On the Efficiency of Markets with Two-sided Proportional Allocation Mechanisms

Transcription

1 On the Efficiency of Makets with Two-sided Pootional Allocation Mechanisms Volodymy Kuleshov and Adian Vetta Deatment of Mathematics and Statistics, and School of Comute Science, McGill Univesity Abstact. We analyze the efomance of esouce allocation mechanisms fo makets in which thee is cometition amongst both consumes and sulies (namely, two-sided makets. Secifically, we exae a natual genealization of both Kelly s ootional allocation mechanism fo demand-cometitive makets [9] and Johai and Tsitsiklis ootional allocation mechanism fo suly-cometitive makets [7]. We fist conside the case of a maket fo one divisible esouce. Assug that maginal costs ae convex, we deive a tight bound on the ice of anachy of about This wost case bound is achieved when the demand-side of the maket is highly cometitive and the suly-side consists of a duooly. As moe fims ente the maket, the ice of anachy imoves to In contast, on the demand side, the ice of anachy imoves when the numbe of consumes deceases, eaching a maximum of in a monosony setting. When the maginal cost functions ae concave, the above bound smoothly degades to zeo as the maginal costs tend to constants. Fo monomial cost functions of the fom C(x = cx 1+ d 1, we show that the ice of anachy is Ω( 1. d 2 We comlement these guaantees by identifying a lage class of two-sided single-aamete maket-cleaing mechanisms among which the ootional allocation mechanism uniuely achieves the otimal ice of anachy. We also ove that ou wost case bounds extend to geneal multiesouce makets, and in aticula to bandwidth makets ove abitay netwoks. 1 Intoduction How to oduce and allocate scace esouces is the most fundamental uestion in economics. 1 The standad tool fo guiding oduction and allocation is a icing mechanism. Howeve, diffeent mechanisms will have diffeent efomance attibutes: no two mechanisms ae eual. Of aticula inteest to comute scientists is the fact that thee will tyically be an inheent tade-off between the economic efficiency of a mechanism (measued in tems of social welfae and its comutational efficiency (both time and communication comlexity. Socially otimal allocations can be achieved using icing mechanisms based on classical VCG esults, but imlementing such mechanisms geneally induces excessively high infomational and comutational costs [13]. In this ae, we study this tadeoff fom the oosite viewoint: we exae the level of social welfae that can be achieved by mechanisms efog imal amounts of comutation. In aticula, we estict ou attention to so-called scala-aametized icing mechanisms. Each aticiant submits only a single scala bid that is used to set a uniue maket-cleaing ice fo each good. Evidently, such mechanisms ae comutationally tivial to handle; moe suisingly, they can oduce high welfae. The chief actical motivation fo consideing scala-aametized mechanisms (both in ou wok and in the existing liteatue is the oblem of bandwidth shaing. Namely, how should we allocate caacity amongst uses that want to tansmit data ove a netwok link? The use of maket mechanisms fo this task has been studied in Asynchonous Tansfe Mode (ATM netwoks [16] and the Intenet [15]. The Intenet is made u of smalle inteconnected netwoks that buy caacities fom each othe, The authos wee suoted in at by NSEC gant In fact, economics is often defined as the study of scacity.

