Coalitions in Routing Games: A Worst-Case Perspective

Transcription

1 Coaitions in Routing Games: A Worst-Case Perspective Gideon Bocq and Arie Orda, Feow, IEEE arxiv: v3 [cs.ni] 27 Mar 206 Abstract We investigate a routing game that aows for the creation of coaitions, within the framework of cooperative game theory. Specificay, we describe the cost of each coaition as its maximin vaue. This represents the performance that the coaition can guarantee itsef, under any (incuding worst) conditions. We then investigate fundamenta soution concepts of the considered cooperative game, namey the core and a variant of the min-max fair nuceous. We consider two types of routing games based on the agents Performance Objectives, namey botteneck routing games and additive routing games. For botteneck games we estabish that the core incudes a system-optima fow profies and that the nuceous is system-optima or disadvantageous for the smaest agent in the system. Moreover, we describe an interesting set of scenarios for which the nuceous is aways system-optima. For additive games, we focus on the fundamenta oad baancing game of routing over parae inks. We estabish that, in contrary to botteneck games, not a system-optima fow profies ie in the core. However, we describe a specific system-optima fow profie that does ie in the core and, under assumptions of symmetry, is equa to the nuceous. Index Terms Atomic Spittabe Routing Games, Non- Transferabe Utiity Coaitiona Games, Cooperative Game Theory, Worst-Case Anaysis, Botteneck Objectives, Additive Objectives, Core, Nuceous. I. INTRODUCTION To date, game theoretic modes have been empoyed in virtuay a networking contexts. These incude contro tasks at the network ayer, such as fow contro and routing (e.g., [] [4]). In particuar, research unti now in routing games has mainy focused on non-cooperative networking games, where the sefish decision makers (i.e., the users or agents) cannot communicate and reach a binding agreement on the way they woud share the network infrastructure. Moreover, the main dynamics that were considered were Best-Response, i.e., each agent woud observe the present state of the network and react to it in a sef-optimizing manner. Accordingy, the operating points of such systems were taken to be some equiibria of the underying non-cooperative game, most notaby Nash equiibria. Such equiibria are inherenty inefficient and, in genera, exhibit suboptima network performance. As a resut, the question of how bad the quaity of a non-cooperative equiibrium is with respect to a centray enforced optimum has received considerabe attention e.g., [4], [5]. However, there is a growing number of networking scenarios where, whie there is competition among sef-optimizing G. Bocq and A. Orda are with the Department of Eectrica Engineering, Technion, Haifa 32000, Israe (e-mais: gideon@tx.technion.ac.i, arie@ee.technion.ac.i) agents, there is aso a possibiity for these agents to communicate, negotiate and reach a binding agreement (see, e.g., [6] [4]). Indeed, in many scenarios, the competition is among business organizations, which can, and often do, reach agreements (e.g., SLAs) on the way that they provide, consume or share the network resources. The proper framework for anayzing such settings is that of Cooperative Game Theory [4], [5]. Such a paradigm transfer, from noncooperative to cooperative games, cas to revisit fundamenta concepts. Indeed, the operating point of the network is not an equiibrium of a non-cooperative game anymore, but rather a soution of a cooperative game. Accordingy, the performance degradation of such systems shoud be considered at new operating points. In the ream of routing games, such an operating point has been proposed in [6], which considered the adoption of the Nash Bargaining Scheme (NBS) [7] as a way of reducing the potentiay high inefficiency of the Nash Equiibrium. Nevertheess, the NBS ony contempates two scenarios, namey the grand coaition (i.e., an agreement reached by a agents) and the disagreement point, i.e., the outcome of the fuy non-cooperative scenario. Thus, whie bargaining between entities is encouraged at the NBS, it might be advantageous for groups to deviate from the agreed strategy and form subcoaitions. Such deviations are to be avoided, since it returns the network to its inefficient non-cooperative scenario. Accordingy, we focus on routing strategies for which deviations of subcoaitions do not occur when agents behave in a rationa manner. A design guideine woud be to have a mediator, e.g., a network administrator, propose to a agents in the network ony to route their fow according to such stabe operating points. More importanty, this gives a credibe guarantee to a agents that the agreed upon soution wi be impemented. To represent the set of stabe operating points, we focus on the core [5] of our game. Since the core might incude severa soutions, we further consider a specific soution in the core with min-max fairness properties, which is a variant of the nuceous [8]. When considering a coaitiona game with N agents, a major question is what cost shoud be attributed to each of the 2 N possibe coaitions. In other words, which scenario can each coaition expect in the network when deviating from the agreed upon soution? In the traditiona game theoretic setting a agents try to minimize their cost function, however in the context of networking, it is often pausibe to contempate scenarios in which some agents do not care about optimizing The NBS has been considered in other networking scenarios, e.g., [6]. However, to the best of our knowedge, [6] is the first to consider the NBS in the context of routing games.

