibria where two auctio sites of uequal size ca coexist. Their solved examples ivolve the uiform ad expoetial distributios of values thus both display a icreasig failure rate. Their coexistece results should be cotrasted to ours uder this assumptio. The di erece is maily due to the edogeous supply feature i our model. I a di eret vei, Neema ad Vula (001) study the coexistece of two forms of trade: direct bilateral egotiatios ad cetralized marets. 11 I their model, buyers with a high willigess to pay ad sellers with low costs prefer to trade through a cetralized maret. This leads to a uravelig of direct egotiatios, so that ultimately, all traders opt for tradig through the cetralized maret. Their aalysis uses a reduced form formulatio ad depeds o the assumptio that there are large umbers of traders. The preset paper is orgaized as follows. I Sectio we describe the basic model igrediets ad itroduce useful de itios ad otatio. I Sectio 3 we study the case of oe auctio site. I Subsectio 3.1 we start with a moopolistic seller usig a uiform price auctio. We focus o the optimal supplied quatity, ad we show that it icreases i the covexity of the distributio fuctio of buyers valuatios. This allows us to derive optimal quatity estimates for large, o-parametric families of distributios (i.e., with mootoe hazard rates, cocave, covex). I subsectio 3. we remai i the sigle auctio-site model, but add competitio amog sellers: each of two ex-ate symmetric sellers provides the auctioeer with several uits of a homogeous good, ad the auctioeer sells the total quatity via a uiform price auctio. With a large umber of buyers, this is equivalet to classical Courot competitio. Usig the potetial game approach (due to Moderer ad Shapley, 1996) we 8 This is a applicatio of a more geeral framewor developed by Elliso ad Fudeberg (00) 9 We could have also preseted our model by assumig that: 1) every uit is owed by a di eret seller; ) the operator of each auctio site restricts etry of sellers i order to maximize total pro t at the site. 10 Aderso et al. (005) retae this aalysis while icorporatig the iteger costraits sice igorig them may lead to icosistet results. 11 Geesove (1995) o ers a isightful study of such a arragemet i the New Eglad sh idustry. 5

6 prove the existece of pure-strategy Nash equilibria for ay costat margial cost. Moreover, we show that a almost symmetric equilibrium, where the supplied quatities di er by at most oe uit, always exists. Fially, we use the above described estimates i the moopoly case to give estimates for the total quatity uder competitio. I Subsectio 3.3 we o er a example illustratig that a merger of auctio sites may ot be pro table if sellers optimize supply before ad after a merger (although a merger is always e ciecy icreasig). This cotrasts a earlier isight due to Schwartz ad Ugo (00) who cosidered sellers with exogeously give quatities. I Sectio 4 we tur to the model with two distict auctio sites, each with oe seller: First, the two sellers simultaeously choose the umber of uits supplied i their respective auctios. Before they lear their valuatios, but after they observe the sellers decisios, buyers choose which auctio to atted. I the last stage, each buyer lears his valuatio ad submits a bid i his selected auctio. I Subsectio 4.1 we study the possibility of coexistece of two auctio sites. A mai result is that if the margial productio cost is low eough, ad if the distributio of buyers values has a icreasig hazard rate the there is o pure strategy equilibrium i which both sellers are active ad mae positive pro ts. The reaso is that each seller cotiually icreases supply i order to satch buyers from the other site. A coexistece equilibrium is possible oly if the margial cost of productio is high eough. The, the uilateral icetives to icrease supply are capped by the productio cost. I Subsectio 4. we study equilibria where oly oe auctio site is active (i.e., the other seller prefers to stay out). Such a equilibrium is possible if the optimal moopolistic quatity produced by the active seller is high eough to deter etry this happes, for example, if the distributio of buyers values is su cietly covex, ad if the margial cost is ot too high. I Sectio 5 we compare the equilibrium quatities ad prices across three di eret models (moopoly, competitio at oe auctio site, ad two competig auctio sites) uder the assumptio that there is a large umber of buyers. We show that, uder a mild coditio, the moopoly model has the highest price ad lowest quatity i equilibrium. Furthermore, if the equilibrium supply is relatively high (e.g., if the productio cost is low ad the distributio of buyers valuatios is su cietly covex), the model with two auctio sites is more competitive tha the model with oe auctio site: the equilibrium price is lower ad the equilibrium quatity is higher i the former model. Fially, if the equilibrium supply is relatively low, the the reverse is true. Sectio 6 cotais several cocludig commets. Appedix A cotais several useful results from the literature o order statistics. Appedix B cotais all proofs. 6