2 and the maket mechanisms we conside ae closely insied by the stuctue of the Intenet. Secifically, we ae esticting ou attention to mechanisms that ae scalable to vey lage netwoks. This euiement fo scalability foces us conside only simle mechanisms, such as those that set a uniue maket cleaing ice. The comutational euiements of moe comlex systems, e.g. mechanisms that efom ice disciation, become imactical on lage netwoks [1]. We emak that uniue ice mechanisms ae also intuitively fai, as evey aticiant is teated eually. This fainess is aealing fom a social and olitical esective, and indeed these systems ae used in many eal-wold settings, such as electicity makets [17]. 1.1 Backgound and Pevious Wok A basic method fo esouce allocation is the ootional allocation mechanism of Kelly [9]. In the context of netwoks, it oeates as follows: each otential consume submits a bid b ; bandwidth is then allocated to the consumes in ootion to thei bids. This simle idea has also been studied within economics by Shaley and Shubik [14] as a model fo undestanding icing in maket economies. In a goundbeaking esult, Johai and Tsitsiklis [5] showed that the welfae loss incued by this mechanism is at most 25% of otimal. Obseve that Kelly s is a scala-aamteized mechanism fo a one-sided maket: evey aticiant is a consume. Johai and Tsitsiklis [7] also exaed one-sided makets with suly-side cometition only. Thee, unde a coesonding single-aamete mechanism, the welfae loss tends to zeo as the level of cometition inceases. We emak that we cannot simly analyze suly-side cometition by tying to model sulies as demand-side consumes [3]. Of couse, cometition in makets tyically occus on both sides. Conseuently, undestanding the efficiency of two-sided maket 2 mechanisms is an imotant oblem. In this wok, we analyze the ice of anachy in a mechanism fo a two-sided maket in which consumes and oduces comete simultaneously to detee the oduction and allocation of goods. This mechanism was fist oosed by Neumaye [10] and is the natual genealization of both the demand-side model of Kelly [9] and the suly-side model of Johai and Tsitsiklis [7]. In ode to exae how the genealized ootional allocation mechanism efoms in a twosided maket, it is imotant to note that thee ae thee imay causes of welfae loss. Fist, the undelying allocation oblem may be comutationally had. In othe wods, in some settings (such as combinatoial auctions, fo examle, it may be had to comute the otimal allocation even when the layes utilities ae known. Secondly, even if the allocation oblem is comutationally simle, the mechanism itself may still be insufficiently sohisticated to solve it. Thidly, the mechanism may be suscetible to gag; namely, the mechanism may incentivize selfish agents to behave in a manne that oduces a oo oveall outcome. As we will see in Section 4, the fist two causes do not aise hee: as long as the uses do not behave stategically, the ootional allocation mechanism can uickly find otimal allocations in two-sided makets. Thus, we ae concened only with the thid facto: how advesely is the ootional allocation mechanism affected by gag agents? That is, the mechanism may be caable of oducing an otimal solution, but how will the agents selfish behaviou affect social welfae at the esultant euilibia? In this ae we ove that the ootional allocation mechanism does efom well in two-sided makets. Secifically, unde uite geneal assumtions, the mechanism admits a constant facto ice of anachy guaantee. Moeove, thee exists a lage family of mechanisms among which the ootional allocation mechanism uniuely achieves the best ossible ice of anachy guaantee. We state ou exact esults in Section 3, afte we have descibed the model and ou assumtions. 2 It should be noted that two-sided maket often has a diffeent meaning in the economics liteatue than the one we use hee. Thee it efes to a secific class of makets whee extenalities occu between gous on the two sides of the maket.

3 2 The Model 2.1 The two-sided ootional allocation mechanism We now fomally esent the two-sided ootional allocation mechanism due to Neumaye [10]. Thee ae Q consumes and sulies in the maket. Each consume has a valuation function V (d, whee d is the amount of the esouce allocated to consume, and each sulie has a cost function C (s, whee s is the amount oduced by sulie. Consumes and sulies esectively inut bids b and b to the mechanism. Doing so, consumes ae imlicitly selecting b - aametized demand functions of the fom D(b, = b, and sulies ae selecting b -aametized suly functions of the fom S(b, = 1 b. We can also inteet a high consume bid as an indicato of high willingness to ay fo the oduct, and a low sulie bid as an indicato of a high willingness to suly (altenatively, a high bid indicates a high cost sulie. The actual choice of constant used fo the suly functions does not affect ou esults, and so we choose it to be 1. Obseve that the aametized demand functions ae identical to the ones in the demand-side mechanism of Kelly [9], and the suly functions ae identical to the ones in the suly-side mechanism of Johai and Tsitsiklis [7]. The eculia