On the Value of Coordination in Network Design

Transcription

1 On the Value of Coordination in Networ Design Susanne Albers Abstract We study networ design games where n self-interested agents have to form a networ by purchasing lins from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well nown that the price of anarchy of these games is as high as n. Therefore, recent research has focused on evaluating the price of stability, i.e. the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first super-constant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of Ω(log W/ log log W) for weighted games, where W is the total weight of all the agents. This almost matches the nown upper bound of O(log W). Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. Department of Computer Science, University of Freiburg, Georges Köhler Allee 79, 790 Freibug, Germany. salbers@informati.uni-freiburg.de Wor supported by the German - Israeli Foundation for Scientific Research & Development, project G /200 and by the German Research Foundation, project AL 464/4-2. This wor was done while on sabbatical leave at Carnegie Mellon University, Pittsburgh, and supported by the ALADDIN project.

2 Introduction Communication networs are pervasive and critical to modern society. Nonetheless, the formation and evolution of large networs is not well understood, a major reason being that these networs typically are not built by a central authority but rather by many economic agents that have selfish interests. For this reason, research on networ design has focused on game-theoretic approaches over the past years, see e.g. [2,, 5, 6, 7, 8, 9,, 4, 22, 2]. We study networ design games that have received a lot of attention recently [2,, 6, 7,, 6] and are simple, yet powerful enough to capture the two most important objectives of agents: connection establishment and cost minimization. Consider a directed or undirected graph G where each edge e has a non-negative cost c(e). There are n agents, each of which has to connect a set of terminals. The agents form a networ by selecting edges. A strategy S i of an agent i is a set of edges connecting the desired terminals. The cost of the edges used by all the agents has to be covered. A fundamental cost sharing mechanism is Shapley cost sharing, which was proposed by Anshelevich et al. [] for networ design games and has been studied with respect to other networing problems as well [2, 5]. In Shapley cost sharing, the cost of an edge is shared in a fair manner among the agents using that edge. In an unweighted game if agents use an edge e in their strategies, then each of these agents pays a share of c(e)/. In a weighted game, each agent i has a weight w i and contributes a share of c(e)w i /W e, where W e is the total weight of agents using e. We are interested in stable networs where no agent has the incentive to deviate from its strategy. Stability is modeled by considering Nash equilibria. A combination S = (S,...,S n ) of strategies forms a Nash equilibrium if no agent has a better strategy with a strictly smaller cost if all the other agents adhere to their strategies. A widely accepted performance measure to evaluate the quality of Nash equilibria is the price of anarchy [9], which is the maximum ratio of the total cost incurred by any Nash equilibrium to the cost spent by the social optimum. Unfortunately, for our networ design games, the price of anarchy is as high as n. As an alternative quality measure, Anshelevich et al. [] proposed the price of stability which is the ratio of the best Nash equilibrium relative to the social optimum. Anshelevich et al. [] proved that the price of stability in unweighted networ design games is O(log n). The scenario described so far assumes that agents are completely non-cooperative, isolated entities. However for long-term decisions such as networ design, given today s communication infrastructure, this assumption is not entirely realistic. It is more natural that agents will discuss possible strategies and, as in other economic marets, form coalitions taing strategic actions that are beneficial to all members of the group. In such cooperative environments we see again stable solutions. In this context, Aumann [4] in 959 introduced the concept of strong Nash equilibria, which ensure stability against deviations by every conceivable coalition of agents. More specifically, no coalition can cooperatively deviate in a way that benefits all its members, taing the actions of the agents outside the coalition as given. With respect to networ design, an important question is if coordination among agents yields strictly better solutions. Is it possible to achieve significant improvements? We prove that this is the case. When coordination is allowed, the price of anarchy of strong Nash equilibria drops from n to O(log n) in unweighted games. Similar improvements show in weighted games. Obviously, any strong Nash equilibrium is a standard Nash equilibrium, which is immune to deviations of single agents. Hence strong Nash equilibria cannot be better than the best standard Nash equilibria. A second natural question is how strong Nash equilibria ran relative to the best standard Nash equilibrium. When coordination is allowed, is the worst-case performance of stable states close to that of the best stable states achievable without cooperation? We answer this question in the affirmative in terms of anarchy and stability measures. For most settings, the price of anarchy of strong Nash equilibria is essentially always as good as the corresponding stability bounds of standard equilibria. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. Previous results: Research on the networ design games defined above was initiated by Anshelevich et

