Competing Auctions Glenn Ellison, Drew Fudenerg, and Markus Moius First draft: Novemer 8, 00 This draft: March 6, 003 This paper studies the conditions under which two competing and otherwise identical markets or auction sites of different sizes can coexist in equilirium, without the larger one attracting all of the smaller one s patrons. We find that the range of equilirium market sizes depends on the aggregate uyer-seller ratio, and also whether the markets are especially "thin." Department of Economics, MIT Department of Economics, Harvard University The authors thank the NF for financial support, Ahijit anerjee, Ed Glaeser, and Dimitri Vayanos for helpful comments, Jonathan Weinstein for research assistance, and Adam zeidl for very careful proof reading. 0
. Introduction Activity in prominent auction markets seems to e concentrated at a few sites. For example, othey s and Christie s have dominated the traditional fine art auction usiness for a century or more, with relatively equal market shares over that period; the comined market share of the two firms is estimated to e greater than 90%. In the online auction market, eay has maintained a dominant position since Yahoo's entry, despite its higher fees. While eay has (often via acquisition) achieved a large share of online auctions in many countries, a notale exception is Japan, where Yahoo entered efore eay and is now reported to have a 95% market share. This paper develops a very simple model of competing auction sites to analyze some forces that may lead auction markets to e concentrated. pecifically, we study when two competing markets or auction sites of different sizes co-exist in equilirium, and when this is impossile ecause the larger auction will attract all of the smaller one s patrons. It is clear that two markets can co-exist if they are geographically distinct and clients face sufficiently large transport costs, and conversely it is clear that otherwise identical markets of different sizes cannot co-exist if larger markets offer a greater variety of goods and uyers have a sufficiently large preference for more diverse markets. This paper astracts away from oth of these issues y examining a model with a single good and no transport costs. In our model, there are ex-ante identical uyers, each with unit demand, and sellers, each with a single unit of the good to sell. At the start of the single period of the model, the uyers and sellers simultaneously choose etween two possile locations. uyers then learn their private values and a uniform-price auction is held at each The two firms achieved prominent positions y the early 9 th century. ee Learmount (985). Each firm had aout $.5 illion in gross merchandise sales in 999; the third largest fine art auction house anticipates gross merchandise sales of aout $50m following its acquisitions of the other two top-five houses. The dominant position of Christie s and othey s is particularly striking given that they were accused of collusion following fee increases in 975 and 99, and convicted of price-fixing in connection with their joint adoption in 995 of a nonnegotiale scale for sellers commissions. Aout $8 illion in merchandise will e auctioned on eay this year. This is perhaps twenty times the volume of trade on Yahoo or Amazon. The second largest auction site for consumer products is actually Uid.com which sells directly to consumers and mostly uses auctions rather than posted prices; it also lists items for sale y other firms.
location. This is a very stark model, ut we hope that some of the insights it provides will e useful and that it can provide a enchmark case for richer and more realistic models. To egin, we show in ection 3 that larger markets are more efficient than smaller ones, holding the seller/uyer ratio constant. The inefficiency of a finite market relative to the continuum-of-traders limit is of order / N, where N is the numer of traders. This efficiency effect makes it harder for the smaller market to survive, ut on its own it does not rule out equiliria with two active markets. Whether two markets can survive depends on the relative magnitude of the efficiency effect and the market impact effect that a trader has on the market price when he or she switches to the other market. Our goal in this paper is to determine the range of equilirium market sizes that is permitted y these two conflicting effects. ection 4 presents some general results. First, we develop some preliminary results: we formalize the efficiency vs. market impact interpretation of the equilirium conditions, and we simplify the equilirium computation y showing that it suffices to consider the constraints under which uyers and sellers are willing to stay in the smaller market. econd, we show that the efficiency and market impact results are oth of order /N, and that as the numer of traders grows the market converges to a well-defined continuum-of-traders limit, where payoffs depend on the relative numers of uyers and sellers. This allows us to appeal to the main theorem of Ellison and Fudenerg [00] to conclude that when the markets are large there is a sustantial plateau of market splits that satisfy the equilirium conditions. To provide a more detailed understanding of the forces at work, ections 5 and 6 specialize the model to two particular distriutions. In section 5 we assume that uyers valuations are uniformly distriuted on [0,]; this might e a reasonale assumption for thinking aout Pokemon cards, eanie aies, and other collectales sold on eay. Here it is typically possile to have a stale two-site equilirium where the small site is onethird or one-fourth as large as the larger site; just how unequal the two markets can e depends on the uyer-seller ratio. We note that this qualitative conclusion is roust in a couple ways: it would suffice for agents on one side of the market rather than oth to recognize that they have a market impact; and similar results still otain if we restrict attention to full equiliria with integer numers of agents in each market. ection 6
