Competition Among Providers in Loss Networks. hal , version 1-22 Aug 2012

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1 Author manuscrpt, publshed n "Annals of Operatons Research (2011)." DOI : /s Noname manuscrpt No. (wll be nserted by the edtor) Competton Among Provders n Loss Networks Patrck Mallé Bruno Tuffn Receved: date / Accepted: date Abstract Communcaton networks are becomng ubqutous and more and more compettve among revenue-maxmzng provders, operatng on potentally dfferent technologes. In ths paper, we propose to analyze the competton of provders playng wth access prces and fghtng for customers. Consderng a slotted-tme model, the part of demand exceedng capacty s lost and has to be resent. We consder an access prce for submtted packets, thus nducng a congeston prcng through losses. Customers therefore choose the provder wth the cheapest average prce per correctly transmtted unt of traffc. The model s a two-level game, the lower level for the dstrbuton of customers among provders, and the upper level for the competton on prces among provders, takng nto account what the subsequent repartton at the lower level wll be. We prove that the upper level has a unque Nash equlbrum, for whch the user repartton among dfferent avalable provders s also unque, and effcent n the sense of socal welfare. Moreover, even when addng a hgher level game on capacty dsclosure wth a possblty of lyng for provders, provders are better off beng truthful, and the unque Nash equlbrum s thus unchanged. Keywords Competton Game theory Wreless Networks Prcng Resource allocaton Mathematcs Subject Classfcaton (2000) 90B10 91A80 P. Mallé Insttut Telecom/Telecom Bretagne, 2 rue de la Châtagnerae CS17607, Cesson-Sévgné Cedex, FRANCE. Tel.: Fax: E-mal: patrck.malle@telecom-bretagne.eu B. Tuffn INRIA Rennes Bretagne Atlantque, Campus Unverstare de Beauleu, Rennes Cedex, FRANCE. Tel.: Fax: E-mal: bruno.tuffn@nra.fr

2 2 1 Introducton Telecommuncaton networks are now managed by commercal servce provders tryng to attract customers n order to maxmze ther revenue. A typcal example s the Internet: The network was ndeed ntally just a connecton of academc and cooperatve stes, but t has now moved to a much broader entty, whose access for customers s enabled by selfsh and compettve provders. Furthermore, nstead of havng a network per applcaton, all applcatons (telephony, emal, web browsng, vdeo, games...) can now be carred out usng any technology, beng the ADSL network, FTTx, 3G wreless networks, WF or WMAX (or LTE), wth heterogeneous qualty of servce (QoS) capabltes. Ths convergence leads to a complex system whch requres to be analyzed from an economcal pont of vew, takng nto account the technologcal specfctes. 1.1 Contrbuton We propose n ths paper to study a competton game among provders wth heterogeneous and non-overlappng capactes (or spectrum f dealng wth wreless). Those provders are modeled by loss networks, such that f demand at a provder exceeds capacty, demand n excess s lost and has to be resent. Congeston prcng s appled by chargng for sent traffc nstead of successfully receved one. More precsely, the more traffc s observed, the more lkely packets are to be lost and then resubmtted (and pad agan for). As a result, the total prce charged per successfully receved packet, named here perceved prce, s an ncreasng functon of demand. Customers are assumed nfntesmal,.e., the strategy of a sngle ndvdual does not have any mpact on others: only a grouped acton of a bunch of customers can affect congeston levels, and thus perceved prces. Therefore, when they act selfshly, ther global repartton wll obey the so-called Wardrop s prncple [22], ntally ntroduced n the (equvalent) transportaton doman: only provders wth the cheapest perceved prce obtan some demand. We show that whatever the access prce at provders, there exsts such a user equlbrum stuaton, and that the (common) perceved prce at all provders wth postve demand s unque. Knowng how customers wll dstrbute themselves for any combnaton of prces, provders try to maxmze ther revenue by playng wth ther prces. We therefore end up wth a two-level Stackelberg game [10], where the provders are the leaders, usng by backward nducton the antcpated decson (the repartton) of customers to determne ther strategy. We show that there exsts a unque Nash equlbrum for the prcng game, and we characterze t explctly. A Nash equlbrum s a prce profle such that no provder can unlaterally mprove ts revenue. We show that ths non-cooperatve case actually and surprsngly leads to the same confguraton than the cooperatve case, when all actors,.e., provders and customers, jontly try to maxmze the sum of ther utltes -also known as socal welfare-. Ths paper s related to [16], where the same prcng tools were appled, but users were assumed to be senstve to ther total submtted traffc, not for receved one. As a consequence, lost packets were somewhat consdered as satsfactory because they were not resubmtted. We consder here the more realstc stuaton where traffc that counts s the successfully transmtted one. Even f the results look smlar to those n [16], that new model requres a reformulaton of the problem and a complete rewrtng of all proofs.