3 al. [2]. In this first paper the authors considered general cost sharing schemes that are not restricted to Shapley mechanisms. Anshelevich et al. studied undirected graphs and first addressed scenarios where each agent has to connect one terminal to a common destination. They designed Nash equilibria whose cost is equal to the cost of the optimum. Furthermore Ashelevich et al. [2] investigated the general scenario that each agent has to connect a set of terminals. In this case there are graphs that do not admit Nash equilibria. The authors therefore studied α-approximate Nash equilibria in which no agent can improve its cost by a factor of more than α, where α >. Anshelevich et al. proved that there always exists a -approximate Nash equilibrium whose cost is equal to that of the optimum. Furthermore, they derived a polynomial time algorithm that gives a ( ǫ)-approximate Nash equilibrium whose cost is twice the optimum. In the following two paragraphs we describe the results nown for networ design games with Shapley cost sharing. The setting was introduced in a second paper by Anshelevich et al. [] who first analyzed unweighted games. Using elegant potential function arguments based on a potential by Monderer and Shapley [20], the authors proved that every directed or undirected graph admits a Nash equilibrium and that the price of stability is upper bounded by H(n). Here H(n) = n i= /i is the nth Harmonic number, which is closely approximated by the natural logarithm, i.e. ln(n + ) H(n) ln n +. The upper bound of H(n) on the price of stability is tight for directed graphs. For undirected graphs Anshelevich et al. [] showed a lower bound of 4/ on the price of stability; the lower bound construction uses two agents that have to establish a connection to a common destination. Additionally, Anshelevich et al [] considered weighted games and showed the existence of Nash equilibria in two-agent games. For directed graphs they gave a lower bound of Ω(max{n,log W }) on the price of stability, where W is the total weight of all the agents. Chen and Roughgarden [7] further investigated weighted games in directed graphs. They showed that there are graphs that do not admit Nash equilibria. Chen and Roughgarden then demonstrated that, for any α = Ω(log w max ), α-approximate Nash equilibria do exist and that the price of stability is O((log W)/α). Here w max is the maximum weight of any agent. These trade-offs are nearly tight. Further wor on unweighted games was presented by Fiat et al. [] and Cheuri et al. [6]. All the above results hold for standard Nash equilibria without coordination. The concept of strong Nash equilibria has been the subject of extensive studies in the game theoretic literature. Recent research in game theory and economics has also investigated strong Nash equilibria in the context of networing problems. A survey article presenting literature on networ formation in cooperative games was written by van den Nouweland [2]. More concretely, the existence of networs that are stable against changes in lins by any coalition is examined in [7]. Furthermore, Andelman et al. [] analyzed strong equilibria with respect to scheduling as well as a different class of networ creation games in which lins may be formed between any pair of agents. For the latter games, strong Nash equilibria achieve a constant price of anarchy. We became aware that, independent of our wor, very recently Epstein et al. [0] studied strong Nash equilibria for unweighted networ design games in directed graphs. They assume that each agent has to connect a pair of terminals and consider Shapley as well as general cost sharing mechanisms. Epstein et al. observe that strong Nash equilibria do not always exist and then present topological characterizations for equilibrium existence. They show that if each agent has to connect a terminal to a common destination, each series parallel graph has a strong Nash equilibrium. If arbitrary terminal pairs are allowed, every extension parallel graph admits a strong Nash equilibrium when Shapley cost sharing is adopted. Furthermore Epstein et al. analyze the quality of strong Nash equilibria, showing a bound of Θ(log n) on the price of anarchy for Shapley cost sharing and a bound of for general cost sharing schemes when each agent has to connect to a common destination. Our contribution: This paper presents an in-depth study of networ design games with Shapley cost sharing when coordination among agents is allowed. We present upper and lower bounds on the price of anarchy achieved by strong Nash equilibria. We study scenarios with unrestricted coordination, i.e. coalitions of any size (or weight) may be formed; we also consider settings where the size (or weight) of a coalition is limited. 2

4 The first part of the paper addresses unweighted networ design games. We first observe that there are graphs that do not admit strong Nash equilibria and then give a sufficient existence condition. More specifically, we show that α-approximate strong Nash equilibria exist in any directed or undirected graph, for any α H(c), if coalitions of size up to c are allowed, c n. Again, H(c) is the cth Harmonic number. An α-approximate strong Nash equilibrium, for α, is one where no coalition (of prescribed size or weight) can deviate such that every member of the coalition improves its cost by a factor of more than α. We next prove that the price of anarchy of strong Nash equilibria is upper bounded by H(n) ln n, allowing coalitions of any size. This upper bound holds for any directed or undirected graph that admits a strong Nash equilibrium. Hence, using coordination, we achieve an exponential improvement in terms of the price of anarchy, compared to non-cooperative environments. We show that the upper bound of H(n) is tight in directed graphs. For undirected graphs we develop a lower bound of Ω( log n) on the price of anarchy. These results can be generalized to α-approximate strong Nash equilibria, for any α. In this case all the upper and lower bounds multiply by a factor of α. For the generalized setting that coalitions of size up to c are allowed, c n, we prove an upper bound of α n c H(c) on the price of anarchy of α-approximate strong Nash equilibria. Again, this bound holds for any directed or undirected graph that admits an α-approximate strong Nash equilibrium, for some α, and not just for the range α H(c). Suppose that α =. If c =, we obtain the anarchy ratio of n achieved by standard equilibria. If c = n, we obtain the best ratio of H(n). Since H(n) is a lower bound on the price of stability of (standard) Nash equilibria in directed graphs [], we conclude that in directed graphs the worst-case performance ratios of strong Nash equilibria are essentially always as good as the performance ratios achievable by the best standard Nash equilibria. In the second part of the paper we extend the above results to weighted networ design games. We first give a sufficient condition for the existence of α-approximate strong Nash equilibria. We then prove that in directed and undirected graphs the price of anarchy of strong Nash equilibria is at most + ln W if the formation of coalitions is not restricted. Here W is the sum of the weights of all agents. For directed graphs we show a matching lower bound of Ω(log W). For undirected graphs we prove a lower bound of Ω( log W). Again, for any α, the results extend to α-approximate strong Nash equilibria, where the lower and upper bounds simply multiply by α. When coordination among agents is limited, we consider two scenarios: () As usual, the number of agents in a coalition might be limited. (2) The sum of the weights of the agents forming a coalition may be limited so that agents of high weight cannot leave agents of low weight in costly configurations. For this general setting we present bounds trading the price of anarchy vs. the coalition size or weight. Furthermore, we prove a lower bound on the price of stability of standard Nash equilibria in undirected graphs. We construct a family of graphs in which the price of stability is Ω(log W/log log W). No super-constant lower bound was nown for undirected graphs, neither for weighted nor for unweighted games. Our lower bound holds even if every agent has to connect only a pair of terminals. However, individual terminal pairs are allowed. Together with the nown lower bound of Ω(log W) for directed graphs [], we conclude that, in undirected as well as directed graphs, anarchy bounds of strong Nash equilibria essentially match the stability bounds of standard Nash equilibria. We remar that our set of results is mostly disjoint from that by Epstein et al. [0]. The results provided in both [0] and this paper are the fact that strong Nash equilibria do not always exist, the upper bound of n c H(c) and the lower bound of H(n) on the price of anarchy in unweighted games. While the upper bound proof by Epstein et al. is based on the potential function by Monderer and Shapley, in our paper we use new combinatorial arguments to establish the result. Generally speaing, our study here is more comprehensive in that we allow each agent to connect set of terminals, consider directed and undirected graphs as well as unweighted and weighted games. Analysis techniques: As mentioned above, our upper bounds on the price of anarchy are achieved using new combinatorial arguments that do not rely on potential functions: Starting from a strong Nash equilibrium, we perform a sequence of specific strategy changes for varying size coalitions. For each strategy change there