examines the model with exponentially distriuted valuations, which might e a reasonale assumption for fine art items. We note some differences in how the model works out, ut the general pattern of the results is the same. Our take on the results is that the efficiency effect can lead agents to concentrate, ut does not seem sufficiently powerful to account for all of the concentration we oserve. ection 7 examines another factor that may support concentration market thinness. One interesting aspect of eay listings is that most of them seem to e unique items; this may e an important common trait of fine art and online auctions. The fact that a size ratio of 3: or 4: is possile is consistent with the long-term coexistence of Christie s and othey s; it is also consistent with the lack of successful large-scale entry into this market. eay is many times larger than any of its competitors. From the viewpoint of our model, the online auction market must thus e thought of as one that has tipped to a single auction, rather than as stale coexistence of multiple auction sites. Our conclusion that a range of size ratios is possile differs from that of past analyzes of the choice of location or market in cases where it is the efficient to have a single market. We discuss this in greater detail in section 8.. The Model In our model, there are uyers and sellers. The uyers values v are identically and independently distriuted according to a cumulative distriution function F, which has a three-times continuously differentiale density f that is positive on its support [0, ], with allowed to e infinite. 3 of n values as kn : v We denote the kth highest order statistic of a draw ; we assume that the expected value Ev is finite, which implies that the expectations of the order statistics are finite too. 4 We write kn : f for the density of v kn :. 3 Propositions,, and 4 of this section extend to distriutions without densities, although some strict inequalities ecome weak, and the notation and proofs ecome a it more complicated due to the need to account for ties. ection 6 considers the exponential distriution, whose support is unounded aove. Only the large-economy approximations of Propositions 3, 6, and 7 use the assumption that f has a third derivative, and it would suffice for these propositions to assume that the third derivative exists only in a neighorhood of one particular point. 4 This is a slight twist on the usual notation in textooks, where first order statistic is the smallest. 3
We model location choice with a two-stage game. In the first stage, which occurs efore the uyers learn their valuations for the good, uyers and sellers simultaneously choose whether to attend market or market. In the second stage, uyers learn their valuations and a uniform price auction occurs in each market. Thus if a market doesn t + : have excess supply, the price there is the +st highest of the uyer values, v, which we will also denote as p. We assume that + < so that if all agents go to a single market there is excess demand and so with proaility one the market price is strictly positive. ince sellers have zero reservation value and are risk neutral, their expected utility is just the expected price in the market they choose, which we denote y p (or p(, ) when we need to track the dependence on the numer of uyers and sellers). The expected utility of a uyer who attends a market with sellers and uyers + : : : (including himself) is u (, ) = E( v v v v )Pr( v v ). 3. The Efficiency of Large Markets Our focus in this paper is on the possiility of equiliria with two active auction sites when uyers and sellers choose etween two sites. To egin, however, we discuss the efficiency and market size in a single market. Consider a market with seller and > uyers. The efficient outcome is for the uyers with the highest values to receive the good, so the expectation of the : : : maximum possile total surplus is Pr( v v )E( v v v ) = E( v v v ). At this outcome, the surplus per seller (or per item sold) is E( v v v ) = E( v v> v + : : ), which we define to e : w (, ) ; note that ( ( ) ) + w(, ) = v f v v > x dv f ( x) dx. Note that 0 x as the market grows, holding the uyer-seller ratio fixed, the efficient outcome converges to a deterministic limit, with the good given to all uyers whose value exceeds the market-clearing price : : p defined y F( p ) =/. Put differently, in the continuum limit the good is allocated to all uyers with ( ) v v F /. Thus welfare per seller converges to the average of the uyer values on the range where value is at least the market price, E [ v v v( / ) ] ; denote this w ( / ). 4
The first two propositions show that our model has the intuitive property that larger markets are more efficient. The first compares a finite market to the continuum limit discussed aove; the second extends this to a comparison of finite markets of different sizes. Proposition : With an efficient allocation of goods to uyers, the expected surplus per seller in a finite market with sellers and uyers is strictly less than the expected surplus per seller in a market with continua of uyers and sellers and the same selleruyer ratio. The idea of the proof is simply that the distriution of realized utility as a function of uyer s value in the large market first-order stochastically dominates the distriution in the small one. In oth markets, uyers have the same proaility of receiving the good, ut in the small market, uyers win in a less efficient way, as they sometimes win when v < v. Proof: ince we are holding the seller-uyer ratio fixed, it will e sufficient to prove that surplus per uyer is higher in the large market. Let cv ( ) e the uyer s proaility of consuming the good in the finite market when his value is v. Then the expected surplus per uyer in the continuum market is E( v v v)pr( v v), and the expected surplus per uyer in the finite market is Evcv ( i ( )). The difference in per uyer surplus etween the large and small market is ( ) ( ) [ ] [ ] E v( c( v)) v v Pr( v v) E( vc( v) v v) Pr( v v) > E v( c( v)) v v Pr( v v) E( vc( v) v v) Pr( v v) = v Pr( v v) E( c( v)) = v / / = 0. The strict inequality follows from the fact that c() v is strictly etween 0 and on for v (0, v ), which in turn follows from the assumptions that > and the cumulative distriution of uyer values is strictly monotone. QED. in market size. The next result extends the previous one y showing that efficiency is monotone 5
