Approximating Congestion + Dilation in Networks via

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Approximating Congestion + Dilation in Networks via ‘Quality of Routing’ Games Costas Busch, Rajgopal Kannan, and Athanasios V. Vasilakos

Abstract—A classic optimization problem in network routing is to minimize C + D, where C is the maximum edge congestion and D is the maximum path length (also known as dilation). The problem of computing the optimal C ∗ + D∗ is NP-complete even when either C ∗ or D∗ is a small constant. We study routing games in general networks where each player i selfishly selects a path that minimizes Ci + Di that reflects the congestion and dilation of the player’s path. We first show that there are instances of this game without Nash equilibria. We then turn to the related quality of routing (QoR) games which always have Nash equilibria. In these games the paths are divided into different classes which do not interfere with each other (for example by using wavelength or frequency division multiplexing). The outcomes of QoR games are evaluated with the price of anarchy which yields a ratio of O(λ · log4 n) between the worst social cost in any Nash equilibrium and the optimal coordinated solution, where λ ≤ min(C ∗ , D∗ ), and n is the number of nodes. Thus, when either C ∗ or D∗ is small (e.g. constant), any Nash equilibrium of a QoR game gives a poly-log approximation to a hard optimization problem. Therefore, the players’ selfishness may help the social welfare.

I. I NTRODUCTION Routing algorithms are important for the functionality of a network because they provide paths on which the packets are sent over the network. Given a set of packets, two important metrics in the literature are considered for measuring the quality of the paths, the congestion C, which is the maximum number of paths that use any edge in the network, and the maximum path length D (dilation). In a seminal result, Leighton et al. [19] showed that there exist packet scheduling algorithms that can deliver the packets along their chosen paths in time very close to C + D. The work on packet scheduling has been extended in [8], [19], [20], [25], [27]. Here, we are concerned with the following optimization problem: Definition 1.1 (C + D-Routing Optimization Problem): Given a network and a set of packets find paths which minimize C + D. Let C ∗ +D∗ denote an optimal solution. In [33] it is shown that there is a constant approximation to this problem for general network topologies. We show that the problem of finding the A preliminary version of this paper appears with title “Quality of Routing Congestion Games in Wireless Sensor Networks”, in the Proceedings of the 4th International Wireless Internet Conference (WICON), November 2008. Dr. Busch is with the Computer Science Department, Louisiana State University, 286 Coates Hall, Baton Rouge, LA 70803, USA; [email protected] Dr. Kannan is with the Computer Science Department, Louisiana State University, 279 Coates Hall, Baton Rouge, LA 70803, USA; [email protected] Dr. Vasilakos is with the Department of Computer Science & Telecommunications Engineering, University of Western Macedonia, Greece 45431; [email protected]

optimal solution is NP-complete even if either (or both) of C ∗ or D∗ is a small constant (equal to 3). Motivated by the selfish behavior of entities in communication networks, we study routing games in general networks where each packet’s path is controlled independently by a selfish player. We model games with N players, where each player has a pure strategy profile from which it selfishly selects a single path from the source to the destination node in an uncoordinated manner. The objective of player i is to select a path that simultaneously minimizes the congestion Ci , which is the maximum number of paths that use any edge in player i’s path, and the path length Di . The player cost is Ci + Di which the player wants to minimize. From the player’s point of view this cost function is a justified metric, since there exists a scheduling algorithm that delivers the player’s packet in time proportional to Ci + Di [4]. We will refer to these games as C + D-routing games. A natural question that arises concerns the effect of the players’ selfishness on the welfare of the whole network measured with the social cost C + D. We examine the consequence of the selfish behavior in pure Nash equilibria which are stable states of the game in which no player can unilaterally improve her situation. The negative effect of selfishness is quantified with the price of anarchy (P oA) [18], [26], which expresses how much larger is the worst social cost in a Nash equilibrium compared to the social cost in the optimal coordinated solution. The price of anarchy provides a metric for evaluating how well do Nash equilibria of C + D-routing games approximate the optimal C ∗ + D∗ of the respective routing optimization problem. Small price of anarchy is preferable. The positive effect of selfishness is quantified with the price of stability (P oS) [1], [2] which expresses how much larger is the best social cost in a Nash equilibrium with respect to the social cost in the optimal coordinated solution. We start by showing that there are C+D-routing games with a small number of players (two or three) which have always Nash-equilibria. However, we also obtain the negative result that there are instances of C+D-routing games with a constant number of players which do not have Nash equilibria. This directed us to examine an alternative type of routing games, the quality of routing games, which have many similarities with the C + D-routing games and three good properties: (i) they always have Nash equilibria, (ii) the price of anarchy is small for interesting instances of the game, and (iii) the outcomes of the games provide approximations to the C + Drouting problem. We are able to connect the outcomes of quality of routing games with C + D-routing problems through the λ parameter.

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Given a set of packets, let Aκ be the set of all path selections with C + D = κ · (C ∗ + D∗ ). Let λκ denote the smallest product κ · min(C, D) among all path selections in Aκ . We define: λ = min λκ . κ

The parameter λ is characteristic for each routing problem. Note that λ ≤ min(C ∗ , D∗ ), since the path selection of the optimal solution belongs to A1 . Intuitively, λ seeks the path selections with smallest congestion or dilation around the optimal solution. The price of anarchy in quality of routing games is closely connected to λ. A. Quality of Routing Games In the quality of routing (QoR) problem there is a set of packets that need to be routed in the network and also there are m routing classes Q1 , . . . , Qm . Each routing class Qj has bQ + D b Q where C bQ is a maximum service cost SQj = C j j j variable service cost that depends on the congestion in the b Q is a fixed service cost for using the particular class, and D j particular class. In the QoR problem each path belongs to exactly one routing class. If it is more beneficial for a packet it can switch routing classes by changing its path. The paths in different routing classes do not affect each other, that is, any two paths in different routing classes do not cause congestion to each other. This non-interfering property can be achieved by selecting the paths of different classes to be edge-disjoint, or by assigning the paths of each routing class to a different transmission channel (e.g. with frequency or time-division multiplexing). Thus, only players within the same routing class bQ can cause congestion to each other and contribute to C j This is analogous to radio networks where only packets on the same frequency channel cause congestion to each other in a frequency multiplexed routing scheme; or analogous to light-wave color multiplexing in optical networks. In any path selection we denote the largest service cost experienced in any used class as S = maxj SQj , where the index j runs only among those classes used by the paths in the path selection. We are interested in the following optimization problem: Definition 1.2 (QoR Optimization Problem): Given a network with m routing classes and a set of packets find a selection of paths that minimizes S. We are now ready to define QoR games. Each player represents a packet which has a pure strategy profile. The player selfishly selects a single path in the network in an uncoordinated manner. For a player i whose paths is in routing bi + D b i , where C bi class Qj the cost function is defined as C is the congestion that the player experiences in the network measured only among the paths that belong to routing class bi = D bQ . Qj , and D j We show that QoR games (including restricted-QoR games) always have pure Nash equilibria, which can be obtained with best response dynamics where players greedily improve their paths whenever possible. We then examine the quality of the Nash equilibria in terms of the price of stability and price of stability. For the price of stability (P oS) we show that every

