The Impact of Collusion on the Price of Anarchy in Nonatomic and Discrete Network Games

Transcription

1 The Impact of Collusion on the Price of Anarchy in Nonatomic an Discrete Network Games Tobias Harks Institute of Mathematics, Technical University Berlin, Germany Abstract. Hayrapetyan, Taros an Wexler recently introuce a framework to stuy the impact of collusion in congestion games on the quality of Nash equilibria. We aopt their framework to network games an focus on the well establishe price of anarchy as a measure of this impact. We rst investigate nonatomic network games with coalitions. For this setting, we prove upper bouns on the price of anarchy for polynomial latencies, which improve upon the current best ones except for ane latencies. Secon, we consier iscrete network games with coalitions. In iscrete network games, a nite set of players is assume each of which controlling a iscrete amount of ow. We present tight bouns on the price of anarchy for polynomial latencies, which improve upon the previous best ones except for the ane an linear case. In particular, we show that these upper bouns coincie with the known upper bouns for weighte congestion games. As we o not use the network structure for any of our results but only rely on variational inequalities characterizing a Nash equilibrium, the erive bouns are also vali for the more general case of nonatomic an weighte congestion games with coalitions. Aitionally, all our results imply bouns on the price of collusion propose by Hayrapetyan et al. Keywors: collusion in network games, congestion games, price of anarchy. Introuction Over the past years, the impact of the behavior of selsh, uncoorinate users in congeste networks has been investigate intensively in the theoretical computer science literature. In this context, network routing games have prove to be a reasonable means of moeling selsh behavior in networks. The basiea is to moel the interaction of selsh network users as a noncooperative game. We are given a irecte graph with latency functions on the arcs an a set of originestination pairs, calle commoities. Every commoity is associate with a eman, which species the rate of ow that nees to be sent from the respective origin to the estination. In the nonatomic variant, every eman represents a continuum of agents, each controlling an innitesimal amount of ow. The latency that an agent experiences to traverse an ars given by a non-ecreasing) function of the total

2 ow on that arc. Agents are assume to act selshly an route their ow along a minimum-latency path from their origin to the estination; this correspons to a common solution concept for noncooperative games, that of a Nash equilibrium. In the iscrete variant, we assume a nite set of players each of which controlling a iscrete amount of ow. In both variants a Nash ow is given when no agent or player can improve his own cost by unilaterally switching to another path. Our focus is to stuy the impact of coalitions of subsets of agents or players, respectively, on the price of anarchy. To stuy such coalitions we aopt the framework of Hayrapetyan et al. [0]: Consier a nonatomic network game an a subset of agents forming a coalition. We assume that this coalition aims at minimizing the average elay experience by this coalition. In this setting, we stuy Nash equilibria: a stable point, where no coalition can unilaterally improve its cost by rerouting its ow. Note that the assumption that a coalition aims at minimizing its average elay oes not imply that all members of the coalition experience lower iniviual cost compare to a Nash equilibrium in which they act inepenently. Still, one might hope that the total cost of the coalition improves. As iscovere by Cominetti et al. [5], even this is not true in general. They presente an instance of a nonatomic network game, where a coalition of nonatomic agents belonging to the same source-estination pair experiences higher cost at Nash equilibrium compare to the case, where every agent acts inepenently. In the following, we review the known positive an negative results in this setting an comment on issues, which are still open.. Relate Work Koutsoupias an Papaimitriou [] initiate the investigation of the eciency loss cause by selsh behavior. They introuce a measure to quantify the ineciency of Nash equilibria which they terme the price of anarchy. The price of anarchy is ene as the worst-case ratio of the cost of a Nash equilibrium over the cost of a system optimum. In a seminal work, Roughgaren an Taros [6] showe that the price of anarchy for network routing games with nonatomic players an linear latency functions is 4/3; in particular, this boun hols inepenently of the unerlying network topology. The case of more general families of latency functions has been stuie by Roughgaren [3] an Correa et al. [6]. For an overview of these results, we refer to the book by Roughgaren [4].) For iscrete network games, where players control a iscrete amount of ow, Roughgaren an Taros examine the price of anarchy for the unsplittable variant [6]. Awerbuch et al. [2], Christooulou an Koutsoupias [4], an Alan et al. [] stuie the price of anarchy for weighte an unweighte congestion games with polynomial latency functions. Closest to our work are the papers by Hayrapetyan et al. [0] an Cominetti et al. [5]. The former presente a general framework for stuying congestion games with colluing players. Their goal is to investigate the price of collusion: the factor by which the quality of Nash equilibria can eteriorate when coalitions form. Their results imply that for symmetric nonatomic loa balancing games with coalitions the price of anarchy oes not excee that of the game without