3 3 1.2 Related work The general framework of the paper s that of non-cooperatve game theory [10]. In telecommuncatons, game theory has been used a lot n the last decade to model the behavor of dstrbuted algorthms, wth potentally selfsh actors (see for example [4] and references theren). We deal here more specfcally wth telecommuncatons prcng, a topc of actve research [6, 7, 18 20]. Remark however that most of the studes are dealng wth a monopoly, whereas we consder here an olgopoly. Olgopoles have been extensvely studed n other areas than telecommuncatons [21], but telecommuncaton networks have specfctes (e.g., congeston effects on QoS) that are not encompassed by most models. Moreover, competton s a realty n the current telecommuncaton world, and needs to be taken nto account, snce t can lead to sgnfcantly dfferent results than monopoly stuatons [11]. For other competton models, wth dfferent assumptons and atomc users, the reader can look at, among others, [8,9,13 15,17]. The case of users dstrbuton followng Wardrop s prncple has been consdered n [5], where prce competton among producers s studed wthout congeston effects on the user sde, but wth a negatve externalty on the supply sde through some producton costs. Our model also has some demand-related costs (that we nterpret as management costs), but we consder that ther level s low wth respect to revenues, as can be expected n wreless networks where most costs come from nfrastructure and are ndependent of demand. We moreover ntroduce a partcular form of negatve externalty on the user level, that s typcal for lmted capacty networks wth losses. Other references [2, 3, 12] apply Wardrop s prncple on the user level to study competton. In all those papers the externalty s the expected delay, not the loss probablty lke here. As descrbed n the prevous subsecton, the present paper s related to one of our prevous works [16], but we now nclude the fact that retransmssons are taken nto account n the demand level. It s actually a more relevant and key new assumpton, that leads to completely dfferent proofs. 1.3 Organzaton of the paper Ths paper s organzed as follows. Secton 2 presents the general model. Secton 3 defnes the socally-optmal stuaton,.e., the cooperatve stuaton wth provders and customers jontly maxmzng socal welfare. In Secton 4, we descrbe and characterze how customers dstrbute themselves, followng Wardrop s prncple, for any fxed profle of provder prces. Usng that user equlbrum, Secton 5 shows that there exsts a unque equlbrum for the prce competton among provders, and that the correspondng outcome s actually socally optmal. Secton 6 studes the potentel nterest for provders to le about ther real capactes n a compettve envronment and then to artfcally ncrease congeston for a potental larger revenue due to resent packets. Fnally, Secton 7 summarzes the contrbutons and presents drectons for future research.

4 4 2 General model We consder a set I := {1,..., I} of I 2 provders n competton at an access pont. Tme s slotted and each provder ( I) can serve C > 0 packets (or unts, seen as a contnuous number) per slot. If demand exceeds capacty at a gven provder, demand n excess s lost. Lost packets are assumed to be chosen unformly over the set of submtted ones. If d s the total demand at provder, the number of served packets s actually mn(d, C ), meanng that packets are actually served wth probablty mn(c /d, 1). Users are assumed to be charged a prce p for each submtted packet nstead of each served one. Ths nduces a congeston prcng to yeld ncentves to lmt demand, the negatve externalty of congeston beng expressed n terms of losses experenced by users. The total ncome of provder s d p and the total servce rate s d mn(c /d, 1). Then the average perceved prce per served traffc unt at provder s therefore p = p / mn(c /d, 1) = p max(d /C, 1). Chargng on sent packets nstead of successfully transmtted ones may seem unrealstc. However, that mechansm can be seen as a volume-based prcng scheme, wth a congeston-dependent charge. Somewhat equvalently, t can also be seen as a consequence of the more frequently used tme-based chargng wth a fxed prce per tme unt. Indeed, when congeston occurs on a network and packets are lost, havng to send them agan multples the total transfer tme (and thus the prce pad) by the mean number of transmssons per packet max(1, d /C ). We assume that total user demand s a functon D( ) of the perceved prce p, and that D s contnuous, dervable, and strctly decreasng wth p on ts support [0, p max) (wth possbly p max = + ), and that lm p + D(p) = 0. We moreover assume that D(0) > C,.e., that there s some congeston: the total resource avalable s not suffcent to satsfy the maxmum demand level. Fnally, we assume that D(0) < + : f the access were free, then the total demand would be fnte. Remark that ths last assumpton can be easly met, by consderng the sendng capacty lmts of user machnes. We also defne the functon v : q nf{p : D(p) q} (wth the conventon nf = 0), that we call the margnal valuaton functon at the q-th unt of demand. From our assumptons on D, v(q) s fnte for all q 0. From an economc pont of vew, v(q) represents the maxmum prce per traffc unt at whch the q traffc unts could be sold. We fnally defne V (q), the overall valuaton, as the sum of the margnal valuatons of the q unts of users wth largest wllngness-to-pay,.e., q V (q) := v(x)dx. x=0 The economc nterpretaton of V (q) s the total value of the frst q served unts of traffc, for users who are wllng to pay the most for the servce. Those margnal and overall valuaton functons wll be useful to characterze the socally-optmal stuaton and the dstrbuton d := (d 1,..., d I ) of customers among provders, obtaned from a gven prce profle. The goal of each provder s, by playng on ts unt prce p, to maxmze ts net beneft R (p 1,... p I ) := p d l (d ),