5 exists one unsatisfied agent whose original cost can be bounded relative to the optimum. From a technical point of view our strongest contribution are the lower bounds for undirected graphs. We present a new recursive framewor for constructing lower bounds in networ design games. Applying the recursive construction for varying parameters, we are able to obtain anarchy as well as stability bounds in both unweighted and weighted games. The protocol could also be applied to derive bounds for directed graphs but simpler constructions wor in the directed case. While the same recursive framewor can be applied to construct graphs for anarchy and stability bounds, the analyses of the graphs differ. To establish anarchy bounds we have to prove that no coalition can deviate, which turns out to be a non-trivial tas because all possible coalitions and strategy changes over the recursive levels must be examined. To establish a stability bound, we have to show that no better Nash equilibria exist. In fact, we will prove that our graphs admit only one Nash equilibrium. 2 Problem statement and definitions Networ design games: Consider a graph G = (V,E,c) with a non-negative cost function c : E IR 0 + defined on the edges. Graph G may be directed or undirected as we will study networ design in both directed and undirected graphs. Associated with G are n selfish agents, each of which has certain connectivity requirements. More specifically, let T i V be the set of terminals that agent i wishes to connect. If G is a directed graph, then for (selected) terminal pairs t,t T i we additionally have to specify which direction between the pair should be established. A strategy of an agent i consists of a set S i E of edges satisfying the connection requirements. If G is undirected, S i is in fact a minimal tree connecting T i. A combination S of strategies is the vector S = (S,...,S n ) of individual agent strategies. Edges used by the agents have to be paid for. We consider Shapley cost sharing mechanisms that split the cost c(e) of an edge e in a fair manner among the agents using that edge. In an unweighted game, if agents use an edge e, then each of the agents pays a share of c(e)/ for that edge. Thus, for a combination S of strategies, the total cost of agent i is equal to cost i (S) = e S i c(e)/ {j : e S j }. In a weighted game each agent i has a weight w i and pays a share proportional to its weight. For any edge e S i, agent i pays a share of c(e)w i /W e, where W e = j:e S j w j is the total weight of the agents j using e in their strategies. Formally, the cost of agent i in a weighted game is cost i (S) = e S i c(e)w i /W e. Strong Nash equilibria: We are interested in stable solutions where agents have no incentive to deviate from their strategies. Previous wor has considered Nash equilibria that are resilient to deviations of single agents. A weaness of Nash equilibria is their vulnerability to deviations by coalitions of agents. To overcome this problem, Aumann [4] defined the notion of strong Nash equilibria. A strong Nash equilibrium is resilient to deviations of coalitions, i.e. there exists no coalition of agents that can jointly change strategies such that every agent in the coalition has a strictly smaller cost. Formally, let I be a non-empty coalition of agents. For a combination S of strategies, let S I be the projection of S on I, i.e. S I are the strategies of agents i I. Similarly, S I represents the strategies of agents i / I. For coalition I, let S I be another choice of strategies. A combination S of strategies forms a strong Nash equilibrium if, for no non-empty coalition I, there exists a strategy change S I such that cost i(s I, S I) < cost i (S), for all agents i I. Note that a standard Nash equilibrium is a strong Nash equilibrium where only coalitions of size one are allowed. In this spirit one can consider generalized settings in which coalitions of size at most c are permitted, c n. As for weighted games we will also be interested in scenarios where the total weight of agents forming a coalition is limited. This ensures that agents of high weight cannot impose too much control on agents outside a coalition. As we shall see, strong Nash equilibria do not always exist. For this reason we relax the notion of stability, calling a combination of strategies stable if agents cannot improve their cost by a factor of more than α. More specifically, for a real value α, a combination S of strategies forms an α-approximate strong Nash equilibrium if, for no non-empty coalition I, there exists a strategy change S I such that cost i(s I, S I) < cost i (S)/α, for all agents i I. Similarly, we can define α-approximate Nash equilibria when the size or 4