3 coalitions. For weighte congestion games with coalitions an polynomial latencies they prove upper bouns of O2 + ). They also presente examples showing that in iscrete network games, the price of collusion may be strictly larger than, i.e., coalitions may strictly increase the social cost. Cominetti et al. [5] stuie the atomic splittable selsh routing moel, which is a special case of the nonatomic congestion game with coalitions. They observe that the price of anarchy of this game may excee that of the stanar nonatomic selsh routing game. Base on the work of Catoni an Pallotino [3], they presente an instance with ane latency functions, where the price of anarchy is.34. Using a variational inequality approach, they presente bouns on the price of anarchy for concave an polynomial latency functions of egree two an three of.5, 2.56, an 7.83, respectively. For polynomials of larger egree, their approach oes not yiel bouns. Fotakis, Kontogiannis, an Spirakis [9] stuie algorithmissues in the setting of atomic congestion games with coalitions..2 Our Results Albeit the above mentione avances in the context of bouning the price of anarchy in nonatomic an iscrete network games with coalitions, several questions remain open. Can the upper boun for the nonatomic network game with coalitions especially for higher egree polynomials be improve? Consier iscrete network games with coalitions. Are there similar counterintuitive phenomena possible as in the nonatomic variant? In particular, oes the price of anarchy in iscrete network games with coalitions excee that of iscrete network games without coalitions? In this paper, we contribute to partially answering these questions. For nonatomic network games with coalitions, we present new bouns on the price of anarchy for polynomial latency functions with nonnegative coecients that improve upon all previous bouns, except for the ane latency case. For iscrete network games with coalitions an latency functions in the sets {lz) : lc z) c lz), c [0, ]}, N, we prove upper bouns on the price of anarchy of Φ +, where Φ is the real solution of the equation x+) = x +. Among others, the above sets of latencies contain polynomials with nonnegative coecients an egree an concave latencies for =. Since classical weighte network games are a special case of the respective games with coalitions, the matching lower bouns for polynomials ue to Alan et al. [] show that our upper bouns are tight. For an overview of the results see Table. We obtain our results by extening the variational inequality approach of Correa et al. [7], Cominetti et al. [5] an Roughgaren [3]. We introuce an aitional nonnegative parameter λ R +, which allows to boun the cost of a Nash ow in terms of λ times the cost of an optimal ow plus an ωλ) fraction of the Nash ow itself. Then, the iea is to vary λ so as to n the best possible λ representation, i.e. one that minimizes the price of anarchy given by ωλ). Alan et al. [] have use a similar technique for bouning the price of anarchy

4 Table. Comparison of our results to [5], [0] an []. We show upper bouns on the price of anarchy for nonatomic network games with coalitions NNGC) an iscrete network games with coalitions DNGC). The matching lower bouns of [] for weighte network games are also vali lower bouns for iscrete network games with coalitions. Consiere are polynomial latency functions with nonnegative coecients an egree. The results in the rst row are with respect to linear latencies {a x : a 0}. The previous best upper bouns for DNGC are O2 + ) prove by [0]. NNGC DNGC LB [5] an [3] Our UB UB [5] Our UB-LB [] , , , 0, , 524, 079, , , 80, 803 Ω) ` ` Φ + in weighte congestion games. Their approach, however, requires polynomial latencies while our technique oes not nee this assumption. As we o not use the network structure for any of our results but only rely on variational inequalities characterizing a Nash equilibrium, the erive bouns are also vali for the more general case of nonatomic an weighte congestion games with coalitions. Aitionally, all our results imply bouns on the price of collusion propose by Hayrapetyan et al. [0]. 2 Nonatomic Network Games with Coalitions In a network routing game we are given a irecte network G = V, A) an k origin-estination pairs s, t ),..., s k, t k ) calle commoities. For every commoity i [k], a eman r i > 0 is given that species the amount of ow with origin s i an estination t i. Let P i be the set of all paths from s i to t i in G an let P = i P i. A ow is a function f : P R +. The ow f is feasible with respect to r) if for all i, P P i f P = r i. For a given ow f, we ene the ow on an arc a A as f a = P a f P. Moreover, each arc a A has an associate variable latency enote by l a ). For each a A the latency function l a is assume to be nonnegative, nonecreasing an ierentiable. If not inicate otherwise, we also assume that l a is ene on [0, ) an that xl a x) is a convex function of x. Such functions are calle stanar [3]. The latency of a path P with respect

