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47
Nearoptimal network design with selfish agents
, 2003
"... We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possi ..."
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Cited by 151 (19 self)
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We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent’s goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NPcomplete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithmtofinda(1+ε)approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65 +ε)approximate Nash equilibrium that does not cost much more.
Intrinsic Robustness of the Price of Anarchy
 STOC'09
, 2009
"... The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium ..."
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Cited by 101 (12 self)
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The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regretminimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regretminimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and BayesianNash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomiallength bestresponse sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worstcase upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with nonpolynomial cost functions, and the first
On the Value of Coordination in Network Design
"... We study network design games where n selfinterested agents have to form a network by purchasing links 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 known that the price of anarchy o ..."
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Cited by 36 (0 self)
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We study network design games where n selfinterested agents have to form a network by purchasing links 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 known 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 superconstant 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 known upper bound of O(log W). Our results imply that, for most settings, the worstcase 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.
Congestion Games with Linearly Independent Paths: Convergence Time and Price of Anarchy
 Theory of Computing Systems
, 2010
"... Abstract. We investigate the effect of linear independence in the strategies of congestion games on the convergence time of best improvement sequences and on the pure Price of Anarchy. In particular, we consider symmetric congestion games on extensionparallel networks, an interesting class of netw ..."
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Cited by 10 (1 self)
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Abstract. We investigate the effect of linear independence in the strategies of congestion games on the convergence time of best improvement sequences and on the pure Price of Anarchy. In particular, we consider symmetric congestion games on extensionparallel networks, an interesting class of networks with linearly independent paths, and establish two remarkable properties previously known only for parallellink games. More precisely, we show that for arbitrary (nonnegative and nondecreasing) latency functions, any best improvement sequence reaches a pure Nash equilibrium in at most as many steps as the number of players, and that for latency functions in class D, the pure Price of Anarchy is at most ρ(D). As a byproduct of our analysis, we obtain that for symmetric congestion games on general networks with latency functions in class D, the Price of Stability is at most ρ(D).
SociallyAware Network Design Games
 IN PROC. OF INFOCOM 2010
, 2010
"... In many scenarios network design is not enforced by a central authority, but arises from the interactions of several selfinterested agents. This is the case of the Internet, where connectivity is due to Autonomous Systems ’ choices, but also of overlay networks, where each user client can decide th ..."
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Cited by 10 (4 self)
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In many scenarios network design is not enforced by a central authority, but arises from the interactions of several selfinterested agents. This is the case of the Internet, where connectivity is due to Autonomous Systems ’ choices, but also of overlay networks, where each user client can decide the set of connections to establish. Recent works have used game theory, and in particular the concept of Nash Equilibrium, to characterize stable networks created by a set of selfish agents. The majority of these works assume that users are completely noncooperative, leading, in most cases, to inefficient equilibria. To improve efficiency, in this paper we propose two novel sociallyaware network design games. In the first game we incorporate a sociallyaware component in the users ’ utility functions, while in the second game we use additionally a Stackelberg (leaderfollower) approach, where a leader (e.g., the network administrator) architects the desired network buying an appropriate subset of network’s links, driving in this way the users to overall efficient Nash equilibria. We provide bounds on the Price of Anarchy and other efficiency measures, and study the performance of the proposed schemes in several network scenarios, including realistic topologies where players build an overlay on top of real Internet Service Provider networks. Numerical results demonstrate that (1) introducing some incentives to make users more sociallyaware is an effective solution to achieve stable and efficient networks in a distributed way, and (2) the proposed Stackelberg approach permits to achieve dramatic performance improvements, designing almost always the socially optimal network.
On strong equilibria in the max cut game
 In: Proc. of WINE 2009, Springer LNCS
, 2009
"... Abstract. This paper deals with two games defined upon well known generalizations of max cut. We study the existence of a strong equilibrium which is a refinement of the Nash equilibrium. Bounds on the price of anarchy for Nash equilibria and strong equilibria are also given. In particular, we show ..."
