Intermediating Autioneers Yuelan Chen Department of Eonomis The University of Melbourne September 10, 2007 Abstrat Aution theory almost exlusively assumes that the autioneer and the owner or the buyer) of the goods are one and the same. In reality, however, most autioneers at as intermediaries between buyers and sellers. Despite the fast growth of these intermediating autioneers in reent years, essentially no studies exist on how they set ommissions. In this paper, we make a first step towards filling in the gap by studying a model with J autioneers, K sellers, N buyers, where J K 2, and eah seller has an item to sell in a standard aution, where buyers have private and independent values. Sellers are loated on the two ends of the unit interval. They have orrelated values drawn from a power distribution. Observing the perentage fees set by the autioneers, eah seller hooses an autioneer to sell her item and sets a reserve prie. Observing the hoies of the sellers, eah buyer hooses at most one item to bid for. We expliitly model the ompetition between autioneers with and without apaity onstraints. We show that when disriminatory priing and rationing are not allowed, equilibrium perentage fees are independent of the number of buyers and the distribution of buyer s values. In the monopoly autioneer ase and the apaity onstraint ase with non-disriminatory rationing, equilibrium perentage fees depend only on the parameter of the power distribution. In the apaity onstraint ase with disriminatory rationing, autioneers play mixed strategies in equilibrium, and perentage fees are lower. Absent apaity onstraints, equilibrium perentage fees are zero due to the Bertrand-type ompetition. JEL Classifiation: D44, L13 Keywords: Aution, intermediaries, ommission fees, reserve prie Postal address: Department of Eonomis, The University of Melbourne, Melbourne, VIC 3010, Australia. Email address: yuelan.hen@gmail.om. I thank Simon Loertsher for valuable advie and very helpful disussions. Finanial support from ARC Eonomi Design Network at the University of Melbourne is gratefully aknowledged. All errors remain mine. 1
PREFACE Thesis Title: Three Essays in Voting, Soial Choie and Intermediation Supervisors: Peter Bardsley, Simon Loertsher, Andrew MLennan This thesis onsists of three independent papers in the related fields of soial hoie theory, politial eonomy, and aution theory. The paper Intermediating Autioneers will be presented at the onferene. The extended abstrats of the other two papers are as follows. 1. Poisson Games, Strategi Candiday and Duverger s Law Abstrat. Duverger s law predits a long-run two-andidate stable outome under a plurality voting system. Duverger 1954) explains the law using the waste vote argument, whih emphasizes voters tendeny to abandon andidates ommonly pereived to have the least support. Palfrey 1989) formalizes Duverger s argument by modeling a three-andidate voting situation as a Bayesian game, and shows that there exists a symmetri Bayesian equilibrium where the vote share of the weakest andidate is asymptotially zero. However, Palfrey s model does not apply to non-generi ases where two weaker andidates have the same expeted vote share. In this paper, we use strategi andiday and the waste vote argument to fully explain Duverger s law. We add unertainty about the number of voters to Palfrey s framework, and model the eletion situation as a two-stage game where andidates make strategi entry deisions in the first stage, followed by plurality voting, whih is modeled as a Poisson game introdued in Myerson 1998, 2000). Using our framework, we suessfully explain both generi and non-generi ases in Palfrey 1989), as well as an anomaly to Duverger s law, viz. in India there persists a strong entral party and two weaker parties with similar strength. 2. Monotoniity and Candidate Stable Voting Correspondenes Abstrat. Dutta, Jakson and Le Breton Eonometria, 2001) initiated the study of strategi andiday. A voting proedure satisfies andidate stability if no andidate has inentives to withdraw her andiday in order to manipulate the voting outome in her favor. Dutta et al. show that a single valued voting proedure satisfying andidate stability and unanimity must be ditatorial if voters have strit preferenes and andidates annot vote. Eraslan and MLennan JET, 2004) extend this result to a framework that allows weak preferenes and multi-valued voting proedures voting orrespondenes). They obtain the existene of a serial ditatorship under a stronger version of andidate stability. We show that voting orrespondenes satisfying strong andidate stability and unanimity are monotoni, that is, if a winning andidate s position is weakly improved in all voters preferene rankings, then the andidate remains a winner. Monotoniity provides a diret link between the standard ditatorship in Dutta et al. and the serial ditatorship in Eraslan and MLennan. Using this partiular property of voting orrespondenes, we provide an alternative proof to the Eraslan and MLennan s result. 2
1 Introdution Aution theory almost exlusively assumes that the autioneer and the owner of the goods or the buyer in the ase of prourement aution) are one and the same. In reality, however, most autions are run by autioneers who do not own nor buy the goods to be sold. Therefore, these autioneers at as intermediaries between buyers and sellers. Examples inlude internet aution sites like ebay, aution houses like Christie s or Sotheby s, and Australian real estate brokers. These intermediating autioneers have been experiening fast growth in reent years 1. However, despite the burgeoning researh on intermediaries as two-sided market platforms 2, essentially no studies exist on how intermediating autioneers set their ommission fees 3. Understanding the behavior of these autioneers is obviously important. Consider, for example, the allegations that Christie s and Sotheby s were ollusively setting their ommission fees above the ompetitive level. Clearly, suh a statement presumes that ollusive fees are higher than ompetitive fees, whih is basially a translation of insights from models of prie setting to models of ommission fee setting. However, it is not lear a priori whether the intuition from the former arries over to the latter, and we shall argue that indeed it does not. That is, we will show that there are very plausible assumptions under whih the ollusive ommission fees are exatly the same as the Nash equilibrium fees. In this paper, we make a first step towards building a theory of intermediating autioneers. Our ontributions are twofold. We introdue a model of intermediating autioneers, and we provide a framework to analyze ompetition between these autioneers. Intermediating autioneers typially harge sellers and/or buyers perentage fees on the sales prie 4. 