QoR game has a very good Nash equilibrium with optimal social cost: P oS = 1. For the price of anarchy we examined restricted-QoR games b Q is at least where the fixed service cost of each class D j the length of each path that the class contains (we can also define the respective restricted-QoR optimization problem in a straightforward manner). Such games are interesting when different routing classes correspond to different ranges of path lengths, as in scenarios where it is more costly to use longer paths than shorter paths. We show that the price of anarchy in any restricted-QoR game is: b · m · log n), P oA = O(λ b is defined below, m is the number of routing classes, where λ and n is the number of nodes in the underlying graph. b in a similar way as λ. For any path The parameter λ b be the worst congestion experienced by any selection let C b the worst fixed service cost experienced by any path and D b and D b may be obtained in different routing path (note that C classes). Let Aκ be the set that contains all path selections bκ denote the smallest product with cost S = κS ∗ . Let λ b b κ·min(C, D) considering all path selections in Aκ . We define b = minκ λ bκ . The parameter λ b explores all path selections λ that can give small congestion or fixed service cost around the optimal solution. B. Relation of QoR games to C + D-routing problems We consider a family of restricted-QoR games which give good approximations to the C +D-routing problem. Each routing class Qj holds paths with lengths in range [2j−1 , 2j − 1]. b Q = 2j − 1, The fixed service cost of routing class Qj is D j which is the largest possible path length in the class. There are m ≤ 1 + lg n routing classes, since the maximum path length cannot exceed n. Let P oS 0 denote how much larger is C + D in the best equilibrium of a QoR game compared to the optimal C ∗ +D∗ . Let P oA0 denote how much larger is C + D in the worst equilibrium of a QoR game compared to the optimal C ∗ +D∗ . We show the following: P oS 0 = O(log n),

P oA0 = O(λ · log4 n).

These bounds are obtained by relating the congestion and fixed service costs in QoR games with the congestion and dilation in the respective C + D-routing problem. In most cases the congestions are within a factor of m from each other in the b = O(mλ). two problems. Further, we show that λ Thus, when λ is small (e.g. constant) any Nash equilibrium of the QoR game provides a very good approximation (within a poly log factor from optimal) to the coordinated C + Drouting problem. For example λ is constant when either C ∗ or D∗ is a constant. Since we also show that it is an NPcomplete problem to find the optimal C ∗ + D∗ when either of C ∗ or D∗ is a small constant, we have that Nash equilibria of QoR games can provide good non-trivial approximations to a hard optimization problem. This is an important result which shows that local optimality and selfishness can provide good outcomes for the social welfare.

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C. Max Games For completeness of our study on C + D related games, we also examine max games where the social cost is max(C, D), and the player cost is max(Ci , Di ). Every max game has at least one pure Nash equilibrium. which can be obtained by best response dynamics. Max game equilibria can provide approximate solutions to the C + D-routing problem since C + D is within a factor of 2 from max(C, D). However, the approximation can only be good when the maximum allowed path length is small. The price of anarchy in max games is O(L + log n), where L is the maximum allowable path length in the network. This bound is worst case optimal, because we provide an example of a ring network where the optimal coordinated social cost is 1, while there is a Nash equilibrium with cost O(L) = O(n). Nevertheless, max games are interesting because they have very good Nash equilibria, where the price of stability is P oS = 1. Thus, there is always a Nash equilibrium that exhibits optimal social cost.

[21] where the authors focus on parallel link networks, but also give some results for general topologies on convergence to equilibria. Bottleneck congestion games have been studied in [3], where the authors consider the maximum congestion metric in general networks with splittable and atomic flow (but without considering path lengths). They prove the existence and nonuniqueness of equilibria in both the splittable and atomic flow models. They show that finding the best Nash equilibrium that minimizes the social cost is a NP-hard problem. Further, they show that the price of anarchy may be unbounded for specific edge congestion functions (these are functions of the number of paths that use the edge). If the edge congestion function is polynomial with degree p then they bound the price of anarchy with O(mp ), where m is the number of edges in the graph. In the splittable case it is shown in [3] that if the users always follow paths with low congestion then the equilibrium achieves optimal social cost.

D. Related Work

Outline of Paper

Congestion games were introduced and studied in [24], [28]. Koutsoupias and Papadimitriou [18] introduced the notion of price of anarchy in the specific parallel link networks model in which they provide the bound P oA = 3/2. Since then, many routing and congestion game models have been studied which are distinguished by the network topology, cost functions, type of traffic (atomic or splittable), and kind of equilibria (pure or mixed). Roughgarden and Tardos [31] provided the first result for splittable flows in general networks in which they showed that P oA ≤ 4/3 for a player cost which reflects to the sum of congestions of the edges of a path. Pure equilibria with atomic flow have been studied in [5], [6], [21], [28], [34] (our work fits into this category), and with splittable flow in [29], [30], [31], [32]. Mixed equilibria with atomic flow have been studied in [10], [9], [11], [13], [14], [15], [17], [18], [22], [23], [26], and with splittable flow in [7], [12]. To our knowledge there is no previous work that considers routing games that optimize congestion and path lengths simultaneously. Most of the work in the literature uses a single cost metric related to the total congestion measured as the sum of congestions of all the edges of the player’s path [6], [15], [30], [31], [32], [34]. Our work differs from these approaches since we adopt the metric Ci + Di for player cost which is directly related to the packet scheduling delay. The vast majority of the work on routing games has been performed for parallel link networks, with only a few exceptions on general network topologies [5], [6], [7], [29], which we consider here. Our work is closer to [5], where the authors consider the player cost Ci and social cost C. They prove that the price of stability is 1. They show that the price of anarchy is bounded by O(L+log n), where L is the maximum allowed path length. They also prove that κ ≤ P oA ≤ c(κ2 + log 2 n), where κ is the size of the largest edge-simple cycle in the graph and c is a constant. Some of the techniques that we use in our proofs are adaptations of the techniques introduced in [5]. Another related result for general networks which has a brief discussion of the convergence of maximum player cost (Ci ) games is

In Section II we give basic definitions. In section III we show that the C +D-routing problem is NP-complete even for small C ∗ or D∗ . We study C + D-routing games in section IV where we show that there are instances of such games without Nash equilibria. We give our results on QoR games in Section V. We examine max games in Section VI. We finish with our conclusions in Section VII. II. D EFINITIONS A. Path Routings Consider an arbitrary graph G = (V, E) with nodes V and edges E. Let Π = {π1 , . . . , πN } be a set of packets such that each πi has a source ui and destination vi . A routing p = [p1 , p2 , · · · , pN ] is a collection of paths, where pi is a path for packet πi from ui to vi . For any routing p and any edge e ∈ E, the edge-congestion Ce (p) is the number of paths in p that use edge e. For any path p, the path-congestion Cp (p) is the maximum edge congestion over all edges in p, Cp (p) = maxe∈p Ce (p). For any path pi ∈ p, we will use the notation Ci (p) = Cpi (p). The network congestion C(p) is the maximum edge-congestion over all edges in E, that is, C(p) = maxe∈E Ce (p). We denote the length (number of edges) of any path p as |p|. For any path pi ∈ p, we will also use the notation Dpi (p) or Di (p) to denote the length |pi |. We will denote by D(p) the maximum path length in routing p, that is D(p) = maxp∈p |p|. Whenever the context is clear, we will drop the dependence on p, for example, we will write b instead of C(p). C B. Routing Games A routing game in graph G is a tuple R = (G, N , P), where N = {1, 2, . . . , N } is the set of players such that each player i has a source ui and destination vS i , and P are the strategies of the players. In particular, P = i∈N Pi , where Pi is the strategy set of player i which a collection of available paths in G for player i from ui to vi . Any path p ∈ Pi is a pure