7 The Model There are two competig sellers ad buyers. The sellers ca each produce several uits of a homogeous good. Each buyer has uit demad. The valuatio of buyer j for a uit of the good is private iformatio to j: Valuatios are draw idepedetly of each other from the iterval [0; T ] ; T 1; accordig to a distributio fuctio F that is commo owledge. We cosider two di eret competitio models amog the sellers: 1. Oe auctio site: Sellers 1 ad simultaeously provide 1 ad uits to the auctioeer, respectively. The the auctioeer sells the ( 1 + ) uits via a uiform price auctio. The reveue of each seller is the product of the equilibrium price ad his supplied quatity.. Two competig auctio sites: I the rst stage, sellers 1 ad simultaeously choose the umber of objects for sale i, i = 1; ; i their ow auctios. I the secod stage, before they lear their valuatios, but after they ow the decisio made by the sellers i the rst stage, buyers choose whether to atted auctio 1 or. I the third ad last stage, each buyer lears his valuatio ad submits a bid i his selected auctio (each of the two auctios is a uiform price auctio). The reveue of each seller is the product of his supplied quatity ad equilibrium price i his ow auctio..1 Notatio We use the followig otatio: X ; deotes -th order statistic out of idepedet valuatios distributed accordig to F (ote that X ; is the highest order statistic, ad so o..). The distributio F ; ad the desity f ; of X ; F ; (x) = f ; (x) = are give by X j= We deote by EX ; the expected value of X ;. F (x) j [1 F (x)] j j! ( 1)!( )! F (x) 1 [1 F (x)] f(x) The failure rate (or hazard rate) of a distributio F is de ed by: (x) = f (x) 1 F (x) A distributio fuctio F has a icreasig failure rate (IF R) if its failure (or hazard) rate, (x) ; is icreasig. Aalogously, F has a decreasig failure rate (DF R) if (x) is decreasig. Note that covexity of F implies IF R, while DF R implies cocavity. The oly distributio that is both cocave ad covex is the uiform, while the oly distributio that is both IF R ad DF R is the expoetial. A distributio fuctio F is star-shaped o [0; T ] if F (x)=x is icreasig i x. Let X(Z) have distributios F (G) such that F (0) = G(0) = 0: Distributio F is star-shaped with respect to G if the fuctio G 1 F (x) is star-shaped. Notice that : 7

8 1. If F (0) = 0 = G (0) ; G 1 F covex implies G 1 F star-shaped.. If G is the expoetial distributio, the G 1 F covex is equivalet to F beig IF R. 3. If G is uiform, the G 1 F covex is equivalet to F is covex. 3 Oe Auctio Site We discuss rst the model of a uiform price multi-object auctio with buyers ad a sigle, moopolist seller with variable supply. The derived isights will be repeatedly used i the aalysis of competitio amog sellers below. 3.1 Moopoly seller The seller decides o the supply, ad the the buyers decide what to bid i a uiform price auctio for uits. We do ot explicitly aalyze here the use of exclusio tools such as reserve prices ad etry fees. Thus all buyers have a icetive to participate i the auctio. Note that i a auctio with edogeous quatity ad large umber of buyers, the margial gai from settig optimal reserve price is small if quatity is optimally set. The ituitio is provided by the fact that the realizatios of X are very close to EX ; if is large. The seller ca cotrol EX ; by varyig the quatity : If the reserve price is bidig ad ehacig reveue, the seller ca always set the quatity to 0 such that, approximately, EX 0 ; = r: After this adjustmet, the reserve price is redudat. We assume rst that the seller has a zero cost of producig the object. If the seller o ers > objects, it is obvious that the equilibrium price is zero, ad the seller has zero pro t. Thus, without loss of geerality, we restrict attetio to the case : I the auctio, each of the highest buyers wis a sigle uit, ad pays the equilibrium price X problem is give by max R (; ) = max EX : ;. I expectatio, the seller s maximizatio The relatio betwee the optimal umber of objects ad the distributio of the buyers valuatio is as follows: Propositio 1 Cosider two distributios G; F leadig to optimal supplies G () ad F (), respectively, whe there are buyers. If G 1 F is star-shaped the G () F (): Propositio 1 implies the followig two corollaries: ; Corollary 1 If F is IFR (DFR), ad is large, the F (); the optimal supply i a auctio with buyers satis es F () () e ; where e is the atural logarithm basis. 8

9 Corollary If F is covex(cocave), the F (); the optimal supply i the auctio with buyers satis es F () () : The followig result cosiders the fuctio F () that relates the optimal supply to the umber of buyers. It shows that the slope of this fuctio is less tha uity if F is IF R (thus, the di erece () wealy icreases i ) Propositio Let F () be the optimal umber of objects i the auctio with buyers. If F is IF R, the, for all ; F ( + 1) F () 1: Remar 1 Whe is large we have the approximatio EX ; F 1 Hece, the seller s reveue i our auctio is R 1 (; ) = EX ; F 1 = F 1 1 Cosider ow a moopolist seller facig buyers, each with uit demad ad valuatio distributio F: The moopolist chooses p to maximize his expected reveue. R (p; ) = p [1 F (p)] Settig = [1 F (p)] ; or equivaletly p = F 1 1 ; we see that the moopoly ad auctio models are equivalet if the umber of buyers is su cietly large. I particular, p ; the limit equilibrium price i a sigle auctio with edogeous quatity ad a su cietly large umber of buyers is give by (p ) = f (p ) 1 F (p ) = 1 p Note that a icrease of the distributio of the buyers valuatios i the hazard rate order (which implies rst-order stochastic domiace) yields a higher equilibrium price. Example 1 Let F (x) = x 1=a (a > 0) with support [0; 1] : Whe is large, the seller s reveue if he sells objects is The optimal is R (; ; a) = EX ; F 1 = F () = 1 a + 1 a ad we have F (; a) < = if a > 1 ad F (; a) > = if a < 1: The limit equilibrium price is p (a) = ( a a+1 )a : 9

