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191
The price of stability for network design with fair cost allocation
 In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS
, 2004
"... Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite differ ..."
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Cited by 208 (28 self)
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Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context — it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widelystudied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fairdivision scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via bestresponse dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form nearoptimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of bestresponse dynamics, and discuss extensions to a weighted game.
A Survey of Models of Network Formation: Stability and Efficiency
, 2003
"... I survey the recent literature on the formation of networks. I provide definitions of network games, a number of examples of models from the literature, and discuss some of what is known about the (in)compatibility of overall societal welfare with individual incentives to form and sever links. ..."
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Cited by 196 (14 self)
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I survey the recent literature on the formation of networks. I provide definitions of network games, a number of examples of models from the literature, and discuss some of what is known about the (in)compatibility of overall societal welfare with individual incentives to form and sever links.
Nearoptimal network design with selfish agents
 IN PROCEEDINGS OF THE 35TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 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 121 (21 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 algorithm to find a (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.
A Theory of BuyerSeller Networks
 American Economic Review
, 2001
"... This paper introduces a new model of exchange: networks, rather than markets, of buyers and sellers. It begins with the empirically motivated premise that a buyer and seller must have a relationship, a “link, ” to exchange goods. Networks buyers, sellers, and the pattern of links connecting them a ..."
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Cited by 78 (0 self)
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This paper introduces a new model of exchange: networks, rather than markets, of buyers and sellers. It begins with the empirically motivated premise that a buyer and seller must have a relationship, a “link, ” to exchange goods. Networks buyers, sellers, and the pattern of links connecting them are common exchange environments. This paper develops a methodology to study network structures and explains why agents may form networks. In a model that captures characteristics of a variety of industries, the paper shows that buyers and sellers, acting strategically in their own selfinterests, can form the network structures that maximize overall
Evolutionary games on graphs
, 2007
"... Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to ..."
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Cited by 54 (0 self)
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Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in nonequilibrium statistical physics. This review gives a tutorialtype overview of the field for physicists. The first four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by nonmeanfieldtype social network structures in general. The next three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner’s Dilemma, the Rock–Scissors–Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
The economics of social networks
 PROCEEDINGS OF THE 9 TH WORLD CONGRESS OF THE ECONOMETRIC SOCIETY
, 2005
"... The science of social networks is a central field of sociological study, a major application of random graph theory, and an emerging area of study by economists, statistical physicists and computer scientists. While these literatures are (slowly) becoming aware of each other, and on occasion drawing ..."
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Cited by 53 (2 self)
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The science of social networks is a central field of sociological study, a major application of random graph theory, and an emerging area of study by economists, statistical physicists and computer scientists. While these literatures are (slowly) becoming aware of each other, and on occasion drawing from one another, they are still largely distinct in their methods, interests, and goals. Here, my aim is to provide some perspective on the research from these literatures, with a focus on the formal modeling of social networks and the two major types of models: those based on random graphs and those based on game theoretic reasoning. I highlight some of the strengths, weaknesses, and potential synergies between these two network modeling approaches.
Network Design with Weighted Players
 In Proceedings of the 18th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA
, 2006
"... We consider a model of gametheoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its p ..."
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Cited by 44 (6 self)
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We consider a model of gametheoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. [2] proved that purestrategy Nash equilibria always exist and that the price of stability—the ratio in costs of a minimumcost Nash equilibrium and an optimal solution—is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has aweightwi≥1, and its cost share of an edge in its path
Designing networks with good equilibria
 In SODA ’08
, 2007
"... In a network with selfish users, designing and deploying a protocol determines the rules of the game by which end users interact with each other and with the network. We study the problem of designing a protocol to optimize the equilibrium behavior of the induced network game. We consider network co ..."
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Cited by 32 (4 self)
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In a network with selfish users, designing and deploying a protocol determines the rules of the game by which end users interact with each other and with the network. We study the problem of designing a protocol to optimize the equilibrium behavior of the induced network game. We consider network costsharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge costsharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal costsharing protocols for undirected and directed graphs, singlesink and multicommodity networks, different classes of costsharing methods, and different measures of the inefficiency of equilibria. One of our main technical tools is a complete characterization of the uniform costsharing protocols—protocols that are designed without foreknowledge of or assumptions on the network in which they will be deployed. We use this characterization result to identify the optimal uniform protocol in several scenarios: for example, the Shapley protocol is optimal in directed graphs, while the optimal protocol in undirected graphs, a simple priority scheme, has exponentially smaller worstcase price of anarchy than the Shapley protocol. We also provide several matching upper and lower bounds on the bestpossible performance of nonuniform costsharing protocols.
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 29 (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.
It’s All About Connections: Evidence on Network Formation
 Journal of Urban Economics
"... We present an economic experiment on network formation, in which subjects can decide to form links to one another. Direct links are costly but being connected is valuable. The gametheoretic basis for our experiment is the model of Bala and Goyal (2000). They distinguish between two scenarios regard ..."
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Cited by 28 (2 self)
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We present an economic experiment on network formation, in which subjects can decide to form links to one another. Direct links are costly but being connected is valuable. The gametheoretic basis for our experiment is the model of Bala and Goyal (2000). They distinguish between two scenarios regarding the flow of benefits through a network, the socalled 1way and 2way flow model. Our main results show that the prediction based on Nash and strict Nash equilibrium works well in the 1way flow model but fails largely in the 2way flow model. We observe a strong learning dynamic in the 1way flow model but less so in the 2way flow model. Finally, costs of a direct link have a positive impact on the occurrence of (strict) Nash networks in the 1way flow model but a negative impact in the 2way flow model. In our discussion on possible explanations for these results we focus on strategic asymmetry and asymmetry with respect to payoffs. We find that the latter asymmetry, i.e., payoff inequity, plays an important role in the network formation process.