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strategy available to player i. A pure strategy profile is any routing p = [p1 , p2 , · · · , pN ], where pi ∈ Pi . The longest path length in P is denoted L(P) = maxp∈P |p|. For game R and routing p, the social cost (or global cost) is a function of routing p, and it is denoted SC(p). The player or local cost is also a function on p denoted pci (p). We use the standard notation p−i to refer to the collection of paths {p1 , · · · , pi−1 , pi+1 , · · · , pN }, and (pi ; p−i ) as an alternative notation for p which emphasizes the dependence on pi . Player i is locally optimal (or stable) in routing p if pci (p) ≤ pci (p0i ; p−i ) for all paths p0i ∈ Pi . A routing p is in a Nash Equilibrium (we say p is a Nash-routing) if every player is locally optimal. Nash-routings quantify the notion of a stable selfish outcome. A routing p∗ is an optimal pure strategy profile if it has minimum attainable social cost: for any other pure strategy profile p, SC(p∗ ) ≤ SC(p). We quantify the quality of the Nash-routings with the price of anarchy (P oA) (sometimes referred to as the coordination ratio) and the price of stability (P oS). Let P denote the set of distinct Nash-routings, and let SC ∗ denote the social cost of an optimal routing p∗ . Then, SC(p) , ∗ P SC

P oA = sup p∈

P oS = inf

p∈ P

SC(p) . SC ∗

III. H ARDNESS OF C + D- ROUTING P ROBLEM Consider a routing problem with N packets in a graph G = (V, E). We prove that it is an N P -complete problem to determine whether C + D ≤ k, for any k ≥ 4. Thus, it is an NP-complete problem to determine the optimal value C ∗ + D∗ when it is at least 4. We extend this result to prove that it is an NP-complete problem to determine the optimal value for any C ∗ ≥ 3 and D∗ ≥ 3. We formulate the problem as follows: Definition 3.1 (k-Routing Problem): Given a graph G, a set of packets Π, and a parameter k, determine whether there is a routing with C + D ≤ k. We prove that the k-routing problem is NP-complete for any k ≥ 4 with a reduction from the following constrained-3SAT problem which is known to be NP-complete [16, page 269]: Definition 3.2 (Constrained-3SAT Problem): Determine whether a CNF formula F is satisfiable, where each clause in F has size at most 3, and for each variable x there are at most three clauses in F that contain a literal of x. We start by showing that the 4-routing problem is N P complete. This result can be extended to larger values of k, and also considering the cases C ∗ ≥ 2 and D∗ ≥ 2. Theorem 3.3: The 4-routing problem is N P -complete. Proof: It is easy to verify that the 4-routing problem is in N P since we can nondeterministically construct any routing and check its C + D value in polynomial time. To prove N P -completeness we give a polynomial time reduction of the constrained-3SAT problem to the 4-routing problem. Consider an arbitrary CNF formula of the form F such that each clause c consists of at most three literals and for each variable x there are at most three clauses of F that contain a literal of x (that is, x or x). Without loss of generality, assume that no variable appears in two or more literals of the same

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clause, and that no variable appears in F as one type of literal only (that is, there are at least two literals for each variable, one with the variable itself and one complemented).1 Given a formula F we construct a graph G = (V, E) with a respective routing problem (see Figure 3). Graph G consists of two kinds gadgets, the variables-gadgets and the clausegadgets (see Figures 1 and 2). For each clause c there is a node c ∈ V , and for each variable x there are two nodes vx1 , vx2 ∈ V . Every literal of each variable x will have a corresponding packet with source node the clause that contains the literal and destination either vx1 or vx2 , Each packet will have two available paths of length at most 3 to its destination, one path through a variable-gadget and the other path through a clausegadget. We now describe the variable-gadgets. Let x be a variable that appears in three different clauses c1 , c2 and c3 . Consider the case where x ∈ c1 , x ∈ c2 , and x ∈ c3 . We add in G the variable-gadget shown in Figure 1.a. We create three packets in G: packet πx1 with source c1 and destination vx1 , packet πx2 with source c2 and destination vx2 , and packet πx3 with source c3 and destination vx1 . The case where x ∈ c1 , x ∈ c2 , and x ∈ c3 is symmetric and is depicted in Figure 1.b; here the packets have the same sources and destinations as in the previous case. The case where variable x appears in two clauses is depicted in Figure 1.c; here we create two packets, one emanating from each clause, so that both packets have destination vx1 (vx2 is not used in this case). For each clause in F we create a clause-gadget. Consider a clause c. Suppose that c contains three literals of variables x, y, and z. From the variable-gadgets constructed above, node c is the source of packets πxi1 , πyj1 , and πzk1 , with respective destination nodes vxi2 , vyj2 , and vzk2 , where 1 ≤ i1 , j1 , k1 ≤ 3 and 1 ≤ i2 , j2 , k2 ≤ 2. We create the clause-gadget shown in Figure 2.a (we do not add any new packets). If c contains two literals, e.g. from variables x and y, then we create the clause-gadget shown in Figure 2.b; here we add a new packet 1 It is easy to convert F in polynomial time to an equivalent formula F 0 that adheres the additional constraints.

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Fig. 3. The graph for formula F = (x ∨ y ∨ z) ∧ (x ∨ y ∨ z) ∧ (x ∨ y) and a routing with C + D = 4 for satisfying assignment x = 1, y = 1, z = 0; dashed lines represent packet paths

1 πc,aux with source in c and destination in a new node uc,aux . If c contains one literal, e.g. from variable x, then we create the clause-gadget shown in Figure 2.c; here we add two new 1 1 and πc,aux with source in c and auxiliary packets πc,aux destination in a new node uc,aux . Next, we continue with showing that F is satisfiable if an only if there is routing in G with C + D ≤ 4. Consider a satisfying assignment of F , we will construct a routing with C + D ≤ 4. If a literal is set to 1 then the respective packet follows the path to its destination through the variablegadget (north path in Figure 3), otherwise the respective packet follows the path to its destination through the clause-gadget (south path in Figure 3). In particular, if variable x = 1, then for every clause that contains literal x we let the respective packet πxi follow the path to the destination through the variable-gadget which has length 3 (north path in Figure 3), and for every clause that contains literal x we let the respective packet πxi follow that path to the destination through the clause-gadget which has length 2 (south path in Figure 3). If variable x = 0, then symmetrically, the packet πxi from each clause that contains x follows the north path, while the packet πxj from each clause that contains x follows the south path. The effect of this is that no more than two packets use the edges of a variable gadget, and the edges they use are disjoint. Thus, the sum of congestion and dilation of the north paths is 4. From every clause emanate at most 2 packets which follow the south path (since otherwise the clause wouldn’t be satisfied and F wouldn’t be satisfied). This causes congestion at most 2 on the edges of the clause-gadgets. Thus, the congestion plus dilation of the south paths is at most 4, namely, C + D ≤ 4. Consider now a routing with C + D ≤ 4; we will construct a satisfying assignment for F . The satisfying assignment is computed in a way that if a packet follows the north path then the corresponding literal is set to true. In order to verify that this is indeed a satisfying assignment we only have to

prove two claims: (i) in each clause at least one packet has to follow the north path, and (ii) only packets of the same kind of literal can follow the north path. Claim (i) has to be true, since otherwise, all the packets of a clause would follow the south path causing congestion 3 on edge e in the scenario depicted in Figure 2.a, which in combination with the length 2 of the south paths gives C + D = 5, a contradiction. The scenarios of Figures 2.b and Figures 2.c. are similar, with the difference that the high congestion on edge e is caused by the 1 2 auxiliary packets πc,aux and πc,aux . For claim (ii), consider a variable x whose literals are x ∈ c1 , x ∈ c2 , and x ∈ c3 ; let πx1 , πx2 , and πx3 be the respective packets with respective destinations vx1 , vx2 , vx3 . Then it is impossible that πx3 follows the north path if either of πx1 or πx2 follows its north path, since this would cause congestion 2 on edges e1 or e2 as shown in Figure 1.a, which in combination with the length 3 of the north paths gives C + D = 5, a contradiction. Note that πx1 or πx2 can follow the north path simultaneously, since their paths do not use common edges. The scenarios of Figures 1.b and 1.c are treated similarly. Thus, for variable x only packets of the same kind of literal can follow the north path. The proof of Theorem 3.3 can be extended to arbitrary krouting problems, where k ≥ 4, by modifying appropriately the routing problem in the reduction. In the reduction graph G it holds that there is a routing with C + D ≤ 4 if and only if F is satisfiable. We can increase the C ∗ + D∗ value in the reduction to any arbitrary value with any of the following two methods. •

•

Create artificial congestion: in every edge we add a artificial packets with source and destination the end points of the edge. Stretch edges: we can increase the path lengths by replacing each edge adjacent to a clause with a path of 1 + b edges.