6 weight of a coalition is limited. We remar that in the context of α-approximate strong equilibria another definition seems reasonable. We could call a combination of strategies an α-approximate strong Nash equilibrium if no coalition can improve its total cost by a factor of more than α, while still requiring that every agent of the coalition performs strictly better than before. Obviously, an α-approximate Nash equilibrium according this this second definition is an α-approximate equilibrium under the former definition, but not vice versa. Thus, our original definition allows for more configurations representing equilibrium states. For this reason and because our first definition requires a sufficiently high benefit for each agent of a coalition to perform a strategy change, we adopt the original definition in this paper. However, all the results that we will present in the following sections also hold for the second definition as well. Performance measures: We are interested in the performance of strong Nash equilibria relative to the social optimum. Let cost(s) = n i= cost i(s) be the total cost of all the agents and let cost(opt) be the cost of the globally optimal solution. We say that strong Nash equilibria achieve a price of anarchy of c if cost(s) max S cost(opt) c, where the maximum is taen over all strong Nash equilibria. The notion can be extended to (α-approximate) strong Nash equilibria with coalitions of limited size or weight. In this paper we will also be interested in the price of stability of standard Nash equilibria where coordination among agents is not allowed. The price of stability is min S, where the minimum is taen over all Nash equilibria. cost(s) cost(opt) Upper bounds for unweighted games We study the existence of strong Nash equilibria and then develop upper bounds on the price of anarchy. The proof of the following proposition is presented in the Appendix. Proposition There exist directed and undirected graphs that do not admit strong Nash equilibria. Theorem In any directed or undirected graph, α-approximate strong Nash equilibria exist, for any α H(c), if coalitions of size up to c are allowed. Proof. We use a classical potential function by Monderer and Shapley [20] to show the existence of α- approximate strong Nash equilibria. Given a graph G = (V,E,c) and a combination S = (S,...,S n ) of strategies, let n e be the number of agents currently using edge e E in their strategies, i.e. n e = {i : e S i }. The potential is defined as Φ(S) = e E c(e)h(n e). We will show that while S does not form an α- approximate strong Nash equilibrium, when allowing coalitions of size up to c, any α-improvement move strictly decreases the potential. An α-improvement move, for a coalition I with I c, is a strategy change S I such that cost i (S I, S I) < cost i (S)/α, for any agent i I. Suppose that we perform a sequence of such α- improvement moves starting from the social optimum. As the potential is upper bounded by H(n)cost(OP T) and lower bounded by 0, the sequence of improvement moves must converge to an α-approximate strong Nash equilibrium. We analyze an α-improvement move, performed by a coalition I with I c. The strategy change S I of I can be viewed as being executed in two steps. () In a first step agents i I drop all the edges used in strategies S i. At this point no agent i I shares the cost of any edge. (2) In a second step agents i I join the edges they want to use in their new strategies S I. Let E be the set of edges dropped in step (), and let E 2 be the set of edges added in step (2). These edge sets need not be disjoint. For any e E, let n e be the number of agents sharing e just after step () and let n 2 e be the number of agents sharing e after step (2). The absolute value of the cost reduction experienced by I due to step () is cost = c(e) n e n e, n e e E 5

7 because e E is dropped by n e n e agents that each paid a share of c(e)/n e. The value of this cost reduction is equal to the cost of I in the original configuration, i.e. cost = i I cost i(s), because after step () the cost of I is 0. The cost increase of I due to step (2) is cost + = c(e) n2 e n e n 2, e E e 2 because e E 2 is bought by n 2 e n e agents i I who pay c(e)/n 2 e each. This cost increase is equal to the cost of I in the new configuration, i.e. cost + = i I cost i(s I, S I), because the cost of I was 0 before step (2) and the strategy change is complete after step (2). Using the definition of an α-improvement move we have i I cost i(s I, S I) < i I cost i(s)/α and hence αcost + cost < 0. () Next we consider the potential change Φ. The potential change stems from edges e E E 2 where cost shares change. Let Φ be the absolute value of the potential drop due to step () of the improvement move and let Φ + be the potential increase due to step (2). We will show Φ cost and Φ + αcost +. This implies Φ = Φ + Φ + cost + αcost + and using () we obtain Φ < 0, which is to be proven. To verify Φ cost we observe Φ = c(e)(h(n e ) H(n e )) c(e) n e n e = cost. n e e E e E The inequality holds because H(n e ) H(n e) = /(n e + ) + /(n e + 2) /n e (n e n e)/n e. It remains to prove Φ + αcost +. The potential increase is given by Φ + = e E 2 c(e)(h(n 2 e) H(n e)). We show that for any e E 2, H(n 2 e ) H(n e ) H(n2 e n e )n2 e n e n 2. (2) e The desired inequality for the potential increase then follows because n 2 e n e c as at most c agents can join any edge in step (2) and H(c) α. The expressions in (2) are H(n 2 e ) H(n e ) = n e + + n e n 2 e H(n 2 e n e) n2 e n e n 2 e = ( n 2 e ) n2 e n e n e n 2. e We compare the th terms of these expressions, for =,...,n 2 e n e, and establish (2) by proving n e+ n 2 e n e. This is equivalent to showing 0 n n 2 e (n2 2 n e ) n e, and this holds because f() = n e (n2 2 n e ) n e e is decreasing in and f(n 2 e n e) = 0. Theorem 2 In any directed or undirected graph and for any α, the price of anarchy of α-approximate strong Nash equilibria is upper bounded by αn c H(c) if coalitions of size up to c are allowed. If there are no restrictions on the coalition size and we are interested in true strong Nash equilibria (i.e. α = ), we obtain: Corollary In any directed or undirected graph the price of anarchy of strong Nash equilibria is upper bounded by H(n). 6

