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

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1 On the Efficiency of Maket with Two-ided Pootional Allocation Mechanim Volodymy Kulehov and Adian Vetta Deatment of Mathematic and Statitic, and School of Comute Science, McGill Univeity Abtact. We analyze the efomance of eouce allocation mechanim fo maket in which thee i cometition amongt both conume and ulie namely, two-ided maket. Secifically, we examine a natual genealization of both Kelly ootional allocation mechanim fo demand-cometitive maket [9] and Johai and Titikli ootional allocation mechanim fo uly-cometitive maket [7]. We fit conide the cae of a maket fo one diviible eouce. Auming that maginal cot ae convex, we deive a tight bound on the ice of anachy of about Thi wot cae bound i achieved when the demand-ide of the maket i highly cometitive and the uly-ide conit of a duooly. A moe fim ente the maket, the ice of anachy imove to In contat, on the demand ide, the ice of anachy imove when the numbe of conume deceae, eaching a maximum of in a monoony etting. When the maginal cot function ae concave, the above bound moothly degade to zeo a the maginal cot tend to contant. Fo monomial cot function of the fom Cx = cx 1+ d 1, we how that the ice of anachy i Ω 1. d 2 We comlement thee guaantee by identifying a lage cla of two-ided ingle-aamete maket-cleaing mechanim among which the ootional allocation mechanim uniquely achieve the otimal ice of anachy. We alo ove that ou wot cae bound extend to geneal multieouce maket, and in aticula to bandwidth maket ove abitay netwok. 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 allocation can be achieved uing icing mechanim baed on claical VCG eult, but imlementing uch mechanim geneally induce exceively high infomational and comutational cot [13]. 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 [16] and the Intenet [15]. The Intenet i made u of malle inteconnected netwok that buy caacitie fom each othe, The autho wee uoted in at by NSEC gant In fact, economic i often defined a the tudy of cacity.

2 and the maket mechanim we conide ae cloely inied by the tuctue of the Intenet. Secifically, we ae eticting ou attention to mechanim that ae 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 [17]. 1.1 Backgound and Peviou Wok A baic method fo eouce allocation i the ootional allocation mechanim of Kelly [9]. 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 [14] 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 analyze 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 an 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 [10] and i the natual genealization of both the demand-ide model of Kelly [9] and the uly-ide model of Johai and Titikli [7]. In ode to examine how the genealized ootional allocation mechanim efom in a twoided maket, it i imotant to note that thee ae thee imay caue of welfae lo. Fit, the undelying allocation oblem may be comutationally had. In othe wod, in ome etting uch a combinatoial auction, fo examle, it may be had to comute the otimal allocation even when the laye utilitie ae known. 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 ee in Section 4, the fit two caue do not aie hee: a long a the ue do not behave tategically, 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 that the ootional allocation mechanim doe efom well in two-ided maket. Secifically, unde quite geneal aumtion, the mechanim admit a contant facto ice of anachy guaantee. Moeove, thee exit a lage family of mechanim among which the ootional allocation mechanim uniquely achieve the bet oible ice of anachy guaantee. We tate ou exact eult in Section 3, afte we have decibed the model and ou aumtion. 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 2 The Model 2.1 The two-ided ootional allocation mechanim We now fomally eent the two-ided ootional allocation mechanim due to Neumaye [10]. Thee ae Q conume and ulie in the maket. Each conume q ha a valuation function V q d q, whee d q i the amount of the eouce allocated to conume q, and each ulie ha a cot function C, whee i the amount oduced by ulie. Conume and ulie eectively inut bid b q and b to the mechanim. Doing o, conume ae imlicitly electing b q - aametized demand function of the fom Db q, = bq, and ulie ae electing b -aametized uly function of the fom Sb, = 1 b. We can alo inteet a high conume bid a an indicato of high willingne to ay fo the oduct, and a low ulie bid a an indicato of 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 [9], 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 the full veion of the ae 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 [10], a well a in [9] and [7]. Futhe jutification fo the mechanim will be ovided by ou eult. Secifically, the ootional allocation mechanim geneally oduce high welfae allocation 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 the uly equal demand equation: Q b q = =1 1 b. The ice theefoe get et to b = q bq+ 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 defined to be b V q q Π q b q = q Q b q + 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 q Q b q + b b b C 1 Π b = q Q b q + b if b > 0 q b q + b C 1 if b = The Welfae atio Given a vecto of bid b, the ocial welfae at the eulting mechanim allocation i defined to be Wb = V q d q b C b If the agent do not tategically anticiate the effect of thei action on the ice, that i if they act a ice-take, we how in Section 4 that the mechanim maximize ocial welfae. Howeve, ince =1