Proposition : If m and n are integers with m<n, then w( m, m) < w( n, n). Proof: ee appendix. 5 Our final efficiency result is an asymptotic approximation to the inefficiency of a large finite market. It is a simple corollary of a result we will give later. We therefore present it here without a proof. Proposition 3: / w ( / ) w( n, n) = + o(/ n). f( v) n The inefficiency of finite markets relative to the continuum-of-players limit is closely related to the following impossiility result: A market of fixed finite size could not survive if agents had the opportunity to attend a market with a continuum of uyers and sellers instead. Let the finite market e market, with numers sellers and uyers, and let market have a continuum of participants with seller-uyer ratio /. The deterministic price p in market is defined y F( p ). A uyer or = / seller moving into the large market will have no effect on the price there, so for oth markets to coexist it is necessary that the expected price in market satisfies p p (or else sellers move to market ) and that u u (or else the uyers move). Proposition 4: There is no equilirium in which trade takes place in oth a continuum market and a finite one. Proof: As efore, let cv () e the uyer s proaility of consuming the good in the small market when her type is v. Then u = E( v )Pr( ) is the uyer s p v p v p 5 We thank Jonathan Weinstein for the proof of this proposition. 6
expected utility in the continuum market, and u = E( v c()) v E( p ) = E(( v p )()) c v is the uyer s expected utility in the finite market. ellers are willing to remain in the finite market only if p p. When this holds, however, we have u = E(( v p ) c( v)) E(( v p ) c( v)) < E(( v p )( v p )) = u, so that uyers are not willing to remain in the finite market. QED 4. General Results on Competing Finite Markets Proposition 4 suggests that there must e a ound on the ratio of the sizes of two markets for oth of them to remain active. In this section, we present some general results on the coexistence of two finite ut large markets. In general, there will e two types of pure-strategy equiliria: equiliria where the market has tipped and all uyers and sellers attend a single market, and equiliria in which oth markets are active. The goal of this section is to characterize the range of market sizes that are possile in equiliria with two active markets. 4A. Equilirium Conditions: The Efficiency and Market Impact Effects functions defined on [ 0, ) [0, ). (We will explain how to construct the continuous () u (, ) u ( +, ) s Although the utility functions u and s us were derived for markets with integervalued numers of uyers and sellers, it will e convenient to think of them as continuous versions from the real ones when we compute the equiliria for specific distriutions.) Oviously, in a pure-strategy equilirium the numer of agents of each type choosing each market must satisfy the following four constraints: () u (, ) u ( +, ) s s () u (, ) u (, + ) () u (, ) u (, + ) ince the variales,,, descrie numer of agents, they should e integer-valued; we will refer to,,, that satisfy the incentive constraints ut are 7
not necessarily integers as quasi-equiliria. Our first step in analyzing the implications of these constraints is to note that they can e rewritten as: u (, ) u (, + ) u (, ) u (, ) ( ) ( ) u (, ) u ( +, ) u (, ) u (, ). s s s s u (, ) u (, + ) u (, ) u (, ) ( ) ( ) u (, ) u ( +, ) u (, ) u (, ). s s s s The left-hand sides of the two "stay-in-market-" conditions ( ) and ( ) measure the detrimental market impact that the agents have when they move to market. The right-hand sides measure the degree to which market is more attractive given the current division of uyers and sellers. ecause larger markets are more efficient, one or oth of the right-hand sides of (') and (') will e positive when market is larger than market. 4. A Characterization of the Quasi-equilirium et The calculation of the quasi-equilirium set can e simplified if the continuous versions of the seller and uyer utility functions satisfy three regularity conditions. We will simply state them and derive the implications here. We will show in sections 5 and 6 that the assumptions are satisfied for the distriutions considered in those sections. First, we assume some oundary conditions: us(, ) > 0 if >, us(, ) = 0 if (A). u (, ) > 0 if > 0, u (0, ) = 0 econd, we assume that the functions are differentiale when > and satisfy the natural monotonicity properties: us us u u (A) < 0, > 0, > 0, and < 0. Finally, recall that we showed in Proposition that holding the uyer-seller ratio constant larger markets are more efficient. In Proposition 4 we showed that either uyers or sellers are worse off in a finite market than in a market with a continuum of sellers, regardless of whether the uyer-seller ratio is different. Our third assumption is that in a comparison of two finite markets, the smaller market is less efficient in a similar sense. 8