A combination of the above two methods produces a new routing problem such that F is satisfiable if and only if there is a routing with C + D ≤ 4 + a + b. This immediately proves that the k-routing problem is NP-complete for any k ≥ 4. It also has implications for C ∗ and D∗ . Note that C ∗ ≤ 3 + a and D∗ = 3 + b, since in the routing problem of Theorem 3.3, C ∗ ≤ 3 and D∗ ≤ 3. Let γ = 3 + a and δ = 3 + b. We obtain the following corollary: Corollary 3.4: The k-routing problem is N P -complete, for any k = γ + δ ≥ 4, where C ∗ ≤ γ, D∗ ≤ δ, and γ, δ ≥ 3. Corollary 3.4 implies that the k-routing problem is NPcomplete even if one (or both) of C ∗ or D∗ is a small constant (equal to 3).

IV. C + D- ROUTING GAMES Let R = (N , G, P) be a routing game such that for any routing p the social cost function is SC(p) = C(p) + D(p), and the player cost function pci (p) = Ci (p) + Di (p). On the positive side, we show that several kinds of games have Nashroutings. On the negative side, we show that there are also games with a small number of players without Nash-routings.

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A. C + D-routing Games with Nash-routings Players whose strategy sets contain at least two paths will be referred to as active players, while players with one path in their strategy sets will be referred to as passive players. By default passive players are locally optimal since they only have one available path. The effect is that passive players create unaltered congestion on the edges. We show that every game with two active players and an arbitrary number of passive players has a Nash-routing. We also show that every game with three players (active or passive) has a Nash-routing. Lemma 4.1: Every C + D-routing game with two active players and an arbitrary number of passive players has a Nashrouting. Proof: Consider any C + D-game instance R = (G, N , P) with two active players and an arbitrary number of passive players. Let the two active players be i and j. Denote pi the smallest length path in strategy set Pi that minimizes player’s i cost if player j was not in the game; let hi = ci +|pi | denote the congestion plus dilation cost of path pi . Similarly, let pj denote the smallest length path in strategy set Pj that minimizes player’s j cost if player i was not in the game; let hj = cj + |pj | denote the congestion plus dilation cost of path pj . Let p be the routing that includes both of paths pi and pj . If p is a Nash-routing then we are done. Thus, consider the case where p is not a Nash-routing. Without loss of generality suppose that player i is unstable in p. Then it has to be that pci (p) = hi + 1, and pi and pj share a common edge e with congestion Ce (p) = ci + 1. There exists an alternative path p0i with cost pci (p0 ) = hi , where p0 = (p0i ; pi− ) (the routing with the new path p0i ). In p0 player i is stable since there cannot be a path for i with cost less that hi . The new congestion c0i in path p0i has to be c0i ≤ ci , since pi was chosen to be a path of smallest length, so |p0i | ≥ |pi |, and thus if c0i > ci then pci (p0 ) > hi . Suppose now that player j is not locally optimal in p0 . Since player j has the same path pj in both p and p0 , pcj (p0 ) = hj + 1, and p0i and pj share a common edge e0 with congestion Ce0 (p) = cj + 1. Thus, c0i ≥ cj + 1. Note that ci + 1 ≤ cj + 1, since pi and pj share edge e with congestion ci + 1. Then c0i ≥ cj + 1 ≥ ci + 1, a contradiction. Thus, player j is also stable in p0 , and p0 is a Nash-routing. Lemma 4.2: Every C + D-routing game with three players (active or passive) has a Nash-routing. Proof: Consider a C + D-routing game instance with 3 players R = (G, N , P), where N = {1, 2, 3}. Let pi denote a shortest path for player i ∈ N in the strategy set Pi . Let p = [p1 , p2 , p3 ], which is a routing with the shortest paths. If p is a Nash-routing then we are done. So, suppose that p is not a Nash-routing. Since there are three players, C(p) ≤ 3. If C(p) = 1 then the paths do not interfere and p is a Nash-routing. Thus, we only need to examine: case (i) with C(p) = 2, and case (ii) with C(p) = 3. First, consider case (i) with C(p) = 2. The player cost is pci (p) ≤ |pi | + 2. The smallest possible cost for each player is |pi | + 1, since pi is a shortest path and whose congestion can never be below 1. Without loss of generality, suppose that player 1 is unstable in p. Then player 1 has a an alternative

path p01 such that pc1 (p0 ) = |pi | + 1, where p0 = (p0i ; pi− ) (the new routing with path p01 ), and player 1 is stable in p0 . Since p1 was a shortest path, |p01 | ≥ |p1 |. Since pc1 (p0 ) = |p01 | + C1 (p0 ) = |pi | + 1, we have that Ci (p0 ) ≤ 1. Thus p01 is also a shortest path in Pi with no common edges with p2 and p3 . In a similar way we can prove that any unstable player in p0 can follow the alternative shortest path with no interference with the other player’s paths. This will result to a Nash-routing. Now, consider case (ii) with C(p) = 3. We have that Ci (p) = 2, and player cost pci (p) ≤ |pi | + 3. Without loss of generality, suppose that player 1 is unstable in p. Then player 1 has a an alternative path p01 such that pc1 (p0 ) ≤ |pi |+2, where p0 = (p0i ; pi− ) (the new routing with path p01 ). and player 1 is stable in p0 . Since p1 was a shortest path, |p01 | ≥ |p1 |. Since pc1 (p0 ) = |p01 | + C1 (p0 ) ≤ |pi | + 2, we have that Ci (p0 ) ≤ 2. Thus, C(p0 ) ≤ 2, since there are only three players. If C(p0 ) = 1 then p0 is a Nash routing. If C(p0 ) = 2 then the problem can be treated similar to case (i) by taking into account the additional scenario that player 1 may use a path length one edge larger than its shortest but with no common edges with the other two paths. B. C + D-routing Games without Nash-routings We give a game instance with a small number of players which does not have have Nash-routings. Lemma 4.3: There is a C + D-routing game instance with three active players and four passive players that has no Nashrouting. Proof: Let R = (G, N , P) denote the routing game instance. The graph G is depicted in Figure 4. There are seven players, namely, N = {1, . . . , 7}. Players 1, 2, and 3, have respective strategy sets P1 = {p1 , p01 }, P2 = {p2 , p02 }, and P3 = {p3 , p03 }; these players are active. In the figure for player i = 1, . . . , 3 the respective source and destination nodes are ui and vi . There are six critical edges denoted e1 , . . . , e6 that the paths use and which are shown in the figure as straight horizontal lines. These edges may have congestion larger than 1. The squiggly part of the paths are assumed to have congestion 1 and their length and their lengths are chosen so that the following relations hold: |p01 | = |p1 | − 2, |p02 | = |p2 | + 3, and |p03 | = |p3 | + 3.