1 In 2006, Sotheby s aution sales totaled a historial reord of $3.75 billion, a 36% inrease over 2005 sales. Christie s art aution sales totaled $4.414 billion, also a 36% inrease. At the online aution sites ebay, registered users inreased by 23% over 2005 to reah 222 million ative users 82 million) at the end of 2006, generating a total of $2.365 billion listings. Also see Luking-Reiley 2000). 2 See Armstrong 2006), Rohet and Tirole 2006). 3 To the best of our knowledge, the only paper that deals with this topi is Ginsburgh, Legros and Sahuguet 2005). However, they only study the welfare effets of an exogenous inrease in ommissions on buyers and sellers. In partiular, they do not address the autioneer s profit maximization problem. 4 For example, Christies s harges a 20% ommission on the first $100,000 and 12% thereafter. Sotheby s harges 20% on the first $200,000 and 12% on anything above. In addition, both aution houses harge buyer s premium, whih typially starting from 25% at Christie s and 20% at Sotheby s. ebay harges a listing fee ranged from $0.20 to $4.80, and 5.25% of the initial sales prie $0.01-$25, plus 3.25% of the next $25.01-$1,000, plus 1.5% of the remaining balane. Amazon harges a $0.99 listing fee, and 6% of the sales prie for omputers, 8% for amera & photo and eletronis, 10% for items in the Everything Else Store, and 15% for all other produt lines. Yahoo offers free servies for using its aution websites Yahoo US and 3
Therefore, it is natural to assume that autioneers ompete in perentage fees. Speifially, we study the following three stage game with J autioneers, K sellers, and N buyers, where J K 2. Eah seller has an item to sell in a standard aution with independent and private values. Sellers are loated on the two ends of the unit interval, with idential values drawn from a power distribution H) = γ, where γ is a positive onstant see e.g. Hörner and Sahuguet 2007)). In eah stage, players ations are taken simultaneously. In the first stage, eah autioneer sets a non-disriminatory perentage fee, i.e. it is the same for both sellers if there are two. In the seond stage, eah seller observes the fees 5. If the seller deides to sell her item, she hooses an autioneer and sets a reserve prie. In the last stage, eah buyer observes the hoies of the sellers, and hooses at most one aution to attend. All partiipating buyers use the same bidding strategies. This setup is general enough to inorporate both ases of monopoly autioneer and ompeting autioneers. For J = 1 and K = 1, our model is redued to a standard aution setting with a broker. For K = 2, we have a Hotelling aution when two differentiated produts on 0 and 1 are autioned off, paralleling the Hotelling-Bertrand model on prie ompetition. The Hotelling aution is onduted by a monopoly autioneer J = 1) or two ompeting autioneers J = 2). In the latter ase, two further subases our, where autioneers may or may not fae binding apaity onstraints. We all these two subases Bertrand-Edgeworth ompetition and Bertrand ompetition between autioneers, respetively. When autioneers are apaity onstrained, rationing rules matter 6. In this paper, we fous on the study of ompeting autioneers, whih has the two monopoly ases as its speial appliations. Both ompetition ases are quite ommonly observed in reality. Christie s and Sotheby s an be onsidered as an example of Bertrand-Edgeworth ompetition between autioneers, whereas ebay and Amazon an fit into Bertrand-type ompetition. Bertrand-Edgeworth ompetition is partiularly relevant for traditional brik and mortar autioneers 7. For example, it Canada autions sites were retired on June 16, 2007). Australian real estate brokerage ommissions typially range from 2.5% to 5%, plus administration fees for advertising, et. 5 For oniseness, we will always use plural form when desribing the setup. It should be understood from the ontext that we refer to singular form when there is only one autioneer and/or one seller. 6 In models of produt market ompetition with prie setting sellers, equilibrium outomes depend on rationing rules see Kreps and Sheinkman 1983), Davidson and Denekere 1986)). The same applies here. 7 See Loertsher 2007) for a study on the prie ompetition among apaity onstrained intermediaries. Our results on ompeting autioneers are in line with his findings that ompetition is substantially softened in the presene of ostly) apaity onstraints. 4
takes onsiderably amount of preparations for autioning a house, inluding property evaluations/appraisals, advertising, pre-sale inspetions, et. Consequently, even a big real estate agent usually holds a limited number of autions at a time. The apaity onstraints are even more prominent when there is some requirement of speialist knowledge on the autioneer s behalf suh as in the art autions. We obtain the following results. In the monopoly autioneer ase and Bertrand-Edgeworth ompetition with non-disriminatory rationing, equilibrium perentage fees depend only on the parameter γ of seller s value distributions. Speifially, eah autioneer sets the same 1 perentage fee equal to. In Bertrand-Edgeworth ompetition with disriminatory ra- γ+1 tioning, autioneers play the same mixed strategy in equilibrium, and the upper bound of the support of the mixed strategy is equal to 1. Absent apaity onstraints, equilibrium γ+1 perentage fees are driven to zero due to Bertrand-type ompetition between autioneers. The Hotelling aution used in our model is a very general form of modelling ompetition between sellers to aution horizontally differentiated produts 8. Thus, our results are readily applied to a wide range of produt differentiation ases. Our results on different rationing rules imposed on apaity onstrained autioneers have interesting poliy impliations as it shows that equilibrium perentage fees are lower under disriminatory rationing rule. There are several papers modelling ompetition among sellers on reserve pries. MAfee 1993), Peters 1997), Peters and Severinov 1997, 2006) all assume a large number of sellers in a perfetly ompetitive market setting. Burguet and Sàkovis 1999) analyzes ompetition between two owners of homogeneous goods. The paper most losely related to the Hotelling aution in our model is Parlane 2005). However, in her model sellers values are assumed to be zero. Assuming positive values in our model allows us to evaluate the effet of perentage fees on sellers equilibrium reserve pries. The only paper that studies perentage fees set by intermediaries is Loertsher and Niedermayer 2007). They show that in the ase of one buyer, one seller and one monopolisti intermediary, perentage fees levied on the prie set by the seller implement the optimal mehanism that maximizes the intermediary s expeted profit if and only if the seller s values are drawn from a generalized power distribution. This provides a rationale for the use of power distributions in our model. 