Analysis of Price Competition in a Slotted Resource Allocation Game

Transcription

1 Analysis of rice Cometition in a Slotted Resource Allocation Game atrick Maillé TELECOM Bretagne 2, rue de la Châtaigneraie, CS Cesson-Sévigné Cedex, FRANCE atrick.maille@telecom-bretagne.eu Bruno Tuffin IRISA/INRIA Camus Universitaire de Beaulieu Rennes Cedex, FRANCE btuffin@irisa.fr Abstract ricing telecommunication networks has become a highly regarded toic during the last decade, in order to coe with congestion by controlling demand, or to yield roer incentives for a fair sharing of resources. On the other hand, another imortant factor has to be brought in: there is a rise of cometition between service roviders in telecommunication networks such as for instance the Internet, and the imact of this cometition has to be carefully analyzed. The resent aer ertains to this recent stream of works. We consider a slotted resource allocation game with several roviders, each of them having a fixed caacity during each time slot, and a fixed access rice. Each rovider serves its demand u to its caacity, demand in excess being droed. Total user demand is therefore slit among roviders according to Wardro s rincile, deending on rice and loss robability. Using the characterization of the resulting equilibrium, we rove, under mild conditions, the existence and uniqueness of a Nash equilibrium in the ricing game between roviders. We also show that, remarkably, this equilibrium actually corresonds to the socially otimal situation obtained when both users and roviders cooerate to maximize the sum of all utilities, this even if roviders have the oortunity to artificially reduce their caacity. I. INTRODUCTION The Internet, and more generally telecommunication networks, have rogressively switched from an academic or monoolistic network to a commercial one with cometitive service roviders. In order to get a return on investment, each rovider has to define a ricing strategy to charge users for the service they exerience. ricing has at the beginning been seen as a way to coe with congestion, to control demand, to deal with and satisfy heterogeneous alications with different quality of service (QoS). It has also been regarded as a way to introduce fairness among users with resect to the traditional flat-rate ricing where light consumers ay as much as big ones. Therefore, there have been many roosals for new ricing schemes motivated by different objectives: the network lanner may want to elicit users to efficiently share the scarce network resources in order to maximize social welfare (see, among others, [1] [4]), to guarantee fairness among users [5], [6], or to maximize revenue [7] [10]; the tyical modeling tool being that of noncooerative game theory [11]. For surveys on ricing in telecommunication networks, the reader is advised to look at [12] [14]. A very large roortion of aers deal with the monoolistic case, where there is only one rovider. Though, telecommunication networks have become highly cometitive and it seems rimordial to us to deal with that cometition in ricing models when defining the otimal rices, since cometition may highly affect the results of rice determination (while ricing in a monoolistic context generally means a single level of game between users, cometition actually introduces an additional level of game, between roviders, resulting in a so-called Stackelberg game [11]). Some tyical illustrations of cometition are: for wired access, DSL users can choose among several cometing roviders to connect to the Internet; the case of wireless access is more flexible. For examle a user wishing to connect to a WiFi hotsot may be located in a zone covered by several wireless access roviders, and can choose which rovider to use for the time of his connection. The same user can/will even be able to choose between different and cometitive transmission latforms: WiFi, WiMA, 3G, Wired oerators, with a ossible combination of all those ones (the so-called multihoming). Our goal in this aer is to introduce a ricing model dealing with cometition. In our model, time is discretized, divided intoslots,andeachroviderhasthecaabilitytoserveagiven number of ackets er slot. We assume that roviders do not share a common limited amount of caacity/bandwidth, but instead each rovider has its own service caacity: it can model for instance cometition between 3G, WiFi and WiMA roviders for instance. Note that it does not corresond to cometition at a WiFi hotsot if roviders share the same bandwidth, but rather to the case where roviders are being oerated on different frequency channels and using different HY modes. In our model, as soon as demand exceeds caacity at a rovider, the excess demand is lost (ackets are selected randomly). We consider a ricing scheme insired by the (monoolist) one introduced in [15], where users are charged for the number of ackets they submit regardless of their being treated or lost, in order to incentivize them to limit their demand. We show existence and uniqueness of a Wardro equilibrium for demand at each rovider for every combination of (fixed) rices, such that each user chooses the /08/$ IEEE 1561