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Cited by 10 (1 self)
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Abstract. This paper deals with two games defined upon well known generalizations of max cut. We study the existence of a strong equilibrium which is a refinement of the Nash equilibrium. Bounds on the price of anarchy for Nash equilibria and strong equilibria are also given. In particular, we show that the max cut game always admits a strong equilibrium and the strong price of anarchy is 2/3. 1
Efficient Graph Topologies in Network Routing Games
"... In this work we explore the topological structure of networks that guarantee that any routing of selfish users is efficient, i.e., any Nash equilibrium achieves the social optimum. We distinguish between two classes of atomic network routing games. In both classes the cost of the social optimum is t ..."
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Cited by 9 (1 self)
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In this work we explore the topological structure of networks that guarantee that any routing of selfish users is efficient, i.e., any Nash equilibrium achieves the social optimum. We distinguish between two classes of atomic network routing games. In both classes the cost of the social optimum is the maximum cost over the players. In the first, network congestion games, the player’s cost is the sum of the latency costs over the edges in its route, while in the second, bottleneck routing games, the player’s cost is the maximum edge cost over the edges in its route. Our interesting results are for the symmetric case of a single source and a single destination (singlecommodity). We show that for network congestion games the efficient topologies are exactly Extension Parallel Graphs, while for bottleneck routing games the efficient topologies are exactly Series Parallel Graphs. For the asymmetric case of multiple sources or destinations (multicommodity), we show that the efficient topologies are very limited and include either trees or trees with parallel edges.
Strong and Pareto Price of Anarchy in Congestion Games
, 2008
"... Strong Nash equilibria and Paretooptimal Nash equilibria are natural and important strengthenings of the Nash equilibrium concept. We study these stronger notions of equilibrium in congestion games, focusing on the relationships between the price of anarchy for these equilibria and that for standar ..."
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Cited by 9 (0 self)
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Strong Nash equilibria and Paretooptimal Nash equilibria are natural and important strengthenings of the Nash equilibrium concept. We study these stronger notions of equilibrium in congestion games, focusing on the relationships between the price of anarchy for these equilibria and that for standard Nash equilibria (which is well understood). For symmetric congestion games with polynomial or exponential latency functions, we show that the price of anarchy for strong and Paretooptimal equilibria is much smaller than the standard price of anarchy. On the other hand, for asymmetric congestion games with polynomial latencies the strong and Pareto prices of anarchy are essentially as large as the standard price of anarchy; while for asymmetric games with exponential latencies the Pareto and standard prices of anarchy are the same but the strong price of anarchy is substantially smaller. Finally, in the special case of linear latencies, we show that the strong and Pareto prices of anarchy coincide exactly with the known value 5 2 for standard Nash, but are strictly smaller for symmetric games.
Approximate Strong Equilibrium in Job Scheduling Games
, 2009
"... A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit sta ..."
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Cited by 9 (0 self)
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A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit stability against coalitional deviations, in that the benefits from a joint deviation are bounded. In this sense, NE approximates strong equilibrium. Coalition formation is a key issue in multiagent systems. We provide a framework for quantifying the stability and the performance of various assignment policies and solution concepts in the face of coalitional deviations. Within this framework we evaluate a given configuration according to three measures: (i) IRmin: the maximal number α, such that there exists a coalition in which the minimal improvement ratio among the coalition members is α, (ii) IRmax: the maximal number α, such that there exists a coalition in which the maximal improvement ratio among the coalition members is α, and (iii) DRmax: the maximal possible damage ratio of an agent outside the coalition. We analyze these measures in job scheduling games on identical machines. In particular, we provide upper and lower bounds for the above three measures for both NE and the wellknown assignment rule Longest Processing Time (LPT). Our results indicate that LPT performs better than a general NE. However, LPT is not the best possible approximation. In particular, we present a polynomial time approximation scheme (PTAS) for the makespan minimization problem which provides a schedule with IRmin of 1 + ε for any given ɛ. With respect to computational complexity, we show that given an NE on m ≥ 3 identical machines or m ≥ 2 unrelated machines, it is NPhard to determine whether a given coalition can deviate such that every member decreases its cost.
The Curse of Simultaneity
"... Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we prop ..."
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Cited by 8 (1 self)
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Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we propose studying the sequential price of anarchy: the quality of outcomes of sequential versions of games whose simultaneous counterparts are prototypical in algorithmic game theory. We study different classes of games with high price of anarchy, and show that the subgame perfect equilibrium of their sequential version is a much more natural prediction, ruling out unreasonable equilibria, and leading to much better quality solutions. We consider three examples of such games: Cost Sharing