4 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. Equivalently, we wih to minimize the welfae lo, 1 WNE. W OPT W OPT 2.3 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. 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 4 that the welfae atio i at leat Concave maginal cot function alo exhibit contant welfae atio, ovided the coeonding total cot function i ufficiently non-linea. Howeve, in the limit a the total cot function become linea, the welfae atio degade to zeo ee Section 5 fo moe detail. Ou main eult thu 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 ae extemely common in both the theoetical and the actical liteatue on indutial theoy [18], o thi aumtion i not aticulaly etictive. In Aumtion 2 we alo et C 0 = 0, but a we how in the full veion of the ae, contant welfae atio till aie wheneve C 0 i bounded below one it cannot be highe than one o the fim i uncometitive. We alo emak that Aumtion 2 wa ued in Johai, Manno and Titikli [4] in thei analyi of the demand-ide ootional allocation mechanim with elatic uly. Mot of the eult of Johai and Titikli [6] and Tobia and Hak [2] on demand-ide Counot cometition with elatic uly alo hold unde the aumtion of convex maginal cot. 3 Ou eult Ou fit eult ae concened with the efomance of the mechanim when the ue act a icetake. Unde Aumtion 1, we ove that: Theoem 1. A unique cometitive equilibium exit fo the two-ided ootional allocation mechanim. The ocial welfae attained at the cometitive equilibium i otimal. Thi oety wa exhibited by Kelly oiginal ootional allocation mechanim, and ha been a featue of all ubequent genealization by Johai and Titikli. It i vey aealing fom a actical oint of view, a in actual netwok, ue ae likely to have little infomation about each othe, making it difficult to maniulate the ytem. In many othe etting howeve, ue will be incentivized to act tategically. In that cae, we need to ue the tonge olution concet of a Nah equilibium to analyze the eulting game. Ou econd eult etablihe the exitence and uniquene of uch equilibia unde Aumtion 1. 3 Fo examle, in maket exhibiting economie of cale.

5 Theoem 2. The two-ided ootional allocation mechanim ha a unique Nah equilibium fo 2. Ou main eult meaue the lo of welfae at that unique Nah equilibium unde Aumtion 2. 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. It follow that the mechanim admit a contant bound on the ice of anachy. Moeove, Theoem 3 allow u to meaue the effect of maket cometition on ocial welfae. The following two coollaie ae concened with that elationhi. Coollay 1. The wot oible ice of anachy i achieved when the uly ide i a duooly = 2. It evaluate numeically to about Coollay 2. When the uly ide i fully cometitive the ice of anachy equal eciely Conequently, a uly-ide cometition inceae, the welfae atio imove. In contat, the welfae atio deceae a demand-ide cometition inceae. Although thi fact may eem uiing at fit, it tun out to have a imle intuitive exlanation. The otimal demand-ide allocation conit in giving the entie oduction to the ue which deive fom it the highet utility. When moe conume ae eent in the maket, they elfihly equet moe of the eouce fo themelve, leaving le fo the mot needy ue and educing the oveall ocial welfae. The bet welfae atio thu aie when thee i only one conume Q = 1, that i, in the cae of a monoony. In the two-ided ootional allocation mechanim, the bet oible ice of anachy ove all oible value of Q and i given by the next coollay. Coollay 3. In a maket in which a monoonit face a fully cometitive uly ide, the ice of anachy equal 3 1, which i about ecall that in the one-ided ootional allocation mechanim fo ulie facing a fixed demand, the welfae lo tend to zeo when the uly ide i fully cometitive [7]. In contat, Coollay 3 imlie that in two-ided maket, that eult no longe hold and that full efficiency cannot be achieved. So fa, ou eult aumed the convexity of maginal cot. Doing that aumtion, we find that the welfae atio equal zeo when the ovide total cot function ae linea. Howeve, the ice of anachy emain bounded fo a cla of concave maginal cot function, and degade moothy to zeo a the total cot become linea. Coollay 4. The welfae atio fo cot function C = c 1+ 1 d whee c > 0 and d 1 i Ω 1 d 2. Like it one-ided veion, the two-ided mechanim can be genealized to multi-eouce maket. An imotant multi-eouce etting i that of bandwidth haed on a netwok of link. The ame guaantee a in the ingle-eouce etting hold fo the netwok veion of ou maket, a well a fo moe geneal multi-eouce maket ee the full veion of the ae fo moe detail. Theoem 4. The welfae atio in netwok equal that of the ingle-eouce model. Theoem 5. The welfae guaantee hold fo moe geneal multi-eouce maket. Finally, we how that the ootional allocation mechanim i otimal in the following way:

6 Theoem 6. 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 ingleided maket by Johai [3], Johai and Titikli [5], [8] and [7], Johai, Manno and Titikli [4], Tobia and Hak [2], and oughgaden [12]. Due to ace limitation, mot of ou eult will be defeed to the full veion of the ae. Hee, we will focu uon the oof of Theoem 3. 4 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 otimal olution and fo equilibium. 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 the demand containt in the ogam, and in Ste VI and VII, we imlify the uly containt. 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.t V q d OPT q d OPT q = =1 0 OPT 1 d OPT q 0 =1 OPT C OPT Since the containt ae linea, thee exit an otimal olution at which the Kauh-Kuhn-Tucke KKT condition hold. A the objective function i concave, the following fit ode condition ae both neceay and ufficient: C OPT λ if 0 < OPT 1 OPT λ if 0 OPT < 1 C V q V q d OPT q λ if d OPT q = 0 d OPT q = λ if d OPT q > 0 We have ued λ to denote the dual vaiable coeonding to the equality containt. 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

7 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 = 1 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, C 1 + NE if 0 < b C NE 1 Fo the conume, V q 0 and V q d NE q 1 dq = if d NE q > 0. if 0 b < Ste III: An otimization oblem. We can now fomulate the welfae atio a an otimization oblem. min Q V qd NE q =1 C NE Q V qd OPT q =1 C OPT 1.t. V q d NE q 1 dne q q.t. d NE q > 0 2 V q d NE q 1 dne q C NE 1 + NE 1 C NE 1 + NE 1 d NE q = =1 q 3.t. 0 < NE 1 4.t. 0 NE < 1 5 NE 6 C OPT λ.t. 0 < OPT 1 7 C OPT λ.t. 0 OPT < 1 8 V q d OPT q λ q.t. d OPT q = 0 9 V q d OPT q = λ q.t. d OPT q > 0 10 d OPT q = =1 OPT 11 d OPT q, d NE q 0 q 12 0 NE, OPT 1 q, 13, λ 0 14 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.

8 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 OPT 1 = OPT and d OPT q = 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 OPT =1 C OPT, and the otimality containt become C OPT 1,.t. 0 < OPT 1 and C OPT 1,.t. 0 OPT < 1. With linea valuation, the new otimality containt enue OPT 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 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 the objective function will be unaffected. So, α q = function: min.t. = 0 and containt 3 doe not hold with equality, we can et α q = and fo all q. Subtituting into the objective d NE 1 + Q q=2 =1 OPT 1 dne 1 C NE d NE q 1 d NE q 1 d NE q / / =1 C NE =1 C OPT 15 = NE 1 C NE 1 + NE 1 d NE q = =1.t. 0 < NE 1 17.t. 0 NE < 1 18 NE 19 C OPT 1.t. 0 < OPT 1 20 C OPT 1.t. 0 OPT < 1 21 d NE q 0 q 2 22 d NE 1 > NE, OPT < 1 25 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.

9 Theefoe, fo any given fixed aignment to the othe vaiable, we mut have d 2 =... = d Q := x. 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, containt 19 become =1 NE / Q 1. Afte ineting containt 16 and the new containt 19, the x = d NE 1 numeato of the objective function 15 become x 1 + Q 1 1 x/ C NE = 1 + Q = 1 + =1 =1 NE =1 NE 1 =1 NE =1 NE d NE 1 d NE 1 1 d NE 1 / Q 1 / Q 1 / Q 1 =1 =1 C NE C NE 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. 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.t C NE =1 OPT 1 + NE =1 NE C NE 1 + NE 1 =1 C NE =1 C OPT 1 26.t. 0 < NE 1 27.t. 0 NE < 1 28 C OPT 1.t. 0 < OPT 1 29 C OPT 1.t. 0 OPT < NE, OPT < 1 32 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 The next te i to 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. In thi ection, we biefly ketch the oof of thi fact and defe the full teatment to the full veion of the ae. Ou oof technique i baed on the wok of Johai, Manno and Titikli on demand-ide maket with elatic uly [4], [6]. The oof conit in 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