8 Proof of Theorem 2. Get G be a graph that admits α-approximate strong Nash equilibria, for some α, and let S = (S,...,S n ) be such an equilibrium state. The basic idea of the proof is to consider all coalitions of size exactly c. For each coalition I we perform a process consisting of exactly c steps in which the agents of I try to buy the edges of the social optimum. At the end of each step exactly one agent will leave the process. Maing use of the fact that in S no coalition of size up to c can improve its cost by a factor of more than α, we will be able to upper bound cost i (S) of the agent i leaving the process relative to the cost of the social optimum. More specifically, we will prove that for any coalition I of size exactly c, cost i (S) αh(c)cost(opt). () e E OPT i i I Let I be the set of all coalitions of size exactly c. Summing () over all the ( n c) coalitions I I, we obtain I I i I cost i(s) α ( n) ( c H(c)cost(OPT). Any fixed agent i, i n, occurs in exactly n ) c coalitions I I. Hence I I i I cost i(s) = ( n c ) cost(s). We conclude cost(s) ( ) ( n c / n ) n c αh(c)cost(opt) = c αh(c)cost(opt), which establishes the stated price of anarchy. Fix an arbitrary coalition I of size exactly c. We will prove (). Let E OPT be the set of edges bought by the social optimum and, for any i I, let Ei OPT be a minimal set of edges necessary to connect the terminals of agent i within the optimal solution. We now start the process mentioned above. Let I := I be the initial coalition consisting of c agents. Suppose that we have already performed steps of the process, where initially =, and let I be the coalition given at the beginning of the th step, where c. The th step proceeds as follows. Starting from initial configuration S, the agents of I perform a strategy change SI in which i I buys set Ei OPT. Let S = (SI, S I ) be the resulting configuration. The new cost of agent i I is cost i (S ) = c(e) {j I : e Ej OPT } {j / I : e S j } c(e) {j I : e E OPT e Ei OPT j }. Since the original configuration S forms an α-approximate strong Nash equilibrium, the strategy change cannot improve the cost of every agent i I by a factor of more than α. Thus there must exist an agent i with cost i (S ) cost i (S)/α and hence cost i (S) α e E OPT i c(e) {j I : e E OPT j }. (4) This agent i leaves the coalition I. If there is more than one agent satisfying the above cost inequality, we select an arbitrary of them. The new coalition at the end of the step is I + := I \ {i }. The process ends after exactly c steps when the coalition is empty. Summing (4) over all the c steps, taing into account that the sequence of agents leaving the process forms I, we find cost i (S) α i I c = e E OPT i c(e) {j I : e E OPT j }. (5) We analyze the right-hand side of the above inequality, which sums edge costs c(e) over edges e E OPT. Consider any fixed edge e E OPT and let n e = {i I : e Ei OPT } be the number of agents in I using e in the described strategy changes. The cost of e contributes to the right-hand side of (5) whenever one of the n e agents leaves the process. The lth time this happens, the contribution is c(e)/(n e l + ), for l =,...,n e. Thus, the cost contribution is c(e)h(n e ) c(e)h(c) and we conclude i I cost i(s) α e EOPT c(e)h(c) = αh(c)cost(opt). 7

9 4 Lower bounds for unweighted games We first present a lower bound for directed graphs. This lower bound implies that if there is no restriction on the coalition size, our upper bound of Corollary is optimal. Theorem In directed graphs and for any α, the price of anarchy of α-approximate strong Nash equilibria is at least αmax{n/c,h(n)} if coalitions of size at most c are allowed. Proof. We modify lower bound graphs that were presented previously in the literature [2]. For the bound of αn/c, consider a simple graph consisting of two vertices s and t that are connected by two parallel edges of cost αn and c + ǫ, respectively, see Figure (a). Associated with the graph are n agents, all of which have to connect terminals s and t. An optimal solution will buy the edge of cost c + ǫ. On the other hand, the configuration in which all the n agents share the expensive edge of cost αn, each one paying a cost of α, represents an α-approximate strong Nash equilibrium: Any coalition of size up to c, when performing a strategy change and buying the edge of cost c + ǫ, incurs a cost of at least + ǫ/c per agent. Hence the agents of the coalition do not save a factor of more than α in cost. t t α α 2 α α n αn s c + ǫ v v 2 v v n w + ǫ (a) (b) Figure : Directed graphs enforcing a high price of anarchy. In order to establish the lower bound of αh(n), we use the graph depicted in Figure (b). There are n vertices v,...,v n, where v i is connected to a vertex t via a directed edge (v i,t) of cost α/i and to a vertex w via a directed edge (v i,w) of cost 0. Additionally, there is a directed edge (w,t) of cost + ǫ. Associated with the graph are n agents, where agent i has to connect v i to t. An optimal solution satisfies the connection requirements by buying the edges of cost 0 and the edge (w, t) of cost + ǫ. The configuration in which agent i connects v i to t using its private edge (v i,t) of cost α/i forms an α-approximate strong Nash equilibrium. Any coalition of size, say c, that performs a strategy change and purchases edge (w, t) incurs a cost of (+ǫ)/c per agent. However, there is at least one agent in the coalition whose original cost was at most α/c and to whom the incentive of changing is not sufficiently high. We next develop a lower bound for undirected networs. Our lower bound construction is quite involved and we therefore concentrate on the most general scenario where there is no limit on the coalition size. Theorem 4 For any α, there exists a family of undirected graphs, each admitting an α-approximate strong Nash equilibrium whose cost is Ω(α log n) times that of the social optimum. Proof. For ease of exposition we first prove the theorem for α = and then show how to adapt the proof for any α >. We present a recursive definition of graphs G. Let n be a positive integer such that log n 2. In this proof logarithms are taen to the base. Let d max = log n. The recursive construction proceeds in d max + steps. At the bottom level of the recursion, i.e. at maximum depth d max, G consists of graphs G dmax 8