2 2 their own cost, but seem to act maiciousy towards others [9] [24]. Such behavior is due to a range of reasons, e.g., hackers or rivaing companies that aim to degrade network quaity. Moreover and perhaps more notaby, it may happen that some agents are not aware of how to optimize their cost, hence they might exhibit seemingy irrationa, thus unexpected, behavior. In ight of such settings, we mode the cost of a coaition to equa its worst-case scenario, in order to investigate what amount of resources it can guarantee under any (incuding worst) conditions. Specificay, we describe the cost of each coaition as its maximin vaue [25], i.e., as the corresponding best-response to the maximin strategy of the agents outside the coaition. 23 This represents the cost that a coaition can guarantee itsef, even under the pessimistic expectation that the agents outside the coaition have unpredictabe or even maicious objectives. More importanty, this way of modeing gives an a priori insight into any agreement proposed by a mediator. Ceary, no agreement can be reached between a agents if a subset of agents can guarantee itsef a better outcome at its worst-case scenario. We concretize our study of coaitions in routing games by considering two types of games based on the structure of the agents performance objectives. The first game considers agents with botteneck objectives, i.e., their performance is determined by the worst component (ink) in the network [26] [29]. Botteneck routing games have been shown to emerge in many practica scenarios. For exampe, in wireess networks, the weakest ink in a transmission is determined by the node with the east remaining battery power. Hence, each agent woud route traffic so as to maximize the smaest battery ifetime aong its routing topoogy. Additionay, botteneck routing games arise in congested networks where it is desirabe to move traffic away from congested hot spots. For further discussion and additiona exampes see [26]. The second type of game considers agents with additive performance measures, where the performance is determined by the sum of its ink performances, e.g., deay or packet oss. Much of the current iterature on networking games has focused on such games, e.g., [] [5], [30] [34], abeit in the traditiona setting of noncooperative agents. Our Contribution We focus our study on the atomic spittabe routing mode [], [26], in which each agent sends its non-negigibe demand to the destination by spitting it over a set of paths in the network. Moreover, agents are abe to cooperate and form coaitions. In particuar, we formuate a Non-Transferabe Utiity Coaitiona Game [5] and describe the cost of each coaition as its maximin vaue. For botteneck routing games we estabish that coaitions with arger aggregated demand 2 When deaing with payoffs instead of costs, this corresponds to the minimax vaue of a coaition [25]. Aternativey, this scenario can be viewed as a Stackeberg game [5], where an adversary, which acts as eader, tries to maximize the cost of the coaition, which acts as foower. Recenty, in [24] the maximin vaue of two-payer games has been studied in the restricted setting of users with M/M/ cost functions. 3 In our considered game, the maximin strategy of the agents outside the coaition does not necessariy equa the minimax strategy of the coaition. receive a smaer cost at their maximin fow profie. Through this resut, we describe a set of fow profies that are stabe 4, i.e., from which it is not worthwhie for any coaition to deviate. In particuar, to represent the set of stabe soutions, we focus on the core and a variant of the min-max fair nuceous. For botteneck games we estabish that any systemoptima fow profie ies in the core. Moreover, at the nuceous we estabish that () a agents route their fow according to the system optimum or (2) ony the smaest agent experiences performance that is worse than at the system optimum. Next, we describe an interesting set of scenarios in which the nuceous is aways system-optima. Specificay, and counterintuitivey, we estabish that in a network where the two smaest agents are of equa size, the nuceous is aways system-optima. A specia case of this scenario is when a agents have equa demands. For additive routing games we focus on the framework of routing in a parae inks network. Beyond being a basic framework of routing, this is the generic framework of oad baancing among servers in a network. It has been the subject of numerous studies in the context of non-cooperative networking games, e.g., [], [3], [5], [30], [32], [34], to name a few. Quite surprisingy, we estabish that at the maximin strategy, the maicious agents outside the coaition act as if they were a continuum of infinitesima (i.e., nonatomic) sefoptimizing agents. With the above structura resut at hand, we estabish that, in contrast to botteneck games, not a systemoptima fow profies are necessariy stabe. Nevertheess, we prove that a particuar system-optima fow profie does ie in the core, namey where a agents send their demand proportionay with respect to the system optimum. Finay, we show that when agents have equa demands, this proportiona system-optima fow profie is aso equa to the nuceous. II. MODEL AND GAME THEORETIC FORMULATIONS A. Mode We consider a set N = {, 2,..., N} of sefish users (or, payers, agents ), which share a communication network modeed by a directed graph G(V, E), as iustrated in Figure. We denote by P the set of a paths in the network. Each Fig.. Communication Network user i N has a traffic demand r i and a users share a common source S and common destination D. Denote the aggregated demand of users in the coaition S N as r S, 4 Formay, the core [5] considers a set of stabe cost aocations, each of which, in our routing game, refers to a set of fow profies. Whie abiding by the mathematica definition of the core, in the context of this paper, we wi aso refer to the corresponding routing strategy fow profies as being stabe.

3 3 and the demand of a the users as R, i.e., R = i N ri. A user ships its demand from S to D by spitting it aong the paths in P, i.e., user i decides what fraction of r i shoud be sent on through each path. We denote by fp, i the fow that user i N sends on path p P. User i can fix any vaue for fp, i as ong as fp i 0 (non-negativity constraint) and p P f p i = r i (demand constraint); this assignment of traffic to paths, f i = {fp} i p P sha aso be referred to as the routing strategy of user i. The (routing strategy) fow profie f is the vector of a user routing strategies, f = (f, f 2,..., f N ). We say that a user s routing strategy is feasibe if its components obey the nonnegativity and demand constraints. We say that a fow profie f is feasibe if it is composed of feasibe routing strategies and we denote by F i the convex and compact set of a feasibe f i s. Aso denote the set F as the set of a feasibe fow profies. Turning our attention to a path p P, et f p be the tota fow on that path i.e., f p = i N f p; i aso denote by fe i the fow that i sends on ink e E, i.e., fe i = p e p f p. i Simiary, the tota fow on ink e E is denoted by f e = i N f e. i For any coaition of users S N the aggregated fow on path p and ink e is denoted by respectivey fp S = i S f p i and fe S = i S f e. i Occasionay, we focus on a the framework of routing in a parae inks network. In such cases G(V, E) corresponds to a graph with parae inks (e.g., communication inks, servers, etc.) L = {, 2,..., L}, L >, and a user ships its demand by spitting it over the inks L, see Figure 2. As observed in [35], it constitutes an appropriate mode for seemingy unreated networking probems. For exampe, in a QoSsupporting network architecture, bandwidth may be separated among different virtua paths, resuting effectivey in a system of parae and noninterfering inks between the source and destination. Additionay, one can consider a corporation or organization that receives service from a number of different network providers. The corporation can spit its tota fow over the various network faciities (according to performance and