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

Transcription

1 On the Efficiency of Maket with Two-Sided Pootional Allocation Mechanim Volodymy Kulehov and Adian Vetta May 10, 2010 Abtact. We analyze the efomance of ingle-aamete mechanim fo maket in which thee i cometition amongt both conume and ulie namely, two-ided maket. Secifically, we examine the ootional allocation mechanim fo two-ided maket. Thi mechanim i the natual genealization of both Kelly ootional allocation mechanim fo demand-cometitive maket [8] and Johai and Titikli ootional allocation mechanim fo uly-cometitive maket [7]. Fit we conide the cae of a maket fo one diviible eouce, fo examle bandwidth on a ingle-link netwok. Unde the tandad aumtion of deceaing maginal valuation fo buye and inceaing maginal cot fo ulie, we how that a Nah equilibium alway exit fo the ootional allocation mechanim, ovided thee ae at leat two ulie. The effectivene of the mechanim in achieving high ocial welfae i deendent uon the cuvatue of the ulie maginal cot function. We meaue thi effectivene in tem of the welfae atio ice of anachy, the wot cae atio of an equilibium againt an otimal ocial olution. In the cae of convex maginal cot, the welfae atio i about Thi atio i tight, thee ae examle achieving the bound the odd looking bound itelf aie fom the olution to a quatic equation. The level of cometition in the maket alo affect the welfae efomance of the mechanim. Fo examle, the wot cae bound aie when the demand-ide i highly cometitive and the uly-ide i a duooly, but imove to 0.64 when the uly-ide i fully cometitive. In ha contat, the welfae atio inceae a demand-ide cometition deceae, eaching a maximum of in the cae of a monoony. Fo concave maginal cot function the welfae atio deend uon the degee of non-lineaity in the coeonding cot function. Thi i illutated by conideation of cot function of the fom Cx = cx 1+ 1 d which oduce a welfae atio of Ω 1 d, fo any 2 fixed d. We comlement thee eult by howing that the ootional allocation mechanim uniquely achieve the otimal welfae atio amongt a cla of two-ided ingle-aamete maket-cleaing mechanim. Futhemoe, ou wot cae bound extend to the cae of multile eouce. In aticula, ou eult extend to the cae of bandwidth maket ove abitay netwok, a well a to geneal multi-item maket. 1. Intoduction How to oduce and allocate cace eouce i the mot fundamental quetion in economic. 1 The tandad tool fo guiding oduction and allocation i a icing mechanim. Howeve, diffeent mechanim will have diffeent efomance attibute: no two mechanim ae equal. Of aticula inteet to comute cientit i the fact that thee will tyically be an inheent tade-off between the economic efficiency of a mechanim meaued in tem of ocial welfae and it comutational efficiency both time and communication comlexity. Socially otimal Deatment of Mathematic and Statitic, and School of Comute Science, McGill Univeity. Suoted in at by NSEC gant volodymy.kulehov@mcgill.ca Deatment of Mathematic and Statitic, and School of Comute Science, McGill Univeity. Suoted in at by NSEC gant vetta@math.mcgill.ca 1 In fact, economic i often defined a the tudy of cacity. 1

2 2 allocation can be achieved uing icing mechanim baed on claical VCG eult, but imlementing uch mechanim geneally induce exceively high infomational and comutational cot [15]. In thi ae, we tudy thi tadeoff fom the ooite viewoint: we examine the level of ocial welfae that can be achieved by mechanim efoming minimal amount of comutation. In aticula, we etict ou attention to o-called cala-aametized icing mechanim. Each aticiant ubmit only a ingle cala bid that i ued to et a unique maket-cleaing ice fo each good. Evidently, uch mechanim ae comutationally tivial to handle; moe uiingly, they can oduce high welfae. The chief actical motivation fo conideing cala-aametized mechanim both in ou wok and in the exiting liteatue i the oblem of bandwidth haing. Namely, how hould we allocate caacity amongt ue that want to tanmit data ove a netwok link? The ue of maket mechanim fo thi tak ha been tudied in Aynchonou Tanfe Mode ATM netwok [18] and the Intenet [17]. In fact, the Intenet i made u of malle inteconnected netwok that buy caacitie fom each othe. Phyical connection ae fomed in cental cleaing houe, called intenet exchange, which act a maketlace fo bandwidth. Indeed, the maket mechanim we conide ae cloely inied by the tuctue of the Intenet. Secifically, we would like ou olution to be calable to vey lage netwok. Thi equiement fo calability foce u conide only imle mechanim, uch a thoe that et a unique maket cleaing ice. The comutational equiement of moe comlex ytem, e.g. mechanim that efom ice dicimination, become imactical on lage netwok [1]. We emak that unique ice mechanim ae alo intuitively fai, a evey aticiant i teated equally. Thi faine i aealing fom a ocial and olitical eective, and indeed thee ytem ae ued in many eal-wold etting, uch a electicity maket [19] Backgound and Peviou Wok. A baic method fo eouce allocation i the ootional allocation mechanim of Kelly [8]. In the context of netwok, it oeate a follow: each otential conume ubmit a bid b q ; bandwidth i then allocated to the conume in ootion to thei bid. Thi imle idea ha alo been tudied within economic by Shaley and Shubik [16] a a model fo undetanding icing in maket economie. In a goundbeaking eult, Johai and Titikli [5] howed that the welfae lo incued by thi mechanim i at mot 25% of otimal. Obeve that Kelly i a cala-aamteized mechanim fo a one-ided maket: evey aticiant i a conume. Johai and Titikli [7] alo examined one-ided maket with uly-ide cometition only. Thee, unde a coeonding ingle-aamete mechanim, the welfae lo tend to zeo a the level of cometition inceae. We emak that we cannot imly analye uly-ide cometition by tying to model ulie a demand-ide conume [3]. Of coue, cometition in maket tyically occu on both ide. Conequently, undetanding the efficiency of two-ided maket 2 mechanim i imotant oblem. In thi wok, we analyze the ice of anachy in a mechanim fo a two-ided maket in which conume and oduce comete imultaneouly to detemine the oduction and allocation of good. Thi mechanim wa fit ooed by Neumaye [11] and i the natual genealization of both the demand-ide model of Kelly [8] and the uly-ide model of Johai and Titikli [7]. In ode to examine how the genealized ootional allocation mechanim efom in a twoide maket, it i imotant to note that thee ae thee imay caue of welfae lo. Fit, the undelying allocation oblem may be comutationally had. Secondly, even if the allocation oblem i comutationally imle, the mechanim itelf may till be inufficiently ohiticated to olve it. Thidly, the mechanim may be ucetible to gaming; namely, the mechanim may incentivize elfih agent to behave in a manne that oduce a oo oveall outcome. A we will 2 It hould be noted that two-ided maket often ha a diffeent meaning in the economic liteatue than the one we ue hee. Thee it efe to a ecific cla of maket whee extenalitie occu between gou on the two ide of the maket.