(A3) If < then us(, ) < us(, ) or u(, ) < u (, ). Checking whether an allocation (,,, ) is an equilirium in principle requires checking four constraints. The main result of this susection is that one need only examine the constraints ensuring that agents do not want to leave the smaller market to determine the market sizes that are possile in a quasi-equilirium. To make the argument, it is helpful to define the loci where the incentive constraints in the small market are satisfied with equality. For fixed values of and, let ( ) e the value of such that () holds with equality at (,,, ). The function ( ) descries the minimal numer of sellers which are necessary in market to prevent uyers from defecting to market. Let ( ) e the value of such that () holds with equality at (,,, ). The function ) descries the maximum of sellers which market can ear efore losing sellers to market. These functions allow us to express the equilirium conditions () and () more conveniently: () is satisfied if and only if ( ( ) and () is satisfied if and only if ( ). Lemma : Given assumptions (A) (A3), function on the domain [ 0, ]. There exist constants min max 0 < < /< < such that min function on the domain [, max ] with min is a well-defined, increasing, differentiale min and max with is a well-defined, increasing, differentiale ( ) = 0and m ax ( ) =. Moreover, ( ) < /+ / and ( )< /when < /. Proof: ee appendix. Proposition 5 is a precise statement of our oservation that one need only consider whether the small market constraints can e satisfied in order to determine whether there is a quasi-equilirium with specified numers of uyers in the two markets. A 9
rough intuition for the proposition is that the fact that large markets are more efficient makes it easier to satisfy () and () than to satisfy () and (). Proposition 5: Fix and with + <. Assume (A) (A3) and that /. Then, there exists an such that (,,, ) is a quasi-equilirium if and only if there exists an such that (,,, ) satisfies the () and () constraints. Proof: ee appendix. Here is a sketch: Fix an allocation, satisfying () and (). Large market efficiency implies that either the uyers or the sellers (or oth) are getting higher utility in the market. If oth, then () and () are oviously satisfied and we are done. If only () is satisfied, then the allocation we started with is not a quasiequilirium. In this case, however, we can add sellers to market until () is just satisfied. At this allocation sellers get higher utility in the small market. The other three constraints therefore hold: () continues to hold ecause we ve made market more attractive; () holds ecause sellers are doing etter in market ; and () holds ecause uyers are doing etter in market (which follows from large market efficiency). 4C. A Lower ound on ize if the Quasi-equilirium et in a Large Economy Ellison and Fudenerg [00] shows that models in which large numers of uyers and sellers simultaneously choose etween two locations have a plateau of quasi-equiliria (with a sustantial degree of market asymmetry eing possile) whenever the utility functions in the economy satisfy the condition elow. Condition A4: There is a non-empty interval Γ = [ γγ, ] (0, ) and twice continuously differentiale functions u Γ s, u, G s, and G on with du s / dγ < 0 and du / dγ > 0 such that the approximations u ( γ, ) = u ( γ) G ( γ)/ + o(/ ) s s s u ( γ, ) = u ( γ) G ( γ)/ + o(/ ) 0
hold uniformly in γ when is large. 6 Ellison and Fudenerg [00] show that this assumption implies that the market impact effects when = γ satisfy s s s u ( +, ) u (, ) = u '( γ)/ + o(/ ) and u (, ) u (, + ) = γu '( γ)/ + o(/ ), and that for = ( α) the efficiency advantages satisfy and u ( γ, ) u ( γ, ) = G ( γ)( α)/( α( α) ) + o(/ ) s s s = α and u ( γ, ) u ( γ, ) = G ( γ)( α)/( α( α) ) + o(/ ). In particular, each effect vanishes at a rate of at least /. Proposition 6: The utility functions in our auction model have extensions to the domain R R that are continuous and satisfy (A4) with Γ any closed su-interval of (0,) and with u ( γ ) = v( γ) F ( γ), u ( γ) = γ E( v v v v), s f ( v) + γ f '( v) G s ( γ) = ( γ) and 3 f( v) f ( v) + γ f '( v) G ( γ) = γ( γ). 3 f( v) Proof: ee appendix. To see why something like this result might e true, consider the formula for the seller s utility u ( γ, ) : As s + : γ (, ) ( ) ( ) γ us γ = E v = γ F v ( F( v) ) vdf γ 0, the integrand ecomes increasingly concentrated around its maximum, and us ( γ, ) tends to F ( γ ) =v ; moreover the tails of the integrand are ounded y an 6 More formally, the assumption on the seller s utility function is that there exists a function m ( ) with s lim m s ( ) = 0 such that u ( ( γ, ) u( γ)) G( ) < m( ) for all γ Γ and all integers. s s s s
exponentially decreasing function of. Thus we can show that the rate at which the integral approaches its limit is determined y the ehavior of F ( y) allows us to use a Taylor expansion of F ( y) around y, and this to determine the rate of convergence. ince the integral of the third-order term of the Taylor expansion is already rate depends only the first and second-order derivatives of F. o, this One implication of Proposition 6 is that, even though larger markets are more efficient, uyers actually prefer smaller ones when f '( v ) is not too negative, at least when the numer of agents is sufficiently large. We will see in the next section that with the uniform distriution (where f '( v ) is identically 0) this conclusion applies for any numer of agents. 7 The result on the asymptotics of the welfare loss we stated earlier without proof (Proposition 3) is an easy corollary of Proposition 6. Theorem (Ellison and Fudenerg [00]): Assume (A4). Then, for any ε > 0 there exists a such that for any integer > and any integer with / Γ, the model with uyers and sellers has a quasi-equilirium with uyers in market for every with / [ ( ), α γ + ε α ( γ ) ε], where γ = / and α ( γ) = max{0, } for r ( γ ) s γ G( γ) G ( ) r ( γ ) = max, ' ' Fs γ + γf( γ) +. ( ) Comining this theorem with Proposition 6 gives Proposition 7: Assume (A4). Then, for any ε > 0 there exists a such that for any integer > and any integer with / [ ε, ε], the model with uyers and 7 Conversely, when f '( v ) is sufficiently negative, sellers prefer the smaller market.