8 Surprisingly, the study on Hotelling aution is almost non-existent in the aution theory literature. To the best of our knowledge, the only paper that studies a Hotelling aution is Parlane 2005). 5
The remainder of the paper is strutured as follows. Setion 2 desribes both the general setting and speifi appliations of the model. Seond 3 briefly summarizes the Hotelling- Bertrand Model, whih losely parallels the Hotelling aution setting. Setion 4 studies Bertrand-Edgeworth ompetition between autioneers, and applies the results to both ases of monopoly autioneer. Setion 5 onsiders Bertrand ompetition. Setion 6 onludes. 2 The Model We first lay out the general model and then briefly disuss four important speial ases. 2.1 The general setting There are K 2 sellers. Eah seller has an item to aution and values her item at. We assume that is distributed on [0, 1] aording to the distribution funtion H given by H) = γ, where γ is some positive onstant, and h) = γ γ 1 is the assoiated density funtion. Notie that the uniform distribution is a speial ase with γ = 1. Sellers are loated on the two ends of the interval [0, 1]. We refer to the seller at 0 as k = 1 and the seller at 1 as k = 2. The autions are onduted by J K autioneers. Autioneer j harges the sellers served by him a perentage fee τ j [0, 1] on the sales prie. That is, we assume that autioneers annot prie disriminate. There are N > 1 buyers, eah bidding for at most one item. Eah buyer has a type θ [0, 1], whih is independently and identially drawn from a ontinuous distribution funtion F with density f. A type θ buyer s valuation for seller k s item is v k θ). We denote the indued umulative distribution funtion of v k θ) by Φ k with density φ k, and define the hazard rate to be λ k v k ) = φ kv k ) 1 Φ k v k. The respetive valuations of a buyer of type θ for the ) two items are 9 v 1 θ) = 1 tθ and v 2 θ) = 1 t1 θ), where t 0, 1] desribes the degree of produt differentiation. Notie when t = 0, v 1 θ) = v 2 θ) = 1, for all θ [0, 1], that is, Φ k would be a degenerated distribution. To obtain 9 We refer to independent values as eah buyer s type being independently distributed aording to F, but notie the valuations of a buyer of type θ for the two items are orrelated. 6
interesting results, we thus assume t > 0. Φ k and φ k satisfy the following onditions 10 : F θ) = 1 Φ 1 v 1 θ)) = Φ 2 v 2 θ)) and fθ) = tφ 1 v 1 θ)) = tφ 2 v 2 θ)). We all this type of autions for two items at 0 and 1 the Hotelling aution beause of its lose relation to the Hotelling-Bertrand model on prie ompetition, whih is briefly disussed in the next setion. We make standard regularity assumptions on buyers type distributions. Assumption 1. For eah θ [0, 1], fθ) > 0, and fθ) F θ) stritly inreasing in θ. Notie Assumption 1 implies that for eah θ [0, 1], θ + F θ) fθ) inreasing in θ. We all this inreasing virtual valuations. is stritly dereasing in θ, fθ) 1 F θ) and θ fθ) 1 F θ) is are stritly The realizations of and θ are private information of sellers and buyers, respetively. H and F are ommon knowledge among all players 11. The aution format for eah item is seond prie sealed bid aution 12. The Game The sequene of the game is as follows. At eah stage of the game, players ations are taken simultaneously. Stage 1. Eah autioneer j announes his perentage fee τ j. Let τ = τ 1, τ 2 ) denote the vetor of perentage fees harged by the two autioneers if J = 2. Stage 2. Eah seller k observes the perentage fees and her value. If she deides to sell her item, she hooses one autioneer to attend if J = 2. Otherwise, she just deides to be ative or inative. If both sellers attend the same autioneer who is apaity onstrained, a rationing rule is used to determine whih seller gets served. We onsider two speifi rules. The disriminatory rationing rule lets the autioneer 10 If x is a random variable with density fx), and y = gx), where g is monotoni, then y has the density 1 given by g g 1 y)) fg 1 y)), where g 1 is the inverse funtion, and g is the derivative. 11 Assuming ommon knowledge of F among all players is stronger than neessary for our results. In fat, exept in Bertrand-Edgeworth ompetition with disriminatory rationing, autioneers need not know the distribution of F. 12 Our results go through to any standard aution that satisfies the revenue equivalene priniple. 7
hoose a seller, and the non-disriminatory rationing rule assigns eah seller to the autioneer with equal probability. A seller with value obtains a utility of if she is inative. If seller k is served by an autioneer, she announes her reserve prie r k. Stage 3. Eah buyer observes whih autioneer hosts whih seller, the reserve pries r k, and his own type θ, and then deides whether to partiipate, whih aution to attend if there are two, and what bidding strategies to use if he partiipates. A buyer obtains a utility of 0 if he does not attend any aution. Eah buyer observes the number of other buyers in the aution he attends. 2.2 Appliations of the general model The above model inorporates the following four situations as speial ases. i) Monopoly autioneer with one seller. The general model is redued to a standard aution setting with the exeption that the autioneer is an intermediary who harges the seller a perentage fee. A type θ buyer s valuation for the seller s item is either v 1 θ) = 1 tθ or v 2 θ) = 1 t1 θ), depending on whether the seller is loated at 0 or 1. v k is distributed on [1 t, 1] aording to Φ k derived from F as follows. For v k [1 t, 1], ) ) 1 v1 v2 1 t) Φ 1 v 1 ) = 1 F and Φ 2 v 2 ) = F. t t We an hek that the property of inreasing virtual valuations arries over from F to Φ k. That is, for eah v k [1 t, 1], v k 1 Φ kv k ) φ k v k ) = Υ k ) is stritly inreasing in v k 13. For the rest of the paper, when K = 1, we let the seller be loated at 1, and drop the subsript k whenever appliable, with the understanding that k = 2. Notie when t = 1, we have v 2 θ) = θ, so this ase would oinide with the standard aution setting with the extra element of an intermediary. 13 To see this, notie θ = 1 v1 t = v2 1+t t, and Υ 1 = 1 t ) ) = θ θ + F θ) fθ) > 0, and Υ 2 v 2 = 1 F θ) θ θ fθ) > 0. Υ 1 v 1 θ + F θ) fθ) ), Υ 2 = 1 t 1 θ )) 1 F θ) fθ). So 8