3 ee in Section 3, the fit two caue do not aie hee: the ootional allocation mechanim can quickly find otimal allocation in two-ided maket. Thu, we ae concened only with the thid facto: how adveely i the ootional allocation mechanim affected by gaming agent? That i, the mechanim may be caable of oducing an otimal olution, but how will the agent elfih behaviou affect ocial welfae at the eultant equilibia? In thi ae we ove the ootional allocation mechanim doe efom well in two-ided maket. Secifically, in quite geneal etting, the mechanim ovide a contant facto welfae atio that i, a contant facto ice of anachy guaantee. Moeove, we will how that thee i lage family of mechanim among which ou uniquely achieve the bet oible ice of anachy guaantee. We tate ou exact eult, in Section 2.4, afte we have decibed the model and ou aumtion. 2. The Model 2.1. The two-ided ootional allocation mechanim. Hee, we fomally eent the two-ided ootional allocation mechanim due to Neumaye [11]. Thee ae Q conume and ulie. Each conume q ha a valuation function V q d q, whee d q eeent the ue allocation, and each ulie ha a cot function C, whee i the ulie oduction quota. The conume and ulie eectively inut bid b q and b to the mechanim. By doing o, conume ae imlicitly electing a b q -aametized demand function of the fom Db q, = b q, and ulie ae electing a b -aametized uly function of the fom Sb, = 1 b. A high bid by a conume indicate a high willingne to ay fo the oduct, and a low bid by a ulie indicate a high willingne to uly altenatively, a high bid indicate a high cot ulie. The actual choice of contant ued fo the uly function doe not affect ou eult, and o we chooe it to be 1. Obeve that the aametized demand function ae identical to the one in the demand-ide mechanim of Kelly [8], and the uly function ae identical to the one in the uly-ide mechanim of Johai and Titikli [7]. The eculia fom of the uly function come fom the inteeting fact that fo mot cala-aametized mechanim, in ode to have a non-zeo welfae atio, the uly function have to be bounded fom above. In othe wod, ulie tategie mut neceaily be contained in ode to obtain high welfae; ee Aendix J fo the ecie tatement of thi fact. Thi ule out, fo intance, Counot-tyle mechanim whee ulie diectly ubmit the quantitie they wih to oduce. Moe detailed jutification fo thi choice of model can be found in [11], a well a in [8] and [7]. Futhe jutification fo the mechanim will be ovided by ou eult. Secifically, the ootional allocation mechanim geneally oduce high welfae olution and, in addition, it i the otimal mechanim amongt a cla of ingle-aamete mechanim fo two-ided maket. Given the bid, the mechanim et a ice b that clea the maket; i.e. that atifie Q b q = P =1 1 b. The ice theefoe get et to b = q bq+p b. Conume q then eceive d q unit of the eouce, and ay d q, while ulie oduce unit and eceive a ayment of. In the game induced by thi mechanim, the ayoff o utility to conume q lacing a bid b q i then defined to be { Vq P b q Π q b q = q Q bq+p b b q if b q > 0 V q 0 if b q = 0 and the ayoff to ulie lacing a bid b i defined a P q Q bq+p b b C 1 Π b = b P q Q bq+p b if b > 0 P q bq+p b C 1 if b = 0 3

4 The Welfae atio. The ocial welfae oduced by the mechanim on bid b i defined a Wb = V q d q b C b If the agent act a ice-take, we how in Section 3 that the mechanim maximize ocial welfae. Howeve, ince the ice i a function of thei bid, each agent i a ice-make. If agent attemt to exloit thi maket owe then a welfae lo may occu at a Nah equilibium. Conequently we ae inteeted in maximizing ove all equilibia the welfae atio moe commonly known a the ice of anachy, W NE W OP T. Equivalently, we wih to minimize the welfae lo, 1 WNE W OP T Aumtion. We make the following aumtion on the valuation and cot function. Aumtion 1. Fo each conume q, the valuation function V q d q : + + i tictly inceaing and concave. Fo each ulie, the cot function C : + + i tictly inceaing and convex. Aumtion 1 coeond to deceaing maginal valuation and inceaing maginal cot. The aumtion i tandad 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 agent to decline on both ide of the maket. In thi ae, we will alo aume that ou function ae diffeentiable ove thei entie domain; thi oety i aumed imaily fo claity and i not eential. Aumtion 1, howeve, i not ufficient to enue a lage welfae atio mall welfae lo. In fact, the welfae atio deend uon the cuvatue of the maginal cot function. Secifically, if the maginal cot function ae convex, then we how, in Section 3, that the welfae atio i at leat Fo concave maginal cot function, we obtain contant welfae atio, ovided the cot function i ufficiently non-linea; ee Section 4. A the maginal cot function tend to contant, the welfae atio tend to zeo; an examle of thi i given in Aendix E ee Neumaye [11] fo anothe examle. Ou main eult concen convex maginal cot function. Fomally, fo mot of the ae, we aume that Aumtion 2. Fo each ulie, the maginal cot function C i convex. Futhemoe, we aume that C 0 = C 0 = 0. Convex maginal cot function fo examle, a often occu with quadatic o highe degee olynomial function ae extemely common in both the theoetical and the actical liteatue on indutial theoy [20], o thi aumtion i not aticulaly etictive. A noted, though, we may alo obtain good guaantee fo concave maginal cot function. In Aumtion 2 we alo et C 0 = 0 but, a we how in Aendix D, contant welfae atio till aie wheneve C 0 i bounded below one it cannot be highe than one o the fim i uncometitive. Finally, we emak that Aumtion 2 wa alo ued in Johai, Manno and Titikli [4] Ou eult. In thi ae we ove that: Theoem 1. A cometitive equilibium exit fo the two-ided ootional allocation mechanim and it otimize ocial welfae. Theoem 2. The mechanim ha a unique Nah equilibium fo 2. Given Aumtion 2, ou main eult i: 3 Fo examle, in maket exhibiting economie of cale. =1

