The Progressive Second Price Auction Mechanism. Center for Telecommunications Research, Columbia University.

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1 The Progressve Second Prce Aucton Mechansm for Network Resource Sharng Aurel A. Lazar and Nemo Semret y Center for Telecommuncatons Research, Columba Unversty New York, NY, , USA. Tel: (212) , Fax: (212) faurel, nemog@comet.columba.edu December 31, 1997 Keywords: resource allocaton, auctons, game theory, mechansm desgn, network prcng. 1 Introducton In the emergng multservce communcaton networks (ATM, Next-Generaton Internet), tradtonal approaches to prcng are not vable. In telephony, the resources allocated to a call are xed, and usage prces are based on the predctablty of the number of actve calls at any gven moment. But the wde and rapdly evolvng range of applcatons (ncludng some whch adapt to resource avalablty) n the new networks makes demand much more dcult to predct. On the other hand, the current Internet practce of prcng by the physcal capacty decouples actual use of resources from the monetary charges, makng the network vulnerable to the well-known \tragedy of the commons". Thus there s a need to develop new approaches to prcng of network resources, whch has led to much research n recent years (see [11, 8, 7, 3, 12, 9, 2] for a representatve sample). 8 th Internatonal Symposum on Dynamc Games and Applcatons, Maastrcht, The Netherlands, July y Correspondng author. 1

2 In [1], we present the Progressve Second Prce (PSP) aucton, an ef- cent mechansm for allocaton of varable-sze shares of a resource among multple users. The PSP rule generalzes Vckrey (\second-prce") auctons [13] whch are for non-dvsble objects, and bears some smlarty to Clarke-Groves mechansms [1, 5]. The derence wth the latter beng that for practcal reasons, we reduce the message (bd) space to two dmensons (prce and quantty), rather than the nnte dmensonal space of valuaton functons (or demand schedules) whch s requred n the drectrevelatons mechansms. Player 's bd s s =(q ;p ) 2S =[;Q] [; 1), meanng he would lke a quantty q at a unt prce p. A bd prole s s =(s 1 ;:::;s I ). The auctoneer follows an aucton rule A to respond wth an allocaton A(s) =(a(s);c(s)), where a (s) and c (s) are respectvely the quantty allocated to and the total charge pad by player. The PSP allocaton rule s: a (s) = q ^ Q (p ;s,) (1) c (s) = X j6= p j [a j (; s,), a j (s ; s,)] (2) where ^ means takng the mnmum, Q (y; s,) = 2 4 Q, X p k y;k6= 3+ q k 5 ; (3) and Q s the total avalable quantty of resource. Q (y; s,) s the quantty avalable for player at a bd prce of y. The rule s computatonally smple { O(I 2 ) where I s the number of players { and can thus be used n real-tme dynamc auctonng. In [1], we show that under elastc demand (concave valuatons), analyzed as a complete nformaton game, the PSP aucton s ncentve compatble and stable, n that t has a \truthful" -Nash equlbrum where all players bd at prces equal to ther margnal valuaton of the resource, for any seller reserve prce p >. PSP s ecent n that the equlbrum allocaton maxmzes total user value to wthn O( p ). The parameter has a natural nterpretaton as a bd fee, and allows a manager to drectly trade-o engneerng and economc ecency (measured respectvely by convergence tme and total user value). In ths paper, we show that the equlbrum holds when PSP s appled by ndependent resource sellers on each lnk of a network wth arbtrary topology, wth users havng arbtrary but xed routes. In ths network 2

3 case, the dstrbuted mechansm has a further ncentve compatblty n that submttng the same bd at all lnks along the route s an optmal strategy for each user, regardless of other players' actons. Thus, PSP constutes a stable decentralzed mechansm for allocatng and prcng capacty for vrtual paths (VPs) and vrtual prvate networks (VPN), and s applcable to programmable ATM networks [4]. 2 Decentralzed PSP Auctonng of Networked Resources 2.1 Formulaton In ths secton, we extend the formulaton of [1] to the network case, where there s a set of resources L = f1;:::;lg, of whch the quanttes are Q 1 ;:::;Q L, and as before, a set of players I = f1;:::;ig. A basc goal s that the mechansm be decentralzed n that the allocatons at any lnk depend only on local nformaton: the resources avalable at that lnk and the bds for that lnk only. Ths makes the mechansm applcable to cases where the varous resources beng auctoned may be owned by derent enttes. Each player s responsble for coordnatng (or not) her bds at the derent lnks on her route n such away that maxmzes her utlty. Let Q l =[;Q l ], and Q = Q l2l Q l. Player 's bd s now s =(s 1 ;:::;sl ) 2 S = Q l2l Sl, where sl =(ql ;pl ) 2Sl = Ql [; 1) s the bd for resource l 2L. At each lnk l 2L,wehave an allocaton rule A l, whch maps a prole s l 2S l = Q 2I Sl to an allocaton Al (s l )=(a l (s l );c l (s l )). Player 's type ncludes a route 1, r L. We wll assume that players only care about the end-to-end \thckness" of ther allocated \ppe" (whch s gven by the thnnest lnk allocaton) and the total charge. Thus player has a valuaton of the resource (:). Thus, for a bd prole of s, under allocaton rule A, player gettng an allocaton A (s) =(a (s);c (s)) has the quaslnear utlty u (s) = (mn l2r a l (s)), c (s); (4) where c (s) = X l2l c l (s): 1 Our analyss wll not requre that r form a contnuous path, or any specc type of subgraph { \route" means any arbtray subset of lnks. 3