10 Assume ow that the seller faces a costat margial cost of producig oe uit, ad deote this cost by c 0: The seller s maximizatio problem is max (; ) = max R (; ) c = max (EX ; c) The margial reveue from a additioal object is give by R( + 1; ) R(; ): The followig result follows from the observatio that the margial reveue is decreasig i the umber of objects if F is IF R. Propositio 3 If F is IF R, the optimal umber of uits F (; c) decreases i c for xed : 3. Competitio amog sellers at a uique auctio site We cosider ow two sellers competig to sell uits to buyers through a sigle, uiform-price auctio. Sellers 1 ad provide 1 ad objects to the auctioeer, respectively. The the auctioeer sells ( 1 + ) through the uiform price auctio. Throughout of this sectio, we assume for simplicity that c = 0: All mai results ca be immediately exteded to the case of positive margial costs. Moreover, the mai result, existece of equilibria of pure strategies geeralizes to a settig with several sellers. If (1; ) is a equilibrium, the for all s f i ; :::; i j g it must hold: i EX i j ; ( i + s)ex i j s; 0, i = 1; (1) The followig example illustrates several iterestig pheomea: 1) Symmetric ad asymmetric pure strategy equilibria may coexist; ) Symmetric pure strategy equilibria may ot exist; 1 3) As the umber of buyers gets large, all equilibria coverge to the symmetric oe. Example Suppose that F is uiform. Give ; rm 1 chooses 1 to maximize R 1 = 1 EX (1+ ); = 1 ( 1 ) + 1 If 1 is optimal, the for ay s = 1 ; :::; 1 Settig s = 1; 1; we get Similarly, if is optimal for rm ; we obtai: 1 ( 1 ) ( 1 + s) ( 1 s) 0 s (s ) 0 1 ( 1) 1 1 ( + 1) () 1 ( 1 1) 1 ( 1 + 1) (3) 1 Amir ad Lambso (000) exibit a su ciet coditio for the existece of a uique equilibrium i the stadard model of Courot competitio. 10

11 Combiig equatio () ad (3), we get 1 1 ( 1) ( + 1) 1 Therefore, ( 1 + 1) 1 ) 1 1 ( 3) 3 1 ( 1 1) + 1 ) 1 1 ( + 3) (4) ) If = 3m there are three possible pure strategy equilibria (see details i the Appedix) 1 = m + 1; = m 1 1 = m 1; = m = = m ) If = 3m + 1; there are also three pure-strategy equilibria 1 = m + 1; = m 1 = m; = m = = m 3) If = 3m + ; there are oly two asymmetric pure strategy equilibria 1 = m; = m = m + 1; = m We tur ow to a proof of existece of pure-strategy Nash equilibria. Sice the strategy spaces are discrete ad sice the payo fuctios are ot cotiuous, it is obvious that the stadard approach caot wor. Fortuately, it turs out that our game is a ordial potetial game i the sese of Moderer ad Shapley (1996). Thus it possesses a Nash equilibrium i pure strategies, correspodig to a maximizer of the potetial fuctio. The ext two results hold also for the case where c > 0. Theorem 1 A pure strategy Nash equilibrium always exists i the competitio model with oe auctio site. By taig a closer loo at the potetial fuctio, we also get the existece of a almost symmetric, pure-strategy Nash equilibrium: De itio 1 A equilibrium (1; ) is quasi-symmetric if j1 j 1: 11

12 Theorem A pure strategy, quasi-symmetric Nash equilibrium always exists i the competitio model with oe auctio site. Remar Cosider ow a large umber of buyers. If the sellers supply 1 ad uits, respectively, the reveues are approximated by i = i EX (i+ j); i F 1 i j = i F 1 1 i + j This is equivalet to a stadard Courot model where the iverse demad fuctio is give by P ( 1 + ) = F I particular, uder the coditios of Amir ad Lambso(000), i the limit as gets large, there exists a uique symmetric equilibrium. We coclude this Sectio with the followig equilibrium characterizatio result: Propositio 4 1) Let F be covex (cocave). The, i ay equilibrium, the total supplied quatity is higher (lower) tha 3: I particular, if the distributio of buyers valuatios is uiform, the the total supplied quatity is 3 uits. ) If F is IF R (DF R), ad if is large, the i ay equilibrium, the total supplied quatity is higher (lower) tha 1+e: I particular, if the distributio of buyers values is expoetial, the supplied quatity is 1+e uits. 3.3 Mergig auctio sites Schwartz ad Ugo (00) cosider a model where two moopolistic auctio houses with separate sets of buyers decide to merge, ad show that such a merger always icreases total reveue (it is obvious that a merger icreases total welfare sice it yields a overall e ciet allocatio of the uits). I their model, the supplied quatities before the merger are exogeous, ad the merged etity simply o ers the sum of the ex-ate quatities to the uited set of buyers. I other words, the sellers do ot optimize their supply, ad, moreover, do ot adjust it after the merger. I the followig example, we loo at the case of two separate auctio sites, each with oe seller. After a merger of the auctio sites, the sellers remai separate ad compete by optimizig the quatity supplied to the merged site. We show that, uless the umber of buyers is small, the post-merger total reveue goes dow, which implies that at least oe of the sellers loses from the merger. Thus, e ciecy gais are per-se, ot eough to explai mergers of auctio sites if we assume that multiple competig sellers use the merged site ad that the site operator ears a xed fractio of reveue. This coforms to the usual ituitio from Courot aalysis, but our results also ideti es the role played i this argumet by a large umber of buyers. Example 3 Assume that F is uiform o [0; 1] ad assume that, before a merger, the two auctio sites face 1 = buyers. For the uiform distributio we ow that the optimal supply at each site is i ; 1