5 Theoem 3. The wot cae welfae atio fo the mechanim involving 2 ulie equal whee i the unique oitive oot of the quatic olynomial γ = Futhemoe, thi bound i tight. Thi how u that the mechanim give a contant facto welfae atio. Moeove, it allow u to examine how the level of cometition in the maket affect welfae. Secifically, we find Coollay 4. The wot cae welfae atio occu fo a duooly = 2 and i Coollay 5. If the uly ide i fully cometitive the welfae atio i Conequently, a uly-ide cometition inceae, the welfae atio imove. Suiingly, the welfae atio deceae a demand-ide cometition inceae. The bet welfae atio aie when thee i only one conume Q = 1, that i, in the cae of a monoony. Fo examle, we obtain Theoem 6. Fo a monoonit facing a fully cometitive uly-ide the welfae atio i ecall that fo a fixed demand welfae lo tend to zeo when the uly ide i fully cometitive [7]. In contat, no uch eult hold in two-ided maket. Thi follow a in two-idedmaket the ituation mot cloely coeonding to a fixed demand i that of a monoony, but Theoem 6 tell u that thee can then be welfae loe. emoving Aumtion 2, the welfae atio i zeo fo linea cot function. The ate at which the welfae atio tend to zeo fo concave maginal cot function i illutated by a cla of olynomial cot function with degee d that give a contant welfae atio fo any fixed d. Theoem 7. The welfae atio fo cot function C = cx 1+ 1 d i Ω 1 d 2. The two-ided maket fo a ingle eouce can be genealized to the cae of multile eouce. An imotant multi-eouce etting i that of bandwidth haed on a netwok of link. A netwok veion of ou maket can be defined, a can moe geneal multi-eouce maket ee Aendix G and ou welfae guaantee till aly. Theoem 8. The welfae atio in netwok equal that of the ingle-eouce model. Theoem 9. The welfae guaantee hold fo moe geneal multi-eouce maket. Finally, we how that the ootional allocation mechanim i otimal in the following way: Theoem 10. In two-ided maket, the ootional allocation mechanim ovide the bet welfae atio amongt a cla of ingle-aamete maket-cleaing mechanim. Ou oof technique ae inied by the aoache and technique develoed to analyze ingle-ided maket by Johai [3], Johai and Titikli [5], [6] and [7], Johai, Manno and Titikli [4], and oughgaden [14]. Due to ace limitation, mot of ou eult will be defeed to the aendice. Hee, in the main text, we will focu uon the oof of Theoem Otimization in Eight Ste The oof of the main eult, Theoem 3, i eented below in eight te. We fomulate the efficiency lo oblem a an otimization ogam in Ste III. To be able to fomulate thi we fit need to undetand the tuctue of otimal olution and of equilibia unde thi mechanim. Thi we do in Ste I and II, whee we give neceay and ufficient condition fo 5

6 6 otimal olution and fo equilibium. Moeove, thee i alway a unique equilibium ovided the uly maket i not a monooly. Thi lead u to an otimization oblem that initially aea lightly fomidable, o we then attemt to imlify it. In Ste IV and V, we how how to imlify eliminate the demand containt in the ogam. Then in Ste VI and VII, we imlify eliminate the uly containt in the ogam. Thi oduce an otimization ogam in a fom moe amenable to quantitive analyi; we efom thi analyi in Ste VIII. Ste I: Otimality Condition. The bet oible allocation i the olution to the ytem: OPT max V q d OP q T C OP T.t d OP T q = =1 =1 OP T 0 OP T 1 d OP q T 0 Since the containt ae linea, thee exit an otimal olution at which the Kauh-Kuhn- Tucke KKT condition hold. The olution i otimal becaue the objective function i concave, the fit ode condition ae thu ufficient. Thee condition imly that C OP T λ if 0 < OP T 1 C OP T λ if 0 OP T < 1 V q V q d OP T q λ if d OP T q = 0 d OP T q = λ if d OP T q > 0 whee λ i the dual vaiable coeonding to the equality containt. We emak that a cometitive equilibia, whee the agent act a ice-take with eect to the mechanim, exit and maximize ocial welfae. A oof of thi that i, of Theoem 1 i given in Aendix C. Of coue, agent ae clealy ice-make unde thi mechanim, o cometitive equilibia ae not ou focu. Conequently, ou goal now i to undetand equilibia given thi fact. Ste II: Equilibia Condition. Hee we decibe neceay and ufficient condition fo a et of bid b to fom Nah equilibium. Fit, obeve that thee mut be at leat two ulie, that i 2. If not, then we have a monoolit k whoe ayoff i i tictly inceaing in b k. Secifically, Π k b k, b k = b k b q C k 1 b q k + q b = q b q C k b q q b q k + q b q Next, we how that if b i a Nah equilibium, then at leat two bid mut be oitive. Suoe fo a contadiction that we have a ulie k and k b = q b q = 0. Then Π k 0 = C k 1, and Π k b k = b k when b k > 0. Fo the econd exeion, we ued the fact that C k x = 0 fo any x 0. Obeve that if b k = 0 then the fim can ofitably deviate by inceaing b k infiniteimally; on the othe hand, if b k > 0 then the fim hould infiniteimally deceae b k. Thu, thee i no equilibium in which eithe all bid ae zeo, o a ingle ulie i the only agent to make a oitive bid. Thu thee mut be at leat two oitive bid at equilibium. Since at leat two bid ae oitive, the ayoff Π k ae diffeentiable and concave, and the following condition ae neceay and ufficient fo the exitence of a Nah equilibium. Fo the ulie,

7 C 1 + NE 1 Fo the conume, V q 0 and V q d NE if 0 < b C 1 + NE 1 q 1 dq = if d NE q > 0. if 0 b < Moeove, if thee ae at leat two ulie then thee i a unique Nah equilibium. A oof of thi that i, of Theoem 2 i given in Aendix D. Ste III: An otimization oblem. We can now fomulate the welfae atio a an otimization oblem min P Q VqdNE q P =1 CNE P Q VqdOP q T P =1 COP T.t. V q d NE q V q d NE q C NE C NE Q dne q 1 dne q 1 dne q 1 + NE 1 + NE = =1 NE q.t. d NE q > 0 q.t. 0 < NE 1.t. 0 NE < 1 C OP T λ.t. 0 < OP T 1 C OP T λ.t. 0 OP T < 1 V q d OP q T λ q.t. d OP q T = 0 V q d OP T Q dop q T q = λ q.t. d OP q T > 0 = =1 OP T d OP q T, d NE q 0 q 0 NE, OP T 1 q,, λ 0 Given the cot and valuation function, the containt 2-6 ae neceay and ufficient condition fo a Nah equilibium by Ste II, and containt 7-11 ae the otimality condition fom Ste I. We now want to find the wot-cae cot and valuation function fo the mechanim. Ste IV: Linea Valuation Function. To evaluate thi intimidating looking ogam we attemt to imlify it. Fit, efficiency lo i wot when each conume ha a linea valuation function. Thi i imle to how uing a tandad tick ee, fo examle, [5]. Thu, we etict ouelve to linea function of the fom V q d q = α q d q. Without lo of geneality, we may aume that α 1 α 2... α Q and that max q α q = 1 afte we nomalize the function by 1/ max q α q. Obeve that thi imlie that d OP 1 T = OP T and d OP q T = 0 fo q > 1. A a eult the objective function become d NE 1 + Q q=2 α qd NE q =1 C NE / =1 OP T =1 C OP T, and the otimality containt become C OP T 1,.t. 0 < OP T 1 and C OP T 1,.t. 0 OP T < 1. With linea valuation, the new otimality containt enue OP T i otimal by etting the maginal cot of each ulie to the maginal valuation, α 1 = 1, of the fit conume. Ste V: Eliminating the Demand Containt. In thi te, we decibe how to eliminate the demand containt fom the ogam. Fit we how that we can tanfom containt 14 7