fom of the suly functions comes fom the inteesting fact that fo most scala-aametized mechanisms, in ode to have a non-zeo welfae atio, the suly functions have to be bounded fom above. In othe wods, sulies stategies must necessaily be constained in ode to obtain high welfae; see the full vesion of the ae fo the ecise statement of this fact. This ules out, fo instance, Counot-style mechanisms whee sulies diectly submit the uantities they wish to oduce. Moe detailed justifications fo this choice of model can be found in [10], as well as in [9] and [7]. Futhe justification fo the mechanism will be ovided by ou esults. Secifically, the ootional allocation mechanism geneally oduces high welfae allocations and, in addition, it is the otimal mechanism amongst a class of single-aamete mechanisms fo two-sided makets. Given the bids, the mechanism sets a ice (b that cleas the maket; i.e. that satisfies the suly euals demand euation: Q b =1 = P =1 (1 b. The ice theefoe gets set to (b = b+p b. Consume then eceives d units of the esouce, and ays d, while sulie oduces s units and eceives a ayment of s. In the game induced by this mechanism, the ayoff (o utility to consume lacing a bid b is defined to be ( b V Π (b = Q b + b b if b > 0 V (0 if b = 0 and the ayoff to sulie lacing a bid b is defined as Q b + b b b C (1 Π (b = Q b + b if b > 0 b + b C (1 if b = The Welfae atio Given a vecto of bids b, the social welfae at the esulting mechanism allocation is defined to be W(b = Q V (d (b C (s (b =1 If the agents do not stategically anticiate the effects of thei actions on the ice, that is if they act as ice-takes, we show in Section 4 that the mechanism maximizes social welfae. Howeve, since =1

4 the ice is a function of thei bid, each agent is a ice-make. If agents attemt to exloit this maket owe, then a welfae loss may occu at a Nash euilibium. Conseuently we ae inteested in maximizing (ove all euilibia the welfae atio, moe commonly known as the ice of anachy, W NE. Euivalently, we wish to imize the welfae loss, 1 WNE. W W 2.3 Assumtions We make the following assumtion on the valuation and cost functions. Assumtion 1 Fo each consume, the valuation function V (d : + + is stictly inceasing and concave. Fo each sulie, the cost function C (s : + + is stictly inceasing and convex. Assumtion 1 coesonds to deceasing maginal valuations and inceasing maginal costs. The assumtion is standad in the liteatue. It cetainly may not hold in evey maket 3, but without it thee will be a natual incentive fo the numbe of agents to decline on both sides of the maket. In this ae, we will also assume that ou functions ae diffeentiable ove thei entie domain; this oety is assumed imaily fo claity and is not essential. Assumtion 1, howeve, is not sufficient to ensue a lage welfae atio. In fact, the welfae atio deends uon the cuvatue of the maginal cost functions. Secifically, if the maginal cost functions ae convex, then we show in Section 4 that the welfae atio is at least Concave maginal cost functions also exhibit constant welfae atios, ovided the coesonding total cost function is sufficiently non-linea. Howeve, in the limit as the total cost functions become linea, the welfae atio degades to zeo (see Section 5 fo moe details. Ou main esult thus concens convex maginal cost functions. Fomally, fo most of the ae, we assume that Assumtion 2 Fo each sulie, the maginal cost function C (s is convex. Futhemoe, we assume that C (0 = C (0 = 0. Convex maginal cost functions ae extemely common in both the theoetical and the actical liteatue on industial theoy [18], so this assumtion is not aticulaly estictive. In Assumtion 2 we also set C (0 = 0, but as we show in the full vesion of the ae, constant welfae atios still aise wheneve C (0 is bounded below one (it cannot be highe than one o the fim is uncometitive. We also emak that Assumtion 2 was used in Johai, Manno and Tsitsiklis [4] in thei analysis of the demand-side ootional allocation mechanism with elastic suly. Most of the esults of Johai and Tsitsiklis [6] and Tobias and Haks [2] on demand-side Counot cometition with elastic suly also hold unde the assumtion of convex maginal costs. 3 Ou esults Ou fist esults ae concened with the efomance of the mechanism when the uses act as icetakes. Unde Assumtion 1, we ove that: Theoem 1. A uniue cometitive euilibium exists fo the two-sided ootional allocation mechanism. The social welfae attained at the cometitive euilibium is otimal. This oety was exhibited by Kelly s oiginal ootional allocation mechanism, and has been a featue of all subseuent genealizations by Johai and Tsitsiklis. It is vey aealing fom a actical oint of view, as in actual netwoks, uses ae likely to have little infomation about each othe, making it difficult to maniulate the system. In many othe settings howeve, uses will be incentivized to act stategically. In that case, we need to use the stonge solution concet of a Nash euilibium to analyze the esulting game. Ou second esult establishes the existence and uniueness of such euilibia unde Assumtion 1. 3 Fo examle, in makets exhibiting economies of scale.