13 i = 1; : Thus total pro t is give by: 1 + = 1= =4 + 1 = 1 ( 1 + 1) After the merger, the uique auctio site faces 1 buyers. For the uiform distributio, we ow that the equilibrium total supply i the competitio model equals 41 3 ; two-thirds the umber of buyers. Thus, total reveue is The reveue di erece is c 1 + c = = ( 1 + 1) c 1 + c ( 1 + ) = ( 1 + 1) 1 ( 1 + 1) = ( 1 + 1) ( 1 + 1) Thus, if 1 4; total reveue after the merger is lower. Our ext result geeralizes the above observatio to ay distributio, as log as the umber of buyers is su cietly large. Propositio 5 Assume that two separate auctio sites, each with oe seller ad i = m buyers, merge, ad assume that the two sellers subsequetly compete i a uique auctio with m buyers. If m is large eough, total reveue goes dow. 4 Competitio amog auctio sites We ow cosider two competig, uiform price, multi-uit auctios. The two sellers simultaeously choose the umber of uits for sale i, i = 1; i their respective auctios. The buyers observe the respective supplies at each site, ad the decide where to bid, ad what bid to place. Sice each auctio is uiform, the biddig part of the decisio is simple, ad of o further cocer: each bidders places a bid equal to his valuatio. Assume the that 1 buyers compete i auctio 1 for 1 uits, while buyers compete i auctio for uits, where 1 + < = 1 + : De itio A pro le ( 1; 1; ; ) is a equilibrium of the competig auctios model if : (i) i maximizes seller i 0 s reveue give i: ; 1; : That is, o seller wishes to adjust supply. (ii) Give (1; ) ad give all other buyers decisios, o buyer h has a icetive to switch to aother auctio site. Note that i a equilibrium with positive pro ts at two auctio sites we caot have 1 + = : I that case, the supply at oe auctio site (at least) is o less tha the umber of buyers, ad the respective price is the zero. The uit price i auctio i is give by EX i i : The followig Lemma gives a coditio o the ; i prices at the two sites that must hold i equilibrium: 13

14 Lemma 1 Cosider ay xed strategy pro le for the two sellers ad for all buyers except h. The, it is optimal for h to joi the auctio with the lower expected price. Moreover, i ay equilibrium, ( 1; 1; ; ) it must hold that EX i i ; i EX j j ; j ) EX i i ; i EX j +1 j ; j +1 I other words, i equilibrium, it caot be pro table for a buyer i the auctio with a higher expected price to move to the other auctio. Thus, while equilibrium prices at the two auctio sites eed ot be strictly equal, there are o arbitrage opportuities. 4.1 Equilibrium coexistece of two auctio sites Our mai result i this sectio is that two auctio sites yieldig positive pro ts for the respective sellers caot coexist i equilibrium if the distributio of buyers values is IF R; ad if the margial productio cost, c; is su cietly low. The proof is based o the followig Lemma which has some idepedet iterest. It shows that, for ay pro le of buyers actios, at least oe of the sellers has a icetive to icrease supply if F is IF R: Lemma 1. Cosider ay co guratio ( 1 ; 1 ; ; ) where 1 ; > 0; 1 ; > 0 ad 1 + < ; ad assume that buyers play a best respose to ay aoucemet of quatities. If EX i i; i EX j j; j ; i; j = 1; ; the seller i ca attract at least oe more buyer by supplyig oe additioal uit.. Assume that F is IF R; ad that c = 0: For ay co guratio ( 1 ; 1 ; ; ) where 1 ; > 0; 1 ; > 0 ad 1 + <, oe of the sellers ca icrease his pro t by supplyig oe additioal uit. 3. If F is IF R; ad if c = 0; the R(; ) = R( + 1; + 1) R(; ), the margial reveue of seller whose supply icreases from to + 1 while the umber of buyers attedig his auctio icreases from to + 1, decreases i for a xed : The Lemma yields: Theorem 3 Assume that F is IF R: The, for su cietly small margial productio costs c; there is o equilibrium i which both sellers are active ad mae positive pro ts. I particular, if c < EX 1; +1 ( 1)EX 1; (5) there is o symmetric equilibrium with positive pro ts. If the IF R coditio is ot satis ed, or if the cost is high eough (i cotrast to the assumptios of the above Propositio) a equilibrium where two auctio sites are active, ad where both sellers 14