5 to a ow f is ene as the sum of the latencies of the arcs in the path, enote by l P f) = l af a ). The cost of a ow f is Cf) = f al a f a ). The feasible ow of minimum cost is calle optimal. In a nonatomic network game, innitely many agents are carrying the ow an each agent controls only an innitesimal fraction of the ow. The continuum of agents of type j traveling from s j to t j ) is represente by the interval [0, j ]. It is well known that for this setting ows at Nash equilibrium exist an their total latency is unique, see [4]. Furthermore, the price of anarchy, which measures the worst case ratio of the cost of any Nash ow an that of an optimal ow is well unerstoo, see Roughgaren an Taros [6], Correa et al. [6, 7], Perakis [2], an Roughgaren [4]. In this paper, we stuy the impact of coalition formation of subsets of agents on the price of anarchy. Consier a iscrete quantity of ow that forms a coalition. We assume that this coalition aims at minimizing the average elay experience by this coalition. Let [C] = {c,..., c m } enote the set of coalitions. Each coalition [C] is characterize by a tuple,..., k ), where every j is a subset of the continuum [0, r j ] of agents of type j. We assume that every agent of type j belongs to exactly one coalition, i.e, the isjoint union of j, i [m] represents the entire continuum of [0, r j ]. The tuple G, r, l, C) is calle an instance of the nonatomic network game with coalitions. Note that this moel inclues the special case, where we have exactly k coalitions each of which controlling the ow for commoity k. This case correspons to the atomic splittable selsh routing moel stuie by Roughgaren an Taros [6], Correa et al. [7], an Cominetti et al. [5]. We enote by f ci any feasible ow of coalition. The cost for coalition is ene as Cf ci ; f ci ) := l a f a ) fa h, j [k] h [j] where f ci enotes the ow of all other coalitions. It is straightforwar to check that at Nash equilibrium, every coalition routes its ow so as to minimize Cf ci ; f ci ) with the unerstaning that coalition optimizes over f ci while the ow f ci of all other coalitions is xe. If latencies are restricte to be stanar, minimizing Cf ci ; f ci ) is a convex optimization problem, see for example Roughgaren [5]. The following conitions are necessary an sucient to characterize a ow at Nash equilibrium for a nonatomic network game with coalitions. We ene fa ci := j [k] h [c f h ij] a to be the aggregate ow of coalition on arc a. Lemma. A feasible ow f is at Nash equilibrium for a nonatomic network game with m coalitions if an only if for every i [m] the following inequality is satise: ) ) la fa + l a fa f ) a f a x ci a ) 0 for all feasible ows x ci. ) The proof is base on the rst orer optimality conitions an the convexity of Cf ci ; f ci ), see Dafermos an Sparrow [8].

6 3 Bouning the Price of Anarchy For every arc a, latency function l a, an nonnegative parameter λ we ene the following nonnegative value: l a f a ) λ l a x a )) x a + l af a ) fa ci x ci a fa ci ) 2 ) ) ωl a ; m, λ) := f a,x a 0 i [m] l a f a ) f a. We assume 0/0 = 0 by convention. For a given class of functions L, we further ene ω m L; λ) := ωl a ; m, λ). Moreover, we ene the following set: l a L Denition. Given a class of latency functions L. The set of feasible λ 0 is ene as ΛL) := {λ R + ω m L; λ) ) > 0}. Theorem. Let L be a class of continuous, nonecreasing, an stanar latency functions. Then, the price of anarchy for the nonatomic network game with coalitions is at most inf λ ΛL) [ ] λ ω m L; λ) ). Proof. Let f be a ow at Nash equilibrium, an x be any feasible ow. Then, Cf) la f a ) f a + ) ) la fa + l a fa f ) a x a fa ci ) ) 3) i [m] = la f a ) x a + l ) a fa f a x ci a fa ci ) ) i [m] = λ la x a ) x a + l a f a ) λ l a x a ) ) x a + l ) a fa f a x ci a fa ci ) ) i [m] λ Cx) + ω m L; λ) Cf). 4) Here, 3) follows from the variational inequality state in Lemma. The last inequality 4) follows from the enition of ω m L; λ). Taking x as the optimal ow the claim is proven. Note that whenever ΛL) = or ΛL) = { }, the approach oes not yiel a nite price of anarchy. Our enition of ω m L; λ) is closely relate to the parameter β m L) in Cominetti, Correa, an Stier-Moses [5] an α m L) in Roughgaren [5] for the atomic splittable selsh routing moel. For λ = we have β m L) = ω m L; ) an α m L) = ω m L; )). As we show in the next section, the generalize value ω m L; λ) implies improve bouns for a large class of latency functions, e.g., polynomial latency functions. The previous approaches with β m L) or α m L)) faile for instance 2)