5 5 where p d s the money earned drectly from demand, and l (d ) represents the cost for provder of managng a demand level d. We assume that for all, l s nondecreasng. Most of our results are vald under the followng assumpton preventng provder management cost functons from beng too steep. Remark that ths assumpton seems reasonable, snce management costs are n general very small wth respect to nfrastructure costs (that are ndependent on current demand, and thus not consdered here), and wth respect to ncomes from customers. Assumpton A The management cost functon l of every provder I s Lpschtzcontnuous on [0, C ] wth a Lpschtz constant κ smaller than the global margnal valuaton of the sum of all provder capactes. In other terms, I, x, y C, ) wth κ v( j I C j. l (x) l (y) κ x y, Remark 1 Remark that Assumpton A s satsfed for example f the functons (l ) are dervable and convex, and such that I, l (C ) p, ) where l s the dervatve of l, and p = v( j I C j. For some results, we wll need a strcter assumpton, that ncludes an elastcty condton on demand: Assumpton( B In addton to Assumpton A, we assume that for unt prces larger ) than p := v j I C j, the demand functon D s suffcently elastc: y p yd (y) D(y) 1 1 κ/y, (1) where κ s the Lpschtz constant for the cost l on [0, C ], κ := max κ, and D s the dervatve of the demand functon D. Remark 2 When management costs are neglgble (.e., κ = 0 for all I), then Assumpton B conssts n demand elastcty beng larger than 1, an assumpton often made n economcs to descrbe stuatons where demand s hghly senstve to prces. 3 Socally optmal stuaton Followng usual vocabulary from economcs, we defne Socal welfare as the sum of utltes of all actors n the game -here, users and provders-. The total user utlty s the overall user valuaton mnus the total prce pad, whle the total provder utlty (revenue) s the total prce pad mnus the total managng cost. Therefore, prces do not drectly appear n the expresson of socal welfare. Proposton 1 For a demand confguraton d := (d 1,..., d I ), socal welfare s expressed by the quantty ( ) SW(d) := mn(d, C ) d V d l (d ). (2)

6 6 Proof The frst term n SW s the total valuaton for the servce experenced by users. Indeed, V (x) s the total user valuaton, f the x users wth largest wllngness-to-pay are served. For a gven demand confguraton, the total quantty served s mn(d, C ). Moreover, when demand exceeds capacty, then not all demand s served: among total demand d, only mn(d, C ) are served, the others gettng no servce and thus havng a zero valuaton. Snce we assume that losses occur regardlessly of user wllngness-to-pay, the actual (per traffc unt) utlty of a user havng (per traffc unt) wllngness-to-pay v s ts wllngness-to-pay tmes the probablty to be served,.e., mn(d, C ) d v. User are assumed nfntesmal, therefore the total user valuaton equals d u=0 mn(d, C ) d v(u)du, the frst term n (2). The second term n (2) s smply the total managng cost for the demand d. In our next result, we characterze the most effcent demand vector d, n the sense of socal welfare. A pror, that demand confguraton may not correspond to users selfshly selectng ther provder. Proposton 2 Under Assumpton A, socal welfare s maxmzed when d = C for each provder. Proof We consder any demand vector d, and we prove that truncatng the demand d to the capacty of each provder I can only ncrease socal welfare. Defnng a new demand vector d n = (mn(d, C )), we have ( ) SW(d n ) = V mn(d, C ) l (mn(d, C )) ( ) mn(d, C ) d V d l (mn(d, C )) SW(d), where the second lne comes from V beng a concave functon wth V (0) = 0, whch mples that αv (x) V (αx) for any x 0 and 0 α 1. The thrd lne smply comes from the nondecreasngness of cost functons (l ). As a result, we can look for an optmal demand profle d opt n the compact convex set C := [0, C ]. The objectve functon beng contnuous, such an optmal profle always exsts. Now compare such an optmal demand d opt to the profle d C := (C 1,..., C I ). Snce both profles are n the set C, we have