8 8 into 0 < 1. Since α q 1, q, we ee that containt 2 imlie that 1. Futhemoe, if = 1, then 2 can neve be atified, and o we mut have d NE q = 0, q. The uly equal demand containt 6 then give NE = 0,. Thi give a contadiction a the eulting allocation i not a Nah equilibium: any ulie can inceae it ofit by oviding a bid lightly malle than emembe that C 0 = 0 by Aumtion 2. Thu < 1. Thi, in tun, imlie that d NE 1 > 0. To ee thi, note that if d NE 1 = 0 then 3 cannot be atified fo q = 1. Conequently, containt 2 and 3 mut hold with equality fo q = 1. In fact, without lo of geneality, containt 2 and 3 hold with equality fo q > 1. If containt 2 doe not hold with equality, we can educe α q, and thi doe not inceae the value of the objective function. If d NE q = 0 and containt 3 doe not hold with equality, we can et α q = and the objective function will be unaffected. So, α q = fo all q. Subtituting into the objective function: 1 d NE q / min.t. dne 1 + P Q q=2 d NE q 1 d NE / P =1 CNE q P =1 OP T P =1 COP T 1 dne 1 = C NE 1 + NE C NE 1 + NE Q dne q = =1 NE.t. 0 < NE 1.t. 0 NE < 1 C OP T 1.t. 0 < OP T 1 C OP T 1.t. 0 OP T < 1 d NE q 0 q 2 d NE 1 > 0 0 NE, OP T 1 0 < 1 Now, obeve that the objective function i convex and ymmetic in the vaiable d 2,..., d Q, when all the othe vaiable ae held fixed. Convexity hold becaue ou function i a um of function d NE q 1 d NE q, q = 2,..., Q, that ae convex on the ange [0, ]; note that dne / q by 6, 12 and 13. Theefoe, fo any given fixed aignment to the othe vaiable, we mut have d 2 =... = d Q := x,, d 1. Othewie, we could ehuffle the vaiable label and obtain a econd minimum, which i imoible by the convexity of the objective function. So, afte elacing evey d q by x,, d 1, containt 19 become x,, d 1 = =1 NE d NE 1 / Q 1. Afte ineting containt 16 and the new containt 19, the numeato of the objective function 15 become x 1 + Q 1 1 x/ C NE = 1 + Q 1 = 1 + =1 =1 NE 1 1 =1 NE =1 NE 1 =1 NE d NE 1 1 d NE 1 d NE 1 / Q 1 / Q 1 / Q 1 =1 =1 C NE C NE

9 whee we have elaced the function x,, d 1 by an odinay vaiable x fo imlicity. Finally, obeve that if we inceae Q by one, the objective function 1 cannot inceae, ince we can et d Q+1 = 0 and at leat kee the ame objective function value a befoe. Theefoe, without lo of geneality, we can take the limit a Q. 4 Note that thi only change the objective function, a all the containt that contained Q have been ineted into the function and can be eliminated. Afte thee change, the otimization oblem become min P =1 CNE P =1 NE P =1 OP T P =1 COP T. t. C NE C NE 1 + NE 1 + NE.t. 0 < NE 1.t. 0 NE < 1 C OP T 1.t. 0 < OP T 1 C OP T 1.t. 0 OP T < 1 0 NE, OP T 1 0 < 1 Hence, we have achieved ou goal and comletely eliminated the demand ide of the otimization oblem. Secifically, all the demand containt have been elaced with an exeion that i a function of the uly-ide allocation. Now we mut find the wot uch allocation. Ste VI: Linea Maginal Cot Function We will now how that, in eaching fo a wot cae allocation, we can etict ou attention to linea maginal cot function of the fom C = β whee β > 0. We do thi by exhibiting, fo any family of cot function C,, two new familie 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 cot, a deied. The cot function ae defined a { C if < NE C = C NE NE if NE and Ĉ = C NE NE whee NE i the Nah equilibium allocation to ulie when the cot function ae C. Obeve that the NE till atify the Nah equilibium condition 27 and 28 fo both the new cot function C and Ĉ. Thu NE = ŝ NE = NE ; conequently, the efficiency at the Nah equilibium i identical in all thee cae. We will now how that the efomance atio fo the C i bette than fo the C. In aticula, the otimal olution can only get bette in the fome cae. To ee thi, obeve that by the convexity of C, we have C C fo NE. Thu OP T C OP T OP T C OP T OP T C OP T =1 =1 =1 =1 whee OP T i the otimal allocation to ulie when the cot function ae C. So ou goal now i to how that uing Ĉ i not bette than uing C. Befoe doing that, we ll need to etablih ome eult. Fit, obeve that we mut have NE > 0 fo all. If NE = 0 fo ome, then by 28, = 0, which can only haen when all bid equal 0. A hown in Ste II, thi i imoible. Similaly, we have OP T > 0 fo all, becaue at OP T = 0, containt 30 i violated. Note that in both cae, we have ued the fact that C 0 = 0. 4 So inceaed cometition on the demand ide cannot imove the welfae atio guaanteed by the mechanim! =1 =1 9

10 10 Next, obeve that fo all we have OP T NE. If OP T = 1, the claim hold tivially, o aume OP T < 1. Then we have C OP T = 1 > C 1+ NE / NE. Hee the equality hold by 29 and 30, the fit inequality follow fom 31 and 32, and the econd one follow fom 27. Since the maginal cot ae tictly inceaing, ou claim hold. By convexity, Ĉ C fo 0 NE. Thu := ĈŝNE C ŝ NE 0; note that i exactly the aea on the gah between Ĉ and C on the inteval [0, NE ]. A NE = NE = ŝ NE, we have =1 OP T =1 NE =1 C NE =1 C OP T =1 ŝne =1 OP T = =1 ŝop T =1 ŝne C =1 ŝ NE C =1 OP T =1 ĈŝNE =1 Ĉŝ OP T To ee the equality, ecall that OP T > NE. Then, becaue C = Ĉ in the ange NE, it follow that OP T = ŝ OP T. Fo ŝ NE, we alo have Ĉ = C +. Thu, the efomance atio of the Ĉ i at mot that of the C. Ste VII: Eliminating the Suly Containt. Auming linea maginal cot function, the otimization oblem become min.t P P =1 NE 1/2 =1 βne 2 P =1 OP T P 1/2 =1 βop T 2 β NE β NE 1 + NE 1 + NE.t. 0 < NE 1.t. 0 NE < 1 β OP T 1.t. 0 < OP T 1 β OP T 1.t. 0 OP T < 1 0 NE, OP T 1 β > 0 0 < 1 with the new vaiable β, = 1,...,. Fom OP T NE, we can then deduce that 34 and 35 hold with equality. Suoe they don t fo ome. Then NE = OP T = 1. Containt 34 i β < 1+1/ < < 1. Hence, β = 1+1/ will be a feaible olution i.e. containt 36 will till be atified. Futhemoe, inceaing β to 1+1/ will only deceae the objective function ince thi i equivalent to ubtacting a oitive numbe fom the numeato and the denominato. We can futhe imlify the ytem by elacing containt 36 and 37 with OP T = min1/β, 1. It i eay to ee that OP T and β atify the equation above if and only if they atify 36 and 37. The educed otimization oblem now become: min P P =1 NE 1/2 =1 βne 2 P =1 OP T P 1/2 =1 βop T 2. t. β NE OP T 1 + NE = = min1/β, 1 0 < NE, OP T 1 β > 0 0 < 1