2 rovider with the least erceived cost, where erceived cost is exressed in terms of charge and dro robability. Remark that this kind of modeling behavior is also of interest in the case of multihoming, when users are able to slit their traffic between roviders in order to kee the lowest overall cost. From this Wardro equilibrium, we show, under mild conditions, that there is a unique Nash equilibrium among cometing roviders for the game consisting of setting rices, and we characterize that equilibrium. Those rices reresent a oint where no rovider can increase its benefits by changing unilaterally its rice. An imortant roerty is that the resulting rices corresond to the socially otimal situation, where the sum of utilities of all agents in the system -including users and roviders- is maximized. This is a very desirable roerty, in favor of the alication of such ricing strategies, since usually noncooeration leads to a loss of efficiency, quantified by the so-called rice of anarchy. Another issue that we address is the interest for a rovider to declare or use only a art of its real caacity. Indeed, it may haen that, due to congestion, serving less users, therefore at a higher market rice, results in a larger revenue. We show here that in this context of cometition, it cannot occur if demand is sufficiently elastic. A. Related work on cometition Studying the imact of rovider cometition on ricing schemes is a quite new toic which is receiving increasing attention in the networking community. The imortance of this field has been highlighted in [16], where it was shown that the very romising aris Metro ricing (M) scheme, which just consists in searating the network into disjoint networks served in the same manner but with different access rices, does not allow service differentiation under cometition at equilibrium. In [17], a model where cometitive roviders lay both on rice and on a QoS arameter is used, and demand at each rovider is driven by an arbitrary function deending on the arameters of all cometitors. We feel that using a unique total demand function that is slit between roviders thanks to Wardro s rincile, i.e. all users choosing the rovider with cheaest erceived cost, is more relevant. [18] studies cometition for e-services, with also a kind of Wardro equilibrium, but where QoS does not deend on demand. In other secific contexts, cometition between wireless oerators in the case of a shared sectrum (more flexible and leading to a more efficient management of the sectrum) has been studied in [19], [20]. In [19], oerators are charged by a central entity for the amount of bandwidth they use, and therefore try to design roer service offers for users. Cometition is shown to increase users accetance robability for offered service. Our analysis is based on a less secific network modeling (and without cometition for caacity between roviders). In [20], the authors discuss a similar cometition roblem, but oerators only lay with the ower of the ilot signals of their base stations, and no ricing is considered. On the other hand, note that there is an increasing bunch of works looking at indeendent and selfish roviders on a ath, that forward traffic of cometitors to ensure end-to-end delivery [21] [23], but do not consider a direct cometition for users between roviders, a different ersective that we adot. The closest works to ours are that in [24], [25], where roviders are reresented by arallel links and the quality of service is delay. The rice of anarchy, measuring the loss of efficiency due to cometition with resect to cooeration, is determined, for fixed demand in [24] and random demand but linear delay in [25]. This is extended to the case of arallel-serial links in [26], for fixed demand. Those models consider that users are sensitive to the sum of the rice charged and a congestion-deendent delay, and roviders are only sensitive to their revenue. Our model is different since we assume that the total cost erceived by a user is the rice charged multilied by a congestion factor, the externality being losses here (which can be considered closer to some wireless environments). Moreover we consider that roviders exerience managing costs that increase with demand, those costs not being erceived by users. Remark that the tye of slotted and caacity-based model we are dealing with can be related to the one in [15], where a similar slotted caacity model is used, but with several riority traffic classes, and in the case of a monooly instead of an oligooly here. We therefore have a different goal: study rice war between roviders instead of rice discrimination among users. B. Organization The aer is organized as follows. In Section