cost considerations), each of which can be represented as a ink in the parae ink mode. More generay, the probem of routing over parae inks is, essentiay, the generic probem of oad baancing among severa servers, and it has been the subject of numerous studies, incuding the semina paper [5] and many others, e.g., [], [3], [30], [32], [34]. Fig. 2. Parae inks network In G(V, E), we associate with each ink a performance function T e ( ), which corresponds to the cost per unit of fow through ink e and ony depends on the tota fow f e. Furthermore, we impose the foowing assumptions on T e (f e ): S T e (f e ) is continuous and stricty increasing in f e. S2 T e : [0, ) [0, ]. S3 For every fow profie f, if not a costs are finite then at east one user with infinite cost (J i (f) = ) can change its own strategy to make its cost finite. Cost functions that compy with the above assumptions sha be referred to as standard. The performance measure of a user i N is given by a cost function J i (f). In botteneck routing games, J i (f) corresponds to the performance of the worstcase ink, and in additive routing games it corresponds to the sum of a ink performances in the system. An N-tupe of positive vaues J = (J, J 2,..., J N ) is said to be a feasibe cost vector if there is a feasibe routing strategy fow profie f F such that, for a i N, J i = J i (f). Denote the set of feasibe cost vectors by J. B. Botteneck routing cost function Foowing [26], we represent the cost of a user i N as the worst performance of any ink in the network that i sends a positive amount of fow on. Thus, the cost of user i equas J i (f) = max T e(f e ). () e E fe i>0 We measure the wefare of the system through the cost of the worst performing ink in the network, i.e., through a ( socia ) cost function J sys defined as J sys = max e E fe>0 T e (f e ). For botteneck routing games we consider users with standard cost functions. C. Additive routing cost function Another important cass of probems is when users are interested additive performance measures, e.g., deay or packet oss. In this case, T e may correspond to the tota deay of ink e. Foowing [], [35], [36] and many others, we consider users whose cost functions are of the foowing form: J i (f) = fp i T e (f e ). (2) p P e p Moreover, we consider T e (f e ) to be convex and continuousy differentiabe. As commony assumed in the iterature (e.g., [3], [4], [30], [32], [33], [35], [36]), for additive routing games the system s cost is equa to the sum of the individua costs of the payers, i.e., J sys = i N J i. The system s cost thus equas: J sys = fpt i e (f e ) = f e T e (f e ). (3) i N p P e p e E D. System Optimization We denote the optima vaue of the system s cost as Jsys, i.e., the minima vaue of J sys over a feasibe routing strategy fow profies. Any fow profie that corresponds to the system optimum, we denote by f = (fp ) p P. For both botteneck and additive routing games, J sys depends ony on the tota fows on the inks. Accordingy, f is an optima vector of ink fows, i.e., Jsys = max e E f e >0 T e (fe ) and Jsys = e E f e T e (fe ) for respectivey botteneck and additive routing games. A recurring fow profie in this study

4 4 is the proportiona fow profie where a users send their fow proportionay with regard to the system optimum: for any system-optima fow profie f, any user i and path p, f i p = ri R f p. We denote the cost vector at the proportiona fow profie as the Proportiona Cost Aocation (PCA). Consequenty, at the PCA, for each user i N with botteneck costs, J i (f ) = J sys, and for users with additive costs, J i (f ) = p P E. Nonatomic users r i R f p e p T e (f e ) = ri R J sys, i N. (4) In this study we focus on a finite set of (nonzero-size) users that can spit their fow among the inks. Nevertheess, some of the foowing resuts state that at times a user may behave as if it were a continuum of infinitesima sef-optimizing users, referred to as a set of nonatomic users. A nonatomic user paces its demand on a singe path ˆp, for which e ˆp T e(f e ) = min p P e p T e(f e ) [37]. If a finite user i behaves as if it were a set of sef-optimizing nonatomic users, it foows that, ˆp, p P: if f iˆp > 0 e ˆp T e (f e ) e p T e (f e ). (5) We wi refer to (5) as the best-response behavior of a set of nonatomic users. F. Coaitiona game We proceed to formaize our coaitiona game, by attributing a set of costs to every coaition S N. Note that our cost functions represent a variety of costs, e.g., deay, which are not considered to be a commodity that users can freey transfer between themseves, hence we define a Non-Transferabe Utiity (NTU) Game [5], as foows. Definition 2.. A NTU coaitiona game consists of a mapping V ( ) that assigns to each coaition S N a set of outcomes V (S) R S, which is non-empty, cosed and convex. 5 A coaition behaves as a singe user controing the fow of its participants and V (S) represents the set of feasibe cost aocations that S can achieve for itsef. III. BOTTLENECK ROUTING GAMES A. Maximin fow profie As expained in the introduction, our goa is to propose stabe and fair routing strategies to a N users. However, once a coaition of users S decides to deviate from our proposed soution, we mode it to incur a worst-case cost. Specificay, we represent the cost of every coaition that deviates (and there are 2 N such possibe coaitions), as its maximin vaue in the game between S and the users in. The maximin vaue of S corresponds to the owest cost S can guarantee against the worst (for S) strategy of the users outside the coaition. In other words, a users in aim to maximize J S by sending 5 In [5] additiona conditions are cited, which triviay foow for our considered game. their fow according to f. Given f, S behaves as a singe user controing the fow of its participants and aims to minimize their combined cost J S. Ceary, by cooperating, the users in can ony increase their damage to S. Therefore, when considering the maximin vaue of S, we represent a users in as a singe maicious user that aims to maximize the cost of S. For each coaition we can now view this as a game between two payers, S and. Given any routing strategy f denote the best-response strategy of coaition S as ˆf S (f ) = arg min{j S (f, f S )}. (6) f S F S The maximin strategy of the users in is equa to ˆf = arg max f F { min f S F S J S (f, f S )} (7) and given ˆf, the coaition S send its demand according to its best-response strategy, i.e., ˆf S (ˆf ) = arg min f S F S {J S (ˆf, f S )}. (8) We refer to (ˆf,ˆf S (ˆf )) as the maximin (routing strategy) fow profie. 6 To simpify the notation, we wi frequenty refer to ˆf S (ˆf ) as ˆf S. From the point of view of the coaition, at the maximin fow profie, the users outside of the coaition act in a maicious manner. Since a coaition S behaves as a singe user controing the fow of its participants, it foows from () that in a botteneck routing game, J S (f, f S ) = max T e (f {e E fe S>0} e + fe S ), (9) where fe S represents the combined fow of a users i S on ink e, i.e., fe S = i S f e. i In order to describe the behavior of a coaition at the maximin fow profie, we first estabish the foowing emma. Lemma 3.. Consider a botteneck routing game. For any two coaitions S, T N with r T r S : J T (ˆf,ˆf T ) J S (ˆf,ˆf S ). Proof. We estabish the emma by constructing a routing strategy f, such that J T (ˆf,ˆf T ) J S ( f,ˆf S ( f )). (0) Then, as a resut of (0) and (7), the emma foows. For every path p, consider f p gp + h N p \S, where gp = ˆf p. Since p P ˆf p = r r, we construct 6 In [38] it has been estabished that, for botteneck routing games, J S (f, f S ) is not necessariy continuous in F S. Thus, at the maximin fow profie we wi consider the set of fow profies (ˆf,ˆf S ) such that J S (ˆf,ˆf S (ˆf )) sup J S (f,ˆf S (f )) δ, f F and given any f J S (f,ˆf S ) inf J S (f, f S ) + ɛ f S F S for some sma enough ɛ, δ 0. Consequenty, due to the compactness of F S and F, it is cear that a maximin profie aways exists, however it is not necessariy unique. For additive routing games, J S (f, f S ) is a continuous function, thus ɛ, δ = 0.