7 7 SW(d opt ) SW(d C ) ( ) ( ) =V d V C + (l (C ) l (d )) ( v (l (C ) l (d )) C ) C ) + (d ( ( ) ) v C κ (d C ) 0. where we used the concavty of V on the second lne, and Assumpton A on the last lne. Ths concludes the proof: the demand vector d C performs as least as well as any other demand vector n terms of socal welfare. 4 Wardrop equlbrum for users Let us nvestgate the necessary and suffcent condtons for a demand vector d to be a user equlbrum followng Wardrop s prncple [22]. That prncple states that users always choose the cheapest optons, so that for a stable stuaton, all users who have the same set of avalable optons end up payng the exact same prce. It was frst ntroduced to model drver route choces n transportaton, but can easly be appled to our problem, yeldng: 1. Provders gettng some demand have the same perceved prce, whch s the cheapest one. Ths can be wrtten as d > 0 p max(1, d /C ) = mn j I p j max(1, d j /C j ). (3) Indeed, f a provder has a postve demand and a larger perceved prce than a compettor, then part of ts customers would churn to the cheapest. 2. The total amount of data that users want to successfully transmt depends on the perceved prce per successful transmsson. Ths wrtes mn(d, C ) = D(p), (4) where p := mn j I p j max(1, d j /C j ),.e., the lowest perceved prce among all provders. The left-hand sde of (4) s the total rate of successful transmsson, that takes nto account the capacty lmtatons of each provder s access network. Ths allows to formally defne the user equlbrum. Defnton 1 For gven capacty C := (C 1,..., C I ) and prce p = (p 1,..., p I ) confguratons, a user equlbrum s a demand confguraton d = (d 1,..., d I ) such that for all, j I,

8 8 d > 0 p max(1, d /C ) p j max(1, d j /C j ), ( ) mn(d k, C k ) = D mn p max(1, d /C ). k I (5) (6) Condton (5) re-expresses (3), the fact that all provders wth postve demand have the same perceved unt prce, otherwse part of the demand wll have nterest n changng provders. Condton (6) s a formulaton equvalent to (4). The assumpton that receved data s the quantty of nterest s by usng r k = mn(d k, C k ) n (5) nstead of d k f we were usng the amount of sent data. Remark that we can equvalently wrte a user equlbrum as a vector d such that (d, p) s a soluton of the system (p max(1, d /C ) p)d = 0, I p max(1, d /C ) p 0, (S) mn(d, C ) D(p) = 0, d 0, p 0. I I In the system (S), p stands for the common value of the perceved prce at all provders that get demand. The followng proposton characterzes the user equlbra correspondng to fxed capactes and prces. Proposton 3 For any capacty and prce confguraton where prces are strctly postve, there exst a (possbly not unque) user equlbrum demand confguraton. Moreover, at a user equlbrum d, the common perceved unt prce p of provders wth d > 0 s unque and equals p = mn{p : D(p) (7) (8) (9) (10) (11) f (p)}, (12) where f (p) := C 1l {p p}, (13) wth 1l X the ndcator functon, of value 1 f condton X s verfed, and 0 otherwse. Remark that we have a mn n (12), snce D s contnuously nonncreasng and f s rght-contnuous and nondecreasng for all I. Proof We follow the same steps as those taken n [1] to establsh the exstence of a soluton for the system (S). But the results of [1] do not drectly apply, due to the dstncton between demand flow d and successful flow r, thus we adapt the proof. We frst show that (S) s equvalent to the nonlnear complementarty problem descrbed