sellers has a quasi-equilirium with uyers in market for every with / [ α ( γ) + ε, α ( γ) ε], where Let α ( γ) = max{0, } for r (γ ) ( f v γ v ) ( ) ( γ) ( ) + f '( ) ( γ) f( v) + γ f '( v) r ( γ ) = max +, f( v) γ f( v) +. Proof: ince u ( ) v ( ) F ( ), s γ = γ = γ us '( γ) = d F ( γ)/ d γ = / f( v). And u = E( v v v v)pr( v v) = (v v) f( v) dv, so f( v) + γ f '( v) f( v) + γ f '( v) ( γ) γ( γ) 3 3 f( v) ( ) = max f v +, + / f( v) γ / f( v) ( γ) f( v) + γ f '( v) ( γ) f( v) + γ f '( v) = max +, f( v) γ f( v) v u '( γ) = ( F( v)) dv( γ )/ dγ = γ / f( v). ustitution of ( ) ( f ( v) + γ f '( v) f ( v) + γ f '( v) G γ = γ γ) and G 3 s ( γ) = ( γ) in the general 3 f( v) f( v) formula for r in the previous theorem yields Gs( γ) G( γ) r ( γ ) = max +, + ' ' γ us ( γ) γu( ) QED ( ) ( ) + To illustrate Proposition 7, consider the uniform distriution. Here, the maximum in the definition of r is attained y the term corresponding to the sellers, so γ r ( γ ) = 3 γ, so α ( γ) = =. In particular, as γ the fraction of (3 γ )) 3 γ uyers that can e in market in a quasi-equilirium converges to (0,) ; as γ 0 the range converges to ( /3,/3). The Proposition gives only sufficient conditions for an 3
equilirium; we will see in the next section that the actual range of equiliria for this case is larger, namely (/4, 3/4). Next consider the exponential distriution, f( v) = exp( v) for v 0. Here e v ( γ ) = γ, so v ( γ ) = lnγ, and v v v v ( e e ) ( e e ) ( γ) γ ( γ) γ r ( γ ) = max +, v v e γ e + = max ( γ,) = γ, so α ( γ) =. ( γ ) As γ the fraction of uyers that can e in market in a quasi-equilirium converges to (0,) as in the uniform case; as γ 0, the range of quasi-equiliria from the theorem is ( /4,3/4). We show in ection 6 that when the seller-uyer ratios are allowed to differ in the two markets, the smallest share of uyers in market in a quasi- γ equilirium is the slightly smaller numer. 4 Proposition 7 has the advantage of applying for any distriution of valuations, ut it has several limitations: it only provides sufficient conditions for the existence of quasiequiliria not necessary ones, it doesn t descrie the equilirium set for small markets, and it doesn t tell us when the requirement that equilirium involves whole numers in each market can e satisfied. To address these questions, we now consider two particular distriutions, the uniform and the exponential. 5. Two Finite Markets: The Uniform Case This section analyzes the uniform distriution, which has a ounded support; the next section riefly analyzes the exponential case, where the distriution of values is unounded. The ounded-support case may e a etter description of markets for goods where there is a readily availale close sustitute that effectively caps the maximum willingness to pay, while the unounded support may e a etter description of models for one-of-a-kind ojects like paintings or rare collectiles. On the technical side, the distinction etween ounded and unounded support is relevant to our analysis ecause 4
with a ounded support, the expected value of the highest value is less sensitive to adding more uyers. 5A. asics and the Quasi-equilirium et Under the uniform distriution on [0,], the ith lowest order statistic out of n i draws is distriuted eta(i, n-i+) and has expectation (see e.g. David [970].) The n + seller s expected utility is equal to the expectation of the price, which we will denote y p. The price is given y the + th highest uyer value, which is the th lowest. Hence, us (, ) = p = +, and p is +. ecause a uyer s valuation conditional + on eing greater than p is uniform on ( p,], a uyer s expected utility conditional on winning the good at p is ( p) /. Each uyer wins the good with proaility /, so ( ) the uyer s expected utility is u (, = p(, ) / = ( + )/ (+ ). Note that holding / the larger market is more efficient. ) constant, uyers are actually etter off in smaller markets, even though Adding a seller to a market causes the expected price to fall y ( + ), irrespective of the numer of sellers in the market, while adding a uyer reduces uyer utility y ( + ), so the market impact of adding another uyer is strongest ( + )( + ) when / is near to. Although the derivation of the utility functions assumes that the numers of uyers and sellers are integers, the utility functions given aove are well-defined for all non-negative real numers. Moreover, it is clear from inspection that these functions satisfy (A) and (A). For (A3) (at least one side is etter off in a larger market) note that we can rewrite the uyer utility as u ( p) = ( p( + )) ; 8 thus if prices are higher in market and is smaller than, then uyers must e etter off in market. 8 To show this, note that / = (+)/(+) + (/ - (+)/(+)). 5
Figures and illustrate the structure of the equilirium and quasi-equilirium sets for the uniform distriution with ten uyers and five sellers. Figure graphs the fractions of sellers in market that make uyers and sellers exactly indifferent etween the two markets against the fraction of uyers in market. The solid curve, which is the locus where uyers utility is equal in oth markets, lies aove the dotted curve, which is the locus where sellers utility is equalized in oth markets, when < /. The unique intersection of the curves is at = /. If uyers and sellers did not adversely affect prices when moving to the other market, the only quasi-equilirium with split markets would e an unstale equilirium with an exactly 50-50 split etween the two markets. Figure graphs the values of for which the (), (), () and () constraints hold with equality for the same utility functions as in Figure. The quasiequilirium set is the parallelogram-shaped region in the center of the figure elow the curves where () and () hold with equality and aove the curves where () and () hold with equality. In this example, quasi-equiliria exist whenever the smaller market has at least % of the uyers (meaning / is at least.). We have placed small stars in the figure at points within the quasi-equilirium set where the numers of uyers and sellers are oth integers. These are the equiliria. In an equilirium the smaller market can have two uyers and one seller or four uyers and two sellers. There is no equilirium with three or five uyers in the smaller market. With three uyers in the smaller market, for example, then there are a range of values of near one-and-a-half which satisfy the quasi-equilirium conditions. None of these allocations, however, satisfy the integer constraints sellers would e unwilling to stay in small market if there were two sellers, while uyers would e unwilling to stay in a market if there was only one seller. To illustrate how markedly the efficiency effect declines with the size of the market, Figure 3 graphs the equal-uyer-utility and equal-seller-utility curves which apply to a model with 30 uyers and 5 sellers (and a uniform distriution of seller valuations as in Figure ). The curves are much closer together than the curves in Figure. ellers are indifferent when prices are equal in the two markets. The closeness of the two curves reflects that efficiency differences are fairly small and can only offset a small 6