5 Theoem 2. The two-sided ootional allocation mechanism has a uniue Nash euilibium fo 2. Ou main esult measues the loss of welfae at that uniue Nash euilibium unde Assumtion 2. Theoem 3. The wost case welfae atio fo the mechanism involving 2 sulies euals s 2 (( ( 1s + 2s 2 ( 1( 1 + 2s whee s is the uniue ositive oot of the uatic olynomial γ(s = 16s 4 +( 1s 2 (49s 24+10( 1 2 s(3s 2 + ( 1 3 (5s 4. Futhemoe, this bound is tight. It follows that the mechanism admits a constant bound on the ice of anachy. Moeove, Theoem 3 allows us to measue the effects of maket cometition on social welfae. The following two coollaies ae concened with that elationshi. Coollay 1. The wost ossible ice of anachy is achieved when the suly side is a duooly ( = 2. It evaluates numeically to about Coollay 2. When the suly side is fully cometitive ( the ice of anachy euals ecisely Conseuently, as suly-side cometition inceases, the welfae atio imoves. In contast, the welfae atio deceases as demand-side cometition inceases. Although this fact may seem suising at fist, it tuns out to have a simle intuitive exlanation. The otimal demand-side allocation consists in giving the entie oduction to the use which deives fom it the highest utility. When moe consumes ae esent in the maket, they selfishly euest moe of the esouce fo themselves, leaving less fo the most needy use and educing the oveall social welfae. The best welfae atios thus aise when thee is only one consume (Q = 1, that is, in the case of a monosony. In the two-sided ootional allocation mechanism, the best ossible ice of anachy ove all ossible values of Q and is given by the next coollay. Coollay 3. In a maket in which a monosonist faces a fully cometitive suly side, the ice of anachy euals 3 1, which is about ecall that in the one-sided ootional allocation mechanism fo sulies facing a fixed demand, the welfae loss tends to zeo when the suly side is fully cometitive [7]. In contast, Coollay 3 imlies that in two-sided makets, that esult no longe holds and that full efficiency cannot be achieved. So fa, ou esults assumed the convexity of maginal costs. Doing that assumtion, we find that the welfae atio euals zeo when the ovides total cost functions ae linea. Howeve, the ice of anachy emains bounded fo a class of concave maginal cost functions, and degades smoothy to zeo as the total costs become linea. Coollay 4. The welfae atio fo cost functions C (s = c s 1+ 1 d whee c > 0 and d 1 is Ω( 1 d 2. Like its one-sided vesions, the two-sided mechanism can be genealized to multi-esouce makets. An imotant multi-esouce setting is that of bandwidth shaed on a netwok of links. The same guaantees as in the single-esouce setting hold fo the netwok vesion of ou maket, as well as fo moe geneal multi-esouce makets (see the full vesion of the ae fo moe details. Theoem 4. The welfae atio in netwoks euals that of the single-esouce model. Theoem 5. The welfae guaantees hold fo moe geneal multi-esouce makets. Finally, we show that the ootional allocation mechanism is otimal in the following way:

6 Theoem 6. In two-sided makets, the ootional allocation mechanism ovides the best welfae atio amongst a class of single-aamete maket-cleaing mechanisms. Ou oof techniues ae insied by the aoaches and techniues develoed to analyze singlesided makets by Johai [3], Johai and Tsitsiklis ([5], [8] and [7], Johai, Manno and Tsitsiklis [4], Tobias and Haks [2], and oughgaden [12]. Due to sace limitations, most of ou esults will be defeed to the full vesion of the ae. Hee, we will focus uon the oof of Theoem 3. 4 Otimization in Eight Stes The oof of the main esult, Theoem 3, is esented below in eight stes. We fomulate the efficiency loss oblem as an otimization ogam in Ste III. To be able to fomulate this we fist need to undestand the stuctue of otimal solutions and of euilibia unde this mechanism. This we do in Stes I and II, whee we give necessay and sufficient conditions fo otimal solutions and fo euilibium. This leads us to an otimization oblem that initially aeas slightly fomidable, so we then attemt to simlify it. In Stes IV and V, we show how to simlify the demand constaints in the ogam, and in Stes VI and VII, we simlify the suly constaints. This oduces an otimization ogam in a fom moe amenable to uantitive analysis; we efom this analysis in Ste VIII. Ste I: Otimality Conditions. The best ossible allocation is the solution to the system: ( max s.t Q =1 Q =1 V (d d = =1 0 s 1 d 0 =1 s C (s Since the constaints ae linea, thee exists an otimal solution at which the Kaush-Kuhn-Tucke (KKT conditions hold. As the objective function is concave, the following fist ode conditions ae both necessay and sufficient: C ( s λ if 0 < s 1 ( s λ if 0 s < 1 C V V ( d λ if d = 0 ( d = λ if d > 0 We have used λ to denote the dual vaiable coesonding to the euality constaint. Ste II: Euilibia Conditions. Hee we descibe necessay and sufficient conditions fo a set of bids b to fom Nash euilibium. Fist, obseve that thee must be at least two sulies, that is 2. If not, then we have a monoolist k whose ayoff is is stictly inceasing in b k. Secifically, Π k (b k, b k = b k b C k (1 b k + b = b C k ( b b k + b

7 Next, we show that if b is a Nash euilibium, then at least two bids must be ositive. Suose fo a contadiction that we have a sulie k and k b = b = 0. Then Π k (0 = C k (1, and Π k (b k = 1 b k when b k > 0. Fo the second exession, we used the fact that C k (x = 0 fo any x 0. Obseve that if b k = 0 then the fim can ofitably deviate by inceasing b k infinitesimally; on the othe hand, if b k > 0 then the fim should infinitesimally decease b k. Thus, thee is no euilibium in which eithe all bids ae zeo, o a single sulie is the only agent to make a ositive bid. Thus thee must be at least two ositive bids at euilibium. Since at least two bids ae ositive, the ayoffs Π k ae diffeentiable and concave, and the following conditions ae necessay and sufficient fo the existence of a Nash euilibium. Fo the sulies, C (s (1 + sne if 0 < b C 1 (s (1 + sne 1 ( Fo the consumes, V (0 and V ( 1 d = if > 0. if 0 b < Ste III: An otimization oblem. We can now fomulate the welfae atio as an otimization oblem. Q =1 V ( =1 C (s NE Q =1 V (d =1 C (s ( (1 s.t. V ( 1 dne s.t. > 0 (2 ( V ( 1 dne C (s NE (1 + sne 1 C (s NE (1 + sne 1 Q =1 = =1 (3 s.t. 0 < s NE 1 (4 s.t. 0 s NE < 1 (5 s NE (6 C (s λ s.t. 0 < s 1 (7 C (s λ s.t. 0 s < 1 (8 V (d λ s.t. d = 0 (9 V (d = λ s.t. d > 0 (10 Q =1 d = =1 s (11 d, 0 (12 0 s NE, s 1, (13, λ 0 (14 Given the cost and valuation functions, the constaints (2-(6 ae necessay and sufficient conditions fo a Nash euilibium by Ste II, and constaints (7-(11 ae the otimality conditions fom Ste I. We now want to find the wost-case cost and valuation functions fo the mechanism.