II, we resent our basic model and the required assumtions. Section III describes the socially otimal allocation when all roviders cooerate, this allocation being used later on to investigate the rice of anarchy for non collaborating roviders. Section IV investigates the demand reartition among roviders for fixed rices, following Wardro s rincile. Section V is devoted to the Nash equilibrium for the ricing game between roviders; the quite long roof of the main result (existence ands uniqueness) is deferred to the aendix. Finally, in Section VI, we consider the interest for roviders to artificially decrease their caacity, and rovide our conclusions and directions for future research in Section VII. II. GENERAL MODEL Consider a model where time is discretized, divided into slots. Assume that there is a set I := {1,...,I} of roviders in cometition at an access oint (I 2), rovider i (i I) having the caacity of serving C i ackets (or units, seen as a continuous number) er slot and asking a rice i er acket. If demand exceeds caacity at a given rovider, demand in excess is lost. We assume that lost ackets are chosen uniformly over the set of submitted ones. If d i is the total demand at rovider i, the number of served ackets is actually min(d i,c i ), meaning that ackets are actually served with robability min(c i /d i, 1). Usersareassumedtobechargedforeachsubmitted acket instead of each served one in order to incentivize them to limit 1562

3 demand. The average erceived rice er served traffic unit at rovider i is therefore = i / min(c i /d i, 1) = i max(d i /C i, 1). Indeed, the total income of rovider i is d i i and the total service rate is d i min(c i /d i, 1), giving the unit erceived rice just above by a direct ratio. In that situation, it can be seen that the negative externality of congestion is exressed in terms of losses exerienced by users. We assume that total user demand is a function D( ) of the erceived rice, andthatd is continuous and strictly decreasing with on its suort [0, max ), with eventually max =+. We moreover assume that D(0) > C i, i.e. there is some congestion: the total resource available is not sufficient to satisfy the maximum demand level. Remark that the demand function can be interreted in two ways (or as a combination of those two effects): it can stem from user heterogeneity in their willingnessto-ay for the service: consider a continuum of infinitesimally small users with fixed demand, whose willingnessto-ay (in terms of unit rice) is distributed according to a given distribution. Then for a given unit rice 0, D() is the amount of flow generated by users with valuation larger than and is naturally decreasing. Likewise, the decreasingness of the total demand function D can also stem from the decreasingness of individual functions. We can also define the function v : q 7 inf{ : D() q} (with the convention inf =0). Since D is continuous, strictly decreasing for < max and null for max for a given max then we simly have v(q) = D 1 (q) if q (0,D(0)) max if q =0 0 if q D(0). The quantity v(q) is then the unit rice that one has to imose on users in order to ensure that total demand will be q (for q D(0)). The function v is called marginal valuation function: v(q) is indeed the maximum unit rice that can be charged for the q th unit of demand without making the demand decrease: it is a nonincreasing function since D is nonincreasing. We finally define V (q) as the sum of the marginal valuations of the q units of users with largest willingness-to-ay, i.e. V (q) := Z q x=0 v(x)dx. As an examle, if all users exerience a fixed unit rice, then only the q = D() units with valuation larger that will subscribe to the service, and the sum of their marginal valuations will be V (q). We refer to this function V as the valuation function: V (q) corresonds to the total rice that the q units of demand with highest marginal valuation are willing to ay to be served (remark that the function V is nondecreasing and concave). For a fixed unit rice, the user surlus then equals V (D()) D(). (1) A first ste of our work will be to study how, for fixed rices i (i I), total demand is slit among roviders. This is described and characterized in Section IV in terms of a Wardro equilibrium. Knowing a riori the distribution d := (d 1,...,d I ) of demand that will result from a rice rofile, the goal of each rovider is, by laying on its unit rice i, to maximize its net benefit R i ( 1,... I ):= i d i i (d i ), where i d i is the money earned directly from demand, and i (d i ) reresents the cost for rovider i of managing a demand level d i. We assume that for all i, i is nondecreasing, differentiable and convex. Notice that contrary to the model considered in [24], this cost is not reflectedhereinthequality of service exerienced by users, but is only erceived by roviders. The rice chosen by a rovider has an imact on demands, and therefore on benefits, of other