9 9 by the system (p max(1, d /C ) p)d = 0 I, p max(1, d /C ) p 0 I, ( ) mn(d, C ) D(p) p = 0, (S ) mn(d, C ) D(p) 0, d 0 I, p 0. (14) (15) (16) (17) (18) (19) A soluton of (S) s obvously a soluton of (S ). Now consder a soluton (d, p) of S : f t s not a soluton of (S), then we necessarly have p = 0 and mn(d, C ) > D(p). Ths last nequalty means that there exsts I wth d > 0, whch mples from (14) that p = p max(1, d /C ) > 0, a contradcton. Therefore the set of Wardrop equlbra corresponds to the set of solutons of (S ), whch we now show s non-empty. Frst defne a constant K 1 < + satsfyng { K 1 > max C v(c ) p K 1 > max C, whch exsts under our assumptons on D, and a constant K 2 < + such that Remark that we then have We defne the functon Φ : R I +1 R I +1 by wth Φ (d, p) = K 2 > max {K 1p /C }. (20) I, D(p K 1 /C ) < C. (21) Φ(d, p) = (Φ 1 (d, p),..., Φ I +1 (d, p)) mn(k 1,[d p max(1, d /C ) + p] + ) f I, mn(k 2,[p +D(p) j, C j )] j Imn(d + ) for = I +1, (22) (23) where [x] + stands for the postve part of x. Snce Φ s a contnuous functon that maps the cube [0, K 1 ] I [0, K 2 ] onto tself, from Brouwer s fxed pont theorem t admts a fxed pont (ˆd, ˆp). We now prove that ths pont s a soluton of (S ),.e., t s a Wardrop equlbrum. Assume that ˆd = K 1 for some I, then from (22) we have p p max(1, K 1 /C ) > 0. Thus (21) mples that D(ˆp) < C = mn( ˆd, C ) j I mn( ˆd j, C j ). Consequently, from (23) and (ˆd, ˆp) beng a fxed pont of Φ, we have ˆp = 0, whch s a contradcton.

10 10 Lkewse, f ˆp = K 2 then from (20), ˆp > p K 1 /C p max(1, ˆd /C ) I, where the second nequalty comes from K 1 > C and ˆd K 1. Ths mples from (22) that ˆd = K 1, I, whch cannot happen as proved just before. As a result, (ˆd, ˆp) s a soluton of the system { ˆd = [ ˆd p max(1, ˆd /C ) + ˆp] + I ˆp = [ˆp + D(ˆp) mn( ˆd, C )] +, whch s exactly equvalent to the system (S ). Thus we have proved the exstence of a Wardrop equlbrum. Now we consder a Wardrop equlbrum, and characterze the mnmum perceved prce p. From Condton (8), p < p d > C, (24) whle from (7) we get p > p d = 0, p = p d C. Usng Inequalty (24), then (9) and fnally (25), we get whch gves (12). 1l {p<p}c D(p) = mn(d, C ) Quanttes D(p) p 1 C 1 p 2 C 2 p 3 C 3 p p 4 C 4 1l {p p}c, Unt prce Fg. 1 Wardrop equlbrum for four provders and a gven prce confguraton: the common perceved prce at each provder wth postve demand (.e., provders 1, 2, 3) s p. Here the ntersecton occurs on an horzontal part of the starstep curve. (25) Remark 3 Fgures 1 and 2 dsplay the two possble confguratons for determnng the Wardrop equlbrum perceved prce p. Ether the starstep curve summng up the capactes at the charged prces crosses the demand functon on a horzontal part, or t happens on a vertcal part. In any case, the exstence and unqueness of p are ensured (because one curve s ncreasng whle the other s strctly decreasng), as shown n Proposton 3.

11 11 Quanttes D(p) C 4 C 2 +C 3 C 1 p 1 p = p 2 = p 3 p 4 Unt prce Fg. 2 Wardrop equlbrum for four provders and a gven prce confguraton: the common perceved prce at each provder wth postve demand (.e., provders 1, 2, 3) s p. Here the ntersecton occurs on a vertcal part of the starstep curve. Remark 4 Total demand served s therefore D(p). For all provders wth prce p p, demand d s then d = 1l {p< p}c p/p. All provders such that p = p (f any) share the remanng demand D(p) j:p j<p d j, all possble sharng wth 0 d C provdng a Wardrop equlbrum. That stuaton s llustrated n Fgure 2. In that sense, there s not always unqueness for the Wardrop equlbrum, and the correspondng revenues for each provder are not necessarly unque. Note nonetheless that the resultng total revenue s always the same. Moreover, we wll see n the followng that when provders are at a Nash equlbrum of the prcng game, then the correspondng user Wardrop equlbrum s unque. 5 Prce competton among provders In ths paper, we consder that provders settng ther prces s the upper stage of a two-level game, where the lower stage corresponds to users reactng accordng to the Wardrop equlbrum descrbed n Defnton 1. We assume that provders are aware of ther advantage of playng frst,.e., they antcpate and take nto account users reacton when determnng ther prce. That common knowledge complcates the competton among provders, and s the purpose of the analyss n ths secton. Our man result s a complete characterzaton of the Nash equlbrum of the prcng game, takng beneft from the above correspondng characterzaton of the Wardrop equlbrum. Proposton 4 Under Assumpton B, there exsts a Nash equlbrum of the prce war among provders, gven by { p = p I,, d = C ( ) where p = v j I C j, that s C = D(p ). (26)