15 mae positive pro ts may the exist. We loo rst at a example where F does ot satisfy the IF R coditio: Example 4 Let F (x) = x 1= ; ad let c = 0: The: R ( i ; i ) = ( i + 1)( i i + 1)( i i ) ( i + )( i + 3) = ( i i )( i i + 1)( i + 1 i ) ( i + 1)( i + )( i + 3) i ( i i + 1)( i i ) ( i + 1)( i + ) Note that R ( i ; i ) 0 if i i+1 : Lettig 1 = = ad 1 = = 3; it is clear that o seller has a icetive to icrease supply. Assume the that seller 1, say, lowers supply to oe uit. The a buyer attedig his auctio has a icetive to switch to the other auctio sice there the price will be EX ;4 < EX ;3 : Therefore o seller has a icetive to deviate, ad we obtai a symmetric equilibrium i which every seller sells two uits ad maes a positive pro t: 13 Next we loo at situatios where the margial cost c is su cietly high, The, sellers do ot wat to icrease their supply above a certai level (sice the price drops below cost), ad a equilibrium may exist. If, i a symmetric equilibrium, a seller sells uits, he should have o icetive to icrease supply to + 1. The, for F which is IF R; we have the ecessary coditio 14 p = EX ; > c > ( + 1)EX ; +1 EX ; (6) where the left iequality idicates that a seller maes a positive pro t by sellig uits to half the umber of buyers, ad the right iequality idicates that a seller has o icetive to icrease the umber of supplied uits from to + 1 if by doig so the umber of buyers attedig his auctio icreases from to + 1. The followig propositio provides a su ciet coditio for existece of a symmetric equilibrium with the maximal supply value for which both sellers mae positive pro ts. Propositio 6 Assume that F is IFR. If ( 1)EX (; +1) ( )EX (; ) > c > EX (1; +1) ( 1)EX (1; ) (7) the the pro le where each seller sells 1 objects to buyers costitutes a symmetric equilibrium. 4. A uique auctio site as the outcome of competitio We ow focus o equilibria where oly oe auctio site is active, ad where oly oe seller maes a positive pro t. Roughly speaig, such a equilibrium may exist if the optimal moopolistic quatity 13 I this example there is also a symmetric equilibrium where each seller supplies three uits, ad where pro ts are zero. 14 If F is IF R; by lemma the RHS of (6) is always positive. 15

16 for the uique active seller is high eough. The, the other seller caot covice eough buyers to switch to his ow auctio, ad thus prefers to stay out. Formally, a equilibrium with the form ( 1 = ; 1 > 0; = 0; = 0) exists if ad oly if: 1. ad 1 = arg max (EX ; c) > 0 (8). For all 1; either the set M = M( ) = fm j m 1 ^ EX m ;m cg is empty or EX e ;e > EX e+1 1 ; e+1 (9) where e = e ( ) is the miimal elemet i M: Coditio (8) requires that seller 1 optimizes his (moopolistic) supply ad maes a positive pro t, give that seller is ot active. I coditio (9), e is the miimal umber of buyers for which sellig uits is pro table for seller. Note that if coditio (9) holds ad if > e, we also get EX ; > EX e ;e > EX e+1 1 ; e+1 = EX +( e +1) 1 ; +( e+1) The rst iequality follows by repeated applicatio of the ow relatio EX (i+1;+1) EX (i;) ; ad the secod iequality is coditio (9). That is, coditio (9) guaratees that if > e buyers atted auctio, at least ( e + 1) buyers will wat to switch to auctio 1. Thus, seller must remai iactive, ad ears zero pro t. Propositio 7 1. Assume that 1 = arg max (EX ; c) is su cietly large (For example, assume that F is su cietly covex ad that c is su cietly small). The there exists a asymmetric equilibrium with a uique active seller.. Assume that 1 = arg max (EX ; c) < : (For example, assume that F is su cietly cocave) The there exists o equilibrium with a uique active seller. The followig examples illustrates the result. Example 5 Let c = 0; = 4 ad let F be uiform o [0; 1]. By Corollary the optimal supply for a moopolistic seller 1 is 1 =. The oly alterative for seller that leads to a positive pro t is to supply oe uit. Sice EX 1; EX 4 1 ;4 1 = EX 1; EX 1;3 > 0, coditio (9) is satis ed. Seller caot attract more tha oe buyer to his auctio, ad he therefore prefers to stay out. 16