8 Ste IV: Linea Valuation Functions. To evaluate this intimidating looking ogam we attemt to simlify it. Fist, efficiency loss is wost when each consume has a linea valuation function. This is simle to show using a standad tick (see, fo examle, [5]. Thus, we estict ouselves to linea functions of the fom V (d = α d. Without loss of geneality, we may assume that α 1 α 2... α Q and that max α = 1 afte we nomalize the functions by 1/ max α. Obseve that this imlies that d 1 = s and d = 0 fo > 1. As a esult the objective function ( becomes 1 + Q =2 α ( =1 C (s NE / =1 s =1 C (s, and the otimality constaints become C (s 1, s.t. 0 < s 1 and C (s 1, s.t. 0 s < 1. With linea valuations, the new otimality constaints ensue s is otimal by setting the maginal cost of each sulie to the maginal valuation, α 1 = 1, of the fist consume. Ste V: Eliating the Demand Constaints. In this ste, we descibe how to eliate the demand constaints fom the ogam. Fist we show that we can tansfom constaint (14 into 0 < 1. Since α 1,, we see that constaint (2 imlies that 1. Futhemoe, if = 1, then (2 can neve be satisfied, and so we must have = 0,. The suly euals demand constaint (6 then gives s NE = 0,. This gives a contadiction as the esulting allocation is not a Nash euilibium: any sulie can incease its ofits by oviding a bid slightly smalle than (emembe that C (0 = 0 by Assumtion 2. Thus < 1. This, in tun, imlies that 1 > 0. To see this, note that if 1 = 0 then (3 cannot be satisfied fo = 1. Conseuently, constaints (2 and (3 must hold with euality fo = 1. In fact, without loss of geneality, constaints (2 and (3 hold with euality fo > 1. If constaint (2 does not hold with euality, we can educe α, and this does not incease the value of the objective function. If the objective function will be unaffected. So, α = function: s.t. = 0 and constaint (3 does not hold with euality, we can set α = and fo all. Substituting into the objective 1 + Q =2 =1 s (1 dne 1 C (s NE 1 1 / / =1 C (s NE =1 C (s (15 = (16 (1 + sne 1 C (s NE (1 + sne 1 Q =1 = =1 s.t. 0 < s NE 1 (17 s.t. 0 s NE < 1 (18 s NE (19 C (s 1 s.t. 0 < s 1 (20 C (s 1 s.t. 0 s < 1 ( (22 1 > 0 (23 0 s NE, s 1 (24 0 < 1 (25 Now, obseve that the objective function is convex and symmetic in the vaiables d 2,..., d Q, when all the othe vaiables ae held fixed. Convexity holds because ou function is a sum of functions 1 /, = 2,..., Q, that ae convex on the ange [0, ]; note that dne by (6, (12 and (13.

9 Theefoe, fo any given fixed assignment to the othe vaiables, we must have d 2 =... = d Q := x. Othewise, we could eshuffle the vaiable labels and obtain a second imum, which is imossible by the convexity of the objective function. So, afte elacing evey d by x, constaint (19 becomes ( =1 sne / (Q 1. Afte inseting constaint (16 and the new constaint (19, the x = 1 numeato of the objective function (15 becomes x (1 + (Q 1 1 x/ C (s NE = (1 + (Q = (1 + =1 ( =1 sne =1 sne ( 1 =1 sne ( =1 sne 1 1 (1 1 / (Q 1 / (Q 1 / (Q 1 =1 =1 C (s NE C (s NE Finally, obseve that if we incease Q by one, the objective function (1 cannot incease, since we can set d Q+1 = 0 and at least kee the same objective function value as befoe. Theefoe, without loss of geneality, we can take the limit as Q. Note that this only changes the objective function, as all the constaints that contained Q have been inseted into the function and can be eliated. Afte these changes, the otimization oblem becomes s.t. (1 2 + C (s NE =1 s (1 + sne =1 sne C (s NE (1 + sne 1 =1 C (s NE =1 C (s 1 (26 s.t. 0 < s NE 1 (27 s.t. 0 s NE < 1 (28 C (s 1 s.t. 0 < s 1 (29 C (s 1 s.t. 0 s < 1 (30 0 s NE, s 1 (31 0 < 1 (32 Hence, we have achieved ou goal and comletely eliated the demand side of the otimization oblem. Secifically, all the demand constaints have been elaced with an exession that is a function of the suly-side allocation. Now we must find the wost such allocation. Ste VI: Linea