12 12 Moreover, f cost functons (l ) are strctly ncreasng, then there s no other Nash equlbrum. In words, the proposton means that at equlbrum, all provders set the same prce, such that demand equals the total capacty of the system. Proof The proof can be decomposed nto two steps: 1. We frst show that f cost functons are strctly ncreasng, only the pont such that d = C and p = p, wth p = v ( C ) can be a Nash equlbrum; 2. then we prove that that pont s ndeed a Nash equlbrum. Remark that we do not need the strct ncreasngness of cost functons for that part. Step 1: Unqueness of the Nash equlbrum. Assume that there exsts a prce confguraton p that s a Nash equlbrum of the prcng game, and decompose the set of provders I nto three dsjont subsets: I = I s I 0 I u, where I s := { I : d > C }, (27) I 0 := { I : d = C }, (28) I u := { I : d < C }. (29) We wll show that I s and I u are empty sets, whch then mples from (7) and (9) that the prce confguraton s p = (p,..., p ). We frst prove that I s =. Assume t s not the case, and consder s I s. From (7), we have p s < p and d s = C s p/p s, leadng to R s = C s p l s ( p C s p s ). (30) Consder provder s unlaterally ncreasng ts unt prce p s to p n s, wth p s < p n s < p. Then from (12) we have D(p) and p < p D(p) > C 1l p p = C 1l p p n C 1l p p C 1l p p n, whch mples (agan from (12), but appled to the new prce profle) that the perceved prce at the new Wardrop equlbrum s unchanged: p n = p. Therefore, snce p n s < p by hypothess, Relaton (30) s stll vald wth new prces, and the revenue change for provder s s ( ) ( ) R n p p s R s = l s C s l s C s p s p n > 0, s due to the strct ncreasngness of l s. Ths contradcts the fact that p s a Nash equlbrum and as a consequence, at a Nash equlbrum, I s =. (31)

13 13 Assume now that I u at a Nash equlbrum prce profle p. Snce we necessarly have I s =, then from (9), D(p) = d < C = D(p ), wth p = v ( C ). Ths mples that p > p. (32) We frst brefly rule out the possblty that D(p) = 0: f t were the case, all provders I would have proft l (0). But any provder I unlaterally changng hs prce to p n = v(c ) would get a total demand C and obtan proft R n R = C v(c ) l (C ) + l (0). Snce v(c ) > v( C ) = p, under Assumpton A, we have R n R > C (p κ ) 0. Thus R n > R, a contradcton. Consequently, at a Nash equlbrum D(p) > 0. Now, the assumpton I u mples that there exsts a provder u such that d u < mn(c u, D(p)). (33) Indeed, there exsts at least a provder n I u, and f that provder does not verfy (33), then he gets all the demand D(p), and therefore every other provder has demand d = 0 < mn(c, D(p)) and verfes (33). Recall that every provder I has p p, from (8) and I s beng empty. We now prove that provder u can strctly mprove ts beneft by changng ts prce from p u p to p ε u := p ε for a suffcently small ε > 0. We dstngush two cases. If C u D(p), then we easly see from (12) that the new perceved prce p ε verfes p ε u = p ε < p ε p. By changng ts prce to p ε, provder u s the only provder wth the lowest declared unt prce, therefore from (7)-(8), ts new demand d ε p u equals C ε u p ε, whch tends to C u when ε tends to 0. If C u > D(p) then for ε suffcently small (such that D(p ε) C u ), provder u gets all the demand,.e., d ε u = D(p ε). When ε tends to 0, that demand tends to D(p) because of the contnuty of the demand functon. Consequently, for a suffcently small ε, the demand for provder u of swtchng from prce p u to prce p ε can be arbtrarly close to y := mn(c u, D(p)) > d u, and the correspondng revenue gan can then be arbtrarly close to p(y d u ) l u (y) + l u (d u ) (p κ ) (y d u ) }{{} >0 (p p )(y d u ) > 0, where the frst and second lne come from y C u and Assumpton A, and the last lne stems from (32). Consequently, provder u can strctly mprove ts net beneft by unlaterally changng ts declared prce, whch contradcts the Nash equlbrum condton and establshes that we necessarly have at a Nash equlbrum, I u =. (34)