10 of order d max, cf. Figure 2(a). A graph G dmax is composed of a stem edge {v,w} of cost s dmax = / dmax and a bridge {u,v} of order d max having cost b dmax = 2/ 2dmax. The bridge and the stem are joined at vertex v. Vertices u and w are connected via an arc {u,w} of order d max having cost a dmax = / dmax. We call u the base and w the tip of G dmax. Associated with G dmax are n dmax = n/ dmax(dmax+) agents of order d max, each having to connect terminals u and w. By the choice of d max we have n dmax. w d = w,d+ G d G 0 G d+ G dmax w stem arc v bridge u G d+ 2 G d+ u,d+ bridge u d arc (a) (b) Figure 2: The recursive construction of graphs G d. (c) Assume that graphs of order d max,d max,...,d + are defined. Then a graph G d of order d, which resides a depth d of the recursion, is constructed as follows, see Figure 2(b). Graph G d consists of three graphs G d+,g d+ 2 and G d+ of order d + that are attached to each other. More specifically, the tip of G d+ and the base of G d+ 2 are merged, i.e. the two vertices are united, and the tip of G d+ 2 is merged with the base of G d+. Let u,d+ be the base of G d+. Attached to this vertex is a bridge {u d,u,d+ } of order d having a cost of b d = 2/ 2d. Let w,d+ be the tip of G d+ and set w d := w,d+. We call u d the base and w d the tip of G d. Additionally, G d contains an arc {u d,w d } of order d connecting the base and the tip. This arc has cost a d = / d. Associated with G d are n d = n/ d(d+) n/ (d+)(d+2) agents of order d, all of which have to connect u d to w d. As we shall see, these agents will govern the connection decisions within G d. The bridge will have the effect that in a strong Nash equilibrium, the order-d agents will establish their connections using the arc of order d instead of routing through the graphs G d+,. The construction proceeds down to a depth d = 0. Associated with graph G 0 are n 0 = n/ 0 n/ 2 = n n/ 2 agents of order 0 that have to connect the outermost vertices of G 0. Graph G := G 0 is the graph we will wor with. A high level setch of G = G 0 is given in Figure 2(c). We start with some observations on G = G 0. First, all the vertices and terminals of the graph are located on a bacbone consisting of all the stem edges and bridges. The nested structure of G 0 contains d subgraphs of order d, for any 0 d d max. Proposition 2 The least expensive path connecting the base and the tip of a graph G d using only edges of G d has a total edge cost of exactly / d, for any 0 d d max. Proof. The statement holds for d = d max as the arc of G dmax has cost a dmax = / dmax while the path crossing the bridge has cost b dmax +s dmax = 2/ 2dmax +/ dmax. Suppose that the statement holds for depths d max,...,d +. In G d the arc of order d has cost a d = / d while, using the inductive hypothesis, any path using the bridge of order d has a cost of at least b d + / d+ = 2/ 2d + / d > / d. 9

11 The total number of agents associated with G 0 and all of its subgraphs is equal to N 0 = d max d=0 d ( n/ d(d+) n/ (d+)(d+2) ) + dmax n/ dmax(dmax+) = n dmax n/ dmax(dmax+) + dmax n/ dmax(dmax+) = n. More generally, in G = G 0 the total number of agents associated with all the order-d graphs G d and the subgraphs therein is, for any d with 0 d d max, N d = d max i=d i ( n/ i(i+) n/ (i+)(i+2) ) + dmax n/ dmax(dmax+) = d n/ d(d+), which is equal to n/ d2 when ignoring ceilings. The social optimum in G buys the bacbone of the graph. As there are dmax graphs of order d max, the total cost of the stem edges is dmax s dmax = dmax / dmax =. There are d graphs of order d, 0 d d max, and hence the total cost of order-d bridges is d b d = d 2/ 2d = 2/ d. Summing over all d we find that the total cost of the bridges is d max d=0 2/d. We conclude that the cost of the social optimum is bounded by 4. Consider the configuration S in which, for any graph G d within G, any order-d agent associated with this graph G d establishes its required connection via the corresponding arc of order d. That is, S buys all the arcs. As we will show in the remainder of this proof, S forms a strong Nash equilibrium. We evaluate the cost of S. As there are d graphs of order d, the total cost of order-d arcs is d a d = d / d =, for any fixed d with 0 d d max. Summing over all d, we obtain cost(s) = d max + log n, and this establishes the desired performance ratio. It remains to show that S is indeed a strong Nash equilibrium. To this end we have to show that no coalition I of agents has an improvement move. We will always consider non-empty coalitions. An improvement move, for a coalition I, is a strategy change S I such that cost i(s I, S I) < cost i (S), for any agent i I. In our graph G, as all the agents have to connect pairs of terminals, a strategy of an agent is a simple path connecting the desired vertices. The property that there exists no improvement move follows from Lemma, which we prove in the sequel. Lemma For d = 0,...,d max, no coalition involving agents of order d has an improvement move. For the proof of Lemma we need Lemma 2 which we prove first. Lemma 2 Consider a fixed d, 0 d d max, and suppose that no coalition involving agents of order smaller than d has an improvement move. Furthermore, assume that no coalition I involving agents of order d has an improvement move in which an order-d agent i I associated with a graph G d (i) chooses a path containing edges outside G d (i). Then no coalition involving agents of order d has an improvement move. Proof of Lemma 2. Let I be a coalition that involves agents of order d. We have to show that I has no improvement move. Based on the assumptions of the lemma, we can restrict ourselves to coalitions I that do not contain agents of order smaller than d. Furthermore, based on the assumptions, we only have to consider strategy changes where each order-d agent i I establishes the required connection within its graph G d (i). Let I I be any maximal sub-coalition of order-d agents that are associated with the same graph G d (I ). We will show that any strategy change S I that consists in choosing connection paths within G d (I ) leads to a strictly higher cost for that sub-coalition, i.e. at least one agent i I has a strictly higher cost and the strategy change is no improvement move. 0