4 In addton, the player can be constraned by abudget b 2 [; 1], so the bd s must satsfy S (s,) =fs 2S : c (s ; s,) b g: (5) Remark: In realty, routng tself s a compettve game, and the decentralzed nature of the aucton makes t possble for players to make the route part of ther strategy and thus vary t n response to other users' actons. In our analyss however, we assume players have obtaned a (xed) route before enterng the aucton game. In the broader context, the aucton game may be nested wthn a larger game whch ncludes routng. 2.2 Equlbrum of Networked PSP Auctons Assume that the allocaton at each lnk l 2Ls performed by a PSP rule,.e. A l sates (1) and (2). Assume further that the demand s elastc: Assumpton 1 For any 2I, () = ; s derentable,, non-ncreasng and contnuous 9 >, 8z, (z) > )8<z; (z) (), (z, ). The key property n the analyss of the network case s that, gven a xed opponent prole, a player cannot do better than place consstent bds,.e, the same bd at all the lnks on her path and bd zero on all lnks not n her path. For each 2I,we dene where for 1 l L, x : S,! S s 7,! x (s) =(z ;y ); z l = 1 r (l) mn m2r a m (s); y l = 1 r (l) max m2r p m : Dene also Q l (y; s l,) = lm &y Q l (; sl,); 4

5 and n o P l (z; s,) = nf y :Q l (y; s,) z : (6) where for each l 2L, Q l s dened by 3. Proposton 1 8s 2S, and 2I, u (x (s); s,) u (s): Moreover, s 2 S(s,) ) x (s) 2 S(s,): Proof: Frst, we prove that, leavng bd prces unchanged, there s no loss of utlty for a player who reduces the bd quanttes to z,.e. u ((z ;p ); s, ) u (s): For any l 2 r, z l a l (s) Ql (pl ;sl ), therefore a l ((z ;p ); s, ) = z l ^ Q l (pl ;sl, )=zl = mn m2r a m (s). Thus, u ((z ;p ); s, ), u (s) = (mn m2r a m ((z ;p ); s, )), (mn m2r a m (s)), c ((z ;p ); s, )+c (s) =, c ((z ;p ); s, )+c (s) XZ a l (s) = P a l ((z l (z; s,) dz ;p );s, ) l2l ; snce P l and al ((z ;p ); s, ) a l (s). Second, for any l 2 r, y l p l, hence Ql (yl ;sl ) Q l (pl ;sl ) z l. Thus, a l ((zl ;yl ); sl, )=zl = al ((z ;p ); s, ). Now snce z l = for l 62 r,wehave c ((z ;y ); s, ) = = = X Z a l l2l Z a l X l2r Z a l X l2r = c ((z ;p ); s, ); ((z ;y );s, ) P l (z; s,) dz ((z ;y );s, ) P l (z; s,) dz ((z ;p );s, ) P l (z; s,) dz hence u ((z ;y ); s, )=u ((z ;p ); s, ); 5

6 whch completes the proof of the rst statement. Now s 2 S(s, ) ) b c (s) c ((z ;y ); s, )=c ((z ;p ); s, ) ) (z ;p ) 2 S(s, ). 2 Thanks to Proposton 1, we can restrct our attenton to consstent strateges only, and stll have feasble best reples 2. Ths forms a \consstent" embedded game wth feasble sets replaced by the consstent strategy set obtaned by applyng x to the feasble strategy set Let ~S (s,) =x (S(s,) fs,g): Dene for 8y; z, 8s 2S, 8 2I, P l (z; s l,) = nf n :Q l (; s,) z ~P (z; s,) = X l2r P l (z; s l,); ~Q (y; s,) = supfz 2 \ ~Q (y; s,) = mn Q l l2r (y; sl,); ~a (s) = q 1 ^ ~ Q (p 1 ;s,); Z ~a (s) o l2r Q l : ~ P (z; s,) <yg; : (7) ~c (s) = ~P (z; s,) dz: Lemma 1 8s, 2S, ; 8s 2 ~ S(s,); 8l 2 r, ~Q (y; s,) = lm &y ~ Q (; s,); ~a (s) = mn a l (s); l2r ~c (s) =c (s); and u (s) = (~a (s)), ~c (s): 2 If >, an even stronger statement holds: a bd can be a best reply only f t results n the same quantty allocaton at all the lnks n the route. 6