7 to generate upper bouns for polynomials of egree 4 since this value excees or is innite). The avantage of Theorem is that we can tune the parameter λ an, hence, ω m L; λ) so as to minimize the price of anarchy. We make use of a result ue to Cominetti, Correa, an Stier-Moses [5]. Theorem 2 Cominetti et al. [5]). The value β m l a ) = ωl a ; m, ) is at most la f a ) l a x a ) ) x a + l af a ) [ x a ) 2 /4 f a x a /2 ) 2 ] /m x a,f a 0 l a f a ) f a. Since the necessary calculations to prove the above claim only aect the last term in 2), which is the same for ωl a ; m, λ) an β m l a ), this boun carries over for arbitrary nonnegative values of λ. Corollary. If λ 0, the value ωl a ; m, λ) is at most la f a ) λ l a x a ) ) x a + l af a ) [ x a ) 2 /4 f a x a /2 ) 2 ] /m x a,f a 0 l a f a ) f a. 3. Linear an Ane Linear Latency Functions Cominetti et al. [5] prove an upper boun of.5 for ane latencies. In the following, we present a stronger result for linear latencies. We also show that for ane latencies the best boun can be achieve by setting λ =. In this case, we have β m L) = ω m L; ). Theorem 3. Consier linear latency functions in L = {a z : a 0} an m 2 coalitions. Then, the price of anarchy is at most ) ) 2 m + 2 m m + ) m m m + ) 2 P m) = 8. m m + ) m + ) Furthermore, lim P m) = 3 m Proof. For proving the rst claim, we start with the boun on ωl a ; m, λ) given in Corollary. ωl a ; m, λ) a f a λ x a ) x a + a xa ) 2 /4 f a x a /2 ) 2 /m ) a f a ) 2. We ene µ := xa f a for f a > 0 an 0, otherwise, an replace x a = µ f a. This yiels ωl a ; m, λ) max ) µ 0 µ 2 m λ 4 m 4 m + µ m+ ) ) m m. For λ > m 4 m this is a strictly convex program with a unique solution given by µ 2 m+) = m λ 4 m. Inserting the solution, yiels ωl a ; m, λ) m+3 4 λ 4 λ m+ m. To etermine an appropriate λ, we also have to analyze { the feasible } set ΛL ). The conition λ ΛL ) is equivalent to λ > max m 4 m, m 2 2 m 2. We ene λ = m + )/m ΛL ). Applying Theorem with this value proves the claim. The proof for ane latencies is similar an leas to Cf) min λ 4 λ 2 λ 4 λ 2 Cx) showing that best boun can be achieve, when λ =.

8 3.2 Polynomial Latency Functions We start this section with bouning the value ωl a ; m, λ) for stanar latency functions. Some of the following results Lemma 2, an Proposition ) are base on results obtaine by Cominetti et al. [5]. In aition to their approach that is base on analyzing the parameter β m L), we nee to keep track of restrictions on the parameter λ. For this reason, we also present complete proofs. We focus in the following on the general case m N { }. Therefore, we ene ωl a ;, λ) := x a,f a 0 la f a ) λ l a x a ) ) x a + l af) x a ) 2 /4 l a f a ) f a. 5) Then, it follows from Theorem 2 that ωl a ; m, λ) ωl a ;, λ), since the square is nonnegative an lim m fa x a /2 ) 2 /m = 0. Lemma 2. If λ an l a f a ) f a is a convex function, then the value ωl a ;, λ) is at most: ωl a ;, λ) 0 x a f a la f a ) λ l a x a ) ) x a + l af) x a ) 2 /4 l a f a ) f a. 6) Proof. Consier the function hx a ) ene as the numerator of the remum in 5). To prove that the solution satises x a f a, we show that h x a ) 0 if x a f a. Using that h x a ) = l a f a ) λ l a x a ) λ x a l ax a ) + xa 2 l af a ), the erivative is negative if an only if l a f a ) + xa 2 l af a ) λ l a x a ) + x a l ax a ) ). By assumption l a f a ) f a is convex, hence, l a f a )+l af a ) f a l a x a )+l ax a ) x a for x a f a. Since furthermore λ, the proof is complete. The following characterization of ωl a ; m, λ) via a ierentiable function sz) can be prove with techniques from Cominetti et al. [5]. Proposition. Let L be a class of continuous, nonecreasing an convex latency functions. Furthermore, assume that λ an l a κ f a ) sκ) l a f a ) for all κ [0, ], where s : [0, ] [0, ] is a ierentiable function with s) =. Then, ωl a ;, λ) max 0 u u λ su) + s ) u/4 ). In the following, we consier polynomial latency functions of the form L := {a x + + a x + a 0 : a s 0, s = 0,..., }. Corollary 2. For latency functions in L,, the price of anarchy is at most [ inf λ max λ ΛL ) R u λ u + u/4 )) ]. 7) 0 u