roviders; consequently our model induces a game between roviders. Since we consider cometitive roviders, the framework is that of noncooerative game theory. We are going to investigate in Section V, under mild conditions, the existence and uniqueness of a Nash equilibrium for the rice game, that is a rice vector := ( 1,..., I ) such that no rovider can increase its own benefit by unilaterally changing its access rice, i.e., i I, 0, R i ( 1,..., I ) R i ( 1,..., i 1,, i+1,..., I ). Most of our results are valid under the following assumtion. Assumtion A: The marginal cost of every rovider when its demand equals its caacity is lower than the global marginal valuation of the sum of all rovider caacities. In other terms, i I, 0 i (C i) v C j, j I or equivalently i I, D( 0 i(c i )) j I C j, where 0 i is the derivative function of i. We will sometimes need a slightly more restrictive assumtion. Assumtion B: For each rovider i I, the following inequality holds: Ã! 0 C i i(c i ) 1 j6=i C v C j. (2) j j I With resect to Assumtion A, Assumtion B is more restrictive esecially if a rovider has a significant ower in terms of caacity, i.e. if C i is not negligible. In j6=i Cj the extreme case of a large number of roviders with the same caacity C i = C, Assumtion B simly resumes to 1563

4 Assumtion A. In general, it is reasonable to consider that the managing costs are small, if not negligible with resect to the rice users are willing to ay for the service. Therefore we exect Assumtion B to hold in networks where cometition occurs. III. SOCIALLY OTIMAL SITUATION In this section, we define social welfare as the sum of the utilities of all agents in the system (i.e. users and roviders), and study the maximal value of that criterion. It is well-known in Game Theory [27] that agent selfishness does not lead in general to a socially efficient situation. The loss of efficiency due to the divergence of user interests, often referred to as the rice of Anarchy [28], is an interesting erformance measure for a game: if social welfare at an equilibrium of the game is close to its maximal value, then letting agents choose their actions selfishly can be referable to introducing costly control or incentive schemes. Definition 1: For a demand configuration d := (d 1,...,d I ), we call social welfare the quantity Ã! SW := V min(d i,c i ) i (d i ). (3) Social welfare SW accounts for the utilities of users (the quantity min(d i,c i ) is the total effective user rate, and the first term of (3) is total user valuation) and of roviders (the second term accounts for the costs associated to the demand configuration). Remark that no monetary exchanges aear in (3): this is due to the fact that such exchanges would be added to the roviders utility and subtracted from the users utility, which would not affect the sum of all agent utilities. We comute here the maximal value of social welfare, that will be used as a reference in the next sections. roosition 1: Under assumtion A, the maximum value of social welfare is reached when d i = C i for each rovider i. roof: The social welfare maximization roblem is exressed by Ã! max V min(d i,c i ) i (d i ). (4) d R I + Notice that since i is strictly increasing for all i, the objective function is strictly decreasing in d i for all i I when d i C i, therefore our otimization roblem is equivalent to Ã! max V d i i (d i ) d R I + subject to d i C i i I. Since V is a concave function and the rovider cost functions ( i ) are convex, this last roblem is a classical convex roblem (maximization of a concave function over a convex set), that can be solved by the Lagrangian method. Denoting by λ i 0 the Lagrange multilier relative to the constraint d i C i,thefirst order conditions imly that for a demand configuration (d 1,...,d I ) maximizing social welfare, we have i I, λ i + 0 i(d i )=, (5) with := v d i (i.e. d i = D( )). Then the comlementary slackness conditions yield i I, min(λ i,c i d i )=0. (6) Let us define I 0 := {i I : d i = C i} and I u := {i I : d i <C i }, and assume that I u 6=. Thenforani 0 I u,(6) imlies that λ i0 =0, and we have C i > d i + C i = d i = D( ) u 0 = D( 0 i 0 (d i 0 )) D( 0 i 0 (C i0 )) C i, where the second line comes from (5), the third one from the convexity of i0 and the nonincreasingness of D, and the last line stems from Assumtion A. We reach a contradiction, which means that I u =, and establishes the roosition. IV. WARDRO EQUILIBRIUM FOR USERS In this section, we investigate how demand is slit among roviders when the rice i erunitofsenttraffic isfixed by each rovider i I. We assume that users are infinitely small and therefore their choices do not individually affect the demand levels (and therefore the erceived costs) of the different roviders: such users are said to be rice-takers since they individually have no influence on rices. The outcome resulting from such user interactions is described by Wardro s rincile [29]: demand is distributed in such a way that all users choose one of the cheaest roviders. As a result, the user erceived rice is the same for all roviders with ositive demand, and is lower than the unit rice i of roviders i with no demand. Since we considered the case of elastic demand, the total demand level must also corresond to the common erceived rice on all roviders that receive some demand. Those roerties are summarized in the next definition. Definition 2: For given caacity C := (C 1,...,C I ) and rice =( 1,..., I ) configurations, a user equilibrium is a demand configuration d = (d 1,...,d I ) such that for all i, j I, d i > 0 i max(1,d i /C i ) j max(1,d j /C j ), (7) Ã! d i > 0 i max(1,d i /C i )=v d k. (8) k I According to Condition (7), all roviders with ositive demand have the same erceived unit rice, otherwise art of the demand will have interest in changing roviders. Condition (8) states that total demand corresonds to that common value of the erceived unit rice via the demand function D. Remark that this definition corresonds to the one rovided in [25], but for a different ricing scheme and different cost functions. The following roosition characterizes the user equilibria corresonding to fixed caacities and rices. 1564

5 roosition 2: For any caacity and rice configuration, there exist a (ossibly not unique) user equilibrium demand configuration. Moreover, at a user equilibrium d, the common erceived unit rice of roviders i with d i > 0 is unique and equals =inf{ : D() f i ()}, (9) C i where f i () :=1l { i }. (10) i roof: Since the erceived unit rice functions for each rovider are continuous and nondecreasing, for any value of the total demand d thereexistademandconfiguration d such that i d i = d and condition (7) is satisfied. Moreover the common value c (d) of the erceived unit rice at roviders with strictly ositive demand is unique when total demand is fixed (this was established in [30]) when erceived unit rices are continuous and nondecreasing functions of demand. It is easy to see that c (d) is a continuous and nondecreasing function of total demand d. The second roerty, given by condition (8), that a user equilibrium must satisfy can be exressed as c (d) =v(d). Since v is a continuous nonincreasing function and c is continuous, nondecreasing and tends to infinity as d becomes large ( c (0) = min i i and for d sufficiently large c is a strictly increasing linear function of d), then if v(0) min i i then there exists a d such that c (d) = v(d). The corresonding value := v(d) is unique due to the resective monotonicity roerties of c and v. We now rove that verifies (9). Since we assumed that v(0) = max min i i, we necessarily have d = D() from (1). Moreover (7) imlies that d i = f i (), so D() = f i(). Relation (9) then comes from max and the strict decreasingness of D on (0, max ). if v(0) < min i i then there is no intersection oint between c and v, since no user is interested in subscribing to the service at a unit rice min i i. Therefore the configuration d = (0,...,0) is the unique user equilibrium. Figure 1 dislays the demand function D and the function f i for a given rice configuration, and illustrates the existence and uniqueness of the Wardro equilibrium erceived rice. Function f i basically reresents the corresonding share of demand 1 that rovider i can get at a given erceived rice. Remark 1: From this roosition, we are able to characterize the unique erceived rice. Total demand is therefore D(). For all roviders with rice i 6=, demandd i is then d i = f i (). All roviders such that i = (if any) share the remaining demand D() i: d i< i, all ossible sharing roviding a Wardro equilibrium. In this sense, there 1 f i is indeed the generalized inverse of the erceived unit rice function d i 7 i max(1,d i /C i ),i.e.f i () =su{d i : i max(1,d i /C i ) }. quantities d 2 d 1 D() d 3 C 3 0 C 2 i f i () C = unit rice Fig. 1. Wardro equilibrium for three roviders and a given rice configuration: the common erceived rice at each rovider with ositive demand (i.e. roviders 1, 2, 3) is = 3. is not always uniqueness for the Wardro equilibrium and the corresonding revenues for each rovider are not necessarily unique. Note nonetheless that the resulting total revenue is always the same. Moreover, we will see in the following that when roviders are at a Nash equilibrium of the ricing game, then the corresonding user Wardro equilibrium is unique. C 4 V. RICE COMETITION AMONG ROVIDERS In this aer, we consider that roviders setting their rices is the first stage of a two-level game, where the second stage corresonds to users reacting according to the Wardro equilibrium described in Definition 2. We assume that roviders are aware of their advantage of laying first, i.e. they take into account users reaction when determining their rice. This common knowledge comlicates the cometition among roviders, and is the urose of the analysis in this