5 5 h p by sending the remaining demand (i.e., r r ) on the paths where ˆf p T > 0, whie abiding by the constraint h N p \S < ˆf p T. This constraint is feasibe since \S p P hn p = r r = r T r S < r T. Denote the best-response strategy ˆf S ( f ) as f S and assume by contradiction that From () it foows that J T (ˆf,ˆf T ) > J S ( f, f S ). () max T e ( ˆf e E ˆf e T >0 e + ˆf e T ) > max T e ( f e E f e S>0 e + f e S ) (2) = max T e ( ˆf e f e S>0 e + h N e \S + f e S ). We now define a new routing strategy f T where for any path p, fp T = h N p \S + f p S. This strategy is feasibe since r T = p P f p T = \S p P hn p + f p S = r T r S + r S. Thus, for any ink e, fe T = h N e \S + f e S. Since ˆf T is the best-response strategy to ˆf, we get that max T e ( ˆf {e E h N e \S + f e S>0} e + h N e \S + f e S ) (3) max T e ( {e E ˆf e T >0} ˆf e + ˆf T e ). As a resut of (2) and (3) it foows that Therefore, max T e ( ˆf {e E h N e \S + f e S>0} e + h N e \S + f e S ) > (4) max T e ( {e E f e S>0} ˆf e + h e + f S e ). max T e ( ˆf {e E h N e \S + f e S>0} e + h N e \S + f e S ) = (5) max T e ( ˆf {e E h N e \S >0, f e S=0} e + h N e \S ). By definition, for any ink e where h N e \S > 0 it hods that > 0. Thus, from (4) and (5) it foows that ˆf T e max T e ( ˆf {e E ˆf e T >0} e + h N e \S ) (6) max T e ( {e E h N e \S >0} ˆf e + h e ) = max T e ( ˆf {e E h N e \S >0, f e S=0} e + h N e \S ) = max T e ( ˆf {e E h N e \S + f e S>0} e + h N e \S + f e S ). Finay, from (6) and (3) we get that max T e ( ˆf {e E ˆf e T >0} e + h N e \S ) max {e E ˆf T e >0} T e ( ˆf e + ˆf T e ), which is a contradiction since h e < ˆf T e for a e E. Thus, (0) foows and the emma is estabished. According to Lemma 3., coaitions with a arger demand receive a smaer cost at their maximin fow profie. In other words, the threat of the users outside the coaition decreases when the coaition s demand increases. B. Worst-Case coaitions We now proceed to describe our coaitiona game as defined in Definition 2.. In particuar, for each coaition S, denote by (ˆf,ˆf S ) a maximin routing strategy fow profie of the game between S and, i.e., the fow profie in which acts according to (7) and S acts according to (8). Aso denote J(ˆf,ˆf S ) J as the cost vector of a users, when sending their demand according to (ˆf,ˆf S ). Furthermore, denote by J S (ˆf,ˆf S ) the projection of J(ˆf,ˆf S ) onto R S, i.e., the cost vector of the users in the coaition S. Note that J S (ˆf,ˆf S ) is not equa to J S (ˆf,ˆf S ), as the former is a cost function and the atter a vector of user costs. We now define our mapping of V (S). Definition 3.. For every coaition S N : V (S) {J S (ˆf,ˆf S )}. Thus, given a coaition of users S, V (S) is a set of a possibe cost vectors in which sends its fow according to (7) and S sends its fow according to (8). It is cear that V (S) satisfies the conditions of an NTU coaitiona game. Cosedness foows from directy from Assumption S2 and foowing [5], [6], it can be shown that V (S) is convex. Having defined the worst-case coaitiona game, we proceed to investigate it through the study of severa (fair and stabe) soution concepts of cooperative game theory. C. The core We consider the game as defined in Definition 3. and continue to describe its core, i.e., a set of soutions that are stabe against coaitiona deviations [5]. Definition 3.2. Given an NTU coaitiona game V ( ), the core of the game is a set of cost vectors J c J such that J J c, S N and for any J J : J V (N). If J i < J i, i S, then J S / V (S). Transated to our scenario, a feasibe cost vector ies in the core if there does not exist any coaition S, such that a users in S stricty decrease their cost when sending their demand according to any maximin fow profie (ˆf,ˆf S ). We continue with the foowing theorem. Theorem 3.. Consider a botteneck routing game. Any system-optima fow profie corresponds to a cost vector that ies in the core. Proof. According to Lemma 3., for any two coaitions S, T N with r T r S it foows that J T (ˆf,ˆf T ) J S (ˆf,ˆf S ). In particuar, for the grand coaition N and any coaition S N it foows that J sys = J N (ˆf N ) J S (ˆf,ˆf S ). Consequenty, for any coaition S N, Jsys { max {e E T e ( ˆf k S e k>0} = max k S { max T e ( e E ˆf e k>0 ˆf e + ˆf S e )} (7) ˆf e + ˆf S e )}.

6 6 Consider a system-optima fow profie f and denote the fow of user i on ink e as fe i. From (7) it foows that for any i N and any S N, max e E f i e >0 T e(f e ) max e E f e >0 T e(f e ) (8) max k S { max T e ( e E ˆf e k>0 Hence, there exists some user j S for which J j (f ) = ˆf e + ˆf S e )}. max T e (f e E fe j e ) max T e ( ˆf >0 e E ˆf e j e + ˆf e S ) >0 = J j (ˆf,ˆf S ). In other words, for any coaition S N, any maximin fow profie (ˆf,ˆf S ) and any system-optima fow profie f, there aways exists some user j S which receives a higher (or equa) cost at (ˆf,ˆf S ) compared to f. Thus, any system-optima fow profie f corresponds to a cost vector that ies in the core. In the proof of Theorem 3. we estabished that for any coaition S N, there exists a user that prefers the systemoptima fow profie over any maximin fow profie. We wi show that the same resut hods with respect to the systemoptima fow profie and any Nash equiibrium. Suppose we consider a different coaitiona game than the one described in Definition 3., in which a deviating coaition S does not incur a maximin cost. Specificay, in this new game the users in the coaition S work together as a singe arge user, whie the users outside the coaition do not aim to maximize the cost of S. Instead, they do not cooperate and their goa is to minimize their own costs. Denote the set of users outside the coaition as {, 2,..., N S }, where N S =. Thus, for each deviating coaition S N, we consider a noncooperative game between the users {S,,..., N S }, where each user i aims to minimize its own cost. The next theorem is proven in Appendix A. Theorem 3.2. Consider users with botteneck objectives and consider an NTU coaitiona game, where for any S N, V (S) is equa to the set of Nash equiibria between S and a individua users in. Any system-optima fow profie corresponds to a cost vector that ies in the core. For the remainder of the paper we ony consider the worst-case NTU coaitiona game as defined in Definition 3.. D. The Routing Game Nuceous Theorem 3. estabishes that in a botteneck routing game, no coaition of users wishes to deviate from any fow