7 Proof: Follows trvally from the dentons and the fact that s 2 ~ S (s, ) ) q l = a l (s) =q1 ; 8l 2 r. 2 Now wthn the feasble sets, the embedded game s dentcal to the sngle node game, wth all elements beng replaced wth the ~ verson. Thus the followng result from [1] holds: for a gven opponent prole s,, an -best reply for player s to bd at a prce equal to the margnal valuaton,.e. set p = (q ). Formally, let T = fs 2S : p = (q )g, the (unconstraned) set of player 's truthful bds, and T = Q T, then: Proposton 2 (Incentve compatblty and contnuty of best-reply) Under Assumpton 1, 8 2I, 8s, 2S,, such that Q ~ (;s,) =, for any >, there exsts a truthful -best reply t (s,) 2T. In partcular, let G (s,) = 8 < : z 2 \ l2r Q l : z ~ Q ( (z);s,) and Z z ~P (; s,) d b 9 = ; : Then wth v = [sup G (s,), = ()]+ and w = (v ), t =(v ;w ) 2 T \ S (s,). Further, t s contnuous n s, on any subset V (P ; P )=fs, 2S : 8z >; P P ~ (z; s,) P g, wth 1 > P P >. In addton, ~a (t ; s,) =v. Proof: See [1]. 2 Fgure 1 llustrates the consstent and truthful best reply for a player wth a two-hop route. Proposton 3 (Network Nash equlbrum) In the network aucton game wth the PSP rule appled ndependently at each lnk, reserve prces p l > ; 8l 2 L, and players descrbed by (4) and (5), f Assumpton 1 holds, then for any >, there exsts a consstent and truthful -Nash equlbrum s 2T. Proof: 8s 2T; 2I; l2l; z >, wehave z>=q l (pl =2;sl,), whch by (7) mples P (z; s, ) p =2 = P. Let P = max k2i k () _ max l2l p l. Then, by Proposton 2, t = (v; w) s contnuous n s on T. From the last statement of Proposton 2), we have ~a (t ; s, )=v. Therefore (z (s);y (s)) def = x (t (s); s, )= 7

8 1 t_ 1 P (z) 2 t_ Q 1 P 2 (z) Q 2 ~ P (z) w u t_ θ c v Q ~ z Fgure 1: Consstent truthful bd for two hop route 8

9 (1 r (l) mn m2r v m (s); 1 r (l) max m2r (vm def )), s contnuous. Let (q; p) = s. Snce s 2T,wehave s =(q; (q)). By Assumpton 1, 8 2I, s contnuous therefore z can be vewed as a contnuous mappng of Q I onto tself. By Brouwer's xedpont theorem (see for example [6]), any contnuous mappng of a convex compact set nto tself has at least one xed pont,.e. 9q = z(q ) 2 [;Q] I. Now wth s =(q ; (q )), we have s = t(s ) 2T. 2 References [1] E. H. Clarke. Multpart prcng of publc goods. Publc Choce, 8:17{33, [2] J. Gong and S. Marble. Prcng common resources under stochastc demand. preprnt { Bellcore, Aprl [3] J. Gong and P. Srganesh. An economc analyss of network archtectures. IEEE Network, pages 18{21, March/Aprl [4] COMET Group. The xbnd project. [5] T. Groves. Incentves n teams. Econometrca, 41(3):617{631, July [6] Hurewcz and Wallman. Dmenson Theory. Prnceton Unversty Press, [7] H. Jang and S. Jordan. Connecton establshment n hgh speed networks. IEEE J. Select. Areas Commun., 13(7):115{1161, prcng.htm. [8] H. Jang and S. Jordan. The role of prce n the connecton establshment process. European Trans. Telecommuncatons, 6(4):421{429, prcng.html. [9] F. P. Kelly. Chargng and accountng for bursty connectons. In L. W. McKnght and J. P. Baley, edtors, Internet Economcs. MIT Press,

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