ii) Monopoly autioneer with ompeting sellers. There are two sellers k = 1, 2 loated on the two ends of the interval [0, 1]. This is the ase desribed in the general setting with J = 1. Sometimes we refer to the two sellers as k and k when there is no need to speify their positions. iii) Competing autioneers with apaity onstraints and ompeting sellers. There are two autioneers j = 1, 2, eah with the apaity of running one aution. If an autioneer attrats both sellers, a pre-speified rationing rule is applied. We study both ases when the rule is disriminatory vs non-disriminatory. The seller who is rejeted an go to the other autioneer or stay out. A seller randomizes with equal probability between the two autioneers if she is indifferent. iv) Competing autioneers without apaity onstraints and ompeting sellers. This is similar to ase iii) exept that eah autioneer has the apaity of running two autions, therefore no rationing rule needs to be speified. Finally, we briefly omment on the Hotelling aution used in our model. Hotelling autions are quite ommon in pratie. For example, in Australia, land and houses are usually sold in autions. Typially, there are several real estate agents, i.e. autioneers, who ompete on ommission fees to attrat sellers. Eah seller usually has one property to sell, and buyers are physially onstrained to attend one aution at a time when several autions are simultaneously onduted at different loations. Indeed, any two or more) autions that sell horizontally differentiated produts, when eah buyer is physially or finanially onstrained to buy only one unit of the good, an be onsidered as some variant of a Hotelling aution. 3 Preliminary: The Hotelling-Bertrand Model Before analyzing buyers and sellers equilibrium behavior, we briefly summarize the Hotelling- Bertrand model also known as the differentiated produts Bertrand model) beause our Hotelling aution setting losely parallels it, where reserve pries instead of pries are the strategi variables of the sellers. In the Hotelling-Bertrand model, there are two sellers 9
k = 1, 2 loated on 0 and 1. They have the same marginal ost [0, 1], and simultaneously set pries p k to maximize their expeted profits. There is a ontinuum of buyers uniformly distributed on [0, 1], eah demanding one unit of the good. A buyer assigns the same value 1 to eah item, and pays the prie for the item purhased plus a transportation ost t [0, 1] per unit distane, where t an be interpreted as a parameter for produt differentiation. As a funtion of and t, there are three regions, eah giving rise to a different type of equilibrium. The regions are illustrated in Figure 1. t 1 2 3 onstrained duopoly loal monopoly t = 1 duopoly t = 2 1 ) 3 0 1 Figure 1: Three regions of equilibria depending on t and Region 1. t > 1 : Loal monopolist sellers. When sellers marginal osts are large, and/or produts are highly differentiated, sellers are loal monopolists in their own markets. That is, for t > 1, and 0, 1], there exist two marginal buyers θ 1, θ 2 with θ 1 < θ 2 suh that all θ [0, θ 1 ] buy from seller 1, all θ [ θ 2, 1] buy from seller 2, and all θ θ 1, θ 2 ) do not buy any item. Buyers θ 1, θ 2 get zero surplus. The equilibrium pries are p 1 = p 2 = 1+, and equilibrium profits are 2 Π 1 = Π 2 = 1 )2 4t. 14 14 When t = 1, and [0, 1], θ 1 and θ 2 onverge to θ, where θ is the buyer who is indifferent between 10
Region 2. t 2 1 ): Duopolist sellers. 3 When t and are low suh that t 2 1 ) and [0, 1], equilibrium pries will be low 3 and there exists a unique indifferent buyer θ who gets positive surplus in equilibrium. All θ [0, θ) buy from seller 1, and all θ θ, 1] buy from seller 2. The equilibrium pries are p 1 = p 2 = t +, and equilibrium profits are Π 1 = Π 2 = t 2. Region 3. 2 3 1 ) < t < 1 : Constrained duopolist sellers15. When t and are in the intermediate range, i.e. 2 1 ) < t < 1 and [0, 1), 3 there are multiple equilibria. In eah equilibrium, there is an indifferent buyer ˆθ who gets zero surplus. That is, p 1 and p 2 are suh that 1 p 1 tˆθ = 1 p 2 t1 ˆθ) = 0. So equilibrium pries satisfy ˆp 1 + ˆp 2 = 2 t, p 1, p 2 [1 t, 1]. Seller 1 gets all θ [0, ˆθ), and seller 2 gets all θ ˆθ, 1]. Figure 2 illustrates sellers best response funtions BR k. Any point on the bold line orresponds to an equilibrium for this onstrained duopoly ase. p 2 2 t 1 BR 1 p 2 ) = BR 2 p 1 ) 1 t 1 t 1 2 t p 1 Figure 2: Multiple equilibria in the onstrained duopoly ase buying from seller 1 and seller 2. 15 This is perhaps the least known type of equilibria, but it obtains generially in this type of model, see e.g. Salop 1979). In partiular, it annot be avoided if, as we assume, is distributed ontinuously on [0, 1]. The standard approah is to fous on the unique symmetri equilibrium, see e.g. Chen and Riordan 2007). 11
