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A Nash bargaining solution for Cooperative Network Formation Games
"... The Network Formation problem has received increasing attention in recent years. Previous works have addressed this problem considering almost exclusively networks designed by selfish users, which can be consistently suboptimal. This paper addresses the network formation issue using cooperative ga ..."
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The Network Formation problem has received increasing attention in recent years. Previous works have addressed this problem considering almost exclusively networks designed by selfish users, which can be consistently suboptimal. This paper addresses the network formation issue using cooperative game theory, which permits to study ways to enforce and sustain cooperation among agents. Both the Nash bargaining solution and the Shapley value are widely applicable concepts for solving these games. However, we show that the Shapley value presents three main drawbacks in this context: (1) it is nontrivial to define meaningful characteristic functions for the cooperative network formation game, (2) it can determine for some players cost allocations that are even higher than those at the Nash Equilibrium (i.e., if players refuse to cooperate), and (3) it is computationally very cumbersome. For this reason, we solve the cooperative network formation game using the Nash bargaining solution (NBS) concept. More specifically, we extend the NBS approach to the case of multiple players and give an explicit expression for users ’ cost allocations. Furthermore, we compare the NBS to the Shapley value and the Nash equilibrium solution, showing its advantages and appealing properties in terms of cost allocation to users and computation time to get the solution. Numerical results demonstrate that the proposed Nash bargaining solution approach permits to allocate costs fairly to users in a reasonable computation time, thus representing a very effective framework for the design of efficient and stable networks.
Greedy Selfish Network Creation
, 2011
"... We introduce and analyze greedy equilibria (GE) for the wellknown model of selfish network creation by Fabrikant et al. [PODC’03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategychanges, (2) GE are outcomes found by ..."
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We introduce and analyze greedy equilibria (GE) for the wellknown model of selfish network creation by Fabrikant et al. [PODC’03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategychanges, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NPhard. Hence, polytime agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the Sum version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3approximate NE on general networks. For the latter we also provide a lower bound of 3 2 on the approximation ratio. For the Max version, where agents attempt to minimize their maximum distance, we show that any GEstar is in 2approximate NE and any GEtree having larger diameter is in 6 5approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the Max version on general networks in GE. This result implies a locality gap of Ω(n) for the metric minmax facility location problem, where n is the number of clients.
Constant Price of Anarchy in Network Creation Games via Public Service Advertising
"... Abstract. Network creation games have been studied in many different settings recently. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without payin ..."
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Abstract. Network creation games have been studied in many different settings recently. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without paying much to construct the network. Many generalizations have been considered, including nonuniform interests between nodes, general graphs of allowable edges, bounded budget agents, etc. In all of these settings, there is no known constant bound on the price of anarchy. In fact, in many cases, the price of anarchy can be very large, namely, a constant power of the number of agents. This means that we have no control on the behavior of network when agents act selfishly. On the other hand, the price of stability in all these models is constant, which means that there is chance that agents act selfishly and we end up with a reasonable social cost. In this paper, we show how to use an advertising campaign (as introduced in SODA 2009 [2]) to find such efficient equilibria in (n, k)uniform bounded budget connection game [10]; our result holds for k = ω(log(n)). More formally, we present advertising strategies such that, if an α fraction of the agents agree to cooperate in the campaign, the social cost would be at most O(1/α) times the optimum cost. This is the first constant bound on the price of anarchy that interestingly can be adapted to different settings. We also generalize our method to work in cases that α is not known in advance. Also, we do not need to assume that the cooperating agents spend all their budget in the campaign; even a small fraction (β fraction) of their budget is sufficient to obtain a constant price of anarchy.
The maxdistance network creation game on general host graphs
 Internet and Network Economics, Lecture
"... Abstract. In this paper we study a generalization of the classic network creation game in the scenario in which the n players sit on a given arbitrary host graph, which constrains the set of edges a player can activate at a cost of α ≥ 0 each. This finds its motivations in the physical limitations ..."
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Abstract. In this paper we study a generalization of the classic network creation game in the scenario in which the n players sit on a given arbitrary host graph, which constrains the set of edges a player can activate at a cost of α ≥ 0 each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its maximum distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of Ω( n/(1 + α)) for any α = o(n). Notice that this implies a counterintuitive lower bound of Ω( n) for very small values of α (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either kregular (for any constant k ≥ 3), or a 2dimensional grid, the PoA is still Ω(1 + min{α, nα}), which is proven to be tight for α = Ω( n). On the positive side, if α ≥ n, we show the PoA is O(1). Finally, in the case in which the host graph is very sparse (i.e., E(H)  = n − 1 + k, with k = O(1)), we prove that the PoA is O(1), for any α.
Productive Output in Hierarchical
"... Abstract. Organically grown crowdsourcing networks, which includes production firms and social network based crowdsourcing applications, tend to have a hierarchical structure. Considering the entire crowdsourcing system as a consolidated organization, a primary goal of a designer is to maximize the ..."