section. Before analysing the ricing game between roviders, we first rove a lemma that establishes some monotonicity results of Wardro equilibria with resect to the rice configuration. In the following, we often comare two situations from the oint of view of one rovider to study the consequences of its rice decisions. We therefore use the suerscrit n torefer to the values (rice, demand, benefit) corresonding to a new situationincontrasttothereferencesituation. Lemma 1: Consider a rice configuration =( 1,..., I ), and i I. If rovider i raises its rice, i.e. chooses n i > i while all other roviders j 6=i kee their rice to j, then the common erceived rice (for roviders with ositive demand) increases: n, if d i > 0 then the demand of rovider i strictly decreases: d n i <d i. roof: Since n i > i then from (10) we have the functional inequality fi n f i. Therefore, (9) imlies that for 1565

6 all ε, 0 <ε, D( ε) >D() j I f j () j I f j ( ε) j I f n j ( ε), which yields n. Now assume that d i > 0 and that d n i d i. The equality = i max(d i /C i, 1) imlies that n >. Therefore the demand on all links j 6=i should strictly increase to reach the same erceived rice n. This would mean that total demand increases, which is in contradiction with n > and the nonincreasingness of total demand D. We can now establish our main result. roosition 3: Assume that i I, i is strictly increasing and convex. Under Assumtion B, there exists a unique Nash equilibrium on rice war among roviders, given by i I, ½ di = C i i =, ³ where = v j I C j, that is C i = D( ). (11) The roof is quite technical and is left to the aendix. The roosition basically states that the only equilibrium is such that demand matches caacity for all roviders, all of them setting the same unit rice. Remark 2: roosition 3 establishes that the demand configuration at the user Wardro equilibrium corresonding to the cometitive Nash equilibrium of the rovider ricing game is exactly the demand ointed out in roosition 1. In other terms, letting roviders setting selfishly their rices and users choosing selfishly their rovider yields the same social welfare as a erfect coordination mechanism would have given. This suggest that when congestion and cometition are sufficient (in order for Assumtion B to hold), no regulation schemes are needed since the market itself determines the right rices and allocations. Remark 3: If Assumtion B is not verified, for examle if one rovider i has a strong ower in the sense that C i > j6=i C j, then roosition 3 does not hold. In that case it is ossible that the dominating rovider take benefit of its osition to imrove its revenue by setting a high rice i >. VI. GAME ON DECLARED CAACITIES In this section, we assume that a rovider i I may voluntarily declare a false value Ci n C i of its caacity C i. Only the declared values Ci n C i are feasible: whereas rovider i caneasilydegradeartificially its service rate, it cannot increase it above its real caacity C i, and a false declaration aimed at increasing one s demand to get a larger benefit would consequently be detected. From the oint of view of a rovider s net benefit, there are two oosite effects of lowering one s caacity: on the one hand the unit selling rice at equilibrium increases and the managing cost decreases because the quantity sold decreases, whereas on the other hand less quantity sold means less revenue. The effect that overcomes the other deends on the elasticity of demand, i.e. the extent to which total demand is affected by variations of unit rice. Recall that when the demand function D is differentiable, the elasticity of demand at a unit selling rice is defined by D 0 () D(). Remark that for all, the elasticity at is a negative number due to the nonincreasingness of D. Since the demand is the inverse function of the marginal valuation v, a small demand elasticity (in absolute value) means that v decreases quickly with the unit rice. In such a case, a small decrease of total demand corresonds to a large rice increase, and the ositive effect on revenue of underdeclaring one s caacity exceeds the negative one. The next roosition gives a sufficient condition on demand elasticity for that situation not to occur: if demand is sufficiently elastic (i.e. absolute value of elasticity larger than 1), then roviders have interest in truthfully declaring their caacity. roosition 4: Under Assumtion B, if the absolute value of demand elasticity is larger than 1 for,i.e. D 0 (), 1, (12) D() then no rovider can increase its revenue by artificially lowering its caacity. roof: Without loss of generality, it is sufficient to rove that the net benefit of rovider 1 always decreases when it underdeclares its caacity. For any value C1 n C 1 of the declared caacity of rovider 1, Assumtion B still holds with declared caacities, and the equilibrium of the rice cometition game is therefore given by roosition 3. If we define C 1 := i6=1 C i, the unit rice n at the rice cometition equilibrium is then n = v(c n 1 + C 1 ), (13) each rovider i 6= 1gets demand C i, and rovider 1 obtains demand C1 n and gets total benefit R n 1 = C1 n n 1 (C1 n ). Notice that n due to the nonincreasingness of the marginal valuation function v. From (13), the total demand should therefore verify C n 1 + C 1 = D( n )=D( )+ Z n D( ) D( ) C 1 Z Z n D() d n 1 d, D 0 ()d where the second line comes from (12), and the third one from the nonincreasingness of total demand and (13) which imly that D() C 1 for all [, n ]. We then obtain C n 1 C 1 C 1 log( n / ):= d 1. (14) 1566