11 We can inet the equality containt 42 and 43 into the objective function 41 to obtain: min.t P =1 NE P =1 min1/β,1 P /2 =1 /2 P =1 NE NE 1+ NE / min1/β, NE / 0 < NE 1 β = NE 1+ NE / 0 < 1 The objective function 47 can be ewitten a: = NE =1 min1/β, 1 /2 NE /2 1+ NE / min1/β,1 2 NE 1+ NE / Conequently, the minimum of the otimization oblem i geate than o equal to min /2 1+/ min 1+/ 1+/, 1 21+/ min, 1 2.t. 0 < 1 0 < 1 We have now educed the ytem to a two-dimenional minimization oblem. The next te i to ty to exlicitly find the minimum. Ste VIII: Comuting the Wot Cae Welfae atio. To obtain Theoem 3 we need to olve the otimization oblem with a a aamete. We how how to do thi in Aendix C. Thu we have oved ou main eult. It ha eveal amification. Fitly, the wot cae welfae atio occu with duoolie, that i when = 2. Thee we obtain = which give a wot cae welfae atio of Moeove, obeve that thi bound i tight. Ou oof i eentially contuctive; cot and valuation can be defined to to ceate an intance that oduce the bound. Secondly, the welfae atio imove a the numbe of ulie inceae. Secifically a, the bound tend to Thu we obtain Coollaie 4 and 5. So, a uly-ide cometition inceae, the welfae atio doe imove. The ooite occu a demand-ide cometition inceae. Secifically, adating ou aoach give Theoem Concave Maginal Cot Function. The welfae atio tend to zeo if the cot function i linea, that i if the maginal cot function i a contant; fo an examle ee Aendix E. We can get ome idea of how the welfae atio tend to zeo fo concave maginal cot function by conideing a cla of olynomial cot function with degee d. Thee function give a welfae atio of at leat 1 d 2, fo any contant d. A oof of thi Theoem 7 i given in Aendix F. 5. Extenion to Netwok and Abitay Maket. So we can analyze two-ided maket fo bandwidth on a ingle-link netwok. We can genealize thee eult to the cae whee bandwidth i haed ove an entie netwok of link. Hee agent can buy and ell bandwidth on numeou link. Each conume q i aociated with a ouce-ink ai and it ayoff i a function of the maximum q, t q -flow it can obtain uing the bandwidth it ha uchaed in the netwok. The welfae guaantee fo the netwok model ae the ame a fo the ingle-link cae. A fomal decition of the netwok model and a oof of Theoem 8 11

12 12 i given in Aendix G. Moeove, if we identify link e E with abitay eouce, then ou eult aly to a geneal cla of maket with any numbe of eouce. The exact definition of thee maket and a oof of Theoem 9 ae alo given in Aendix G. 6. Smooth Maket-Cleaing Mechanim It wa hown in [3] and [6] that in ingle-ided maket, the ootional allocation mechanim uniquely achieve the bet oible welfae atio within a boad cla of o-called mooth maketcleaing mechanim. Thi family ha a natual extenion to the cae of two-ided mechanim, and we how that, given a ymmety condition, the two-ided ootional allocation mechanim i otimal amongt that cla of ingle-aamete mechanim. A decition of mooth maketcleaing mechanim and a oof of Theoem 10 i given in Aendix H. efeence [1] A. Guta, D. Stahl and A. Whinton, The economic of netwok management, Communication of the ACM, 429, 57-63, [2] T. Hak and K. Mille, Efficiency and tability of Nah equilibia in eouce allocation game, Poceeding of the Intenational Confeence Of Game Theoy in Netwok GameNet, , [3]. Johai, Efficiency Lo in Maket Mechanim fo eouce Allocation, Ph.D. Thei, Maachuett Intitute of Technology, [4]. Johai, S. Manno and J. Titikli, Efficiency lo in netwok eouce allocation game: the cae of elatic uly, IEEE Tanaction on Automatic Contol, 5011, , [5]. Johai and J. Titikli, Efficiency lo in netwok eouce allocation game, Math. Oeat. e., 293, , [6]. Johai and J. Titikli, Efficiency of cala-aameteized mechanim, Oeation eeach, 574, , [7]. Johai and J. Titikli, Paameteized uly function bidding: equilibium and welfae, eint, [8] F. Kelly, Chaging and ate contol fo elatic taffic, Euo. Tan. Telecommun., 8, 33-37, [9] P. Klemee and M. Meye, Suly function equilibia in oligooly unde uncetainty, Econometica, 576, , [10] H. Moulin, The ice of anachy of eial, aveage and incemental cot haing, Economic Theoy, 363, , [11] S. Neumaye, Efficiency Lo in a Cla of Two-Sided Mechanim, M.Sc. Thei, Maachuett Intitute of Technology, [12] N. Nian, T. oughgaden, E. Tado and V. Vaziani ed, Algoithmic Game Theoy, Cambidge, [13] J. oen, Exitence and uniquene of equilibium oint fo concave N-eon game, Econometica, 33, 520Ð534, [14] T. oughgaden, Potential function and the inefficiency of equilibia, Poc. Intl. Conge of Mathematician, Vol. III, , [15] N. Semet, Maket Mechanim fo Netwok eouce Shaing, PhD thei, Columbia Univeity, [16] L. Shaely and M. Shubik, Tade uing one commodity a a mean of ayment, Jounal of Political Economy, 855, , [17] S. Shenke, D. Clak, D. Etin and S. Hezog, Picing in comute netwok: ehaing the eeach agenda, Telecommunication Policy, 203, , [18] D. J. Songhut, edito, Chaging Communication Netwok: Fom Theoy to Pactice, Elevie Science, Amtedam, The Netheland, 1999 [19] S. Stoft, Powe Sytem Economic: Deigning Maket fo Electicity, IEEE Pe, Picataway, New Jeey, [20] J. Tiole, The Theoy of Indutial Oganization, MIT Pe, 1988.