profie which is optima from a system s perspective. Yet, out of a the stabe fow profies in the core, which one shoud be proposed to the users in the network? We proceed to consider a further refinement of the core with additiona fairness guarantees. In particuar, we focus on a routing strategy fow profie which preserves stabiity whie adding min-max fairness properties to the users. For that, we consider a variant of the we studied Nuceous [8]. Athough the nuceous is originay defined for Transferabe-Utiity games, we sha adapt its definition to fit our mode. Due to this change, certain attributes such as its uniqueness are not guaranteed, uness proven. Moreover, we wi need to prove that the nuceous ies in the core of the game. Consider a feasibe cost vector J(f) J. We define the excess of each coaition S N at J(f) as e S (J(f)) J S (f) J S (ˆf,ˆf S ), (9) where J S (f) corresponds to the coaition s cost at fow profie f. For botteneck routing games, J S (f) corresponds to (9), thus the excess of a coaition S is equa to e S (J(f)) max T e (f e ) max T e ( e E fe S>0 e E ˆf e S>0 = max i S { max T e(f e )} max e E fe i>0 i S { max T e ( e E ˆf e i>0 ˆf e + ˆf S e ) (20) ˆf e + ˆf S e )}. For ease of notation we sha refer to e S (J(f)) as e S (J). The excess can be interpreted as the dissatisfaction of a coaition S at a cost vector J(f). The smaer the excess, the more a coaition S is satisfied. Note that even though the maximin fow profie is not necessariy unique, due to (7), J S (ˆf,ˆf S ) is equa for a maximin fow profies. Denote the excess vector e(j) = [e S (J)] S N, and denote e (J) as a permutation of the entries of e(j) arranged in non-increasing order. Definition 3.3. We define the Routing Game Nuceous (RGN) as: {J J J J, e ( J) xm e (J)}, where xm means that it is smaer in the exicographica sense. Hence, the RGN is the cost vector that minimizes the excess of the coaitions in the exicographic ordering, i.e., it treats the wefare of coaitions in a min-max fair manner. We denote the vector of user costs at the Routing Game Nuceous as J RGN. Proposition 3.. Consider a botteneck routing game. The Routing Game Nuceous ies in the core. Proof. From Lemma 3. it foows that for any coaition S N, J sys J S (ˆf,ˆf S ). Therefore, due to (9), the excess of any coaition S N at J [J sys] i N is equa to e S (J ) = J sys J S (ˆf,ˆf S ) 0. (2) Consequenty, from Definition 3.3 it foows that for any coaition S N, e S (J RGN ) max S N e S (J ) 0. Denote the fow profie at the RGN as f. It foows from (20) that for any coaition S N, ˆf e max { max T e ( + ˆf S i S e E ˆf e i>0 e )} max { max T e(f e )}. i S e E fe i>0 (22) As a resut of (22), for any coaition S there exists some user k S for which J k (ˆf,ˆf S ) = max i S { max i S { max T e ( e E ˆf e i>0 ˆf e + ˆf S e )} max T e(f e )} J k e E fe i>0 RGN. In other words, for any coaition S there exists some user k S that receives a ower cost at the nuceous compared to any maximin fow profie. Thus, the nuceous ies in the core.

7 7 We now continue to describe the RGN. Before doing so, we bring the foowing definition from [39]. Definition 3.4. A fow profie f is referred to as baanced if for any two paths p, p 2 P with f p > 0, it hods that max e p T e (f e ) max e p2 T e (f e ). In other words, at a baanced fow profie, every possibe path with positive fow has the same cost for its worst-case ink. In [39], the next emma is proven. Lemma 3.2. [Lemma in [39]] Consider a botteneck routing game. Any system-optima fow profie is baanced. For the foowing theorem denote the user with the smaest demand as j, i.e., r j = min k N r k. Theorem 3.3. Consider a botteneck routing game. (): J RGN = [J sys] i N, and/or (2): J j RGN > J sys, and for a k N \j, J k RGN J sys. Proof. Consider the coaition K N \j and the cost vector J [J sys] i N. According to Lemma 3., for any coaition S N, J K (ˆf N \K,ˆf K ) J S (ˆf,ˆf S ) and e K (J ) e S (J ). (23) Denote the fow profie at the RGN as f. Due to (23) and Definition 3.3 it foows that e K (J RGN ) e K (J ). Hence, { } max i K max T e(f e ) e E fe i>0 max T e(f e E fe >0 e ). (24) Assume by contradiction that there exists some user k K for which J k RGN > J sys. Thus, max T e (f e ) > max T e(f e E fe k>0 e E fe >0 e ), which is a contradiction to (24). Hence, there does not exist any k j for which J k RGN > J sys. Now suppose that J j RGN < J sys and for a k j, J k RGN J sys. According to Lemma 3.2, any system-optima fow profie is baanced. Thus, every path with positive fow has the same cost for its worst-case ink. It foows that, max e E f i e >0 T e(f e ) = max e E f e >0 T e(f e ), i N (25) i.e., for any i N, J i (f ) = J sys. As a resut, there does not exist a system-optima fow profie f for which J j (f ) < J sys. Consequenty, there must exist some user k j for which J k RGN > J sys, which is a contradiction. Therefore, at the Routing Game Nuceous, either (): J i RGN = J sys for a i N, and/or (2): J j RGN > J sys and J k RGN J sys for a k N \j. Theorem 3.3 estabishes that at the nuceous of a botteneck routing game, there are two possibe scenarios. (): A users send their fow according to the system optimum and/or (2): Ony the user with the smaest demand sends its fow on the worst-case ink in the system, whie the other users receive a simiar or better performance than at the system optimum. This resut is quite counterintuitive, and combined with Lemma 3.2, Theorem 3.3 iustrates that the smaest user in the system is never better off at the nuceous than at any system-optima fow profie. In the foowing exampe, we iustrate two different networks, where each corresponds to one of the settings of Theorem 3.3. Fig. 3. Nuceous of a botteneck routing game Exampe 3.. Consider the network in Figure 3, where for each ink e E\{n}, T e (f e ) = f e. Moreover, consider two users, r = 2 and r 2 = 20. At the system-optima fow profie, both users spit their fow equay among the paths {S, A, D} and {S, B, D}. We first describe the maximin fow profie of user 2. User maximizes the cost of user 2 by pacing a of its fow on the path {S, A, B, D}. As a bestresponse, user 2 paces 2/3 on ink n and spits the rest of its fow equay among the paths {S, A, D} and {S, B, D}. Thus, J 2 (ˆf,ˆf 2 ) = 2 3 and e 2(J (f )) = Jsys J 2 (ˆf,ˆf 2 ) = 2 3 = 2 3. We now consider the excess of user. If user 2 paces an amount of 5 on ink n and sends the rest on path {S, A, B, D}, then the best-response