difference in price. Figure 4 graphs the four curves that ound the quasi-equilirium set in this case. One interesting thing to note is that the range of market sizes in the quasiequilirium set is very similar to that in Figure : Here a quasi-equilirium exists whenever at least % of the uyers are in the small market, as compared to the % in Figure. The quasi-equilirium set looks much flatter in the -dimension. This reflects that the market impact is much smaller and hence uyer and seller utility (the latter of which is equal to the price) have to e more nearly equal in the two markets in equilirium. The stars in the figure illustrate that there are nonetheless a sustantial numer of true equiliria. Proposition 8: Fix and with > +. When uyer values have the uniform distriution, there is an unique [0, / ] for which there is an such that () and () oth hold with equality at (,,, ). There exists an such that (,,, ) is a quasi-equilirium if and only if [, ]. Moreover, + 5 > 4 and lim ( ) = lim. 4 4 Proof: ee appendix Here is an intuition for the role of the aggregate seller/uyer ratio = γ in the limiting value of. When oth markets are large, the crowding effects are small, so the seller/uyer ratios in each market must e aout the same. ince oth () and () hold with equality, oth uyers and sellers are etter off in the larger market; a computation shows that on a per-uyer asis, the size of this efficiency advantage is.5 γ ( γ / ).To have a quasi-equilirium, this efficiency advantage must e )(/ offset y the market impacts that uyers and sellers have when moving to the larger market. As noted aove, the seller s market impact is / ( + ), and the uyer s market impact is approximately γ /. This is consistent with the size of the efficiency 7
advantage if ( + γ ) /( γ ) ( ) /, so that can e larger when γ is larger. It is, in fact, equivalent to our conclusions that for large, can e aout (3 + γ ) / 4. 5. One-sided Market Impact One feature of the model that some may find unintuitive is that uyers and sellers consider their market impact even when they are very small relative to the total market size. For example on eay, where most uyers are casual consumers and most sellers are small and not-so-small usinesses, it might e more plausile that sellers would consider the market impact effect than that uyers would. 9 Possile reasons for this would e that the market impact of a typical uyer is so small that the uyer might round it off to zero, or that uyers do not think aout things enough or have enough experience to learn aout the effect. In this susection, we note that it is not necessary to assume that oth uyers and sellers recognize that they have a market impact to otain our conclusions. It would suffice for one side to do so. The proposition elow estalishes that there are still quasiequiliria with sustantially different market sizes if we add the restriction that sellers must e exactly indifferent etween the two markets (as one would want to if only uyers recognized the market impact effect.) The minimum possile fraction of agents in the smaller market is increased from aout to y this change. We have 4 4 chosen to add a seller indifference condition rather than a uyer indifference condition only ecause the algera is simpler that way. Which side of the market recognizes that there is a market impact is not important to the qualitative conclusion. Proposition 9: Fix and with > +. For every partition (, ) with [, + ], there is a quasi-equilirium (,,, ) with u (, ) = u (, ). s s 8
Proof: ee appendix. 5C. Integer-valued Equilirium o far we have een ignoring the constraint that the numers of uyers and sellers in each marker should e an integer. With a small numer of traders it may e that only a few ratios of markets sizes are possile. However, one would expect these integer prolems to ecome less important in large markets, and the next result shows that the any ratio of market sizes in the interval given in Proposition 9 can e approximated y an integer-valued equilirium when the numer of traders is sufficiently large. ince given target ratios of to and to can only e approximated y integers, the statement of the result uses α as the target level of Proposition 0: For any target market ratios α, γ with and γ as the target level of /. 0< γ < γ γ α, +, and any ε > 0, there exists such that for all > there is an equilirium (,, ) with + =, / α < ε, and / γ < ε., and Proof: ee appendix. The proof first constructs a quasi-equilirium with equal prices that approximates the target ratios, ut where only and are guaranteed to e integers; we then use this partition to construct an integer-valued partition where all of the incentive constraints are satisfied ut prices are only approximately equal. Proposition 0 proves that when the numer of uyers is large there exist equiliria throughout the range of market sizes for which Proposition 9 shows that quasi-equiliria with equal seller utility exist. 9 A recent New York Times article (Guernsey, 000) reports that one-quarter of one percent of eay s registered users are responsile for over two-thirds of the items listed. These sellers had an average of 9
6. Exponentially Distriuted Values To test the roustness of our results we now consider another tractale distriution, the exponential with f( v) = exp( v) for v 0. As we remarked earlier, the exponential has an unounded support, which might e appropriate for thinking aout rare art ojects. ecause of this unounded support, we would expect that adding more uyers has a greater effect on the size of the highest order statistic, and this is indeed the case. Here the mean of the r th highest of n draws is µ rn : n = i, so the expected price in a market i= r with uyers and sellers is p(, ) = i = i i. The uyer s expected utility is i= + i= i k : : E( µ µ + ) = i i = i = i k= k= i= k i= + k= i= k i= k= The uyer s utility function extends to the positive reals y setting i= =. u (, ) / = if > and u (, ) = if. The seller s utility function can e extended to noninteger values of and y setting us (, ) = 0 for and u (, ) =Ψ( + ) Ψ( + ) for >, where Ψ ( x) is the digamma function. (The s property of the digamma function that makes this a natural extension is that x Ψ ( x) = η + i when x is an integer, where η 0.577 is Euler s constant. 0 ) The i= digamma function has an asymptotic expansion of the form Ψ ( x+ ) = ln( x + C. When and are large this gives the approximation ) k k x k = x seventy items each in the process of eing auctioned at any point in time. 0 The standard definition of the digamma function is Ψ ( x) = Γ ( x)/ Γ( x) where gamma function. 0 x t x t e dt 0 Γ ( ) = is the