13 7. Aendice 7.1. Aendix A: Cometitive Equilibia and Otimal Welfae. Hee we ove Theoem 1; that i, we how that a cometitive equilibium exit and achieve maximum welfae. A ai b, whee b 0 and > 0 i a called a cometitive equilibium if: Π k b k, Π k b k, fo all b k 0, fo evey k = 1,..., + Q q b = b q + b Thu, in a cometitive equilibium, the agent ae maximizing thei ayoff unde the aumtion that they ae ice-take. We will how that the neceay and ufficient condition unde which the demand and uly allocation d, induced by b, fom a cometitive equilibium ae identical to the condition unde which d, i an otimal olution to OPT. We have aleady found the otimality condition in Section 3. So let calculate the equied condition fo a cometitive equilibium. Fo a ulie, the ayoff a a function of b i: Π b = 1 b C 1 b Since thi i diffeentiable and concave fo all b 0 ecall that we aume that the ight deivative exit at b = 0, the maximum occu when the following condition ae atified: C 1 b if 0 < b C 1 b if 0 b < To ee thi, note that when 0 < b, we have Π b 0. If Π b < 0 then we can inceae Π by infiniteimally deceaing b and on 0, ] we can alway chooe a malle b. The fit condition then follow. The econd condition i deived analogouly. Alo obeve that we can let b, ince at b > 0 ulie will eceive a negative ayoff. Similaly, we can wite the ayoff of conume q a a function of b q : Π q b q = V q b q b q Thi i again concave by aumtion, and the condition ae V q bq = if b q > 0 V q 0 if b q = 0 Next, define b = 1 λ and b q = d q λ. Then it can be checked that the equilibium condition above ae atified with = λ. We mut have λ 0 becaue V q d q 0 by aumtion, and λ V q d q fo all q. Thu thee exit a cometitive equilibium and it i an otimal olution to OPT. Finally, uoe b, i a cometitive equilibium. Then define = 1 b and d q = bq. Then we can check that the KKT condition ae atified, and the allocation i an otimal olution to OPT. Thi comlete the oof of Theoem 1. 13

14 Aendix B: The Exitence of Nah Equilibia. To ove Theoem 2 we how that the Nah equilibium condition obtained in Section 3 ae exactly the neceay and ufficiency condition fo the following ytem. NASH max ˆV q d q Ĉ whee ˆV q = Ĉ =.t. =1 d q = 0 1 d q 0 =1 1 d q V d q + 1 ˆ dq V q zdz C 1 1 ˆ C zdz 1 By aumtion, the ˆV q and Ĉ ae diffeentiable and tictly concave and tictly convex, eectively. The KKT condition fo NASH then give: 1 + C λ if 0 < C λ if 0 < d q V q d q λ 1 d q V q d q λ if 0 < d q whee λ i the dual vaiable coeonding to the equality containt. Thee ae the neceay and ufficient condition fo an otimal olution. Since the objective function of NASH i tictly convex, the otimal olution i unique. We now claim that an otimal olution to NASH i a Nah equilibium fo the two-ide ootional allocation mechanim. ecall that fo thee ice-anticiato, a Nah equilibium i a vecto b 0 uch that fo evey agent k = 1,..., Q +, Π k b k, b k Π k b k, b k, fo all b k 0 A befoe, define b q = d q λ and b = 1 λ. Then one can check that the Nah equilibium condition ae atified with = λ. We mut have λ 0 becaue V q d q 0 by aumtion and λ V q d q fo all q. Thee ae at leat two oitive bid becaue othewie the uly equal demand containt of cannot be atified. Conveely, we claim that any Nah equilibium i an otimal olution to NASH. To ee thi take any Nah equilibium b,. Then define = 1 b and d q = bq. Then we can eaily check that the KKT condition ae atified, and the allocation i an otimal olution to NASH. Thi comlete the oof of Theoem 2. 0

15 7.3. Aendix C: Comuting the Wot Cae Welfae atio. So we have to olve the otimization oblem We will conide a a aamete. At the minimum of 51, we will have eithe / o 55 When 54 hold, we olve the following ytem: min 1+/ 1+/ /2. t. 1 + / 1 0 < 1 0 < / 1 Now et W = 1 and let g be the objective function 60. Thu 1 2 g = 2W W + + W + W 2W + 2 Diffeentiating we obtain g = 2 W 2 W W + 2W and thi i tictly negative fo any W,. Theefoe, the otimal value of mut occu at one of the boundaie of the feaible egion. Since g i tictly deceaing in, we mut have 1 + /W =. Thu it i enough to conide a vaiant of the econd cae 55 in which the inequality i no longe tict. Obeve that the oint at which the minimum of i achieved i alo at of the feaible egion of the following new otimization oblem: min /W 21+/W. t. 1 + /W 0 < 1 0 < 1 Let f denote the objective function 60. We have f = W W W 2W + 2 and at = 1 thi become f 1, = W + 1W 2 2W 2 2 which i alway oitive ince each tem in the numeato i negative. But by the KKT condition, f/ mut be negative at = 1 if that oint i the minimum, and theefoe we mut ule out that oibility. Similaly, when containt 61 i tight, f/ become f = 2 2 W W 42 W 23 W 2 + W 2 1 W + It can be checked that thi i alway negative. But, a befoe, if containt 61 i tight at the minimum, f/ mut be oitive, and we again have to ule out that oibility. Finally, we > 1 15

16 16 cannot have 0 at the minimum, ince the objective function would go to one in that cae. Thu, at the minimum, we have f = S S S 2S + 2 = 0 Solving thi equation fo, we find that ± = S ± S γ 2S + 2 whee γ = S S S When we lug eithe + o into the objective 60, we find quite uiingly that the objective function educe to the ame imle exeion in both cae. The otimization oblem thu become: min 2 S 2 + 4S SS + 2. t. 0 < 1 0 ± < /S ± ± It i not had to how that the objective function 64 i tictly inceaing in. The minimum value i thu achieved at the mallet feaible. Fo condition 68 to be atified, we mut have γ 0. One can check that γ ha only one zeo on [0, 1], with γ 0 fo. Futhemoe, at, ± = S S + 2 > 0 and containt 66 hold. Finally, one can check uing a comute that 67 hold fo all. Since i the mallet feaible oint in ou domain, we conclude that the minimum i achieved at that oint. Thi finally ove the main claim of Theoem 3.

17 7.4. Aendix D: Poitive Maginal Cot at Zeo. Hee we conide the cae whee C 0 > 0 and how that the welfae atio emain a contant ovided C 0 i bounded below one. Secifically, aume C = 1 2 β 2 +γ whee 0 γ 1. Then we can follow the oiginal oof and find that the welfae atio i the olution to the following oblem: minimize / γ γ min 1 γ β, γ 2β min β, 12 γ min 1 γ β, 1.t γ 1 17 whee β = 1 + / 1 γ 1 A befoe, we have two cae to conide. Fit uoe 1 γ β. minimize.t / γ γ / γ 1 γ 1 1 γ 1 + / 1 γ γ 1 We claim that fo any fixed γ < 1, the olution to thi oblem i tictly oitive. Fit, we can do the / γ 1 tem fom the denominato, a thi can only deceae the value of the objective function. We will alo let in the numeato. The objective function then become minimize.t γ 2 1 γ 1 γ γ / 1 γ 1 0 If thee i a ai, that make the objective function go to zeo, it mut atify γ 2 = = 21 2 γ Obeve that ince 0, we mut have γ >. But by the fit containt, we have γ, a contadiction. So the oint at which the objective function i outide ou domain, and ince that et i cloed, we alo cannot get abitaily cloe to that oint.