p1 p02

e1 p1 u1

p1 p03

p1 p2 p3

e3

e2

v3

p3

p3

p2

p01 p2 (+3)

v2

p2

(+1)

p1

u3

e4 Fig. 4.

p03

p03

p02

u2

p01

p02

v1

p3 p2 p01

p01 p3

e5

(+3)

e6

p01

(+4)

The graph of a C + D-routing game without Nash-routings

7 Routing [p1 , p2 , p3 ] [p01 , p2 , p3 ] [p01 , p02 , p3 ] [p01 , p02 , p03 ] [p1 , p02 , p03 ] [p1 , p2 , p03 ] [p1 , p02 , p3 ] [p01 , p2 , p03 ]

C1 4 5 5 5 2 3 3 5

D1 10 8 8 8 10 10 10 8

pc1 14 13 13 13 12 13 13 13

C2 4 5 1 1 2 4 2 5

D2 7 7 10 10 10 7 10 7

pc2 11 12 11 11 12 11 12 12

C3 4 5 5 1 2 2 4 1

D3 7 7 7 10 10 10 7 10

pc3 11 12 12 11 12 12 11 11

Unstable player 1 player 2 player 3 player 1 player 2 player 3 player 2 player 2

TABLE I A LL POSSIBLE ROUTINGS WITH COSTS AND UNSTABLE PLAYERS

Players 4 to 7 are passive, in the sense that they have only one path in their strategy sets, and their sole purpose is to create additional congestion on edges e3 , e4 , e5 , e6 (the paths of these players are not shown explicitly in the figure). In particular, the passive players cause additional congestion 1 to edge e3 , additional congestion 3 to e4 and e5 , and additional congestion 4 to e6 . The additional congestion is depicted in the figure inside a parenthesis under each edge. Since the only active players are 1, 2, and 3, and each player has two path choices, there are eight possible different routings. We examine each routing and prove that it is not a Nash-routing. We use the vector [p1 , p2 , p3 ] to denote a routing where the ith position of the vector contains the path choice of user i. We set the specific lengths as: |p1 | = 10, |p01 | = 8, |p2 | = 7, |p02 | = 10, |p3 | = 7, and |p03 | = 10. For a player i ∈ {1, 2, 3} and routing p we define the complementary routing to be the one where player i chooses the alternative path. For example, for player 2 the complementary routing of [p1 , p2 , p3 ] is [p1 , p02 , p3 ]. A player is stable (locally optimal) if the complementary routing does not give a lower cost for the player. Using Table I it is easy to determine whether a player is stable or not by examining the respective costs in the complementary routings. In this way, we find the nonlocally players which are shown in the rightmost column of the table. We find that: player 1 is unstable in routings [p1 , p2 , p3 ] and [p01 , p02 , p03 ]; player 2 is unstable in routings [p01 , p2 , p3 ], [p1 , p02 , p03 ], [p1 , p02 , p3 ], and [p01 , p2 , p03 ]; and player 3 is unstable in routings [p01 , p02 , p3 ] and [p1 , p2 , p03 ]. V. Q UALITY OF ROUTING G AMES Here we describe the quality of routing congestion games (QoR games). An instance of a QoR game is defined by a tuple R = (G, N , P, Q), where in addition to the regular components of a routing game there is also the set of routing classes Q = {Q1 , . . . , Qm }. Each path p ∈ G belongs to exactly one routing class, denoted Q(p). Each routing class b Q to each path that uses it, Qj imposes a fixed service cost D j bQ ≤ D b Q , for any 1 ≤ i < j ≤ m. such that 1 ≤ D i j Consider now a routing p. We will denote by Q(p) the class that path p belongs to. Let Qj (p) denote the set of paths in p that use class Qj . For any path pi ∈ p, we will b i (p) = D b p (p) = D b Q(p ) . Let D(p) b use the notation D = i i b maxi∈N Di (p) denote the maximum fixed service cost that b any player has to pay in routing p. Note that D(p) is also equal to the fixed service cost of the largest index routing class used in routing p.

Only paths in the same class can cause congestion to each other. For an edge e and routing class Qj we define b(e,Q ) (p), the congestion of edge e at class Qj , denoted C j as the number of paths in p that belong to class Qj and use edge e. The largest congestion of edge e at any class b(e,Q ) (p). For any path be (p) = maxj∈[m] C is denoted by C j bp (p) (alternatively, pi ∈ p we define the path congestion C i player i’s congestion), to be the maximum congestion at class bp (p) = Q(pi ) of any of the edges of path pi ; namely, C i b(e,Q(p )) (p). We will also denote C bi (p) = C bp (p). maxe∈pi C i i b Let CQj (p) denote the congestion at routing class Qj , which is the maximum congestion experienced by any player in that bQ (p) = maxp ∈Q C bp (p). Let C(p) b class; namely, C = j i j i b maxi∈N Ci (p) denote the maximum congestion experienced b by any path. Note that C(p) is also equal to the maximum congestion in any service class. The player cost pci is defined as the sum of the congestion and fixed service cost of its chosen path; namely, bi (p) + D b i (p). If Qj (p) 6= ∅, the maximum pci (p) = C player cost experienced by a player in class Qj is denoted as SQj (p) = maxi∈N pci (p). The maximum service cost experienced by any player in any class is denoted as S(p) = maxj:(j∈[m])∧(Qj (p)6=∅) SQj (p). The social cost of routing p is SC(p) = S(p). Note that the social cost expresses the maximum cost experienced by a player in routing p, that is, b SC(p) = S(p) = maxi∈N pci (p). Note that C(p) < S(p), b b b D(p) < S(p), and S(p) ≤ C(p) + D(p); thus, b b S(p) ≤ C(p) + D(p) ≤ 2S(p).

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A. Existence of Nash-routings in QoR Games We show that QoR games have Nash-routings and we also estimate the price of stability. The existence of Nash routing relies on finding an appropriate potential function and a total order of the routings. We prove that given an arbitrary initial state any greedy move of a player can only give a new routing with smaller order. Thus, repeated greedy moves converge either to the smallest order routing or to a routing where no player can improve further. In either case, a Nash-routing is reached. Let R = (G, N , P, Q) denote a QoR game. The potential function is defined as follows. Let r = N + S(Qm ) (this is largest possible player cost). For any routing p we define the routing vector M (p) = [m1 (p), . . . , mr (p)], where mi (p) is the number of players in p whose cost is i. Note that if k is the largest player cost in p then mk 6= 0 and mk0 = 0 for all k 0 > k. The routings can be totally ordered lexicographically. Let p and p0 be two routings, with M (p) = [m1 , . . . , mr ] and M (p0 ) = [m01 , . . . , m0r ]. We say that M (p) = M (p0 ) if mi = m0i for all 1 ≤ i ≤ r, and M (p) < M (p0 ) if there is a j, 1 ≤ j ≤ r, such that mk = m0k for all k > j, and mj < m0j . We order the routings p and p0 according to the order of their respective vectors, that is p ≤ p0 if and only if M (p) ≤ M (p0 ). Note that for any two p and p0 it either holds that p = p0 or p < p0 , that is, the routings are totally ordered. Consider an arbitrary routing p. If p is not a Nash-routing, there is at least one user i which is unstable. Then a greedy