4 Bertrand-Edgeworth Model In this setion, we analyze the full game with two apaity onstrained autioneers. As in the Hotelling-Bertrand model, let θ k be the type who is indifferent between attending aution k and not partiipating when in equilibrium sellers are loal monopolists. Here aution k refers to the aution for seller k s item. Let θ be the type who is indifferent between attending aution k and k when in equilibrium sellers are duopolists, and ˆθ be the indifferent type when sellers are onstrained duopolists. 4.1 The buyers subgame Eah buyer hooses a partiipation strategy not partiipating, attending aution k, or attending aution k, and a bidding strategy b to maximize his expeted utility, where b : [0, 1] R + is a mapping from the type spae to the set of bids. We assume that buyers follow the same bidding strategy. As is well known, it is a weakly dominant strategy for eah buyer to bid his true valuation in seond prie autions. That is, a buyer s bidding strategy is bθ) = v k θ) if he attends aution k. As is standard, we fous on the equilibrium where buyers play weakly dominant strategies, and do not further mention this restrition. Buyers partiipating strategies depend on sellers reserve pries and the degree of produt differentiation. If t is high, buyers are ompletely separated in two submarkets. That is, there is no buyer who expets non-negative utility from both autions if he bids truthfully. The ondition and the haraterization of this equilibrium is given in the following proposition. Proposition 4.1. Suppose that reserve pries and t are suh that t > 2 r 1 r 2. Then there exists a unique equilibrium in the buyers subgame where all buyers with θ θ 1 = 1 r 1 t attend aution 1, all buyers with θ θ 2 = r 2 1+t t attend aution 2, and all buyers with θ θ 1, θ 2 ) do not attend any aution. Proof. Sine buyers bid truthfully in equilibrium, they attend aution k if and only if their valuations for item k are no less than the reserve prie r k. Note that r k = v k θ k ), k = 1, 2, and t > 2 r 1 r 2 θ 1 < θ 2. For all θ θ 1, v 1 θ) r 1, v 2 θ) < r 2. So all these buyers attend aution 1. Similarly, all buyers with θ θ 2 attend aution 2. For all θ θ 1, θ 2 ), 12
v k θ) < r k, k = 1, 2, so these buyers do not partiipate. When t = 2 r 1 r 2, the equilibrium in Proposition 4.1 remains true with the slight modifiation that θ 1 = θ 2, and this is the unique type indifferent between attending the two autions. When produts are less differentiated and/or reserve pries are lower, sellers markets overlap in the sense that some buyer s valuation for either item is higher than the assoiated reserve prie. This buyer attends the aution that gives him higher expeted utility. In partiular, there exists some indifferent type who obtains the same expeted utility from attending either aution. This result is given in the following proposition. Proposition 4.2. Suppose that reserve pries and t are suh that t < 2 r 1 r 2. Then there exists a unique θ r 2 1+t, 1 r 1 t t ) suh that in equilibrium, all buyers with θ < θ attend aution 1, all buyers with θ > θ attend aution 2, and θ is indifferent between attending the two autions, where θ is suh that Φ N 1 1 v 1 θ))v 1 θ) r 1 ) = Φ N 1 2 v 2 θ))v 2 θ) r 2 ). 1) Proof. Let θ 1 = 1 r 1, θ t 2 = r 2 1+t. Then r t k = v k θ k ), k = 1, 2, and t < 2 r 1 r 2 θ 2 < θ 1. Define the funtion : [ θ 2, θ 1 ] R by θ) = Φ N 1 1 v 1 θ))v 1 θ) v 1 θ 1 )) Φ N 1 2 v 2 θ))v 2 θ) v 2 θ 2 )), whih measures the differenes in a buyer s expeted surplus from attending the two autions. It is ontinuous and dereasing in θ on its domain, and θ 2 ) > 0, θ 1 ) < 0. Therefore, there exists a unique θ θ 2, θ 1 ) suh that θ) = 0, that is, 1) holds. So θ is indifferent between attending the two autions. It is straightforward to hek that all θ [0, θ 2 ) attend aution 1, all θ θ 1, 1] attend aution 2. For all θ [ θ 2, θ 1 ], v k θ) r k. Furthermore, for all θ [ θ 2, θ), θ) > 0, that is, the expeted surplus is higher in aution 1, therefore, all these buyers attend aution 1. For all θ θ, θ 1 ], θ) < 0, so these buyers attend aution 2. 4.2 The sellers subgame Eah seller first determines whih autioneer to attend if she deides to sell her item after observing the perentage fees. If seller k is hosted by some autioneer, she sets a reserve 13
prie r k to maximize her expeted profit. In the one seller ase, all buyers with valuations larger than the reserve prie attend the aution. The same needs not be true in the ase of ompeting sellers beause some buyers with valuations larger than r k may still want to attend aution k sine their valuations for k s item are larger. Consequently, the minimum bid in aution k an be stritly larger than r k. Denote by x k the minimum equilibrium bid in aution k, whih in general depends on r k and r k. The problem of seller k hosted by autioneer j an be written as max r k Π k, τ j, r k, r k ) = 1 τ j )NM k Φ k, r k, r k ) + Φ N k x k ) 2) where M k Φ k, r k, r k ) is the ex ante expeted payment of a buyer in aution k, whih is alulated as follows. 1 M k Φ k, r k, r k ) = r k 1 Φ k x k ))Φ N 1 k x k ) + N 1) y1 Φ k y))φ N 2 k y)dφ k y) x k 3) The first term represents the ase when there is only one buyer attending aution k, and he pays the reserve prie r k. The seond term is when there are at least two buyers in aution k, and the winner pays the seond highest bid. Sellers partiipation strategies are as follows. First, a seller with value attends autioneer j only if 1 τ j sine the maximum payment she an get from a buyer is 1. If both perentage fees are no greater than 1, her first hoie is always the autioneer with lower fee, sine her equilibrium expeted profit is dereasing in fees. Seond, it is a weakly dominant strategy for the seller to go to the other autioneer instead of staying out if she is rejeted by the first. Let us now solve sellers equilibrium reserve pries. Notie that if an autioneer sets perentage fee at 1, then only sellers with zero value will attend him, whih is an event of zero probability. When analyzing sellers priing strategies, we therefore assume that both τ 1 and τ 2 are stritly less than 1. 4.2.1 Loal monopolist sellers We have shown that buyers equilibrium behavior depends on the degree of produt differentiation and sellers reserve pries. The latter, in turn, depend on sellers values and perentage fees, as well as on t via buyers equilibrium bidding strategies. When produts are suffiiently 14