strategy of users is to spit its entire demand on the paths {S, A, D} and {S, B, D}. Thus, e (J (f )) = Jsys J (ˆf 2,ˆf ) 6 = 5. However, if we propose a different fow profie, f, where user paces an amount of on ink n and spits the rest of its demand together with user 2 on the paths {S, A, D} and {S, B, D}, we get that e 2 (J(f)) = = 6 e 2 (J (f )) and e (J(f)) 2 6 = 4. Therefore, J [Jsys] i N is not equa to the RGN. Now consider a network as iustrated in Figure 3, however without ink n. It is straightforward that J is equa to the RGN. We continue to focus on a specia case in which there exist at east two smaest users in the system, i.e., there exist users j, j 2 such that r j = r j2 = min k N r k. Theorem 3.4. Consider a botteneck routing game where there exist users j, j 2 such that r j = r j2 = min k N r k. The Routing Game Nuceous is equa to [J sys] i N. Proof. Denote the grand coaition without user j as K N \j and the grand coaition without user j 2 as K 2 N \j 2. Ceary, r K = r K2 r S for any coaition S N. As a resut of Lemma 3. for any S N, J K (ˆf N \K,ˆf K ) = J K2 (ˆf N \K2,ˆf K2 ) J S (ˆf,ˆf S ). Therefore, the excess of K and K 2 at J [Jsys] i N is equa to e K (J ) = e K2 (J ) = max T e(f e E fe >0 e ) J K (ˆf N \K,ˆf K )

8 8 e S (J ), S N. (26) Moreover, from Definition 3.3 and (26), it foows that e K (J RGN ) e K (J ) and e K2 (J RGN ) e K2 (J ). (27) Denote the fow profie at the RGN as f and assume by contradiction that there exists a user k for which JRGN k > J sys. Since user k ies in K or K 2 (or both) it hods that: { } max max T e(f e ) > J i K e E fe i>0 sys { } and/or max i K 2 max T e(f e ) e E fe i>0 > J sys. Therefore, e K (J RGN ) > e K (J ) and/or e K2 (J RGN ) > e K2 (J ), which is a contradiction to (27). Now assume by contradiction that there exists a user k for which JRGN k < J sys. According to Lemma 3.2, any system-optima fow profie f is baanced. Thus, there does not exist a system-optima fow profie f and a user k for which J k (f ) < Jsys. It foows that at the RGN there must exist another user i for which JRGN i > J sys, which is a contradiction. The conditions of Theorem 3.4 describe a range of scenarios for which the nuceous is aways equa to the system-optima fow profie. For exampe, in a network where users have a finite number of different demands, e.g., arge, medium and sma, the conditions of Theorem 3.4 are satisfied for a network with at east two sma users. Of course Theorem 3.4 aso hods in a network where a users have equa demands, i.e., for a i, j N, r i = r j. We denote such users as symmetrica. Coroary 3.. Consider a botteneck routing game with symmetrica users. The Routing Game Nuceous is equa to [J sys] i N. Through Theorem 3. we estabished that a fow profie which is optima from a system s perspective is aso stabe against coaitiona deviations. Moreover, Proposition 3. and Theorem 3.3 estabishes that the min-max fair nuceous is (i) stabe and (ii) system-optima or disadvantageous for the smaest user in the system. Finay, Theorem 3.4 describes an interesting set of scenarios where the RGN is a desirabe soution, from a fairness perspective, from a socia perspective, and easy to impement due to its stabiity. IV. ADDITIVE ROUTING GAMES A. Maximin respresentation In contrast to botteneck routing games, we now focus on users that are interested in additive performance measures, such as deay or packet oss. Moreover, we focus on the parae inks mode as described in Section II. Thus, from [], f is a unique vector of ink fows and the system-optima proportiona fow profie (4) is unique in the individua ink fows. Moreover, since a coaition S behaves as a singe user controing the fow of its participants, it foows from (2) that J S (f) = i S J i (f) = L f S T (f ). (28) Consequenty, as stated in (7), the maximin strategy of the users in M is equa to ˆf M = arg max f M F M { min f S F S L f S T (f M + f S )} (29) and given ˆf M, the coaition S send its demand according to ˆf S (ˆf M ) = arg min f S T ( ˆf M + f S ). (30) f S F S L The (maicious) user M aims to send its demand according to (29) and sends an amount of fow ˆf M on each ink L, such that, when S responds according to (30), its cost is maximized. To describe the behavior of M at the maximin fow profie, we first estabish severa emmas. In each emma we start out with a fow profie f = (f M,ˆf S (f M )), where ˆf S (f M ) denotes the best-response routing strategy of S given f M, i.e., given f M the coaition S sends its demand according to (6). We then change the strategy of the maicious user to g M, whereafter the coaition S responds with a new best-response strategy ĝ S (g M ). Finay, we draw concusions on the cost of S at (g M, ĝ S (g M )). Denote the best-response strategy of S to f M and g M as, respectivey ˆf S, and ĝ S. Consider the fow profie f = (f M,ˆf S ) and order the inks such that T (f M + ˆf S )... T L (fl M + ˆf L S ). We define the set of the inks with the owest cost per unit of fow at f as L { L T (f M + ˆf S ) = min n L T n(f M n + ˆf S n )}. The first emma estabishes that when M decreases the amount of fow it sends on the inks with the owest cost, it aso owers the cost of S. Lemma 4.. Consider a fow profie f = (f M,ˆf S ), the set of inks L and ink q / L. Moreover, consider a new routing strategy g M and an amount of fow L > 0 such that for any ink L g M = f M, g M q = f M q + and L\{L q}, g M = f M. Then J S (g M, ĝ S ) < J S (f M,ˆf S ). (3) Proof. In order to estabish the emma, we construct an aternative routing strategy for S, ḡ S. By definition, the cost of S when appying ĝ S wi be ess or equa to its cost at ḡ S, i.e., J S (g M, ĝ S ) J S (g M, ḡ S ). Thus, it is sufficient to construct a feasibe routing strategy ḡ S, which satisfies We consider two cases: ) > / L ˆf S 2) / L ˆf S. J S (g M, ḡ S ) < J S (f M,ˆf S ). (32) Consider Case. We construct ḡ S by sending a of r S randomy on the inks L, whie abiding by the constraint that ḡ S + ˆf S for a L. According to the conditions of Case, > r S ˆf L S. Thus, r S = ḡ S < [ + ˆf S ]. L L