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move is available to player i in which the player can obtain lower cost by changing the path from pi to some other path p0i with lower cost. In other words, the greedy move takes the original routing p = (pi ; p−i ) to a routing p0 = (p0i ; p−i ) with improved player cost pci (p0 ) < pci (p). We show now that any greedy move gives a smaller order routing: Lemma 5.1: If a greedy move by player i takes a routing p to a new routing p0 , then p0 < p. Proof: Let pi and p0i denote the paths of i in routings p bi (p) + D b i (p) = z1 and and p0 , respectively. Let pci (p) = C 0 0 0 b b pci (p ) = Ci (p ) + Di (p ) = z2 . Since player i decreases its cost in p0 , z2 < z1 . Consider now the vectors of the routings M (p) = [m1 , . . . , mr ] and M (p0 ) = [m01 , . . . , m0r ]. We will show that M (p) < M (p0 ). Let Y denote the set of players whose cost increases in p0 with respect to their cost in p. Then, for any j ∈ Y it has to be Q(pj ) = Q(p0i ), where pj is the path of player j in routing p. We show that pcj (p0 ) ≤ pci (p0 ). Suppose, for the sake of contradiction, that pcj (p0 ) > pci (p0 ). Then, bj (p0 ) + D b j (p0 ) > C bi (p0 ) + D b i (p0 ). Since Q(pj ) = Q(p0 ), C i 0 0 b b bj (p0 ) > we have that Dj (p ) = Di (p ), which implies that C bi (p0 ). The increase in congestion of pj in p0 can only be C caused by p0i due to a common edge e with pj that has conb(e,Q(p0 ) (p0 ) = C bj (p0 ). Therefore, C bi (p0 ) ≥ C bj (p0 ), gestion C i 0 0 a contradiction. Consequently, pcj (p ) ≤ pci (p ). Consequently, in vector M (p0 ) all the entries in positions z2 + 1, . . . , r do not increase with respect to M (p). Further, since player i switches paths from cost z1 to z2 with z1 > z2 , we obtain mz1 > m0z1 . Thus, M (p) < M (p0 ), as needed. Since there are only a finite number of routings, Lemma 5.1 implies that starting from arbitrary initial state, best response dynamics converge in a finite time to a Nash-routing, where every player is locally optimal (stable). Thus, we have: Theorem 5.2: In any QoR game, starting from an arbitrary initial routing, best response dynamics converge to a Nashrouting. Since the routings are totally ordered, there is a routing pmin which is the minimum, that is, for any routing p, pmin ≤ p. The minimum routing is also a Nash-routing, since no greedy move can improve from it. The minimum routing pmin achieves social cost within a factor of 2 from optimal. This implies that the price of stability is 2. The details are in the following result. Theorem 5.3: For any QoR game the price of stability is P oS = 1. Proof: It suffices to show that the minimum routing pmin achieves optimal social cost. Suppose for the sake of contradiction that there is a routing p ≥ pmin with SC(pmin ) > SC(p). Let k1 denote the largest player cost in p and let k2 denote the largest player cost in pmin . From the definition of social cost in QoR games we have that k1 = SC(p) and k2 = SC(pmin ). Therefore, k2 > k1 . In the vector M (p) = [m1 , . . . , mr ] it holds that mk1 6= 0 and mk = 0 for k > k1 . Similarly, in the vector M (pmin ) = [m01 , . . . , m0r ] it holds that m0k2 6= 0 and m0k = 0 for k > k2 . Therefore, M (pmin ) > M (p), contradicting the fact that p ≥ pmin .

B. Price of Anarchy in restricted-QoR Games We will consider restricted-QoR games where each routing class has the restriction that for any path p it holds that b Q(p) , namely, the path lengths in a routing class do |p| ≤ D not exceed the fixed cost of the class. Consider from now on a restricted-QoR game R = (N , G, P, Q), where G has n nodes, and Q has m routing classes. We will bound the price of anarchy of R. Let p∗ be the routing with optimal social cost. We denote SC(p∗ ) = S ∗ . Let p0 be a routing that provides the value of b with C b 0 = C(p b 0 ), D b 0 = D(p b 0 ), S 0 = S(p0 ), S 0 = κ·S ∗ and λ, 0 b0 b b λ = κ · min(C , D ). From Theorem 5.2, game R has at least one Nash-routing. Let p be a Nash-routing with S = S(p), b = C(p) b b = D(p). We bound the price of anarchy C and D by finding a relation between S and S ∗ through S 0 . We begin with the following result. bi ≥ Lemma 5.4: In Nash-routing p, for any player i with C 0 b b b C − x, where x ≥ 0, it holds that Di ≤ Di + x + 1. Proof: Suppose for the sake of contradiction that there bi ≥ C b − x and D bi > D b 0 + x + 1. Then, is a player i with C i 0 b b b b b b pci = Ci + Di > C −x+ Di +x+1 = C + Di0 +1. If user i was b b 0 , since p0 to switch to path p0i its cost would be pc0i ≤ C+1+ D i i b has congestion at most C before player i switches its path, and b after player the congestion of path p0i increases to at most C+1 0 0 b b i switches to it. Therefore, pci ≤ C +1+ Di < pci . Thus, in p player i would not be locally-optimal, which is a contradiction, bi ≤ D b 0 + x + 1, as since p is a Nash-routing. Therefore, D i needed. For each edge e ∈ G let Πe (p) denote the set of players whose paths in routing p use edge e and belong to the be (p). For any edge e ∈ E same class that contributes to C and i ∈ Πe (p), we define F (e, i) to be the set that conb(e0 ,Q(p0 )) (p) ≥ C bi − D b 0. tains all edges e0 ∈ p0i with C i Let F (e)S= ∪i∈Πe (p) F (e, i), and for any set of edges X, F (X) = e∈X F (e). We have: Lemma 5.5: In Nash-routing p, for every edge e with be (p) ≥ 1 and player i ∈ Πe (p) it holds that |F (e, i)| ≥ 1. C Proof: Suppose that |F (e, i)| = 0. Then for every edge b(e0 ,Q(p0 )) (p) < C bi − D b 0 . Therefore, e0 ∈ p0i it holds that C i 0 bp0 (p) < C bi − D b . If player i was to choose path p0 its cost C i i bp0 (p) + 1 + D b0 < C bi − D b0 + 1 + D b0 = would be pc0i ≤ C i i 0 bi + 1 ≤ C bi + D b i = pci (p). Thus, pc < pci (p) which C i implies that player i is not locally-optimal in Nash-routing p, a contradiction. Lemma 5.6: Let Z be the set that contains all edges e with be (p) ≥ C b − 2D b 0 · lg n. If C b > 2D b 0 · lg n, then congestion C there is a set of edges X ⊆ Z with |F (X)| ≤ 2|X|. Proof: We recursively define sets of edges E0 , . . . , E2 lg n , such that Ei = Ei−1 ∪ F (Ei−1 ), and set E0 contains all the be (p) = C. b We show now that there edges e with congestion C is a j, 0 ≤ j ≤ 2 lg n, such that |F (Ej )| ≤ 2|Ej |. Suppose for contradiction that such a j does not exist. Thus, for all j, 0 ≤ j ≤ 2 lg n, it holds that |F (Ej )| > 2|Ej |. In this case, it it straightforward to show that |Ek | > 2|Ek−1 |, for any 1 ≤ k ≤ 2 lg n. Since |E0 | ≥ 1, it holds that |E2 lg n | > 22 lg n = n2 . However, this is a contradiction, since the number of edges in G cannot exceed n2 . Thus, there is a j, 0 ≤ j ≤ 2 lg n, with