18 18 Suoe now 1 γ β < 1. Then the otimization oblem i minimize.t / γ γ 1 γ < β γ β 0 γ 1 and the denominato can be ewitten a 1 γ2 1 γ γ1 γ γ2 β γ 1 γ β 1+/ γ If the objective function goe to zeo, then eithe the denominato goe to infinity, which can only haen when β 0, o when the numeato goe to zeo. The fit cae cannot occu becaue of the 1 γ < β containt. The econd cae can be teated like above: if the numeato i zeo, we mut have 21 2 = γ which imlie γ >, but fom the 0 < 1 γ < β containt we get that > γ.

19 7.5. Aendix E: Contant Maginal Cot Function and Total Welfae Lo. Hee we how that if the maginal cot ae contant then the welfae atio tend to zeo. Conide linea valuation and cot function with loe α q and β, eectively, V q d q = α q d q C = β The bid coeonding to thi allocation at a Nah equilibium can then be defined accoding to the equilibium condition develoed in Section 3. We will aametize the allocation and the function by a vaiable 0 < x < 1 and then let x 0. So, let = x µ = 1 x Q 19 β = 1 x Q 1 + x d q = x Q q Obeve that α q = 1 q d q = x Q = x = and that 0 < < 1 and d q > 0. Thi how the above allocation i a feaible olution to NASH. Next we will how that the KKT condition hold at, d with λ = µ. Thi will imly that the allocation, d coeond to a Nah equilibium. We have V q d q 1 d q = 1 1 d q =1 = 1 x Q = Fo the fim, we have C = β 1 + = = 1 x Q 1 + x x 1 Thu the KKT condition ae atified, and o thee allocation fom a Nah equilibium. The exeion fo the welfae atio can be witten a Q V qd NE q =1 C NE Q V qd OP q T =1 C OP T = Q α qd NE q =1 β NE =1 β

20 20 Thi equality hold becaue evey ulie that ha NE > 0 ha a loe β < 1. To ee thi, obeve that by the KKT condition, wheneve d q > 0, V q d NE µ µ q = > µ > = C 1 dq 1 + NE and ecall that µ < 1. Alo, if β 1, that ulie will not oduce anything neithe at the Nah equilibium no at the otimal olution. So, without lo of geneality, we can emove that ulie away and, thu, conide only the cae whee β < 1. Since fo any ulie the maginal cot of oducing one unit i alway le than one, in an otimal olution eveyone oduce = 1 and give it to the conume with loe α 1 = 1. Now utting ou definition in the fomula, we obtain: Q α qd NE q =1 β Q NE 1 x Q =1 β = =1 1 x Q x 1+ x x = The eult follow by imly letting x 0. = x 1 x Q 1+ x =1 1 x Q 1 x Q 1+ x 1+ x x

21 7.6. Aendix F: Concave Maginal Cot Function. Suoe C = 1 1+1/d 1+1/d fo ome fixed d. The educed otimization oblem i: minimize min1/β, 1.t whee β = 1/d 1 + / 1 Fit uoe that 1/β > 1. The otimization oblem i: minimize t 1/d /d 1+/ β 1+1/d min1/β, 11+1/d 1+1/d 1+/ 1+1/d 1 1/d 1+/ We can lowe bound the objective function by etting the denominato to one. We can alo let in the objective function. Then the objective function become /d and the only way thi can go to zeo i if 0, 1. But it not had to ee that fo any d, thi i imoible by the fit containt. Suoe now that 1/β < 1. The otimization oblem i: minimize.t < β /d 1+/ 1/β 1 1+1/d 1/β1/d Thi goe to zeo eithe when the denominato goe to infinity, which can only haen when β 0, o when the numeato goe to zeo. The fit cae cannot occu becaue of the 1 < β containt. The econd cae can be teated a above. A welfae atio of Ω 1 d 2 can then eaily be hown. 21

22 Aendix G: Extenion to Netwok and Abitay Maket. The cae of conume and ulie imultaneouly cometing ove a ingle link can be natually extended to the cae of a netwok of link. Thi netwok model i baed uon the wok of [3]. We will model the netwok a a gah G = V, E with L edge, called link. Thee ae Q ue, each chaacteized by a ouce-taget ai of node q, t q V, q = 1,..., Q, a well a ovide, each caable of oviding evice ove a ubet L of link l 1,..., l L, whee = 1,...,. Each conume wihe to end flow fom it ouce to it taget. Fo thi eaon, we define P to be the et of ath in the gah G, and we ay that a ath P belong to q Q denoted q when connect q, t q. We aume without lo of geneality that each ath belong to only one q. If ome ath k i haed by q 1 and q 2, then we imly define two ath q1, q2 in P that contain the ame link. Futhemoe, we let f q denote the flow that conume q end ove the ath, and let f q = q f q = q f denote the total flow ent by q. In the lat equality, f i the total flow on ath, and we have f = f q ince ath only belong to conume q. Each conume ha a valuation function V q and eceive a value of V q f q fo ending a flow of f q. We will make the following aumtion on the valuation function: Aumtion 3. Fo all q, the valuation function V q f q : + + ae tictly inceaing and concave. Ove f q > 0, the function ae diffeentiable. At f q = 0, the ight deivative exit, and i denoted V q 0. In tun, each oduce can uly bandwidth at a ubet L of the netwok link, and we denote thi l fo l L. We alo aume that the cot of oviding bandwidth at one link ae indeendent of the cot at any othe link. Fomally, each oduce ha L cot function C l, l = 1,..., L atifying the following aumtion: Aumtion 4. Fo all, l, thee exit a continuou, convex, and tictly inceaing function l t : + + uch that l 0 = 0, and fo all l 0 we have: C l l = ˆ l 0 l tdt and fo l, 0 we have C l l = 0. The cot function C l : + ae thu continuou, tictly convex, tictly inceaing, and diffeentiable ove thei entie domain. Let l be the bandwidth ulied by oduce at link l L, and let = 1,..., L be the coeonding vecto. The total cot to i then defined to be C = l L C l l. The ocial objective in thi model i again to allocate flow and uly allocation o a to maximize the aggegate ulu. Fomally, we want to come a cloe a oible to the eult of the following otimization oblem: maximize V q f q C =1 uch that f q = f q q f l l = :l :l 0 f q q 0 l 1 Definition of the Mechanim. Conume and ulie now ubmit vecto of bid w q and w eectively. We define w q := w q1,..., w q L whee w ql i the bid to link l. The vecto w i defined in the ame way. The collection of all the bid i denoted w = w 1,..., w Q, w Q+1,..., w Q+.