9 Proof. All assumptions of Proposition are satise with sf) = f. Therefore, s ) = an ωl a ;, λ) max u λ u + u/4 ). 8) 0 u Applying Theorem yiels the claim. In Table, we present upper bouns for latency functions in L,..., L 0. The results have been obtaine by optimizing the expression in 7) over the parameter λ ΛL ) R. An asymptotic approximation for general is provie in the next theorem. Theorem 4. Given are latency functions in L, 2. Then, the price of anarchy is at most ) Proof. We ene λ) as follows λ) := ) The proof +) +) procees by proving a claim, which yiels a boun on ωl a ;, λ)) for l a L. Claim. max 0 u [ T u) := u λ) u + u 4 )] = +, for all 2. Proof. To prove the claim it is convenient to write λ) as λ) = u ) + 4 u ) 4 / + ) ) / 4 u ) +) with u ) := 2/ + ). Then, the claim is proven by verifying the following facts:. T u )) = 0, T u )) < 0 an T u) has at most one zero in 0, ) 2. T 0) = 0, T ) 0 an T u )) = +. Before we prove these facts, we show how it implies the claim. The rst fact implies that u ) is the only local maximum of T u) in the open interval 0, ). Then, by comparing T u )) to the bounary values T 0) an T ) it follows that T u )) = / + ) is the global maximum. We start by proving the rst fact. The rst erivative T u) evaluates to T u )) = u ) u ) u ) 2 4))/4 u )). Then, it is easy to check that u ) solves the equality x x x 2 4 = 0 proving T u )) = 0. Furthermore, T u )) = + ) 4 + u ) T u )) < 0, it is sucient to show that + ) 4 + Inserting the enition of u ) an rewriting yiels ) +) u ) + 2 u ) 4 > 2 2 ). For +) u ) > 2 2., which hols for all. Furthermore, it is straightforwar to check that T u) has at most one zero in 0, ), so we omit the etails. Using the above claim we can now invoke Corollary 2, which implies that ωl a ;, λ)) is boune by +. Hence, λ) ΛL ) so we can use Theorem to obtain the claime boun of + ) λ). In the following we analyze the growth of the erive upper boun for large, 4). Th proof consists of stanar calculus an is omitte. Corollary 3. ) for 4.

10 4 Discrete Network Games with Coalitions In this section, we stuy iscrete network games with coalitions. We aopt the same notation as in the previous section an only point out the ierences. The main istinction between nonatomic network games an iscrete network games is that in the latter one we are given a nite set of players enote by [P ] = {,..., P }. Each player p [P ] controls a iscrete amount of ow, which has to be route along a single path. We call a player p of type j [k] if player p controls a fraction of commoity j. A coalition [C] is characterize by a tuple,..., k ), where every j is a subset of players of type j. As before, we assume that every player p of type j belongs to exactly one coalition. The tuple G, r, l, P, C) is calle an instance of the iscrete network game with coalitions. We enote by f p an f ci the ow of player p an coalition i, respectively. The cost for coalition i is given by Cf ci ; f ci ) = l a f a ) fa p. p [] We ene fa ci = p [c f p i] a to be the aggregate ow of coalition i on arc a. It is straightforwar to check that at Nash equilibrium every coalition routes its ow so as to minimize Cf ci ; f ci ). Note that while the coalition etermines a ow minimizing C ci f ci ; f ci ), still all iniviual ows f p have to be route along a single path. The following lemma states that at Nash equilibrium every coalition minimizes C ci f ci ; f ci ), i.e. C ci f ci ; f ci ) C ci x ci ; f ci ) for any other feasible ow x ci for the emans of coalition i. Lemma 3. Let f be a feasible ow at Nash equilibrium for a iscrete network game with coalitions C. Then, for every [ ] [C], i [m] the following inequality is satise: ) l a fa f a l a fa p ) + x ci a x a for all feasible ows xci. 9) p [P ] p / [] Using a similar technique as in the previous section, we ene for a given latency function lz), λ, an nonnegative values f, x, the following values { lf+x) λ lx) ) x lf) f if lf) f > 0 ωl; λ) := x,f 0 0 if lf) f = 0. 0) For a class L of latency functions, we further ene ωl; λ) := l L ωl; λ). Figure shows an illustration of ωl; λ). Denition 2. For latency functions in L, the feasible scaling set for λ is ene as ΛL) := Λ L) Λ 2 L), where Λ L) := {λ : ωl; λ) > 0} an Λ 2 L) := {λ : lf + x) λ lx)) x 0 for all f, x R +, l L, lf) = 0}.