10 C C = C NE NE if < 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 C and Ĉ. Thu NE = ŝ NE = NE. The heat of the oof conit in howing that the otimal welfae can only imove when going fom one family to the next. Ste VII: Eliminating the Suly Containt. Auming linea maginal cot function, the otimization oblem become min.t =1 NE 1 2 =1 OPT 1 2 β NE 1 + NE 1 β NE 1 + NE 1 =1 β NE 2 =1 β 33 OPT 2.t. 0 < NE 1 34.t. 0 NE < 1 35 β OPT 1.t. 0 < OPT 1 36 β OPT 1.t. 0 OPT < NE, OPT 1 38 β > < 1 40 with the new vaiable β, = 1,...,. Fom OPT NE, we can then deduce that 34 and 35 hold with equality. Suoe they don t fo ome. Then NE = OPT = 1. Containt 34 i β < 1+1/ 1 < < 1. Hence, β = 1+1/ 1 will be a feaible olution i.e. containt 36 will till be atified. Futhemoe, inceaing β to 1+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 OPT = min1/β, 1. It i eay to ee that OPT and β atify the equation above if and only if they atify 36 and 37. The educed otimization oblem now become: min.t =1 NE 1 2 =1 OPT 1 2 β NE 1 + NE 1 =1 β NE 2 =1 β 41 OPT 2 = 42 OPT = min1/β, < NE, OPT 1 44 β > < 1 46

11 We can inet the equality containt 42 and 43 into the objective function 41 to obtain: min.t =1 NE 2 =1 =1 min1/β, 1 2 =1 NE 1+ NE min1/β,1 2 NE 1+NE / 1 / < NE 1 48 β = NE 1 + NE 49 / 1 0 < 1 50 The objective function 47 can be ewitten a: = NE =1 min1/β, 1 2 NE 2 1+ NE / 1 min1/β,1 2 NE 1+NE / 1 Conequently, the minimum of the otimization oblem i geate than o equal to min / 1 51 min 1+/ 1 1+/ 1, 1 21+/ 1 min, 1 2.t. 0 < < 1 53 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 the full veion of the ae. 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 1 and 2. So, a uly-ide cometition inceae, the welfae atio doe imove. The ooite occu a demand-ide cometition inceae. Secifically, adating ou aoach give Coollay 3. 5 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 the full veion of the ae. 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 Ω 1 d 2, fo any contant d. A oof of thi Coollay 4 i given in the full veion of the ae. See Neumaye [10] fo anothe examle of inefficiency in the eence of linea cot function. 6 Extenion to Netwok and Abitay Maket. We can genealize ou eult fo bandwidth maket ove a ingle netwok connection to the cae whee bandwidth i haed ove an entie netwok. In that model, each conume q i aociated with

12 a ouce-ink ai, and ovide at aociated with edge of the netwok at which they can offe bandwidth. A conume 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 4 i given in the full veion of the ae. Moeove, if we identify link e E with abitay eouce, then ou eult extend to a geneal cla of maket with any numbe of eouce. The exact definition of thee maket and a oof of Theoem 5 ae alo given in the full veion of the ae. 7 Smooth Maket-Cleaing Mechanim It wa hown in [3] and [8] that in one-ided maket, the ootional allocation mechanim uniquely achieve the bet oible welfae atio within a boad cla of o-called mooth maket-cleaing 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 maket-cleaing mechanim and a oof of Theoem 6 i given in the full veion of the ae. efeence 1. Guta, A., Stahl, D.O., Whinton, A.B.: The economic of netwok management. Commun. ACM 429, Hak, T., Mille, K.: Efficiency and tability of nah equilibia in eouce allocation game. In: GameNet 09: Poceeding of the Fit ICST intenational confeence on Game Theoy fo Netwok IEEE Pe, Picataway, NJ, USA Johai,.: Efficiency lo in maket mechanim fo eouce allocation. Ph.D. thei, MIT Johai,., Manno, S., Titikli, J.N.: Efficiency lo in a netwok eouce allocation game: The cae of elatic uly. Mathematic of Oeation eeach 29, Johai,., Titikli, J.N.: Efficiency Lo in a Netwok eouce Allocation Game. Mathematic of Oeation eeach 293, Johai,., Titikli, J.N.: A calable netwok eouce allocation mechanim with bounded efficiency lo. IEEE Jounal on Selected Aea in Communication 245, Johai,., Titikli, J.N.: Paameteized uly function bidding: Equilibium and welfae Johai,., Titikli, J.N.: Efficiency of cala-aameteized mechanim. Oe. e. 574, Kelly, F.: Chaging and ate contol fo elatic taffic. Euoean Tanaction on Telecommunication 8, Neumaye, S.: Efficiency Lo in a Cla of Two-Sided Mechanim. Mate thei, MIT oen, J.B.: Exitence and uniquene of equilibium oint fo concave n-eon game. Econometica 333, oughgaden, T.: Potential function and the inefficiency of equilibia. Poceeding of the Intenational Conge of Mathematician ICM 3, Semet, N.: Maket mechanim fo netwok eouce haing. Ph.D. thei, Columbia Univeity, New Yok, NY, USA Shaley, L.S., Shubik, M.: Tade uing one commodity a a mean of ayment. Jounal of Political Economy 855, Octobe Shenke, S., Clak, D., Etin, D., Hezog, S.: Picing in comute netwok: ehaing the eeach agenda. SIGCOMM Comut. Commun. ev. 262, Songhut, D.J.: Chaging communication netwok: fom theoy to actice. Elevie Science Inc., New Yok, NY, USA Stoft, S.: Powe ytem economic : deigning maket fo electicity. IEEE Pe, Picataway, NJ Tiole, J.: The Theoy of Indutial Oganization, vol. 1. The MIT Pe, 1 edn. 1988