17 5 Compariso amog models How do the models aalyzed i this paper (ad i particular the competitio models with either oe or two auctios sites) compare i terms of equilibrium prices ad quatities? I this sectio we aswer this questio uder the assumptio that the umber of buyers is large. We cosider the followig: Moopoly (M): oe auctio site with oe seller ad buyers; Competitio amog sellers at oe auctio site (C1): two competig sellers at oe auctio site with buyers; Competig auctios (C): two sellers ruig separate auctios, competig to attract buyers. For the results below we assume that a symmetric equilibrium exists i this model. Propositio 8 Assume that F; the distributio of buyers values, is IF R. Assume total equilibrium supply i all three models is at least 3 + : (This happes, for example, if either F is covex eough with respect to the expoetial distributio, or if the margial cost is ot too high). The the followig hold: (1) The equilibrium price uder moopoly is higher tha the prices i symmetric equilibria of the competitio models. () If total supply is higher (lower) tha + ; the the equilibrium price i the competitio model with oe auctio site is higher (lower) tha the equilibrium price i the competitio model with two auctio sites. 6 Cocludig remars We have studied competitio amog two sellers who optimize their respective supply i marets with a ite, ot ecessarily large, umber of buyers. We studied a model where all trasactios tae place at price determied i oe auctio, ad aother where there two separate auctio sites may operate side by side. A ui ed perspective o both models ca be obtaied by cosiderig buyers switchig costs. Whereas a equilibrium where both sellers mae pro ts always exists i the rst model (oe auctio site), this is ot the case i the secod model (two auctio sites) uder ubiquitous coditios o the demad fuctio (decreasig margial reveue) ad o the supply fuctios (su cietly low margial costs). I marets with siig productio (or tradig) costs, our results suggest a movemet towards domiat maret places featurig several competig suppliers. Ebay is a obvious example. Istead of a competitio mode characterized by poachig busiess from competitors, we curretly observe a move towards cosolidatio also i the tradig of securities. 15 For example, o October 17th 006 the Chicago Mercatile Exchage agreed to buy the Chicago Board of Trade, thus aimig to create the 15 See "The Big Squeeze", The Ecoomist, October 7th, 006, pp

18 world s biggest acial maretplace. 16 Similarly, the Deutsche Börse has log argued for the eed of a pa-europea exchage, ad tried (usuccessfully) to buy the Lodo Stoc Exchage which is ow o NASDAQ s buyig list. Fially, we hope that the mathematical methods used i this paper will prove to be useful i a variety of other competitio models with a small umber of traders. 7 Appedix A Barlow ad Porscha (1966) proved the followig 5 results about order statistics: Lemma 3 Assume that F (0) = 0: If F is covex (cocave) the ( + 1) EX i; =i is decreasig (icreasig) i i ad icreasig (decreasig) i. Lemma 4 Let G 1 F be star-shaped o the support of F ad assume F (0) = G (0) = 0: The EX i =EY i is decreasig i i: Lemma 5 If F is IF R(DF R) ad if F (0) = 0; the P r 1 ( i + 1)(X i; X i 1; ) is stochastically icreasig (decreasig) i r: Lemma 6 If F is IFR (DFR) ad if F (0) = 0; the ( i + 1) (X i; X i 1; ) is stochastically icreasig (decreasig) i i for xed i: Lemma 7 If F is IFR (DFR) ad if F (0) = 0; the ( i + 1) (X i; X i 1; ) is stochastically decreasig (decreasig) i i for xed : 8 Appedix B Proof of Propositio 1: If the distributio of buyers valuatios is F; the seller s reveue whe he sells uits is Suppose that r = r 0 = 0 < r ; we obtai EX ; = ( r) EX r; where r F () is optimal: sellig F () is always better tha sellig 0 > F (): Lettig ( r ) EX r ; ( r 0 ) EX r 0 ;, EX r ; EX r 0 ; r0 r (10) 16 See "Chicago Bulls", i The Ecoomist, October 1th, 006. pp. 88. I aother istace, also i October, three North Europea share exchages ui ed their listigs. 18

19 For distributio is G; the seller s reveue whe he sells uits is ( ) EY ; = ( r) EY r; where r Suppose that G () uits are ow optimal. I order to show that G () F (); it is su ciet to show that uder distributio G; the seller s reveue of sellig F () uits is always larger tha whe sellig 0 > F (): That is, we eed to show that, for r > r 0 ( r ) EY r ; ( r 0 ) EY r0 ;, EY r ; EY r0 ; r0 r Give coditio (10) ; it is su ciet to show that EY r ; EY r0 ; EX r ; EX r0 ;, EX r ; EY r ; EX r 0 ; EY r0 ; By Lemma 4 i Appedix A, EX i =EY i is decreasig i i. Sice r > r 0 ; the last iequality holds. Q.E.D. Proof of Corollary 1: Whe is large, the optimal umber of objects G () for the expoetial distributio G is G() arg max EX ; arg max G 1 = arg max l ) G() e F (0) = G (0) = 0 ad G 1 F covex imply together that G 1 F star-shaped. If G is the expoetial distributio, G 1 F covex (where ite) is equivalet to F beig IF R. Thus, applyig propositio 1, we obtai that F () =e if F is IF R: The reverse iequality holds for F which is DF R. Q.E.D. Proof of Corollary : If G is uiform distributio, the optimal umber of objects G () is G() arg max EX ; = arg max ) G() = ( ) + 1 Applyig propositio 1, we obtai that F () () if F is covex (cocave) Q.E.D. Proof of Propositio : By Lemma 6 i Appedix A, if F is IF R(DF R); the ( i+1)(ex i; EX i 1; ) is stochastically icreasig (decreasig) i i for xed i: Thus, if F is IF R we obtai: ( i + )(EX i;+1 EX i 1;+1 ) > ( i + 1)(EX i; EX i 1; ) 19