differentiated, even at = τ 1 = τ 2 = 0, equilibrium reserve pries are already high enough to separate the two markets. Eah seller is a loal monopolist in her own market. This intuition is formalized in Propositions 4.3. Before we state the results, we introdue and larify a few notations. Denote as the highest value for a seller to be willing to attend either autioneer for a given τ. Notie = 1 max{τ 1, τ 2 }. Denote sellers marginal revenues for any given θ and t as MR 1 θ, t) = 1 t θ + F θ) ) and MR 2 θ, t) = 1 t 1 θ 1 F θ) )) fθ) fθ) In the following analysis of equilibrium reserve pries, we refer to τ k as the perentage fee harged on seller k, and as seller k s effetive marginal ost. Proposition 4.3. There exists a unique t suh that for any t > t, τ 1, τ 2 [0, 1), the following is the equilibrium outome in the sellers and buyers subgame. 1. For <, sellers set reserve pries satisfying r 1, τ, t) = 1 τ 1 + t F θ 1 ) f θ 1 ) and r 2, τ, t) = 1 τ 2 + t 1 F θ 2 ) f θ 2 ) 4) where θ 1 = 1 r 1, θ t 2 = r 2 1+t. t 2. All buyers with θ θ 1 attend aution 1, all buyers with θ θ 2 attend aution 2, and all buyers with θ θ 1, θ 2 ) do not attend any aution. At t = t, and = 0, reserve pries satisfy 4) with θ 1 = θ 2. Proof. We proeed in three steps. First, we show the existene and uniqueness of t with the property stated in the proposition. Seond, we show that for any given τ, and t > t, the ondition in Proposition 4.1 holds for <. This then implies that buyers behavior stated in 2) onstitutes an equilibrium. Third, we show that sellers reserve pries stated in 1) are quilibrium pries. First, we show that there exists a unique t suh that MR k θ, t ) = 0, for some θ [0, 1]. For any given t, we an solve MR 1 θ, t) = MR 2 θ, t), and the solution θ satisfies θ + F θ) ) = f θ) 1 θ 1 F θ). This implies that θ ) + F θ) = 1 1 + 1 > 1. Notie θ is independent f θ) f θ) 2 f θ) of t. Now fix θ and onsider MR k θ, t) as a funtion of t. MR k θ, t) is dereasing in t, and 15
MR k θ, 0) = 1 > 0, MR k θ, 1) = 1 that MR k θ, t ) = 0. ) θ + F θ) < 0. So there is a unique t 0, 1) suh f θ) Seond, we show that for any t > t, τ 1, τ 2 [0, 1), the ondition in Proposition 4.1 holds for <, i.e. reserve pries given by 4) satisfy t > 2 r 1 r 2. Notie 4) are equivalent to MR 1 θ 1, t) = and MR 2 θ 2, t) = 5) 1 τ 1 1 τ 2 At t = t, and = τ 1 = τ 2 = 0, MR k θ, t ) = 0. Now for any given t > t, and τ 1, τ 2 [0, 1), we must have θ 1 < θ, θ < θ 2 for 5) to hold, due to Assumption 1. So θ 1 < θ 2 t > 2 r 1 r 2. Buyers equilibrium stated in 2) follows from Proposition 4.1. Third, we prove that reserve pries given in 4) are equilibrium pries. Sine all buyers with valuations larger than or equal to r k attend aution k, we have x k = r k. So seller k s expeted profit is where Π k, τ k, r k ) = 1 τ k )NM k Φ k, r k )) + Φ N k r k ), M k Φ k, r k ) = r k 1 Φ k r k ))Φ N 1 k r k ) + N 1) The first order ondition is 1 y1 Φ k y))φ N 2 k r k y)dφ k y). 0 = Π k r k = NΦ N 1 k r k )1 Φ k r k ))1 τ k )1 r k λ k r k )) + λ k r k )). The optimal reserve prie satisfies r k 1 λ k r k ) = 2 Π k r 2 k. Rearranging, we have 4). We show that the seond order ondition for a maximum is satisfied. Sine 2 Π k rk 2 < 0 follows immediately from Π 1 θ 1 = N1 F θ 1 )) N 1 f θ 1 ) Π 2 θ 2 = NF θ 2 ) N 1 f θ 2 ) MR 1 θ 1, t) MR 2 θ 2, t) 1 τ 1 ) 1 τ 2 ) = 0 2 Π 1 θ 2 1 = 0 2 Π 2 θ 2 2 < 0 < 0 = t 2 2 Π k θ, k 2 due to Assumption 1. So seller k s expeted profit obtains the maximum at r k given by 4). We are left to hek that given monopoly reserve prie r 2, seller 1 has no inentive to lower her reserve prie to overlap seller 2 s market. Suppose that seller 1 does deviate, then the best 16
she an do is to get all the buyers in the overlapping market segment. But in that ase she remains the monopolist in her own market, and the solution to her maximization problem is given by 4). This shows that sellers behavior stated in 1) onstitutes an equilibrium. We all the equilibrium outome in Proposition 4.3 the loal monopoly outome. The following proposition shows that this remains the equilibrium outome for high value sellers in the ase of less differentiated produts. For the rest of the paper, t defined as in Proposition 4.3 refers to the ritial value that separates the high and low produt differentiation ases. Proposition 4.4. For any given τ and t < t, there exists a unique suh that for, ), the loal monopoly outome is the equilibrium outome in the sellers and buyers subgame, where is the solution to 4) for θ 1 = θ 2. Proof. We first prove the existene and uniqueness of that satisfies MR 1 θ ), t) 1 τ 1 = MR 1 θ ), t) 1 τ 1 θ k ) is suh that MR k θ k ), t) = 0. Then, we show that for, ), we have θ 1 ) < θ 2 ), where = 0, for k = 1, 2. This is equivalent to saying that equilibrium reserve pries satisfy the ondition in Proposition 4.1. The rest of the proof parallels the argument in Proposition 4.3. The idea of the proof is illustrated in Figure 3. = < < θ ) θ θ 1 ) ) θ 2 ) MR 2 1 τ 2 MR 1 1 τ 1 Figure 3 MR 1 1 τ 1 MR 2 1 τ 2 Let τ and t < t be given. First, we show that there exists a unique suh that MR k θ, t) = 0 for some θ [0, 1]. Assume τ 1 τ 2. The proof for the ase τ 1 > τ 2 )) follows a very similar argument.) Let = 1 τ 1 ) 1 t 1 + 1. For any [, ], f1) define funtions k : [0, 1] R by k θ, ) = MR k θ, t) 17, k = 1, 2. Let θ, ) =
1 θ, ) 2 θ, ). The following three steps show that for any [, ], there exists a unique θ 0, 1] suh that θ, ) = 0. 1. θ, ) is ontinuous and dereasing in θ. ) 2. At θ = 0, 0, ) = t 1 + 1 + f0) 1 τ 2 1 τ 1 > 0. 3. At θ = 1, 1, ) 0 sine 2 1, ) = 1 1 τ 2 0, 1 1, ) = 1 t ) 1 + 1 f1) 1 τ 1 0. Now, onsider k θ), ) as a funtion of. We show that there exists a unique suh that k θ ), ) = 0, k = 1, 2. Notie any given pair t and τ will be suh that =, or <. In the former ase, let = 1 τ 2. In the latter ase, there is a unique, ). This is shown in the following three steps. 1. 1 θ), ) is dereasing in on [, ]. Taking the derivative with respet to of the equation θ), ) = 0, we have θ) MR1 θ, t) θ MR ) 2θ, t) = 1 1. θ 1 τ 1 1 τ 2 Sine the RHS is non-positive due to τ 1 τ 2, and MR 1θ,t) θ θ) 0. Therefore, < 0, MR 2θ,t) θ > 0, we have 1 θ), ) = MR 1θ, t) θ) 1 < 0. θ 1 τ 1 2. For = 1 τ 2, 1 θ ), ) = 2 θ ), ) = MR 2 θ ), ) 1 < 0. The strit inequality holds sine < 1, ) < 0 θ) 0, 1). )) 3. For = 1 τ 1 ) 1 t 1 + 1, f1) 1 θ)) = 1 t = t 1 + F 1) f1) θ) + F θ)) f θ)) ) t ) 1 τ 1 θ) + F θ)) f θ)) ) > 0. Therefore, there exists a unique MR k θ ), t) = 0, k = 1, 2., ) suh that 1 θ )) = 0. This implies that Seond, we show that reserve pries given by 4) satisfy t > 2 r 1 r 2 for, ). To see 18