12 Graph G d (I ) has n d order-d agents associated with it. Let f be the fraction defecting, i.e. f = I /n d. In the original configuration S, when routing through the arc of order d, sub-coalition I paid a cost of fa d = f/ d. When changing strategy and choosing a different connection route within G d (I ), each i I selects a path P i that crosses the bridge of order d and then, if d = d max, traverses the stem edge of G dmax (I ) (see Fig. 2(a)). If d < d max, path P i then traverses the order-(d + ) graphs G d+ (I ),, located within G d (I ) (see Fig. 2(b)). If d = d max, then the total cost of edges on P i is b dmax + s dmax ( + 2/ d )/ d. If d < d max, then the total cost is at least b d + / d+ ( + 2/ d )/ d because, by Proposition 2, the least expensive path traversing an order-(d + ) graph has cost / d+. In both cases we have the same lower bound on the cost, expressed in terms of d. The cost of P i is not shared by agents of order smaller than d, as they are not part of the original coalition I, nor is the cost shared by order-d agents associated with other graphs G d G d (I ). The cost of P i can only be shared by agents of order larger than d, and there exist N d+ such agents if d < d max. If d = d max, the cost is not shared by other agents. If d = d max, we are done because the new cost of I is ( + 2/ d )/ d, while the original cost was f/ d / d. If d < d max, then at best all the N d+ agents of order larger than d support the edges traversed by I and the new cost of I is at least cost fn I d fn d +N d+ ( + 2 ). We will to show that cost d d I is higher than the original cost of f/ d n, which is equivalent to proving d fn d +N d+ ( + 2 ) >. Since 0 < f it d suffices to show n d ( + 2 n d + N d+ d) >. Using the definition of n d, eliminating ceilings, we find n d > n/ d(d+) n/ (d+)2 +d = n d (d+)2 n d (d+)2. ( 2 (d+)(d+2) n ) n d (d+)2 ( ) d2+d+2 2d n The second inequality holds because / 2d. For the third inequality note that d < d max and d max = log n imply (d + )(d + 2) log n and hence (d+)(d+2) n. Moreover, we have N d+ < 2n/ (d+)2. We conclude n d n d + N d+ ( + 2 d) > d d + 2 ( + 2 d d) = d + 2 d + 2 d =. Proof of Lemma. We prove the lemma inductively for increasing values of d. For d = 0, the statement follows immediately from Lemma 2 as the assumptions of that lemma are trivially satisfied: There are no agents of order smaller than 0 and an agent of order 0 cannot connect its terminals using edges outside G 0. Suppose that the statement of Lemma holds for depth 0,...,d. We prove that no coalition I involving order-d agents has an improvement move in which an order-d agent i I associated with a graph G d (i) chooses a path using edges outside G d (i). The inductive step then follows from Lemma 2. So consider a coalition I involving agents of order d and a corresponding strategy change S I in which at least one order-d agent chooses edges outside its order-d graph to connect the desired terminals. We show that the strategy change is not an improvement move. By the inductive hypothesis, we can restrict ourselves to coalitions I not involving agents of order smaller than d. Thus I only contains agents of order d or larger. Let I I be the maximum sub-coalition of order-d agents i choosing connection paths outside their graph

13 G d (i). As d, each such graph belongs to a graph G d in the nested structure of G 0. Consider all graphs of order d containing at least one G d (i), i I, and number these order-(d ) graphs in an arbitrary way. Let J be the resulting index set. Each graph G d,j, j J, contains three graphs G d,j, Gd,j 2, Gd,j of order d. For =,2,, let f j be the fraction of the order-d agents associated with Gd,j that are member of I, i.e. f j = {i I : i is order-d agent associated with G d,j } /n d. Recall that n d is the number of agents associated with an order-d graph. We have 0 f j and fj + fj 2 + fj > 0. In the original configuration S, coalition I pays a total cost of cost I = j J = fj a d because the order-d arcs are bought. This expression holds even if some of the f j are zero. We show in the following that the new cost cost I of I is strictly higher than cost I. Hence at least one agent in I does not improve its cost and the strategy change is no improvement move. In order to estimate cost I consider an agent i I and let G d,j, with j J, be the graph where i s graph G d (i) is located. First suppose that G d (i) = G d,j, cf. Figure (a). After the strategy change, i connects the base and the tip of G d,j on a path P i that uses edges outside G d,j. Since strategies are simple paths, all the edges of P i are outside G d,j. Starting at the base of Gd,j, path P i has to traverse the bridge of order d in G d,j. To reach the tip of G d,j, path P i has to travel to the tip of G d,j. This can be done using the arc of order d or using another subpath outside G d,j. After having reached the tip of G d,j, path P i traverses G d,j and G d,j 2, reaching the desired terminal. Ignoring edges visited between the base and the tip of Gd,j, agent i has to pay a share for the order-(d ) bridge and for the subpaths of P i within G d,j 2 and G d,j. The total cost of edges traversed within graph G d,j, {2,}, is at least /d by Proposition 2. The cost may be shared with other agents. G d,j G d,j G d,j G d,j G d,j G d,j G d,j G d,j G d,j 2 P i G d,j 2 P i G d,j 2 P i G d,j 2 G d,j G d,j G d,j G d,j P i (a) (b) (c) (d) Figure : The paths traversed after strategy change. If G d (i) = G d,j 2 or G d (i) = G d,j, the situation is similar, see Figures (b) and (c), respectively. In the first case, P i has to traverse G d,j and the order-(d ) bridge. From there it has to travel to the tip of G d,j and pass through G d,j. Edges used within a graph Gd,j, {,}, have a total cost of at least /d ; cost sharing may occur. If G d (i) = G d,j, path P i passes through G d,j 2 and G d,j. After traversing the bridge of order d it connects to the tip of G d,j, which is also the tip of G d,j representing the desired terminal. Next let C j be the total number of agents i I that are not associated with G d,j, Gd,j 2 or G d,j but choose edges of these graphs when performing the strategy change. In order to use such edges, the new path P i of i 2