20 This implies that ( i + 1)EX i;+1 ( i + )EX i 1;+1 + EX i;+1 > ( i)ex i; ( i + 1)EX i 1; + EX i; Sice EX i;+1 < EX i; we obtai ( i + 1)EX i;+1 ( i + )EX i 1;+1 > ( i)ex i; ( i + 1)EX i 1; Lettig i = i the iequality yields ( + 1)EX +1 (+1);+1 ( + )EX +1 (+);+1 > EX +; ( + 1)EX (+1); Thus, if it is pro table to decrease the umber of objects from + 1 to whe the umber of buyers is ; the it is also pro table to decrease the umber of objects from + to + 1 whe the umber of buyers is + 1: I particular, F ( + 1) F () + 1: Q.E.D. Proof of Propositio 3: By Lemma 7 i Appedix A, if F is IF R, the ( i + 1)(EX i; EX i 1; ) is stochastically decreasig i i for xed : Thus, if F is IF R we obtai: ( i + 1)(EX i; EX i 1; ) > ( (i + 1) + 1)(EX i+1; EX i; ) Lettig = i i the the last iequality yields: EX ; ( + 1)EX (+1); + EX ; > ( 1)EX ( 1); EX ; + EX ( 1); Sice EX ; < EX ( 1); we obtai EX ; ( 1)EX ( 1); > ( + 1)EX (+1); EX ; The last iequality idicates that the margial reveue of a seller is decreasig i the umber of objects ad this property implies the desired result. Q.E.D. Derivatios for Example : 1) If = 3m; where m is a iteger, iequality 4 becomes, m 1 1 m + 1 ) 1 = m 1; m; or m + 1 Suppose 1 = m 1; the = ( 1 ) + 1 = (m + 1 ) 3m + 1 ) = m; or = m + 1 But if = m; the best respose of seller 1 is 1 = m: Therefore, 1 = m equilibrium pro le. If = m + 1; the best respose for rm 1 is 1 = m; or m 1 ad = m is ot a 1: Thus, there are 0

21 three possible pure strategy equilibria 1 = m + 1; = m 1 1 = m 1; = m = = m ) If = 3m + 1; where m is a iteger, iequality 4 becomes, Suppose 1 = m; the = ( 1 ) + 1 Suppose 1 = m + 1; the = ( 1 ) + 1 m 3 1 m ) 1 = m; or m + 1 = (3m + 1 m ) + 1 = (3m + 1 m 1 ) + 1 Agai, there will be three pure-strategy equilibria 1 = m + 1; = m 1 = m; = m = = m 3) If = 3m + ; where m is a iteger, iequality 4 becomes Suppose 1 = m; the = ( 1 ) + 1 = (m + 1 ) 3m + = (m ) 3m + m m ) 1 = m; or m + 1 = (3m + m ) + 1 Therefore, there will be two asymmetric pure strategy equilibria Q.E.D. 1 = m; = m = m + 1; = m = (m + ) 3m + ) = m or m + 1 ) = m ) = m + 1 Proof of Theorem 1: Sice pro ts are zero if the produced quatity is larger tha the umber of bidders, we ca assume without loss of geerality that the sellers choose quatities out of a ite set of itegers: = f0; 1; ; :::; Kg ; K < 1: Give ; seller 1 chooses 1 to maximize the payo fuctio R 1 ( 1 ; ) = 1 EX (1+ );: 1

22 ad give 1 ; seller chooses to maximize the payo fuctio R ( 1 ; ) = EX (1+ ); Let deote the above described two-perso game, ad de e P ( 1 ; ) = 1 EX (1+ ); We ow verify that P is a ordial potetial of : 17 We eed to chec that: 8 > 0; R 1 ( 1 ; ) R 1 ( 0 1; ) > 0, P ( 1 ; ) P ( 0 1; ) > > 0; R ( 1 ; ) R ( 1 ; 0 ) > 0, P ( 1 ; ) P ( 1 ; 0 ) > 0 The above coditios become 1 EX (1+ ); 0 1EX ( ); > 0, 1 EX (1+ ); 0 1 EX ( ); > 0 EX (1+ ); 0 EX ( ); > 0, 1 EX (1+ ); 1 0 EX ( ); > 0 which trivially hold. Sice both 1 ad are chose from a ite set ; it is obvious that the potetial P has a maximum o : (Note that the maximum eed ot be uique). The existece result follows ow directly from the followig result: Lemma 8 (Lemma.1, Moderer ad Shapley, 1996) Let P be a ordial potetial fuctio for : The the equilibrium set of (R 1 ; R ) coicides with the equilibrium set of (P; P ) : That is, ( 1 ; ) is a equilibrium poit for if ad oly if for every i f1; g ; P ( i ; i ) P (i; 0 i ) for every i 0 : Cosequetly, if P admits a maximal value i ; the (R 1 ; R ) possesses a pure strategy equilibrium. Q.E.D. Proof of Theorem : By the above proof, the potetial P has a maximum. Suppose that this maximum is achieved at (1; ) : We eed to show that j1 j 1: Assume, by cotradictio, that the opposite holds: j1 j > 1, j1 j 17 If the rms face a margial cost c > 0; de e the potetial P ( 1 ; ) = 1 (EX (1 + ); c): This ad the ext proof proceed the i a aalogous fashio yieldig the respective results.