Maginal Cost Functions The next ste is to show that, in seaching fo a wost case allocation, we can estict ou attention to linea maginal cost functions of the fom C (s = β s whee β > 0. In this section, we biefly sketch the oof of this fact and defe the full teatment to the full vesion of the ae. Ou oof techniue is based on the wok of Johai, Manno and Tsitsiklis on demand-side makets with elastic suly [4], [6]. The oof consists in exhibiting, fo any family of cost functions C (s,, two new families Ĉ ( and C ( with the oety that the C have a bette efomance atio than the C which, in tun, have a a bette efomance atio than the Ĉ. Futhemoe, the Ĉ will be a family with linea maginal costs, as desied. The cost functions ae defined as

10 C (s C (s = C (s NE s NE s if s < s NE if s s NE and Ĉ (s = C (s NE s NE s whee s NE is the Nash euilibium allocation to sulie when the cost functions ae C (s. Obseve that the s NE still satisfy the Nash euilibium conditions (27 and (28 fo both C and Ĉ. Thus s NE = ŝ NE = s NE. The heat of the oof consists in showing that the otimal welfae can only imove when going fom one family to the next. Ste VII: Eliating the Suly Constaints. Assug linea maginal cost functions, the otimization oblem (26-(32 becomes s.t. (1 2 + =1 sne 1 2 =1 s 1 2 β s NE (1 + sne 1 β s NE (1 + sne 1 =1 β (s NE 2 =1 β (33 (s 2 s.t. 0 < s NE 1 (34 s.t. 0 s NE < 1 (35 β s 1 s.t. 0 < s 1 (36 β s 1 s.t. 0 s < 1 (37 0 s NE, s 1 (38 β > 0 (39 0 < 1 (40 with the new vaiables β, = 1,...,. Fom s s NE, we can then deduce that (34 and (35 hold with euality. Suose they don t fo some. Then s NE = s = 1. Constaint (34 is β < 1+1/( 1 < < 1. Hence, β = 1+1/( 1 will be a feasible solution (i.e. constaint (36 will still be satisfied. Futhemoe, inceasing β to 1+1/( 1 will only decease the objective function since this is euivalent to subtacting a ositive numbe fom the numeato and the denoato. We can futhe simlify the system by elacing constaints (36 and (37 with s = (1/β, 1. It is easy to see that s and β satisfy the euation above if and only if they satisfy (36 and (37. The educed otimization oblem now becomes: s.t. (1 2 + =1 sne 1 2 =1 s 1 2 β s NE (1 + sne 1 =1 β (s NE 2 =1 β (41 (s 2 = (42 s = (1/β, 1 (43 0 < s NE, s 1 (44 β > 0 (45 0 < 1 (46

11 We can inset the euality constaints (42 and (43 into the objective function (41 to obtain: s.t. (1 2 + =1 sne 2 =1 =1 (1/β, 1 2 =1 s NE 1+s NE (1/β,1 2 s NE (1+sNE /( 1 /( 1 (47 0 < s NE 1 (48 β = s NE (1 + s NE (49 /( 1 0 < 1 (50 The objective function (47 can be ewitten as: =1 ((1 2 + s NE ( =1 (1/β, 1 2 s NE 2 1+s NE /( 1 (1/β,1 2 s NE (1+sNE /( 1 Conseuently, the imum of the otimization oblem (47-(50 is geate than o eual to (1 2 + s 2 s 1+s/( 1 (51 ( s(1+s/( 1 s(1+s/( 1, 1 2s(1+s/( 1 (, 1 2 s.t. 0 < s 1 (52 0 < 1 (53 We have now educed the system (33-(40 to a two-dimensional imization oblem. The next ste is to ty to exlicitly find the imum. Ste VIII: Comuting the Wost Case Welfae atio. To obtain Theoem 3 we need to solve the otimization oblem (51-(53 with as a aamete. We show how to do this in the full vesion of the ae. Thus we have oved ou main esult. It has seveal amifications. Fistly, the wost case welfae atio occus with duoolies, that is when = 2. Thee we obtain s = which gives a wost case welfae atio of Moeove, obseve that this bound is tight. Ou oof is essentially constuctive; costs and valuations can be defined to to ceate an instance that oduces the bound. Secondly, the welfae atio imoves as the numbe of sulies inceases. Secifically as, the bound tends to Thus we obtain Coollaies 1 and 2. So, as suly-side cometition inceases, the welfae atio does imoves. The oosite occus as demand-side cometition inceases. Secifically, adating ou aoach gives Coollay 3. 