14 14 Relatons (31) and (34) mply that at a Nash equlbrum, d = C for all I. Then the demand relaton (9) mples that p = p, whle (7) gves p = p for all I. At a Nash equlbrum, each provder necessarly declares unt prce p = p. Step 2: p = p, s a Nash equlbrum. We now consder the prce profle p such that p = p,. For that prce profle, we have d = C for all I, and p = p. Frst note that all provders I get a revenue larger than l (0): R + l (0) = p C (l (C ) l (0)) C (p κ ) 0, (35) where the nequaltes come from Assumpton A. Let us now prove that no provder has an ncentve to change hs prce f all the others keep ther prce to p. Wthout loss of generalty, consder a possble move of provder 1 from p to p n 1 p. We dstngush two cases. If p n 1 < p, then D(p ) = C = C 1l {p p n } and p < p D(p) > C C 1l {p p n }, whch from (12) means that p n = p. Therefore, (8) and (7) mply that d n 1 = C 1 p /p n 1. The revenue dfference for provder 1 s thus R1 n p R 1 = l 1 (C 1 ) l 1 (C 1 p n ) 0, 1 where the last nequalty comes from the nonncreasngness of l 1. Remark that R n 1 < R 1 f l 1 s strctly ncreasng. If p n 1 > p, then p p D(p) > 1 C C 1l p p n. Moreover, snce all provders I have a prce p n = p p n 1, then D(p n 1) D(p ) = 1l {p n 1 p n } C. As a result, from (12) the new perceved prce p n s such that p < p n p n 1. If p n < p n 1 then d n 1 = 0 from (7), and R n 1 = l 1 (0) R 1 from (35). If p n = p n 1 then (8) mples that d n 1 C 1 and d n > C for all 1. Therefore(9) mples that d n 1 = D(p n 1) 1 C,

15 15 and the revenue change for provder 1 s R1 n R 1 =p n 1d n 1 l 1 (d n 1) p C 1 + l 1 (C 1 ) p n 1 C p C1 D(p n 1 ) 1 +κ 1 ( C D(p n 1)) <p n 1D(p n 1) p D(p ) + κ 1 (D(p ) D(p n 1)) =(p n 1 κ 1 )D(p n 1) (p κ 1 )D(p ). (36) where the second lne comes from Assumpton A, and the thrd one from p n 1 > p and D(p ) = C. Now consder the functon g(y) := (y κ 1 )D(y) (p κ 1 )D(p ). (37) We have g(p ) = 0. Moreover, g s dervable on [p, p n 1], and ts dervatve has the same sgn as yd (y) 1 +, whch s nonpostve under Assumpton B. Consequently, g(p n 1) g(p ), and gong back to (36) we have R1 n < R 1, concludng the D(y) 1 κ 1/y proof. 6 Can provders le on ther capactes? In the prevous sectons, we assumed that the total capactes (C ) were common knowledge of all partcpants. Whle ths may not be true n realty, we may consder that provders be asked to declare ther capacty level at the very begnnng of the nteracton,.e., before choosng ther prces, or that the used capactes have been learnt. A queston that then naturally arses, snce provders are stll assumed to be selfsh, s related to the capacty declaraton strategy: s there an nterest to le on one s capacty? In ths secton, we answer negatvely to that queston, by provng that truthfulness s a domnant strategy for provders under Assumpton B. As a result, even f we add a thrd level -a game on declared capactes- on the consdered nteracton -game on prces plus user choces-, there s stll a unque equlbrum, that s socally effcent. To establsh that result, we assume now that each provder I has to declare ts capacty value C, and denote by C dec the value that t chooses to declare. Frst, remark that only the declared values C dec C are feasble: whereas provder can easly artfcally degrade ts servce rate, t cannot ncrease t above ts real capacty C : a false declaraton amed at ncreasng one s demand to get a larger beneft would be detected. We assume that the capacty declaraton occurs before the provders set ther prce,.e., they commt to a certan servce rate C dec. Then from Proposton 4, provders know that prce competton wll lead to a unque Nash equlbrum where all provders declare the same unt prce ( ) p NE = v, (38) C dec