us (, ) ln + o(/ ) o(/ ). The u and us functions clearly satisfy (A); the 00 working paper version of this paper (Ellison and Fudenerg [00a]) shows that (A) and (A3) are satisfied when the economy is sufficiently large. Proposition : For any ε > 0 and any γ (0,) there exists a such that for all > the model with uyers, = γ sellers and exponentially distriuted values satisfies γ 3+ γ (a) If ε, ε +, then there exists an for which 4 4 (,,, ) is a quasi-equilirium; and γ 3+ γ () If ε, ε + 4 4, then there is no for which (,,, ) is a quasi-equilirium. Proof: ee appendix. As with the uniform case, we can get intuition for the role of γ u ( γ, ) u ( γ, ) + = γ γ γ + γ = ). Using the in the range of equilirium market sizes y comparing the seller s market impact and efficiency effects. (uyers are indifferent etween the two markets when the uyer-seller ratios are equal.) The price impact of adding a seller is proportional to/( approximation u ( γ, ) ln(/ γ) + (/ γ / ) /, the efficiency advantage of the large market is approximately s s. The coefficient C is the k th ernoulli numer divided y k. The first two values for these coefficients k are C = / and C = /0. As a result, the approximation is quite accurate even for relatively small 4 values of and. For example, the error is approximating u (, ) is less than 0.0 for any if is at least 3. s
When γ is larger, the efficiency advantage is smaller for fixed and, and thus can e increased without violating the constraint that the efficiency effect must e smaller than the market impact. Remarks:. Proposition says that when is large quasi-equilirium requires that the fraction of uyers in the small market e at least aout ( γ ) / 4. The numer of uyers need not e very large for the aout in this statement to e practically unimportant. For example, we ve examined this numerically and found that when γ = 0. conclusions (a) and () of the proposition will e true forε = 0.0 if is at least eleven. For γ = 0.8, they will e true forε = 0.0 if is at least eighteen.. Note that oth for the uniform distriution and for the exponential distriution we have shown that when there are many agents, there is an equilirium consistent with a given share of agents in the smaller market if and only if this share is at least aout ( γ ) / 4. 3. In section 4 we showed that with the exponential distriution there are quasi- equiliria in which the share of uyers in the small market is aout α ( γ) =. ( γ ) Proposition shows that the share of uyers in the smaller market can in fact e γ ( γ ) smaller than this. The difference is due to our considering allocations with (4 γ ) unequal uyer-seller ratios in Proposition. 7. Thin Markets We have seen that a sustantial range of market sizes is possile in oth the uniform and exponential cases. We now test the roustness of that conclusion to the possiility that markets are thin, in the sense of there eing very few items availale for trade. pecifically, we suppose that each seller only has the good with a proaility q<, and investigate the effect of varying q. We suppose that uyers and sellers choose markets efore either the uyer s uncertainty or the seller s is resolved; we think of q as
the proaility that the seller has a good of the appropriate type to sell in the current period. Thus when is the numer of potential sellers who enter a market, the numer of actual units for sale is a random variale that is distriuted inomial (q, ). The exponential model is more tractale when this sort of uncertainty is considered, so this is the only case we analyze here. As noted aove, the expected utility of a uyer in a market with uyers and units for sale is simply. Thus when supply is random, if we let e the random numer of units that are availale for sale, we have that uyer expected utility is q E =. ince sellers only care aout the price in the event that they have a unit of the good to sell, their expected utility conditional on having a good to sell from eing in a market with uyers and - other sellers is i E i i= i=. If q 0, the proaility that any other seller has a unit for sale also goes to 0, and so all sellers prefer to e in the market with more uyers. More precisely, for large there cannot e an equilirium with two active markets and if q >. This is intuitive: when q is very small, each seller expects to e a monopolist, and so prefers to e in the larger market; even if it has more sellers. However, when q 0 there are vanishingly few ojects offered for sale, so this is a fairly extreme version of a thin market. We next we consider a somewhat less extreme version, with exactly three sellers, each of whom has an oject to sell (so we go ack to q =.) Here there can e two active markets even when the numer of uyers is very large. To see this in the exponential case, suppose is a multiple of 3 and let =, =, = /3 and = /3. Then uyers receive exactly the same utility in oth markets and strictly prefer not to switch. The seller s payoff in market is /3 i, i= which is approximately ln ( / 3) + η ; the seller s payoff in market is i 3/, ( ) which is approximately ln / 3 + η 3/. o the sellers in the large market get a /3 i= 3