9 9 Consequenty, there exists some ink r for which ḡ S r < r + ˆf S r. Therefore, it foows that J S (g M, ḡ S ) = L ḡ S T (f M + ḡ S ) (33) < L ḡ S T (f M + ˆf S ) L ˆf S T (f M + ˆf S ) = J S (f M,ˆf S ), where the ast inequaity of (33) is due to the fact that L, n / L, T (f M + ˆf S ) < T n(f M n + ˆf S n ). For Case 2, we consider two subcases. 2a. ˆf S q. 2b. > ˆf S q. Consider Subcase 2a. We start at ˆf S and construct ḡ S by sending from ink q to a the inks L such that ḡ S = + ˆf S. It foows that for a L, T (g M + ḡ S) = T (f M + ˆf S) and T q(gq M + ḡq S ) = T q (fq M + ˆf q S ). Therefore, J S (g M, ḡ S ) = ( ˆf S + )T (f M + ˆf S ) L + ( ˆf q S )T q (fq M + ˆf q S ) ( + ˆf S T f M + ˆf ) S L\{L q} < L ˆf S T (f M + ˆf S ) = J S (f M,ˆf S ), since L, T (f M + ˆf S) < T q(fq M + ˆf q S ). Now consider Subcase 2b. We start at ˆf S and construct ḡ S by sending an amount of fow ˆf q S from ink q to the inks in L and randomy consider a different ink q / L. If ˆf q S ˆf q S we finish constructing ḡs by sending ˆf q S from ink q to the inks in L. Otherwise we send ˆf q S from ink q to the inks in L and again consider a different ink q / L. Through this process, we construct ḡ S by sending a tota amount of fow equa to from a set of inks with a higher cost per unit of fow than the inks in L. Since L is finite and ˆf / L S, we wi reach a ink n / L for which ˆf k F k S ˆf n S, where F denotes the set of inks we considered before reaching ink n. We finish to construct ḡ S by sending a tota fow of from the inks {F n} to the inks in L. Observe that for any L, T (g M + ḡ S ) = T ((f M ) + ( ˆf ) S + ) (34) Consequenty, from (34), J S (g M, ḡ S ) = = T (f M + ˆf S ) < T r (f M r + ˆf S r ), r / L. ( ˆf S + )T (f M + ˆf S ) + ḡnt S n (fn M + ḡn) S L + ˆf S T (f M + ˆf S ) L\{L n F} < L ˆf S T (f M + ˆf S ) = J S (f M,ˆf S ). Hence, given g M, we constructed a routing strategy ḡ S that satisfies (32). We continue to estabish that when M increases the amount of fow it sends on the inks with the owest cost, it raises the cost of S. Lemma 4.2. Consider a fow profie f = (f M,ˆf S ), the set of inks L and ink q / L. Moreover, consider a new routing strategy g M and an amount of fow L > 0 such that for any ink L g M = f M +, gq M = fq M, L\{L q}, g M = f M, and { } L T (g M + ĝ S ) = min T n(gn M + ĝn) S = L. (35) n L Then J S (g M, ĝ S ) > J S (f M,ˆf S ). Proof. The constraint in (35) impies that the set of inks with minima cost per unit of fow are equa in both (f M,ˆf S ) and (g M, ĝ S ). Assume by contradiction that J S (g M, ĝ S ) J S (f M,ˆf S ). Consider a new routing strategy h M where for a L h M = g M, h M q = g M q + and \{L q}, h M = g M. Due to (35) and Lemma 4., it foows that J S (h M, ĥ S ) < J S (f M,ˆf S ). However, h M = f M, which is a contradiction. Through Lemma 4.2, we have proven that it is in the maicious user s interest to send more fow on the inks with the owest cost per unit of fow. We are now abe to estabish the foowing theorem. Theorem 4.. Consider an additive routing game and a coaition S N. At the maximin fow profie sends its fow according to (5), i.e., sends its fow according to the best-response behavior of a set of non-atomic users with an aggregated demand of r. Proof. Consider a maximin fow profie (ˆf,ˆf S ) and denote the set { } ˆL L T ( ˆf + ˆf S ) = min n( ˆf n + ˆf n S ). n L Assume by contradiction that there exists a ink q / ˆL, for which ˆf q > 0. We construct a feasibe routing strategy g at which the maicious user moves a sma amount of fow from ink q to the inks in ˆL, such that 0 < < ˆf q and { L T ( ˆf + + ˆf S ) = min T n( n L ˆf n + n + ˆf n S )} = ˆL,i.e., such that the set of inks in ˆL does not change. According to Lemma 4.2, this increases the cost of S. Thus, ˆf does not maximize (7), which is a contradiction. Hence, at its optima routing strategy, wi send its fow according to (5). According to Theorem 4., for any coaition S N at its maximin strategy, the users in act as if they were a set of sef-optimizing non-atomic users. Moreover, S represents a singe finite user and sends its fow according to its best-response strategy (30). This specific scenario has been investigated in [34], [40] among others. In [40] it has

10 0 been estabished that, for additive routing games, the setting of sef-optimizing non-atomic users together with a singe finite user S, admits a unique equiibrium. Therefore, as a resut of Theorem 4., we concude that the maximin profie is unique for any coaition. Further, in [34] the foowing emma is estabished. Lemma 4.3. [Theorem 5.2 in [34]] Consider an additive routing game. For any two coaitions S, T N with r T r S : r T J T (ˆf,ˆf T ) r S J S (ˆf,ˆf S ). (36) In other words, Lemma 4.3 impies that the average cost of a deviating coaition decreases when its aggregated demand increases. Note that this resut is simiar to Lemma 3. for botteneck routing games, athough in Lemma 4.3 the costs are averaged. B. The core We continue to describe the core of our coaitiona game with additive costs. In the foowing proposition we estabish that in genera, not every system-optima aocation ies in the core. This is in contrast to Theorem 3., which considers botteneck routing games. Proposition 4.. Consider an additive routing game and suppose there exists a coaition S and a maximin fow profie (ˆf,ˆf S ), which is not system-optima. Then there exists a system-optima fow profie whose corresponding cost vector does not ie in the core. Proof. Our goa is to construct a system-optima fow profie f such that for some coaition S N with maximin fow profie (ˆf,ˆf S ) it hods that, J i ( f) > J i (ˆf,ˆf S ), i S. (37) Consider a coaition S N and a maximin fow profie (ˆf,ˆf S ) f as described in the proposition. Aso consider the two compementary sets: L + = { L ˆf f } and L = { L ˆf < f }. We construct a new routing strategy for S, f S in which it fis up the inks in L according to the system optimum, starting from ink L upwards. Since L f + ˆf 0, it foows that L f ˆf r S. Therefore, after the fiing process, S reaches a ink, K L for which f K S f K ˆf K and for any ink > K, L, f S = f ˆf. At the new fow profie, S s cost is equa to J S (ˆf, f S ) = [f ˆf ]T (f ) >K L + f KT S K ( f K S + ˆf K ) > J S (ˆf,ˆf S ), (38) because ˆf S is the best-response strategy to ˆf and from [] it foows that the best-response strategy is unique. We now change the routing strategy of and construct a fow profie f ( f, f S ) that is system-optima and for which J S ( f, f S ) J S (ˆf, f S ). By doing so we have constructed a feasibe system-optima fow profie for which J S ( f, f S ) > J S (ˆf,ˆf S ). We define the strategy f as foows. On any ink L, sends an amount f such that f + f S = f. S Since for any ink, f f, this new routing strategy is feasibe. Thus, on any