this, note that for 4) to hold at, ), we must have θ 1 ) < θ ) and θ ) < θ 2 ), due to Assumption 1. Therefore, θ 1 ) < θ 2 ) t > 2 r 1 r 2. The rest of the proof is the same as in Proposition 4.3. The preeding two propositions onsider ases where the realizations of sellers values are suh that both prefer attending some autioneer to staying out. The following proposition haraterizes the situation where one autioneer s perentage fee is too high at a partiular realization of to attrat either seller. In this ase, both sellers first attend the autioneer with the lower fee, and the one who gets rejeted stays out. The seller in the aution is the monopolist in the whole market, and her equilibrium behavior remains the same as in the loal monopoly ase. Proposition 4.5. Suppose that 1 τ j < 1 τ j. Then in equilibrium both sellers go to autioneer j, and the one who is rejeted stays out. If seller k gets served, she sets reserve prie r k satisfying r k 1 λ k r k ) = 1 τ k. A buyer of type θ attends aution k if and only if v k θ) r k. 4.2.2 Duopolist sellers When produts are less differentiated and sellers values are low, sellers ompete for buyers in the overlapping market segment by setting reserve pries low. In doing so, sellers leave positive partiipation rents to the buyer who is indifferent between attending the two autions. When sellers values inrease, they raise reserve pries and the overlapping region beomes smaller. Consequently, the partiipation rents for the indifferent type derease. In partiular, there exists a ritial value suh that the indifferent type gets exatly zero rents. However, even at that point, sellers effetive marginal osts are smaller than marginal revenues. This distinguishes the ase from the loal monopoly one. The following proposition haraterizes the equilibrium outome and the relevant region of sellers values for this outome. Proposition 4.6. For any given τ and t < t, there exists a unique < suh that for any <, the following is the equilibrium outome in the sellers and buyers subgame. 19
1. Sellers set reserve pries satisfying r k, τ, t) = v k θ) 1 N ψ k θ, t) 1 τ k ) where ψ 1 θ, t) = MR 1 θ, t) t F θ) F θ) N 1, 1 F θ) ψ2 θ, t) = MR fθ) 1 F θ) 2 θ, t) t fθ) and θ satisfies 1 F θ)) N 1 ψ 1 θ, t) 1 τ 1 ) ) = F N 1 θ) ψ 2 θ, t) 1 τ 2 ) 6) ) 1 F θ) N 1, F θ) 2. All buyers with θ < θ attend aution 1, all buyers with θ > θ attend aution 2, and θ is indifferent between attending the two autions. is suh that = 1 τ k )ψ k θ ), t) if ψ k θ0), t) > 0, and = 0 otherwise. 7) Proof. We proeed in three steps. First, we show the existene and uniqueness of with the property stated in the proposition. Seond, we show that for <, the ondition in Proposition 4.2 holds. This then implies that buyers behavior stated in 2) onstitutes an equilibrium. Third, we show that sellers reserve pries stated in 1) are equilibrium pries. First, we show that there exists a unique 0, ) suh that ψ k θ, t) = 0, for some θ 0, 1). Sine the event = has zero probability, we assume for the following analysis that <. Now for any given <, define funtions k : [0, 1] R by 1 θ, ) = 1 F θ)) ψ N 1 1 θ, t) 1 τ 1 ), and 2 θ, ) = F N 1 θ) ψ 2 θ, t) 1 τ 2 ). Our objetive is to find a θ 0, 1) suh that 1 θ, ) = 2 θ, ) = 0. This is done in the following four steps. Figure 4 illustrates the idea of the proof. Notie at = 0, we have either θ 1 0) θ 2 0), in whih ase we set = 0, or θ 1 0) > θ 2 0), in whih ase we show the existene and uniqueness of in 0, ) as follows. 1. For any given <, there exist a unique θ k ) with k θ k ), ) = 0. Sine 1 0, ) = 1 1 τ 1 > 0 and 1 1, ) = t f1) < 0, there is a unique θ 1 ) 0, 1) suh that 1 θ 1 ), ) = 0. Similarly, 2 0, ) = t f0) < 0, 21, ) = 1 there is a unique θ 2 ) 0, 1) suh that 2 θ 2 ), ) = 0. 2. θ1 0) θ 2 0) > 0. This is the ase to be onsidered here. 20 1 τ 2 > 0, so
= 0 = 0 = = θ 20) θ 10) θ θ 1 ) θ2 1 0) θ2 0) θ 1 ) = θ 2 ) ). Set = 0 0, ) Figure 4: Proof of the existene of 3. θ1 ) θ 2 ) < 0. Reall that Proposition 4.4 shows MR k θ ), t) =. Sine ψ k θ, t) < MR k θ, t), k θ k )) = 0 θ 1 ) < θ ), θ ) < θ 2 ) θ 1 ) < θ 2 ). 4. θ 1 ) θ 2 )) < 0. For θ 0, 1), taking the derivative with respet to of the equation ψ k θ k )) 0, we get θ 1 ) < 0 and θ 2 ) > 0. = Combining 1)-4) implies that there exists a unique 0, ) suh that θ 1 ) = θ 2 ) = θ )). This shows that k θ ), ) = 0, for k = 1, 2. Sine θ ) 0, 1), this implies that ψ k θ ), t) = 0. Seond, we show that for <, reserve pries given by 6)-7) imply θ 2 ) < θ 1 ). It then follows diretly from Proposition 4.2 that buyers behavior stated in 2) onstitutes an equilibrium. To see θ 2 ) < θ 1 ) for <, simply notie that θ 1 ) θ 2 ) is dereasing in, and θ 1 ) θ 2 ) = 0. Third, we show that for <, equilibrium reserve pries are given by 6)-7). Seller k solves 2) with x k > r k. The first order ondition is 0 = Π k = NΦ N 1 k x k )φ k x k ) x k r k r k 1 τ k ) x k Ω )) λ k x k ) Nx k r k ) where Ω = 1 + ) 1 Φk x k ) N 1. Φ k x k ) The optimal reserve pries are given by ) r k = x k 1 N x k Ω λ k x k ) 1 τ k 8) 21
Rearranging, we get 6). The indifferent type θ is haraterized by 1 F θ)) N 1 v 1 θ) r 1, τ, t)) = F N 1 θ)v 2 θ) r 2, τ, t)) Using 6) to replae v k θ) r k, τ, t) with 1 N ) ψ k θ, t) yields 7). The seond order ondition for a maximum is satisfied, as shown below. ) 2 2 Π k = NΦ N 1 xk rk 2 k x k )φ k x k )1 τ k ) ) 1 λ k x k ) + 1 r k x k λ k x k ) x k +NΦ N 1 k x k )φ k x k )1 τ k ) x k N 1) x ) k N r k r k The first term on the RHS is negative sine 1 λ k x k ) x k also negative sine 0 < x k r k overlapping, the indifferent type satisfies 1): where v k θ) = x k. We have θ r 1 x 1 = 1 + θ r 1 r 1 θ x 2 r 2 = 1 + θ r 2 < 0 and x k < 0. The seond term is < 1. To see this, reall that for r 1, r 2 suh that markets are Φ N 1 1 v 1 θ))v 1 θ) r 1 ) = Φ N 1 2 v 2 θ))v 2 θ) r 2 ) θ < 0, θ r 2 > 0. So x 1 r 1 = t θ r 1 > 0, x 2 r 2 = t θ r 2 > 0, and ) N 1 ) N 1 ) F θ) F θ) x 2 r 2 ) + t < 1 1 F θ) 1 F θ) ) N 1 ) N 1 ) 1 F θ) 1 F θ) x 1 r 1 ) t < 1 F θ) F θ) Finally, we hek that given seller 2 s equilibrium reserve prie, seller 1 has no inentive to deviate. Suppose that given r 2, seller 1 inreases her reserve prie to v 1 θ) suh that markets are separated. Then seller 1 loses buyers in [θ, θ], and the marginal revenue that she an get from eah of those buyers is higher than her effetive marginal ost. Therefore, it is not optimal for seller 1 to deviate. 4.2.3 Constrained duopolist sellers We are left with the ase where sellers values are in the intermediate range, that is,, ). In this ase, sellers values are neither high enough for them to set monopoly reserve pries, nor suffiiently low for them to overlap eah other s market. Indeed, if eah seller simply responds to her inreasing values by raising her reserve prie, there would be a range 22