7 From the convexity of 1 and Assumtion A (imlied by Assumtion B), we have R n 1 n d1 1 ( d 1 ). By truthfully declaring C 1, rovider 1 would get a net benefit R 1 = C 1 1 (C 1 ). The gain of underdeclaring one s caacity can thus be uerbounded: R n 1 R 1 n d1 C (C 1 ) 1 ( d 1 ) n d1 C 1 +(1 C 1 /C 1 ) (C 1 d 1 ) = C 1 ( n ) + log( n / )( (C 1 C 1 ) n C 1 ). where the second line stems from the convexity of 1 and Assumtion B, and the third one from (14). Define x := n / 1. We can thus write R n 1 R 1 g(x), (15) where g : x 7 C 1 (x 1)+ log(x)((c 1 C 1 ) xc 1 ). It is then straightforward to check that g is concave, g(1) = 0, g is derivable and g 0 (1) = 0. Those three oints imly that g(x) 0 for all x 1, which from (15) means that R n 1 R 1 and gives the roosition. VII. CONCLUSION In this aer, we have analyzed a ricing game among service roviders with fixed caacities, such that each rovider has an access rice it can lay with. We have characterized how demand will be naturally slit between those roviders, following Wardro s rincile, and determined the existence and uniqueness of the Nash equilibrium of the ricing game. We have shown that this equilibrium corresonds to the socially otimal oint (meaning that the rice of anarchy is 1), and discussed the interest for roviders to voluntarily reduce their caacity. There are different ways to extend the results we have obtained. First of all, we could investigate the case where roviders share, at least artially, their caacities: does it lead to a rice war? Another interesting issue concerns the caacity exansion game. Indeed, caacity could also be an imortant arameter roviders can lay with, at the same time as rices: what would be the resulting equilibrium? Finally, it would be of interest to extend the game to a multiclass system with different riority levels at each rovider similarly to what was done in [15] for the monoolistic game. ACKNOWLEDGMENT The authors would like to thank New-Zealand ISAT rogram for igniting this research activity, and acknowledge the suort of Euro-FGI Network of Excellence through the secific research roject CA (Cometition Among roviders), as well as WINEM ANR roject for the second author. AENDI We rove here roosition 3. roof: The roof can be decomosed into two stes: 1) We first show that the oint such that d i = C i and i = i, with = v C i defines a Nash equilibrium; 2) then we rove that no other oint can be a Nash equilibrium. Ste 1: i =, i is a Nash equilibrium. Note firstthatinordertohaver i 0,i.e., i d i i (d i ) 0 at the equilibrium oint, means that C i i (C i ) 0, i.e.here i (C i ) C i v j I C j. This last inequality indeed holds under Assumtion A in the realistic case when i (0) = 0, due to the convexity of i for all i. We begin by establishing that the rice configuration with i = for all i where is given by (11) is indeed a Nash equilibrium of the ricing game. First notice that in this situation, the common erceived rice for users is,andthe demand of each rovider is d i = C i. We need to rove that in such a rice configuration, no rovider can imrove its revenue by unilaterally changing its suggested rice. Without loss of generality, consider a ossible move of rovider 1 from to n 1 6=. We distinguish two cases. If n 1 < then alying Lemma 1 to a change from to n 1 we get that the new erceived rice n at the Wardro equilibrium is lower than the original erceived rice: n, because of a smaller function f 1 while the others remain unchanged. the new allocation d n 1 at rovider 1 is strictly ositive, therefore from Lemma 1 it is strictly above the original one: d n 1 > C 1, which imlies that d n 1 = C 1 n / n 1 from (8). Consequently, the revenue gain of rovider 1 for lowering its rice is R n 1 R 1 = n 1 d n 1 C (C 1 ) 1 (d n 1 ) = ( n ) C (C 1 ) 1 (d n 1 ) {z } {z } <0 <0 < 0. If n 1 > then from Lemma 1 we have n and d n 1 < C 1. First notice that if n 1 inf{ : D() j6=1 C j/ } (i.e. the situation dislayed in Figure 2), then d n i =0and rovider 1 gets a null rofit, that is lower than the rofit yielded by laying as noticed at the beginning of the roof. Therefore only the case when d n 1 > 0 needs to be roved. Actually since 0 <d n 1 <C 1 then we have = n 1 >, which from (8) imlies that 1567