ink L, > K, user does not increase its fow, i.e., f = ˆf. Furthermore, on ink K, f K ˆf K. As a resut, = >K L J S ( f, f S ) (39) [ ] f T (f ) + f KT S K ( f K S + f K ) >K L [ f f ] ˆf T (f ) + f KT S K ( f K S + ˆf K ). Hence, from (38) and (39), J S ( f, f S ) J S (ˆf, f S ) > J S (ˆf,ˆf S ). Now that we have constructed a systemoptima fow profie f, we can decide on how to spit its cost among the individua users. Specificay, consider the system-optima cost vector where a users in S send their fow proportionay to f S. Thus, for every user i S, f i = ri f S r S and the cost of every user i S is equa to J i = ri J S ( f, f S ). Furthermore, denote by r Ĵ i the cost of S user i S when sending its fow proportionay to (ˆf,ˆf S ), i.e., Ĵ i = ri J S (ˆf,ˆf S ). Ceary, it is in every users interest r S to deviate from the proposed system-optima cost vector and send their fow proportionay to the maximin fow profie, i.e., for a i S, J i > Ĵ i, which satisfies (37). It foows from Definition 3.2 that the system-optima profie f, at which users in S proportionay spit their fow, does not ie in the core. According to Proposition 4., for additive routing games, in genera not every system-optima cost aocation ies in the core. Nevertheess, we focus on a particuar cost aocation that does aways ie in the core. Theorem 4.2. Consider an additive routing game. The Proportiona Cost Aocation ies in the core. Proof. Assume by contradiction that the system-optima Proportiona Cost Aocation (4), does not ie in the core. Hence, there exists a coaition S for which it is worthwhie to deviate from the proportiona fow profie and incur its cost at the maximin fow profie (ˆf,ˆf S ), as defined in Definition 3.. Therefore, it foows that for a users i S r i R J sys > J i (ˆf,ˆf S ). (40) However, it foows from Lemma 4.3 that S N : R J N (ˆf N ) = R J sys r S J S (ˆf,ˆf S ), which turns into r S R J sys J S (ˆf,ˆf S ), S N. (4) r Since, S R J sys = i S ri R J sys and J S (ˆf,ˆf S ) = i S J i (ˆf,ˆf S ), inequaity (4) impies that if any coaition S N deviates from the PCA, there must exist some

11 user i S, for which ri R J sys J i (ˆf,ˆf S ). This is a contradiction to (40). Therefore, the Proportiona Cost Aocation ies in the core. Theorem 4.2 iustrates that the proportiona fow profie is not ony system-optima, but aso stabe against coaitiona deviations. Moreover, as a resut of Proposition 4., in an additive routing game stabiity is a specific property of ony certain system-optima fow profies, in particuar the proportiona fow profie. Note that Theorem 4.2 and Proposition 4. consider users with standard cost functions. In Appendix B we provide an exampe of a network where the users cost functions are not standard. In that exampe, users have different performance objectives, and the proportiona fow profie does not ie in the core. This iustrates that Theorem 4.2 heaviy depends on the setting of the system. Nevertheess, Theorem 4.2 states that, for users with standard additive performance objectives, if a users send their demand according to the proportiona routing strategy, then (i) the system is optima and (ii) none of the 2 N possibe coaitions woud benefit by deviating from this strategy. Thus, for such users the proportiona fow profie is easy to impement and highy desirabe from a socia perspective. C. The Routing Game Nuceous We now continue to describe the RGN of our additive routing game. Due to the cost function in (28), for additive routing games we define a sighty different excess for each coaition, which is normaized in the coaition s demand. Specificay, for each coaition S N and a cost vector J(f), e S (J(f)) [ ] r S J S (f) J S (ˆf,ˆf S ). (42) Hence, we consider the average dissatisfaction of a coaition S at a cost vector J(f). Simiary to botteneck routing games, we first prove that the RGN ies in the core. Proposition 4.2. Consider an additive routing game. The Routing Game Nuceous ies in the core. Proof. Denote the Proportiona Cost Aocation as J. From Lemma 4.3 it foows that for any coaition S N, rs R J sys J S (ˆf,ˆf S ). Therefore, due to (42), the excess of any coaition S N at J equas e S (J ) = R J sys r S J S (ˆf,ˆf S ) 0. (43) Consequenty, from Definition 3.3 it foows that for any coaition S N, e S (J RGN ) max S N e S (J ) 0. Denote the fow profie at the RGN as f. It foows from (42) that for any coaition S N, r S i S L f i T (f ) r S i S L ˆf i T ( ˆf + ˆf S ). Hence, for any coaition S there exist a user k S for which J k RGN = L f k T (f ) L ˆf k T ( ˆf + ˆf S ) = J k (ˆf,ˆf S ). Consequenty, the nuceous ies in the core. We now continue to describe the nuceous for symmetrica users. Theorem 4.3. Consider an additive routing game with symmetrica users. The Routing Game Nuceous is equa to the system-optima Proportiona Cost Aocation 7. Proof. Consider an additive routing game and denote the Proportiona Cost Aocation as J. Since a users have equa demands, for any two coaitions S, T N with T > S it foows that r T > r S. Moreover, according to Lemma 4.3, r Ĵ T (ˆf,ˆf T ) T r Ĵ S (ˆf,ˆf S ) and e S T (J ) e S (J ). Simiary, if T = S it foows that e T (J ) = e S (J ). This impies that, at the PCA, any coaition of size S = N has the argest excess. Atogether, there exist ( N N ) = N different combinations of coaitions for which S = N and their excesses are equa. Now assume by contradiction that there exists some cost vector J( f) J for which e ( J) xm e (J ). Consider a coaition K for which K = N. From Definition 3.3 it foows that e S ( J) max S N e S (J ) = e K (J ). Therefore, it foows from (42) that for any coaition K with size S = N : r K [ J K Ĵ K (ˆf N \K,ˆf K ) ] R J sys r K Ĵ K (ˆf N \K,ˆf K ). Hence, for any coaition K with size K = N : i K As a resut of (44), we get that K N i K K =N K N K =N J i K N K =N J i rk R J sys. (44) r K R J sys = K N K =N N N J sys. (45) On the other hand we have J i = (N ) J i (N )Jsys (46) i K i N = K N K =N N N J sys. Therefore, J by combining (45) with (46) it foows that i = Jsys. However, since J J, there exists a user i N k for which J k < rk R J sys. Thus, i N \k J i = Jsys J k > R rk R J sys = rn \k R J sys, which is a contradiction to (44). Therefore, there does not exist any J J for which e ( J) xm e (J ) and the proportiona routing strategy fow profie is equa to the Routing Game Nuceous. 7 In Appendix C we provide an exampe of non-symmetrica users for which the PCA does not equa the RGN.