23 23 At each link l, a conume q bid elect a demand function of the fom { wql 69 c ql µ l = µ l if w ql > 0 0 if w ql = 0 and oduce bid elect a uly function of the fom { 1 w l 70 l µ l = µ l if w l > 0 1 if w ql = 0 Then at each link, the mechanim then et a ice µ l w l that clea the maket by etting the ice to q:l q µ l w l = w ql + :l w l l whee l i the numbe of oduce cometing at link l. Conume q then get allocated a caacity of c ql, and ay fo it µ l c ql, while oduce ovide l unit of bandwidth and eceive a ayment of µ l l. Once all caacitie have been allocated, conume q olve a max-flow oblem on G with edge caacitie et to c ql, and with the ouce and taget being q and t q. Hence we have f q w = max-flowg, c q w, q, t q Thi mechanim induce a game whee the ayoff to conume q i π q w q ; w q = V q f q w q ; w q l q w ql and the ayoff to oduce i π w ; w = l µ l w l Exitence of a Nah equilibium. w l l C l 1 Definition. A game i aid to be concave if the following hold: w l q:l q w ql + :l w l 1 Evey joint tategy, viewed a a oint in the oduct ace of the individual tategy ace, lie in a convex, cloed, and bounded egion in the oduct ace. 2 Each laye ayoff function i continuou and concave in it own tategy. Theoem 11. oen [13] Evey concave n-eon game ha an equilibium oint. We now aly thi to how that a Nah equilibium exit in ou game. Theoem 12. The extended eouce allocation game ha a Nah Equilibium Poof. We have to how that ou game i concave. Fit let look at condition 1. Fix a conume q, and a eone vecto w q. By definition, w q 0. Alo obeve that π q w q ; w q π q w q ; 0 fo all w q. Alo, ince the allocation to q at link l i l fo any w ql > 0 a long a w ql = 0, thee i a W q.t. fo w q > W, π q w q ; 0 < 0. Thu we can etict without lo of geneality the conume bid on [0, W ] whee W = max q W q. The ame thing can be hown fo oduce. Fix a oduce. Obeve that π w ; w = µ l w l C l l l l l = q w ql + q w ql w l C l l l l l l l

24 24 Suoe w l > l C l l fo all l. Then π w ; w = q w ql + q w ql l C l l l l l < q w ql + q w ql C l l l l l = π 0; w and the oduce could ofitably deviate to 0. It not had to aly thi eaoning eaately to each link, and deduce that we mut have w l l C l l l. Thi how that we can aume without lo of geneality that the ayoff ae choen within [0, M] whee M = max max,l C l 1, W. Next, we have to how that the ayoff ae continuou and concave on ou comact et. Obeve that π q and π ae dicontinuou when the eone vecto w q, w equal 0. Fo thi eaon, ick two oduce 1 and 2 abitaily and etict thei tategy ace to [ɛ, M] while keeing the othe tategy ace at [0, M]. We will fit how that the game eulting fom thi etiction i concave, and then we will exlain why that etiction can be made without lo of geneality. Given the etiction, the ayoff π q and π ae continuou in w q, w ince the eone vecto i alway non-zeo. It i alo taightfowad to ee that π i continuou and concave in w. To ee that π q i continuou, obeve that evey function in the comoition V q f q c q w q, aticulaly the max-flow function, i continuou in w q. Alo, ince c q i continuou and f q i non-deceaing, we have that f q i concave. Then it follow taightfowadly that V q i concave in w q. By oen theoem, we can then conclude that a Nah Equilibium exit. The Welfae Bound. Theoem 13. In the extended eouce allocation game, we have V q fq NE C NE 2 S 2 + 4S inf V q fq NE S SS + 2 =1 whee i the unique oitive oot of the olynomial γ = S S S Thi value can be numeically evaluated to aoximately Futhemoe, thi bound i tight. C NE The idea i to fit chaacteize the Nah equilibium in tem of caacity allocation to conume and uly allocation to oduce. Then we can lineaize the utilitie like in a ingle-link cae, and eentially et fo each conume a eaate valuation function fo each link. Doing o will educe the extended game to L ingle-link game, to which we will aly ou exiting bound. Fit, we need to how that given a eone tategy w q, fo evey caacity c ql thee exit a bid w ql that enue q get that caacity. Lemma 14. The joint tategy w i a Nah equilibium if and only if the following hold: 1. At evey link l, thee ae at leat two ulie 1, 2 that have oitive bid w 1, w Fo each q, c q w ag max c V q f q c q w 1 ql c ql, w q l q =1

25 whee c q i defined accoding to 69 and w 1 c ql, w 1 i a function that atifie c ql y, w q = x if and only if w 1 ql x, w q = y 3. Fo each, w ag max µ l w w l C l 1 w l l l w l q:l q w ql + :l w l Poof. Suoe w i a Nah equilibium. The fit claim follow a befoe. To etablih the exitence of w 1, we obeve that ince w q 0, c ql conume q caacity at link l i a continuou function of w ql conume q bid at link l. Futhemoe, c ql x, w ql i tictly inceaing in x, c ql 0, w ql = 0, and lim x c ql x, w ql =. It then follow that w 1 exit. If w q i at of a Nah equilibium, then by definition we mut have w q ag max w π q w, w q = ag max w V q f q c q w; w q l q The econd claim follow fom thi fact. Suoe that it doe not hold. Then, given w q, thee i a c that eult in a highe ayoff. But then uing the function w 1 ql we can find a vecto of bid that will eult in a highe value fo π q, which i a contadiction. Finally the thid claim hold by definition of a Nah equilibium. The othe ide of the imlication can be etablihed by eveing the agument above. Since V q i comoed with the max-flow function, which may not be diffeentiable, we need to ue the following tool fom convex analyi. Definition. We ay that γ i a ubgadient of a convex function f : n at x 0 if fo all x n we have fx fx 0 + γ x x 0 The et of all ubgadient of f at x 0 i called the ubdiffeential and i denoted fx 0. The uegadient and the uediffeential ae the equivalent of a ubgadient and a ubdiffeential fo a concave function. Lemma 15. Let w be a Nah equilibium. Then fo evey conume q thee exit a vecto α q uch that the lineaized valuation function atifie the following elationhi: 71 c q w ag max c V q c := α q c c q w + V q c q w Vq c w 1 ql c ql, w ql l q In othe wod, w q i alo a Nah equilibium fo the game whee V q ha been elaced by V q. Moeove, the ayoff at w in the new game emain the ame. Poof. We can aume without lo of geneality that V q c q i a convex function of c q L by extending it aoiately to the entie domain. Alo obeve that c ql = w ql /µw l i concave in w ql, which imlie that w 1 ql c ql, w ql i convex. ecall that the ayoff to q a a function of it allocated caacitie i π q c q, w q = V q f q c q l q w 1 ql c ql, w q w ql 25