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|F (Ej )| ≤ 2|Ej |; we set X = Ej . It only remains to show that X ⊆ Z. We show the stronger be (p) ≥ C b − kD b 0 , where claim that for any Ek and e ∈ Ek , C 0 ≤ k ≤ 2 lg n. We prove this claim by induction on k. For be (p) = C b=C b−0· k = 0 we have that every edge in e has C 0 b D , thus the claim trivially holds. For the induction hypothesis, suppose that the claim holds for any k = t < 2 log n. In the induction step we will prove that the claim holds also for k = t + 1. We have that Et+1 = Et ∪ F (Et ). By induction be (p) ≥ C b − tD b 0 . Therefore, for hypothesis, for any e ∈ Et , C 0 b b b b b 0 · lg n and any e ∈ Et , Ce (p) ≥ C − (t + 1)D . Since C > 2D b t + 1 ≤ 2 lg n, Ce (p) ≥ 1. Thus, for any e ∈ Et there is at bi ≥ C be (p) ≥ C b − tD b 0 . By least one player i ∈ Πe (p) with C 0 definition, for player i it holds that every edge e ∈ F (e, i) has bi −D b 0 ≥ C−(t+1) b b 0 . Thus, C be0 (p) ≥ C− b b(e0 ,Q(p0 )) (p) ≥ C D C i b 0 . From the definition of F (Et ), it follows that for any (t+1)D be0 (p) ≥ C b − (t + 1)D b 0 . By considering the edge e0 ∈ F (Et ), C union Et+1 = Et ∪ F (Et ), we obtain that for any e ∈ Et+1 , be (p) ≥ C b − (t + 1)D b 0 , as needed. C b ≤ 10C b0 · D b0 · Lemma 5.7: In Nash-routing p it holds that C lg n. b ≤ 2D b 0 · lg n, then the claim holds immeProof: If C b > 2D b 0 · lg n. From Lemma 5.6, diately. So, assume that C there is a set of edges X with |F (X)| ≤ 2|X|, where for be (p) ≥ C b − 2D b 0 · lg n. each edgeSe ∈ X it holds that C P b Let Π = e∈X Πe (p). Let M = e∈X Ce (p) be the total utilization of the edges in X by the players in Π. We have b − 2D b 0 · lg n). that M ≥ |X|(C bi (p) ≥ C b −2D b 0 ·lg n. From For any player i ∈ Π we have C 0 0 b b b b 0 · lg n. Lemma 5.4, we obtain Di ≤ D + 2D · lg n + 1 ≤ 4D 0 Let L denote the maximum path length of any player in Π. Since R is a restricted-QoR game, for every player i ∈ Π we b i , which implies that L0 ≤ 4D b 0 · lg n. have that |pi | ≤ D The parameter M can also be bounded as M ≤ L0 |Π| which is an upper bound on the total utilization of the edges in X by b − 2D b 0 · lg n) ≤ L0 |Π|, the players in Π. Consequently, |X|(C which gives: 0 b0 b ≤ L |Π| + 2D b 0 · lg n ≤ 4D |Π| lg n + 2D b 0 · lg n C |X| |X|

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Since from Lemma 5.5, |F (e, i)| ≥ 1, in routing p0 the path of each user in Π has to use at least one edge in F (X). Thus, in routing p0 , the edges in F (X) are used at least |Π| times. Therefore, there is an e ∈ F (X) which in p0 is used by at least |Π|/|F (X)| players. Since there are m service classes, the congestion of e in one of those service classes is at least b 0 ≥ |Π|/(|F (X)| · m). Since |Π|/(|F (X)| · m). Therefore, C |F (X)| ≤ 2|X|, we obtain: b 0 · |X| · m. |Π| ≤ 2C

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b ≤ 8C b0 · D b0 · By Combining Equations 2 and 3, we get: C b 0 lg n ≤ 10C b0 · D b 0 · m lg n. m lg n + 2D Finally, we obtain the main result on the price of anarchy of restricted-QoR games: Theorem 5.8: For any restricted QoR game, the price of anarchy is: b · m lg n). P oA = O(λ

Proof: Suppose that p is the worst Nash-routing with maximum social cost. Then, using Equation 1 we get: b + D) b S κS 2κ(C SC(p) = ∗ = 0 ≤ . ∗ b0 + D b0 SC(p ) S S C We examine two cases: b > D/4: b • C b = O(C), b and from Lemma 5.7, C b ≤ 10C b0 · D b0 · Since D m lg n, we get: ! Ã b0 · D b 0 · m lg n κ·C P oA = O b0 + D b0 C P oA =

= = •

b0 , D b 0 ) · m lg n) O(κ · min(C b · m lg n). O(λ

b ≤ D/4: b C Let i be the player with maximum cost in p. Clearly, bi + D b i ≥ D. b Further, 0 ≤ C bi ≤ C b ≤ D/4. b pci (p) = C b b b b b b b Thus, Di ≥ D − Ci ≥ D − D/4 = 3D/4. Since Ci ≥ b − C, b Lemma 5.4 gives Di ≤ D0 + C b+1 ≤ 0 = C i 0 0 b + D/4 b + 1. Therefore, 3D/4 b ≤D b + D/4 b + 1. Thus, D b ≤ 2(D b 0 + 1) ≤ 3D b 0 . In order words, D b = O(D b 0 ). D b = O(D), b we obtain: Since C Ã ! b0 κD b P oA = O = O(κ) = O(λ). b0 + D b0 C

By combining the two above cases we obtain the desirable result. C. Relation to C + D-routing problems Consider a restricted-QoR game R = (N , G, P, Q) where each routing class Qj holds paths with lengths in range b Q = 2j − 1, which [2j−1 , 2j − 1]. The fixed service cost is D j is the largest possible path length in the class. Thus, there are m ≤ 1 + lg n routing classes. b + D)/2 b Lemma 5.9: For any routing p it holds: (C m0kmax , since when the path switches to p0i , 1 mk1max = ak1max +bk1max decreases by at least one because either ak1max decreases by one (if the new path has lower congestion) or bk1max decreases by one (if the new path has lower length). Since k2max < k1max , M (p) > M (p0 ) implying that p > p0 . Since there are only a finite number of routings, Lemma 6.1 implies that starting from arbitrary initial state, every best response dynamic converges in a finite time to a Nash-routing, where every player is locally optimal. Therefore we have the following result: Theorem 6.2: In any max game, starting from an arbitrary initial routing, best response dynamics converge to a Nashrouting. Since the routings are totally ordered, there is a routing pmin which is the minimum, that is, for all routings p, pmin ≤ p. Clearly, the minimum routing is also a Nash-routing. The minimum routing pmin achieves also optimal social cost which gives price of stability 1. The details are in the following theorem. Theorem 6.3: For any max game the price of stability is P oS = 1. Proof: It suffices to prove that the minimum routing pmin achieves optimal social cost, that is, SC(pmin ) ≤ SC(p), for any other routing p ≥ pmin . Suppose for contradiction that there exists a routing p ≥ pmin with SC(p) < SC(pmin ). Let SC(p) = max(C(p), D(p)) = k1 , and SC(pmin ) = max(C(pmin ), D(pmin )) = k2 . Clearly, k1 < k2 . Therefore, in the vector M (p) = [m1 , . . . , mr ] it holds that mk1 6= 0 and mk = 0 for k > k1 . Similarly, in the vector M (pmin ) = [m b 1, . . . , m b r ] it holds that m b k2 6= 0 and m b k = 0 for k > k2 . Therefore, M (pmin ) > M (p), contradicting the fact that p ≥ pmin . B. Price of Anarchy in Max Games Consider a max routing game R = (N , G, P), where G has n nodes. Theorem 6.2 implies that there is at least one Nash-routing. Consider a Nash-routing p. Denote C = C(p) and D = D(p). Let p∗ be the optimum (coordinated) routing with minimum social cost. Denote C ∗ = C(p∗ ) and D∗ = D(p∗ ). Note that each payer i ∈ N has a path pi ∈ p and a corresponding optimal path p∗i ∈ p∗ from the player’s source to the destination. Let L = L(P) denote the maximum allowed path length in the strategy sets.