higher utility y approximately ln / > 0. Despite this difference in utility, the partition is still an equilirium: ecause of the crowding effect, the seller in the small market does not wish to move to the large one. If she did, the numer of sellers there would increase to 3, and she would receive approximately ( ) less than her current utility ecause ln.693 < 5 / 6 ln / 3 + η / 6, which is. A similar exercise shows that =, =, = ( )/5 and = (3+ ) / 5 is an equilirium of the model with uniformly distriuted uyer values whenever is a multiple of five. 8. Related Work There are several sets of papers that develop other sorts of models of multiple markets. Our model astracts away oth the preference for more varieties at a site and the exogenous difference in market locations; these issues have een studied in a numer of papers. For example, in Gehrig [998], firms choose etween locations; all firms at given location are equally spaced on a circle that represents product characteristics, as in alop [979], and in equilirium all firms at a given location charge the same prices. Consumers are located on the line a la Hotelling and pay a transportation cost to visit the markets, which are located at the endpoints. Additionally, as in tahl [98] and Wolinsky [983], consumers prefer larger markets ecause they have a finer grid of availale varieties; in Gehrig s model the externality arises ecause a finer grid of varieties reduces the expected distance etween the closest availale good and the consumer s preferred type. The papers of Caillaud and Julien [998, 00] have a related model of a preference for more varieties: there are two types of agents, whose utility depends only on whether or not they are matched; agents prefer to participate in larger markets ecause their proaility of meeting their unique positive-value match in the other population is proportional to the mass of agents in the other population. These papers study different price-setting games etween two "competing matchmakers; " they suppose there is a continuum of agents of each type, so they do not address our question of the alance etween the market impact and efficiency effects. There is a literature on competing auctions in large markets, for example McAfee [993] and Peters and everinov [997]. In these papers, as in our model, all goods are identical, and all agents are risk-neutral. However, these models assume that 4
each market has a single seller, and they study only equiliria where each market has the same size. There is also a literature on the asymptotic efficiency of doule auctions and other exchange mechanisms, e.g. Gresik and atterthwaite [989], atterthwaite and Williams [989], and Tatur [00]. These papers study trade in settings of two-sided incomplete information, where we know from the Myerson-atterthwaite theorem that no (incentive-compatile) mechanism can e ex-post efficient, and derive ounds on the rate at which the ex-post inefficiency disappears as the economy grows. Our welfare results concern a different sort of inefficiency: trade in a given market is always ex-post efficient in that market, ut as Proposition shows, ex-ante welfare is increasing in market size. Finally, there is a literature on choice etween financial markets, with prices in each market determined y a rational expectations equilirium, as in Pagano [989]; we discuss Pagano s model in some detail in Ellison and Fudenerg [00]. 9. Ideas for Future Work We would like to develop a model that incorporates adverse selection in the market-participation decision. Our casual empiricism suggests that a major reason that the Amazon and Yahoo auction sites have struggled is that they tried to compete y having zero listing fees. This led to their listings eing filled up with products eing offered y non-serious sellers with very high reserve prices. If we suppose that there is a cost to reading we pages, or to investigating the quality of a good and/or its seller, then uyers will prefer to frequent sites with a high percentage of good listings- listings y reputale sellers who have high-quality goods and are willing to sell them at a reasonale price. In this case, a market with too many ad sellers might collapse. However, two markets might e ale to co-exist if sites have some ackground flow of captive traffic from people who click in from Yahoo or Amazon without considering another auction site. In Peters and everinov, each seller runs a separate second-price auction with a reserve price; uyers oserve the reserve prices and then choose a single auction in which to id. The paper looks at symmetric equiliria in which all sellers attract an equal numer of uyers. McAfee analyzes a related model that supposes players ignore the fact that a change in one seller s mechanism may change the distriution of uyers participating in other mechanisms, and thus alter uyers incentives to participate in them. 5
The issue of reserve prices poses a prolem for a would-e new market site: On the one hand, when the market is new, sellers may not expect to get competitive ids, and so e unwilling to participate unless they can protect themselves with a reserve price. However, while the imposition of a uniform reserve price in a market can increase the payoff of sellers for any fixed uyer-seller ratio, it lowers the overall efficiency of the market, and so we would expect it to reduce the viaility of the new market. In some applications, the competing auction sites may e run y entities that can compete in prices. A two-stage game in which sites choose listing fees in the first stage, and uyers and sellers then choose etween sites, has equiliria with two active markets as in our model, ut price levels cannot e pinned down without additional assumptions aout how uyers and sellers coordinate on and off the equilirium path. For example, we can make the sites charge any common listing fee etween zero and the expected seller surplus and have uyers and sellers split etween two sites as in any of the equiliria descried in this paper y assuming that any deviation to a higher or lower listing fee y one site leads uyers and sellers to coordinate on the other site. With other pricing games, however, one might e ale to say more. Caillaud and Jullien [00] show that the aility to charge positive and negative fees on oth sides of the market can reduce equilirium multiplicity in a matching model with a continuum of agents. 3 Finally we would like to note that while the paper has analyzed competition etween two markets, its analysis also applies to the study of M markets, M smaller and M larger. uch a configuration will e an equilirium provided that it is an equilirium for M =. This shows that any tendency to have only two markets, as opposed to more, must e due either to relatively small numers of participants, or to agglomerative forces not captured y our model. One reason why such configurations may e less common in practice is that our model suggests they could e quite fragile if two or more of the markets merge, the merged entity may e sufficiently large relative to the others so as to attract all of the patrons of every small market. 3 Their model has no market-impact effect so it cannot e analyzed with the techniques we developed here and in Ellison and Fudenerg [00]. 6