of buyers whose partiipation rents are negative and hene would not attend any aution. Therefore, sellers are better off setting reserve pries at one of these buyers valuations for their items respetively, so long as their marginal revenues from that buyer is no less than their effetive marginal osts. There are typially multiple equilibria in this ase. We fous on the equilibrium that maximizes sellers joint expeted profits without further mentioning this seletion riteria for the rest of the paper. Proposition 4.7. For any given τ and t < t, for any, ), the following haraterizes all equilibrium outomes in the sellers and buyers subgame. 1. Sellers set reserve pries satisfying r k, τ, t) = v k ˆθ) 9) for any ˆθ Θ = [ θ 1, θ 2 ] [ θ 2, θ 1 ], where ψ k θ k ) =, MR k θ k ) =. 2. All buyers with θ < ˆθ attend aution 1, all buyers with θ > ˆθ attend aution 2, and ˆθ is indifferent between attending the two autions. In partiular, there is an equilibrium where sellers maximize their joint expeted profits. The indifferent type in that equilibrium is given by 1 F ˆθ)) N 1 1 τ 1 )MR 1 ˆθ, t) ) = F N 1 ˆθ)1 τ 2 )MR 2 ˆθ, t) ) 10) Proof. In any of the equilibria speified in this proposition, sellers set reserve pries suh that the indifferent type ˆθ gets zero rents. All buyers with θ < ˆθ expet non-negative profits from attending aution 1, and all θ > ˆθ expet the same from aution 2. This shows that buyers behavior stated in 2) onstitutes an equilibrium. We now show that sellers strategies form a Nash equilibrium. First, note that in any equilibrium, sellers hoose the indifferent type ˆθ from Θ. To see why this must be true, notie for, ), i) MR k θ k ) = θ 1 > θ 2. Both sellers are willing to attrat buyers from [ θ 2, θ 1 ] sine eah seller s marginal revenue from a buyer in that region is greater than her effetive marginal ost. If θ / [ θ 2, θ 1 ], one seller s marginal revenue from θ is less than effetive 23
marginal ost, hene it is not optimal for that seller to set reserve prie at θ s valuation for her item. ii) ψ k θ k ) = θ 1 < θ 2. Seller 1 will not set r 1 = v 1 θ), θ < θ 1, sine she an get all buyers from [θ, θ 1 ] if she were to lower reserve prie to v 1 θ 1 ). Similarly, seller 2 will not set r 2 = v 2 θ), θ > θ 2. Combining i) and ii) shows that if equilibrium reserve pries satisfy 9), then ˆθ Θ. Next, suppose that seller 2 sets r 2 = v 2 ˆθ), for some ˆθ Θ. We show that seller 1 s best response is to set r 1 = v 1 ˆθ). Seller 1 setting r 1 = v 1 θ 1 ) > v 1 ˆθ) is not optimal sine she simply loses buyers in θ, ˆθ). We prove by ontradition that seller 1 setting r 1 = v 1 θ 1 ) < v 1 ˆθ) is also not a best response to r 2. Suppose the ontrary is true. Then, Proposition 4.2 implies that there exists a type θ ˆθ, θ 1 ) who is indifferent between the two autions, and gets positive rents. So r 1 satisfies r 1 = v 1 θ) 1 ψ N 1 θ, t) 1 τ 1 ), subjet to the onstraint ψ 1 θ, t) 1 τ 1 > 0. Sine r 2 = v 2 ˆθ) is fixed, seller 1 should hoose r 1 suh that θ s rents F N 1 θ)v 2 θ) v 2 ˆθ)) is minimized, in other words, θ ˆθ. But sine we assume θ 1 > ˆθ, this minimum annot be ahieved, whih is a ontradition to r 1 being a best response. The above argument implies that r 1 = v 1 ˆθ) and r 2 = v 2 ˆθ) form a Nash equilibrium. Finally, we onsider the equilibrium where sellers hoose the indifferent type to maximize their joint expeted profits. The first order ondition to maximize Π 1,2 = Π 1 + Π 2 is 0 = Π 1,2 θ = 1) k tnφ N 1 k r k )1 Φ k r k ))1 τ k )1 r k λ k r k )) + λ k r k )) k=1,2 = Nf θ) 1 F θ)) N 1 1 τ 1 )MR 1 θ, t) ) F θ) N 1 1 τ 2 )MR 2 θ, t) ) ) Rearranging, we have 10). The seond order ondition for a maximum is satisfied sine 2 Π 1,2 θ 2 < 0 due to Assumption 1. Let us summarize the above results. For any given τ and t < t, there are two ritial values and whih divide [0, ] into three regions, eah giving rise to a different type of equilibrium, whih losely parallels the Hotelling-Bertrand model. Region 1. : Loal monopolist sellers. 24
When <, sellers are loal monopolists in their respetive markets. They set reserve pries r k = v k θ k ), where θ k satisfies MR k θ k ) =. A buyer of type θ attends aution k if and only if v k θ) r k. This is illustrated in Figure 5, where the red and blue lines refer to buyers who attend aution 1 and 2 respetively. When =, reserve pries are suh that r k = v k θ), where θ is the type indifferent between the two autions, and satisfies MR k θ) =. Sellers are still loal monopolists, and the market [0, 1] is fully overed. This is illustrated in Figure 6.. 0 θ 1 θ2 1 Figure 5: <. 0 θ 1 = θ = θ 2 Figure 6: = 1 Region 2. 0 : Duopolist sellers. When 0 <, sellers have low values and ompete fierely. Reserve pries are set low suh that the partiipating rents are positive for the type θ who is indifferent between attending the two autions. Speifially, reserve pries are r k = v k θ k ), where θ k satisfies ψ k θ k ) =. This is illustrated in Figure 7. When =, reserve pries are higher, given by r k = v k θ), where θ satisfies ψ k θ) =. That is, the indifferent type θ gets zero rents. This is illustrated in Figure 8. Note that although the market is fully overed, MR k θ) > reserve pries.. 0 θ 2 θ θ1 1 Figure 7: 0 <, so r k are not monopoly Region 3. < < : Constrained duopolist sellers. 25
. 0 θ 2 = θ = θ 1 Figure 8: = 1 When seller s values are in this intermediate range, there are multiple equilibria. In eah equilibrium, sellers set reserve pries r k = v k ˆθ), where ˆθ is the type indifferent between the two autions, and satisfies ψ k ˆθ) Figure 9. ˆθ. 0 θ 1 θ 1 θ 2 1 θ 2 multiple equilibria MR k ˆθ). This is illustrated in Figure 9: < < 4.3 The full game Autioneer j sets a perentage fee τ j [0, 1] to maximize his expeted profit antiipating whih seller to host and the seller s equilibrium reserve prie. Different rationing rules in general lead to different antiipations, whih we will disuss soon after speifying autioneers problem. For a given rationing rule, Let p jk τ ) be the probability that autioneer j hosts seller k when the perentage fees are τ. Autioneer j hooses τ j [0, 1] to maximize Π j = p jk τ ) j k=1,2 0 τ j NM k Φ k, r k, r k )dh) 11) where j = 1 τ j is the highest value for sellers to be willing to partiipate in the aution subjet to perentage fee τ j. In the next two subsetions, we derive equilibrium perentage fees for apaity onstrained autioneers. Depending on the rationing rule, autioneers may have inentives to ompete for the good seller. That is, assume one seller generates more expeted revenue for a given perentage fee. Suh inentives do not exist under non-disriminatory rationing rule sine an autioneer is expeted to host eah seller with equal probability regardless of the other autioneer s fee. However, when disriminatory rationing rule is used, autioneers always 26