14 must pass through the base and the tip of G d,j, see Figure (d). The path between these two vertices crosses the bridge of order d and must consist of subpaths within G d,j, =,2,. We are ready to lower bound the new cost cost I of I. To this end we will only consider the cost spent in graphs G d,j, and j J, and on order-(d ) bridges in Gd,j. Fix a j J. Graph G d,j is traversed by exactly (f j 2 + fj )n d + C j agents from I, each using edges of cost at least / d. The cost is shared by at most (f j 2 + fj )n d + C j + ( f j )n d + N d+ agents. Here ( f j )n d is the number of order-d agents associated with G d,j that establish their connection within this graph and N d+ is the total number of agents of order larger than d that may participate in the strategy change and reside in the entire coalition I. If d = d max, the we set N d+ = 0. Thus sub-coalition I spends a cost of at least d in graph G d,j. Similarly, in Gd,j 2 and G d,j the costs are (f j 2 + fj )n d + C j ( f j + fj 2 + fj )n d + N d+ + C j d (f j + fj )n d + C j ( f j 2 + fj + fj )n and d + N d+ + C j d (f j + fj 2 )n d + C j ( f j + fj + fj 2 )n d + N d+ + C. j Finally the bridge of order d has cost b d = 2/ 2(d ), which is shared by (f j +fj 2 +fj )n d+n d+ +C j agents and I 2 incurs a cost of ((f j 2(d ) + fj 2 + fj )n d + C j )/((f j + fj 2 + fj )n d + N d+ + C j ). Note that for any {,2,} the other two indices from that set can be expressed as = mod + and = ( + ) mod +. We conclude cost I ( (f j + f j )n d + C j j J = ( f j + fj + f j )n d + N d+ + C j d ) (f j + + fj 2 + fj )n d + C j (f j + fj 2 + fj )n d + N d+ + C 2 j 2(d ). Ratios of the form (x + c)/(y + c) are increasing in c if x y. Hence we can drop the terms C j and obtain cost I ( ) (f j + f j )n d j J ( f j + fj + f j )n d + N d+ d + (f j + fj 2 + fj )n d 2 (f j + fj 2 + fj )n d + N d+ 2(d ), = Reordering the expression in the bracets, by focusing on one particular f j in the numerators, we find cost I (( ) f j n d f j j J = ( + f j fj + f j + n d )n d + N d+ ( + f j + fj f j )n d + N d+ d f j + n ) d 2 (f j + fj 2 + fj )n d + N d+ 2(d ). To simplify the last expression we observe that for any real values x, y and c inequality x y+c + x+y+c 2 x+c holds, which we apply for x = f j n d and y = (f j f j )n d as well as c = n d + N d+. Furthermore, f j n d (f j + fj 2 + fj )n d + N d+ f j n d (f j + 2)n d + N d+ 2 f j n d ( + f j )(n d + N d+ ).

15 Hence ( cost I j J = ( > j J = 2f j n d ( + f j )n d + N d+ d + f j n d 2f j n d ( + f j )(n d + N d+ ) ( + 2 d) d ( + f j )(n d + N d+ ) ). ) 2(d ) n d As shown at the end of the proof of Lemma 2, n d +N d+ (+ 2 ) > if d < d d max. If d = d max, then N d+ = 0 and the inequality is also satisfied. In each case cost I > j J = 2f j + f j d j J f j d = and the new cost of I is strictly larger than the original cost of I in configuration S. This completes the proof of Theorem 4 for α =. We finally show how to adapt the proof for any α >. In the construction of the graphs G d only the costs of the arcs change. An arc of order d now has cost αa d. All other costs remain the same. This increases the cost of configuration S by a factor of α, i.e. cost(s) α log n while the cost of the social optimum remains the same. This establishes a performance ratio of Ω(α log n). In the statements of Lemmas and 2, the term improvement move has to be replaced by α-improvement move. An α-improvement move, for a coalition I, is a strategy change S I such that cost i(s I, S I) < cost i (S)/α, for any agent i I. In the proof of Lemma 2 we considered any coalition I involving agents of order d or larger and investigated strategy changes where order-d agents establish connections with their respective graph of order d. We identified a sub-coalition I, with f = I /n d, incurring a new cost of cost I > f/ d. This cost inequality still hold in our modified graph as edge costs did not decrease. Since cost I > f/ d = (αf/ d )/α and αf/ d is the original cost of I in the scaled graph, the strategy change is no α-improvement move. In the proof of Lemma 2 we studied coalitions I involving agents of order d or larger. We analyzed strategy changes in which order-d agents buy edges outside their graph of order d and identified a sub-coalition I of order-d agents incurring a new total cost of cost I > j J = fj /d, where G d,j were the graphs the agents i I are associated with. Again, when arcs are scaled by a factor of α, this cost inequality still holds. As the original cost of I in the scaled graph is j J = αfj /d, the strategy change is no α-improvement move. 5 Weighted games In this section we study weighted networ design games where each agent i has a positive weight w i. We scale the weights such that the minimum weight is equal to and hence w i for all agents. Let W = n i= w i be the total weight of all the agents. If agents are allowed to coordinate their strategies, two scenarios are of interest. In a first setting we assume that coalitions of size up to c are allowed, for any c n. In this case let W c be the maximum total weight of any coalition having size at most c. In a second setting we assume that the total weight of a coalition is upper bounded so that agents of high weight cannot impose too much control on agents of low weight. In this case let Wmax c be the maximum total weight any coalition may have. We extend our results shown for unweighted games. 4