Calculating the Price of Anarchy for Network Formation Games (original) (raw)
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The Calculation and Simulation of the Price of Anarchy for Network Formation Games
2011
We model the formation of networks as the result of a game where by players act selfishly to get the portfolio of links they desire most. The integration of player strategies into the network formation model is appropriate for organizational networks because in these smaller networks, dynamics are not random, but the result of intentional actions carried through by players maximizing their own objectives. This model is a better framework for the analysis of influences upon a network because it integrates the strategies of the players ...
Network creation games: structure vs anarchy
2017
We study Nash equilibria and the price of anarchy in the classical model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price a > 0 in order to get connected to the network formed by all the n agents. In this setting, the reformulated tree conjecture states that for a > n, every Nash equilibrium network is a tree. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of anarchy would be constant for a > n. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. Up to now the last conjecture has been proven in (i) the lower range, for a = O(n1-o?) with o? = 1 and (ii) in the upper range, for a > 65n. In ?log n contrast, the best upper bound known for the price of anarchy for the remaining range is 2O(vlog n). In this paper we give new insights into the structure of the Nash equilibria for different ranges of ...
Proceedings of the twenty-second annual symposium on Principles of distributed computing - PODC '03, 2003
We introduce a novel game that models the creation of Internet-like networks by selfish node-agents without central design or coordination. Nodes pay for the links that they establish, and benefit from short paths to all destinations. We study the Nash equilibria of this game, and prove results suggesting that the "price of anarchy" [4] in this context (the relative cost of the lack of coordination) may be modest. Several interesting extensions are suggested.
New Insights into the Structure of Equilibria for the Network Creation Game
ArXiv, 2020
We study the sum classic network creation game introduced by Fabrikant et al. in which n players conform a network buying links at individual price α. When studying this model we are mostly interested in Nash equilibria (ne) and the Price of Anarchy (PoA). It is conjectured that the PoA is constant for any α. Up until now, it has been proved constant PoA for the range α = O(n1−δ1) with δ1 > 0 a positive constant, upper bounding by a constant the diameter of any ne graph jointly with the fact that the diameter of any ne graph plus one unit is an upper bound for the PoA of the same graph. Also, it has been proved constant PoA for the range α > n(1 + δ2) with δ2 > 0 a positive constant, studying extensively the average degree of any biconnected component from equilibria. Our contribution consists in proving that ne graphs satisfy very restrictive topological properties generalising some properties proved in the literature and providing new insights that might help settling the...
Anarchy Is Free in Network Creation
Lecture Notes in Computer Science, 2013
The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by Fabrikant et al. In it, each of n agents corresponds to a vertex, which can create edges to other vertices at a cost of α each, for some parameter α. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent's cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices.
A game-theoretic network formation model
2014
We study the dynamics of a game-theoretic network formation model that yields large-scale small-world 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 game-theoretic network formation models in which links are formed due to strategic behaviors of individuals, rather than based on probabilities. Inspired by Even-Dar 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 earlier model, not only the existence of equilibrium networks is guaranteed in our model, but also these networks coincide with the outcomes of pairwise Nash equilibrium in network formation. Furthermore, we provide a network formation simulation that generates small-world networks. We also analyze the impact of locating players in a hierarchical structure by constructing a strategic model, where a complete b-ary tree is the seed network.
On the Price of Anarchy for High-Price Links
Web and Internet Economics, 2019
We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other n − 1 players at a prefixed price α > 0. The player's goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is constant for all α. This conjecture has been confirmed for α = O(n 1−δ) with δ ≥ 1/ log n and for α > 4n − 13. The best known upper bound on the price of anarchy for the remaining range is 2 O(√ log n). We give new insights into the structure of the Nash equilibria for α > n and we enlarge the range of the parameter α for which the price of anarchy is constant. Specifically, we prove that for any small > 0, the price of anarchy is constant for α > n(1 +) by showing that any biconnected component of any non-trivial Nash equilibrium, if it exists, has at most a constant number of nodes.
Exact and approximate equilibria for optimal group network formation
Theoretical Computer Science, 2011
We consider a process called Group Network Formation Game, which represents the scenario when strategic agents are building a network together. In our game, agents can have extremely varied connectivity requirements, and attempt to satisfy those requirements by purchasing links in the network. We show a variety of results about equilibrium properties in such games, including the fact that the price of stability is 1 when all nodes in the network are owned by players, and that doubling the number of players creates an equilibrium as good as the optimum centralized solution. For the general case, we show the existence of a 2-approximate Nash equilibrium that is as good as the centralized optimum solution, as well as how to compute good approximate equilibria in polynomial time. Our results essentially imply that for a variety of connectivity requirements, giving agents more freedom can paradoxically result in more efficient outcomes. * A preliminary version of this paper appeared in ESA 2009. Recall that a (pure-strategy) Nash equilibrium is a solution where no single player can switch her strategy and become better off, given that the other players keep their strategies fixed.
Locality-based network creation games
Proceedings of the 26th ACM symposium on Parallelism in algorithms and architectures - SPAA '14, 2014
Network creation games have been extensively studied, both from economists and computer scientists, due to their versatility in modeling individual-based community formation processes, which in turn are the theoretical counterpart of several economics, social, and computational applications on the Internet. In their several variants, these games model the tension of a player between her two antagonistic goals: to be as close as possible to the other players, and to activate a cheapest possible set of links. However, the generally adopted assumption is that players have a common and complete information about the ongoing network, which is quite unrealistic in practice. In this paper, we consider a more compelling scenario in which players have only limited information about the network they are embedded in. More precisely, we explore the game-theoretic and computational implications of assuming that players have a complete knowledge of the network structure only up to a given radius k, which is one of the most qualified local-knowledge models used in distributed computing. To this respect, we define a suitable equilibrium concept, and we provide a comprehensive set of upper and lower bounds to the price of anarchy for the entire range of values of k, and for the two classic variants of the game, namely those in which a player's cost-besides the activation cost of the owned links-depends on the maximum/sum of all the distances to the other nodes in the network, respectively. These bounds are finally assessed through an extensive set of experiments.