8 D() i f i () D() i f i () quantities i C i i =1 C i C 1 quantities i C i i =1 C i dn 1 0 n 1 unit rice Fig. 2. Wardro equilibrium if layer 1 switches from to n 1 inf{ : D() j6=1 C j / }. 0 = n 1 unit rice Fig. 3. Wardro equilibrium if layer 1 switches from to n 1 < inf{ : D() j6=1 C j / }. i 6= 1,d n i = C i n 1 /,and d n 1 = D( n 1 ) n 1 C j / j6=1 D( ) n 1 C j /, (16) j6=1 where the first line second line comes from (11) and the nonincreasingness of the demand function. This is illustrated in Figure 3. In that case, the revenue gain for rovider 1 is R n 1 R 1 = n 1 d n 1 C (C 1 ) 1 (d n 1 ) n 1 dn 1 C (C 1)(C 1 d n Ã!! 1 Ã ) n1 d n 1 1 C 1 j6=1 C j {z } >0 C1 2 j6=1 C j j6=1 C j ( n 1 ) 2 < 0, where the second and third line resectively stem from the convexity of 1 and Assumtion B, and the fourth line is obtained after uer-bounding d n 1 by the exression in (16) and erforming some simlifications. Ste 2: uniqueness of the Nash equilibrium. Now knowing that there exists a Nash equilibrium under Assumtion B, we establish that this equilibrium is unique. Indeed, the roof of the uniqueness only needs Assumtion A to hold, as we see below. To rove uniqueness, consider a Nash equilibrium of the ricing game, i.e. a rice configuration, and decomose the set of roviders I into three disjoint subsets: I = I s I 0 I u, where I s := {i I : d i >C i }, (17) I 0 := {i I : d i = C i }, (18) I u := {i I : d i <C i }. (19) It is sufficient to show that I s and I u are emty sets, since only the rice configuration = (,...,) can lead to the demand configuration with d i = C i for all i. We first rove that I s =. Assume it is not the case, and consider i s I s.westudytheinfluence of rovider i s increasing its unit rice is to n i s = is + ε for ε>0, all other roviders keeing their rice unchanged. From the continuity of the erceived unit rice at Wardro equilibrium in terms of the rice configuration, there exists ε>0 such that n > n i s, which imlies that d n i s = C is n / n i s from the relation between demand and erceived rice. For this ε, the net benefit of rovider i is Ri n s = n C is is (d n i s ). Its gain in net benefit with resect to the initial situation is therefore R n i s R is = C is ( n )+ is (d is ) is (d n i s ) > 0, where the strict ositivity comes from Lemma 1 and the strict increasingness of is. We have established that any rovider in the set I s can strictly increase its net benefit by increasing its rice, which imlies that at a Nash equilibrium, I s =. (20) Remark that I s = imlies that d i C i for all i I, thus d = D() i C i and consequently v( C i). Now we assume that I u 6=, and rove that at least a rovider i u I u can strictly increase its net benefit by decreasing its rice. We still consider a Nash equilibrium rice configuration, and denote by the user erceived rice for that rice configuration. The total demand should therefore be D(). Since we reviously roved that I s =, then D() < C i = D( ),with = v C i, therefore >. (21) 1568

9 When there are at least two roviders, at a Nash equilibrium with I u 6= then for all i we have d i min(c i,d()), since I s = and total demand equals D(). Moreover, there necessarily exists a rovider i u for which d i <D(): indeed there exists at least a rovider in I u, and if this rovider gets all the demand then it imlies that all the other roviders have demand 0. Thus consider i u such that d iu < min(c iu,d()). (22) We now rove that rovider i u can strictly imrove its benefit by changing its rice from iu to ε i u := ε for a sufficiently small ε>0. We distinguish two cases. If C iu D(), then we easily see from (9) that the new erceived rice ε verifies ε i u = ε< ε. By changing its rice to ε, rovider i u is the only rovider with the lowest declared unit rice, therefore from Definition 2 its demand d ε i u equals C ε iu ε,which tends to C iu when ε tends to 0. If C iu >D() then for ε sufficiently small (such that D( ε) C iu ), rovider i u gets all the demand, i.e. d ε i u = D( ε). When ε tends to 0, that demand tends to D() because of the continuity of the demand function. Consequently, for a sufficiently small ε, the demand for rovider i u of switching from rice iu to rice ε can be arbitrarily close to y := min(c iu,d()) >d iu, and the corresonding revenue gain can then be arbitrarily close to (y d iu ) iu (y)+ iu (d iu ) ( 0 i u (y)) (y d iu ) {z } >0 ( 0 i u (C iu ))(y d iu ) ( )(y d iu ) > 0, where the first and second lines come from the convexity of iu and y C iu, the third one from Assumtion A, and the last line stems from (21). Consequently, rovider i u can strictly imrove its net benefit by unilaterally changing its declared rice, which contradicts the Nash equilibrium condition and establishes that we necessarily have at a Nash equilibrium, I u =. (23) Relations (20) and (23) together with the demand relation d i = D() imly the uniqueness of the Nash equilibrium: a Nash equilibrium is necessarily such that each rovider i declares unit rice i =. 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