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For each edge e ∈ G, denote Πe (p) the set of players whose paths in routing p use edge e. We define H to be a set that contains all edges e ∈ G with congestion Ce (p) ≥ D + 2. Consider an edge e ∈ H. Let i ∈ Πe (p) be a player whose path pi in routing p uses edge e. We define F (e, i) to be a set that contains all edges e0 ∈ p∗i with Ce0 (p) ≥ Ce (p) − 1. It holds that |F (e, i)| ≥ 1, since in routing p player i prefers path pi instead of p∗i because there is at least one edge e0 ∈ p∗i with Ce0 (p) ≥ Ce (p) − 1 > D. Let F (e) = ∪i∈ΠS F (e, i). e (p) For any set of edges X ⊂ H, we define F (X) = e∈X F (e). Lemma 6.4: Let Z be the set that contains all edges e with congestion Ce (p) ≥ C − 2 log n. If C ≥ D + 2 lg n + 2, then there is a set of edges X ⊆ Z such that |F (X)| ≤ 2|X|. Proof: We recursively construct a sets of edges E0 , . . . , E2 lg n , such that Ei = Ei−1 ∪ F (Ei−1 ), and set E0 contains all the edges e with congestion Ce (p) = C. From the construction of those sets it holds that for any e ∈ Ej , Ce (p) ≥ C − j, where 0 ≤ j ≤ 2 lg n. Thus, Ej ⊆ Z, for all 0 ≤ j ≤ 2 lg n. (Note that Z ⊆ H.) We can show that there is a j, 0 ≤ j ≤ 2 lg n, such that |F (Ej )| ≤ 2|Ej |. Suppose for contradiction that such a j does not exist. Thus for all j, 0 ≤ j ≤ 2 lg n, it holds that |F (Ej )| > 2|Ej |. In this case, it it straightforward to show that |Ek | > 2|Ek−1 |, for any 1 ≤ k ≤ 2 lg n. Since |E0 | ≥ 1, it holds that |E2 lg n | > 22 lg n = n2 . However, this is a contradiction, since the number of edges in G do not exceed n2 . Lemma 6.5: If C ≥ D+2 lg n+2, then C < 2LC ∗ +2 lg n. Proof: From Lemma 6.4, there is a set of edges X ⊂ Z with |F (X)| ≤ 2|X|. P For each e ∈ Z it holds that Ce (p) ≥ C − 2 lg n. Let M = e∈X Ce (p) ≥ |X|(C − 2 lg n), where M denotes the total utilization S of the edges in X by the paths of the players in Π. Let Π = e∈X Πe (p), that is, Π is the set of players which in routing p their paths use edges in X. By construction, the congestion in routing p in each of the edges of X is caused only by the players in Π. Since path lengths are at most L, each player in Π can use at most L edges in X. Hence, M ≤ L · |Π|. Consequently, |X|(C − 2 lg n) ≤ L · |Π|, which gives: C ≤ (L · |Π|)/|X| + 2 lg n. By the definition of F (X), in the optimal routing p∗ each user in Π has to use at least one edge in F (X). Thus, edges in the optimal routing p∗ , the edges in F (X) are used at least Π times. Thus, there is some edge e ∈ F (X) with Ce (p∗ ) ≥ |Π|/|F (X)|. Therefore, C ∗ ≥ |Π|/|F (X)|. Since |F (X)| ≤ 2|X|, we obtain |Π| ≤ 2C ∗ · |X|. Therefore: C ≤ 2LC ∗ + 2 lg n. Theorem 6.6: For any max game, the price of anarchy is P oA = O(L + log n). Proof: Suppose that p is the worst Nash-routing with maximum social cost. We have P oA = SC(p)/SC(p∗ ). If C ≥ D + 2 lg n + 2, then SC(p) = C. From Theorem 6.5, P oA ≤ C/ max(C ∗ , D∗ ) ≤ (2LC ∗ + 2 lg n)/C ∗ ≤ 2L + 2 lg n. If C < D + 2 lg n + 2, then SC(p) < D + 2 lg n + 2; thus P oA ≤ L + 2 lg n + 2. Hence, in both cases P oA = O(L + log n). There is a max game that shows that the result of Theorem 6.6 is tight in the worst case. Consider a ring network with n nodes and n edges. Give the same orientation to the

edges, so that each edge has one left node and one right node. For each edge ei , there is a corresponding player i whose source is the left node and the destination is the right node of the edge. The strategy set of each player has two paths: path pi which is only the edge ei , and path p0i which goes around the ring. Note that p = [p1 , p2 , . . . , pn ] is a Nash-routing with social cost 1. However, p0 = [p1 , p2 , . . . , pn ] is also a Nash routing with social cost n − 1. Thus, the price of anarchy is O(n) = O(L). VII. C ONCLUSIONS In this work we provided the first study (to our knowledge) of bicriteria routing games, where the players attempt to simultaneously optimize two parameters: their path congestion and path lengths. The motivation is the existence of efficient packet scheduling algorithms which deliver the packets in time proportional to the social cost. We examined C + D games, QoR games, and max games. The C +D have games instances that do not stabilize. QoR games stabilize and their equilibria provide good approximations to the C + D optimization problem. Max games stabilize and have price of stability 1, the best possible, however the price of anarchy may be large. Several open problems remain to examine. We studied two particular functions of the bicriteria, namely, the C + D and max(C, D) functions. It would also be interesting to study variation of these functions by adding additional criteria, such as a time component for dynamic traffic scenarios. It would also be interesting to explore splittable flow, or mixed Nash equilibria. The original C + D sum games do not stabilize in general, however, there exist interesting instances which stabilize. For example, it can be easily shown that the games where the available paths have equal lengths always stabilize. It would be interesting to find a general characterization of the game instances that stabilize. Another interesting problem is to provide time efficient algorithms for finding equilibria in our games. R EFERENCES ´ Tardos, Tom [1] Elliot Anshelevich, Anirban Dasgupta, Jon Kleinberg, Eva Wexler, and Tim Roughgarden. The price of stability for network design with fair cost allocation. SIAM Journal on Computing, 38(4):1602–1623, 2008. ´ [2] Elliot Anshelevich, Anirban Dasgupta, Eva Tardos, and Tom Wexler. Near-optimal network design with selfish agents. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages 511–520, San Diego, California, USA, June 2003. [3] Ron Banner and Ariel Orda. Bottleneck routing games in communication networks. IEEE Journal on Selected Areas in Communications, 25(6):1173–1179, 2007. (Also published in INFOCOM’06.). [4] P. Berenbrink and C. Scheideler. Locally efficient on-line strategies for routing packets along fixed paths. In 10th ACM-SIAM Symposium on Discrete ALgorithms (SODA), pages 112–121, 1999. [5] Costas Busch and Malik Magdon-Ismail. Atomic routing games on maximum congestion. Theoretical Computer Science, 410(36):3337– 3347, August 2009. [6] George Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 67–73, Baltimore, MD, USA, May 2005. ACM. [7] Jos´e R. Correa, Andreas S. Schulz, and Nicol´as E. Stier Moses. Computational complexity, fairness, and the price of anarchy of the maximum latency problem. In Proc. Integer Programming and Combinatorial Optimization, 10th International IPCO Conference, volume 3064 of Lecture Notes in Computer Science, pages 59–73, New York, NY, USA, June 2004. Springer.

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