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Abstract. Organically grown crowdsourcing networks, which includes production firms and social network based crowdsourcing applications, tend to have a hierarchical structure. Considering the entire crowdsourcing system as a consolidated organization, a primary goal of a designer is to maximize the net productive output of this hierarchy using reward sharing as an incentive tool. Every individual in a hierarchy has a limited amount of effort that they can split between production and communication. Productive effort yields an agent a direct payoff, while the communication effort of an agent improves the productivity of other agents in her subtree. To understand how the net output of the crowdsourcing network is influenced by these components, we develop a game theoretic model that helps explain how the individuals trade off these two components depending on their position in the hierarchy and their shares of reward. We provide a detailed analysis of the Nash equilibrium efforts and a design recipe of the reward sharing scheme that maximizes the net productive output. Our results show that even under strategic behavior of the agents, it is sometimes possible to achieve the optimal output and also provide bounds on the achievability when this is not the case. 1
Incentives for Information Sharing and Outcome Efforts in Networks
"... discussions during the preparation of this paper. In social or organizational networks, it is often observed that different individuals put different levels of production effort depending on their position in the network. One possible reason is reward sharing, which incentivizes particular agents t ..."
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discussions during the preparation of this paper. In social or organizational networks, it is often observed that different individuals put different levels of production effort depending on their position in the network. One possible reason is reward sharing, which incentivizes particular agents to spend effort in sharing information with others and increasing their productivity. We model the effort level in a network as a strategic decision made by an agent on how much effort to expend on the complementary tasks of information sharing and production. We conduct a gametheoretic analysis of incentive and information sharing in both hierarchical and general influencerinfluencee networks. Our particular interest is in understanding how different reward structures in a network influence this decision. We establish the existence of a unique purestrategy Nash equilibrium in regard to the choice made by each agent, and study the effect of the quality and cost of communication, and the reward sharing on the effort levels at this equilibrium. Our results show that a larger reward share from an influencee incentivizes the influencer to spend more effort, in equilibrium, on communication, capturing a freeriding behavior of well placed agents. We also address the reverse question of designing an optimal reward sharing scheme that achieves the effort profile which maximizes the system output. In this direction, for a number of stylized networks, we study the Price of Anarchy for this output, and the interplay between information and incentive sharing on mitigating the loss in output due to agent selfinterest. 1
Topological Implications of Selfish Neighbor Selection in Unstructured PeertoPeer Networks
, 2009
"... Current peertopeer (P2P) systems often suffer from a large fraction of freeriders not contributing any resources to the network. Various mechanisms have been designed to overcome this problem. However, the selfish behavior of peers has aspects which go beyond resource sharing. This paper studies t ..."
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Current peertopeer (P2P) systems often suffer from a large fraction of freeriders not contributing any resources to the network. Various mechanisms have been designed to overcome this problem. However, the selfish behavior of peers has aspects which go beyond resource sharing. This paper studies the effects on the topology of a P2P network if peers selfishly select the peers to connect to. In our model, a peer exploits locality properties in order to minimize the latency (or response times) of its lookup operations. At the same time, the peer aims at not having to maintain links to too many other peers in the system. By giving tight bounds on the price of anarchy, we show that the resulting topologies can be much worse than if peers collaborated. Moreover, the network may never stabilize, even in the absence of churn. Finally, we establish the complexity of Nash equilibria in our game theoretic model of P2P networks. Specifically, we prove that it is NPhard to decide whether our game has a Nash equilibrium and can stabilize.
Greedy Selfish Network Creation∗ (Full Version)
"... We introduce and analyze greedy equilibria (GE) for the wellknown model of selfish network creation by Fabrikant et al. [PODC’03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategychanges, (2) GE are outcomes found by ..."
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We introduce and analyze greedy equilibria (GE) for the wellknown model of selfish network creation by Fabrikant et al. [PODC’03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategychanges, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NPhard. Hence, polytime agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the Sum version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3approximate NE on general networks. For the latter we also provide a lower bound of 32 on the approximation ratio. For the Max version, where agents attempt to minimize their maximum distance, we show that any GEstar is in 2approximate NE and any GEtree having larger diameter is in 65approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the Max version on general networks in GE. This result implies a locality gap of Ω(n) for the metric minmax facility location problem, where n is the number of clients. 1
A gametheoretic network formation model
, 2014
"... We study the dynamics of a gametheoretic network formation model that yields largescale smallworld networks. So far, mostly stochastic frameworks have been utilized to explain the emergence of these networks. On the other hand, it is natural to seek for gametheoretic network formation models in ..."
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We study the dynamics of a gametheoretic network formation model that yields largescale smallworld networks. So far, mostly stochastic frameworks have been utilized to explain the emergence of these networks. On the other hand, it is natural to seek for gametheoretic network formation models in which links are formed due to strategic behaviors of individuals, rather than based on probabilities. Inspired by EvenDar and Kearns (2007), we consider a more realistic model in which the cost of establishing each link is dynamically determined during the course of the game. Moreover, players are allowed to put transfer payments on the formation of links. Also, they must pay a maintenance cost to sustain their direct links during the game. We show that there is a small diameter of at most 4 in the general set of equilibrium networks in our model. Unlike