23 Without loss of geerality assume that 1 + : (11) Cosider ow ( 0 1; 0 ) i de ed by 0 1 = = + 1 ad observe that = 1 + : We obtai the that P ( 0 1; 0 ) P ( 1; ) = ( 1 1) ( + 1) EX ( 1 + ); 1 EX ( 1 + ); = ( 1 1) EX ( 1 + ); > 0 where the last iequality follows by (11). This yields a cotradictio to the assumptio that P achieves a maximum at ( 1; ) : Thus, j 1 j 1 as desired. Q.E.D. Proof of Propositio 4: 1) Suppose F is covex. Give ; seller 1 chooses 1 to maximize R 1 = 1 EX ( ) 1; By Corollary, if F is covex, 1 1 ( ) : Aalogously, we have 1 ( 1) : Combiig the two iequalities, it follows that The opposite iequality holds for cocave F: ) Now suppose F is IF R ad is large. Give ; seller 1 chooses 1 to maximize R 1 = 1 EX ( ) 1; By Corollary 1, we have 1 1 e ( ) : Similarly, we have 1 e ( 1) : Therefore, The proof for the DF R case is aalogous. Q.E.D e Proof of Propositio 5: merger is Let arg max F 1 m m : Suppose that m = 1 = is large eough. Total reveue before the m 1 + = max F 1 m 3

24 I the post merger competitio the reveues are give by: m c 1 = c = max i F 1 i i i m We focus o the (quasi) symmetric equilibrium i the post merger games (which always exists by Propositio ). The potetial fuctio is P ( 1 ; ) = 1 F 1 m 1 m Sice j c 1 j c 1, the pro le (1; c ) c that maximizes the potetial must satisfy i c arg max () m F 1 ; i = 1; m Oe implicatio is that The chage i total reveue is: m 1 c 6 arg max F 1 m c 1 + c ( 1 + ) m = 1F c 1 c 1 m < 0 m F 1 m because arg max F 1 m m : Thus, a merger reduces total reveue. Q.E.D. Proof of Lemma 1: Fix a strategy pro le for all agets except buyer h; ad assume that i this pro le the respective quatities ad umber of bidders at each site are 1 ; ; 1 ; ; where 1 + = 1: Deote by X the radom variable represetig bidder h s valuatio. If h jois site i; i = 1; ; his expected payo is give by PrfX EX i+1 i; i+1ge[x EX i+1 i; i+1 j X EX i+1 i; i+1]; i = 1; It is immediate that the above expressio is higher for i = arg mi fex 1+1 1; 1+1; EX +1 ; +1g; thus it is optimal for bidder h to joi the site with the lower expected price. The secod argumet follows the immediately by the de itio of equilibrium. Q.E.D. Proof of Lemma : 1) Suppose without loss of geerality that EX 1 1; 1 EX ;. Sice EX 1+1 ( 1+1); 1+1 = EX 1 1; 1+1 < EX 1 1; 1 we obtai EX 1+1 ( 1+1); 1+1 < EX ; 4

25 The last iequality provides a su ciet coditio for a buyer to move from auctio to auctio 1 if seller 1 icreases the supplied quatity by oe uit while all other agets stay put. ) Let 1 = m: By Lemma 5 i Appedix A, if F is IF R the P r 1 (m i + 1)(EX i;m EX i 1;m ) is stochastically icreasig i m r: Let r = 1; the mex 1;m < (m + 1)EX 1;m+1 Sice EX 1;m+1 < EX 1;m ; we have (m 1)EX 1;m < mex 1;m+1 Similarly, by Lemma 5 EX 1;m + (m 1)EX ;m < EX 1;m+1 + mex ;m+1 Sice EX ;m+1 < EX ;m ; we have (m )EX ;m < (m 1)EX ;m+1 By iductio, for all 1 i < m; (m i)ex i;m < (m i + 1)EX i;m+1 Lettig i = m ; we get EX m ;m < ( + 1)EX m ;m+1 The last iequality implies that a seller icreases his reveue by attractig oe more buyer from the other site. 3) Suppose that the umber of buyers i the auctio is m: By Lemma 6, if F is IF R(DF R) the (m i + 1)(EX i;m EX i 1;m ) is stochastically icreasig (decreasig) i m i for xed i: Thus, (m i + )(EX i;m+1 EX i 1;m+1 ) > (m i + 1)(EX i;m EX i 1;m ) This implies that (m i + )EX i;m+1 (m i + 1)EX i;m > (m i + )EX i 1;m+1 (m i + 1)EX i 1;m Sice EX i;m+1 < EX i;m we obtai (m i + 1)EX i;m+1 (m i)ex i;m > (m i + )EX i 1;m+1 (m i + 1)EX i 1;m Lettig i = m ; we get ( + 1)EX m ;m+1 EX m ;m > ( + )EX m 1;m+1 ( + 1)EX m 1;m 5