11 lf + x) λ lx) lx) lf) l ) 0 0 f x f + x Fig.. Illustration of the value ωl; λ) in 0) with < λ < lf+x). The light-gray lx) shae area correspons to the numerator of ωl; λ) an the ark-gray shae area to the enominator. Theorem 5. Let L be a class of continuous an nonecreasing latency functions. Then, the price of anarchy for a iscrete network game with coalitions is at most [ ] inf λ ΛL) λ ωl; λ) ). Proof. Let f be a ow at Nash equilibrium, an x be any feasible ow. Then, Cf) = ) l a fa f a i [m] l a i [m] p [P ] p [] fa p ) + x ci a x a ) l a f a + x a ) x a 2) = λ l a x a ) x a + l a f a + x a ) λ l a x a ) ) x a λ Cx) + ωl; λ) Cf). 3) Here, ) follows from Lemma 3. The secon inequality 2) is vali since latency functions are nonecreasing. The boun involving ωl a ; λ) hols because of the following: If l a f) = 0 then it follows l a f +x) λ l a x)) x 0 since λ Λ 2 L). Thus, the boun is true. If f = 0, then l a f) λl a x)) 0 since λ an l a ) is nonecreasing. The case l a f) f > 0 follows from the enition. Taking x as the optimal oine solution an since λ ΛL), the claim is proven. Note that our only assumptions on feasible latencies are continuity an monotonicity. As an implication of Theorem 5, we show that the price of anarchy for continuous an nonecreasing latencies satisfying lc z) lz) c for all c [0, ] is at most Φ +, where Φ is the solution to the equation x + ) = x.

12 Theorem 6. Let L be a class of continuous an nonecreasing latency functions satisfying lc z) lz) c for all c [0, ]. Then, the price of anarchy of a iscrete network game with coalitions is boune by Φ +, where Φ is the solution to the equation x + ) = x +. Proof. We present a complete prove for = an sketch the proof for larger. Key for proving the claim is the following lemma. Lemma 4. Let L be a class of continuous an nonecreasing latency functions satisfying lc z) lz) c for all c [0, ]. Then, ωl; λ) 4 λ ). Proof. We assume f > 0 an x > 0 since otherwise the claim is trivially true. We consier two cases: ) f x. In this case we ene µ = x f with µ [0, ]. Then, we have ) l + µ) f) λ lµ f) µ f ωl; λ) = lf) f f 0, µ [0,] + µ) lf) λ µ lf) ) µ f f 0, µ [0,] lf) f ) = + µ λ µ µ. µ [0,] This problem is a concave program with optimal value 4 λ ). The secon case is similar: 2) f x. In this case we ene µ = f x with µ [0, ]. Then, using similar arguments we arrive at ωl; λ) = x 0, µ [0,] l + µ) x) λ lx) ) x lµ x) µ x µ 0 + µ λ µ 2. This problem is again a concave program with optimal value 4 λ ). By Lemma 4, ωl; λ) is boune by 4 λ ) λ 4 λ ). Thus, Theorem 5 implies that the price of anarchy is boune by 4 λ 5. Choosing λ = as the minimizer of the price of anarchy, the claim for = is proven. For general N, the iea is to boun ωl; λ) by µ [0,] + µ) λ µ ) µ. Then, we can write + µ) as κ + µ κ) for some κ 0, ) an solve the remum. Finally, we ene λ an κ as a function of Φ leaing to the esire boun. Corollary 4. For continuous, nonecreasing an concave latency functions, the price of anarchy of a iscrete network game with coalitions is boune by Remark. As weighte network games are a special case of iscrete network games with coalitions, the matching lower bouns for polynomials ue to Alan et al. [] show that our upper bouns are tight.

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