13 8 Aendice 8.1 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 4. 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.

14 8.2 Aendix B: The Exitence of Nah Equilibia To ove Theoem 2 we how that the Nah equilibium condition obtained in Section 4 ae exactly the neceay and ufficiency condition fo the following ytem. NASH max ˆV q d q Ĉ =1.t. d q = =1 0 1 d q 0 whee ˆV q = 1 d q V d q + 1 dq V q zdz 0 Ĉ = 1 + C 1 1 C zdz 1 0 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.

15 8.3 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 1 + / o When 54 hold, we olve the following ytem: 1 + / 1 > 1 55 min 1+/ 1 1+/ t. 1 + / < < 1 59 Now et W = 1 and let g be the objective function 60. Thu Diffeentiating we obtain g = 2W 1 2 W + + W + W 2W + 2 g = 2 W W + 3 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 60. t. 1 + /W 61 0 < < 1 63 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.

16 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 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 whee ± = S ± S γ 2S + 2 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 < ± < /S ± 67 ± 68 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 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. > 0

17 8.4 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 1+/ 1 γ γ min 1 γ β, γ 2β min β, 12 γ min 1 γ β, 1.t γ 1 whee β = 1 + / 1 γ 1 A befoe, we have two cae to conide. Fit uoe 1 γ β. minimize / 1 γ γ / 1 γ 1 γ 1.t. 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 / 1 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 γ 2 1 γ.t. 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 Suoe now 1 γ β < 1. Then the otimization oblem i minimize t. 1 γ < β γ β 0 γ 1 1+/ γ2 β γ 1 γ β and the denominato can be ewitten a 1 γ2 1 γ γ1 γ 2 γ γ 1+/ 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 8.5 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 4. 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 β = d q = x Q α q = 1 1 x Q 1 + x 1 q q Obeve that 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 1 d q = 1 x Q = Fo the fim, we have C 1 + = β 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 OPT q =1 C OPT = Q α qd NE q =1 β NE =1 β

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 q = µ 1 dq > µ > µ = C 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 NE =1 β = The eult follow by imly letting x 0. Q 1 x Q x = = x =1 1 x Q 1+ x 1 1 x Q 1+ x 1 =1 1 x Q x 1 x Q 1+ x 1+ x 1 1 x

21 8.6 Aendix F: Concave Maginal Cot Function Suoe C = 1 1+1/d 1+1/d fo ome fixed d. The educed otimization oblem i: whee minimize min1/β, 1.t β = 1+1/d 1+/ 1 β 1+1/d 1/d 1 + / 1 Fit uoe that 1/β > 1. The otimization oblem i: minimize t 1/d min1/β, 11+1/d 1+1/d 1+/ 1 1+1/d 1 1/d 1+/ 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 /d 1 1/β 1+1/d 1/β1/d.t 1 < β / 1 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.

22 8.7 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 = 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 uch that f q = q f q =1 :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+. f l

23 At each link l, a conume q bid elect a demand function of the fom c ql µ l = { wql µ l if w ql > 0 0 if w ql = 0 and oduce bid elect a uly function of the fom { 1 w l µ l µ l = l if w l > 0 1 if w ql = 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 1. 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 7. oen [11] Evey concave n-eon game ha an equilibium oint. We now aly thi to how that a Nah equilibium exit in ou game. Theoem 8. 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 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 9. In the extended eouce allocation game, we have V q fq NE C NE 2 S 2 + 4S Q inf V q fq NE C 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. 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 1. 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 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 =1

25 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 2. 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 2. Let w be a Nah equilibium. Then fo evey conume q thee exit a vecto α q uch that the lineaized valuation function V q c := α q c c q w + V q c q w w ql atifie the following elationhi: c q w ag max c Vq c w 1 ql c ql, w ql 71 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 and it i concave. Then by the oetie of ubdiffeential we have π q c q, w q = V q c q + l w 1 ql c ql, w ql