16 16 and each provder gets demand C dec. Provders should therefore use that knowledge when choosng the capacty level to declare. Focusng on the net revenue of a provder, there are two opposte effects of declarng a falsely low capacty C dec < C nstead of the real capacty C : snce the total avalable capacty decreases, from (38) the unt sellng prce at equlbrum ncreases, and the managng cost decreases because the quantty sold decreases; on the other hand, less quantty sold means less revenue. The next proposton gves a suffcent condton for the latter effect to overcome the former. Proposton 5 Consder that provders can artfcally lower ther capacty. Under Assumpton B, truthfully declarng one s real capacty s a domnant strategy for each provder. Moreover, all provders truthfully declarng ther capactes s the only Nash equlbrum of the capacty declaraton game, and s a strct equlbrum. Frst recall from Game Theory that a strct Nash equlbrum s a strategy profle such that each player s strctly worse off by any unlateral devaton. Proof Wthout loss of generalty, we prove that provder 1 strctly decreases ts revenue by declarng a capacty C1 U < C 1, when each compettor I \{1} declares C dec C. Frst remark that due to the nonncreasngness of v, Assumpton B stll holds wth declared capactes and the correspondng prce p = p NE. Therefore, the equlbrum of the prce competton game s unque and gven by Proposton 4 wth those declared capactes. In other words, f we defne C 1 dec := 1 Cdec, the unt prce p NE at the prce competton equlbrum s then p NE = v(c dec 1 + C dec 1 ). (39) Each provder I gets demand C dec, and gets total beneft R = C dec p NE l 1 (C1 dec ). Notce that p NE p. Now let us compare any untruthful declaraton C1 dec = C1 U < C 1, leadng to provder 1 revenue R1 U, to the truthful declaraton C1 dec = C 1, wth provder 1 revenue R1 T. We have R U 1 R T 1 =C U 1 v(c U 1 +C dec 1 ) C 1 v(c 1 +C dec 1 ) + l 1 (C 1 ) l 1 (C U 1 ) =(D(p U ) C dec 1 )p U +(D(p T ) C dec 1 )p T +l 1 (C 1 ) l 1 (C U 1 ), where p U := v(c1 U + C 1 dec ), and p T := v(c 1 + C dec D(p U ) = C1 U + C 1 dec nonncreasng. From Assumpton A, we have 1 ). The second equalty comes from and D(p T ) = C 1 + C 1 dec. Remark that p U > p T snce D s R U 1 R T 1 (D(p U ) C dec 1 )p U + (D(p T ) C dec 1 )p T + κ 1 (C 1 C U 1 ) = (p U p T ) C 1 dec + p U D(p U ) p T Dp T + κ 1 (C 1 C1 U ), }{{}}{{} <0 =D(p T ) D(p U ) (p U κ 1 )D(p U ) (p T κ 1 )D(p T ). (40)

17 17 The last lne, taken as a functon of p U, s of the same form as n (37), and s therefore nonncreasng for p U p T under Assumpton B. Snce t s null at p T, then p U > p T yelds R1 U R1 T, whch proves that truthful declaraton s a domnant strategy. As a result, all provders beng truthful s a Nash equlbrum of the prce declaraton game. Remark that as soon as one compettor declares a non-zero capacty, then C 1 dec > 0, and the nequalty n (40) s strct. Therefore, the (truthful) Nash equlbrum s a strct equlbrum, and the only possble other equlbrum would consst n all provders declarng a null capacty,.e., C dec = 0 for all I. We now exclude that possblty. Consder provder 1, and assume all ts compettors declare a null capacty,.e., C 1 dec = 0. By declarng a null capacty C1 dec = 0, provder 1 would get a total revenue R1 U = l 1 (0). However declarng ts true capacty C 1 and settng ts prce to v(c 1 ) would yeld a revenue R1 T = C 1 v(c 1 ) l 1 (C 1 ). Under Assumpton A, the revenue gan s therefore R1 T R1 U C 1 (v(c 1 ) κ 1 ) > 0, where the strct nequalty comes from the strct decreasngness of v. As a result, provder 1 has an nterest to devate from the stuaton where provders declare zero capacty, whch rules out that stuaton for beng a Nash equlbrum, and concludes the proof. As a consequence of Proposton 5, even f provders have the possblty to artfcally reduce ther servce capacty before fxng ther prces, the fnal outcome of the competton game stll corresponds to the socally effcent stuaton ponted out n Proposton 2. 7 Concluson Ths paper provdes an analyss of a prcng game among compettve telecommuncaton servce provders wth potentally dfferent but fxed capactes. Accordng to the prce profle, we have been able to defne and characterze the demand repartton for selfsh nfntesmal users, applyng Wardrop s prncple. Usng the knowledge of what ths repartton would be, provders can play wth ther prce n order to maxmze ther revenue. We have proved the exstence of a unque Nash equlbrum for that game, where all provders set the same prce, for whch demand exactly meets the sum of capactes. We have also establshed that provders have no ncentve to artfcally create some congeston by declarng a falsely low capacty. It turns out that those nteractons among selfsh agents (provders and customers) lead to an outcome that maxmzes socal welfare,.e., avalable network resources are optmally used. As extensons of ths work, we would lke to nvestgate the vablty of (or to defne rules to make vable) scenar that mght be more specfc to wreless. For nstance when a provder s a vrtual operator leasng capacty to a compettor ownng a lcense. Other scenar of nterest would regard cogntve networks,.e., the case when unused capacty can be used by secondary users. In general, consderng a capacty expanson game s also an nterestng ssue. Indeed, capacty can be an mportant parameter provders can play wth, at the same tme as prces: what would the resultng equlbrum be? Acknowledgements The authors would lke to acknowledge the support of Euro-NF Network of Excellence, and the French Natonal Research Agency through the CAPTURES project and the WINEM project for the second author.

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