5 Concave Maginal Cost Functions. The welfae atio tends to zeo if the cost function is linea, that is if the maginal cost function is a constant; fo an examle see the full vesion of the ae. We can get some idea of how the welfae atio tends to zeo fo concave maginal cost functions by consideing a class of olynomial cost functions with degee d. These functions give a welfae atio of Ω( 1 d 2, fo any constant d. A oof of this (Coollay 4 is given in the full vesion of the ae. See Neumaye [10] fo anothe examle of inefficiency in the esence of linea cost functions. 6 Extensions to Netwoks and Abitay Makets. We can genealize ou esults fo bandwidth makets ove a single netwok connection to the case whee bandwidth is shaed ove an entie netwok. In that model, each consume is associated with

12 a souce-sink ai, and ovides at associated with edges of the netwok at which they can offe bandwidth. A consume s ayoff is a function of the maximum (s, t -flow it can obtain using the bandwidth it has uchased in the netwok. The welfae guaantees fo the netwok model ae the same as fo the single-link case. A fomal descition of the netwok model and a oof of Theoem 4 is given in the full vesion of the ae. Moeove, if we identify links e E with abitay esouces, then ou esults extend to a geneal class of makets with any numbe of esouces. The exact definition of these makets and a oof of Theoem 5 ae also given in the full vesion of the ae. 7 Smooth Maket-Cleaing Mechanisms It was shown in [3] and [8] that in one-sided makets, the ootional allocation mechanism uniuely achieves the best ossible welfae atio within a boad class of so-called smooth maket-cleaing mechanisms. This family has a natual extension to the case of two-sided mechanisms, and we show that, given a symmety condition, the two-sided ootional allocation mechanism is otimal amongst that class of single-aamete mechanisms. A descition of smooth maket-cleaing mechanisms and a oof of Theoem 6 is given in the full vesion of the ae. efeences 1. Guta, A., Stahl, D.O., Whinston, A.B.: The economics of netwok management. Commun. ACM 42(9, ( Haks, T., Mille, K.: Efficiency and stability of nash euilibia in esouce allocation games. In: GameNets 09: Poceedings of the Fist ICST intenational confeence on Game Theoy fo Netwoks IEEE Pess, Piscataway, NJ, USA ( Johai,.: Efficiency loss in maket mechanisms fo esouce allocation. Ph.D. thesis, MIT ( Johai,., Manno, S., Tsitsiklis, J.N.: Efficiency loss in a netwok esouce allocation game: The case of elastic suly. Mathematics of Oeations eseach 29, ( Johai,., Tsitsiklis, J.N.: Efficiency Loss in a Netwok esouce Allocation Game. Mathematics of Oeations eseach 29(3, ( Johai,., Tsitsiklis, J.N.: A scalable netwok esouce allocation mechanism with bounded efficiency loss. IEEE Jounal on Selected Aeas in Communications 24(5, ( Johai,., Tsitsiklis, J.N.: Paameteized suly function bidding: Euilibium and welfae ( Johai,., Tsitsiklis, J.N.: Efficiency of scala-aameteized mechanisms. Oe. es. 57(4, ( Kelly, F.: Chaging and ate contol fo elastic taffic. Euoean Tansactions on Telecommunications 8, ( Neumaye, S.: Efficiency Loss in a Class of Two-Sided Mechanisms. Maste s thesis, MIT ( osen, J.B.: Existence and uniueness of euilibium oints fo concave n-eson games. Econometica 33(3, ( oughgaden, T.: Potential functions and the inefficiency of euilibia. Poceedings of the Intenational Congess of Mathematicians (ICM 3, ( Semet, N.: Maket mechanisms fo netwok esouce shaing. Ph.D. thesis, Columbia Univesity, New Yok, NY, USA ( Shaley, L.S., Shubik, M.: Tade using one commodity as a means of ayment. Jounal of Political Economy 85(5, (Octobe Shenke, S., Clak, D., Estin, D., Hezog, S.: Picing in comute netwoks: eshaing the eseach agenda. SIGCOMM Comut. Commun. ev. 26(2, ( Songhust, D.J.: Chaging communication netwoks: fom theoy to actice. Elsevie Science Inc., New Yok, NY, USA ( Stoft, S.: Powe system economics : designing makets fo electicity. IEEE Pess, Piscataway, NJ ( Tiole, J.: The Theoy of Industial Oganization, vol. 1. The MIT Pess, 1 edn. (1988