The Calculation and Simulation of the Price of Anarchy for NetworkFormation Games (original) (raw)
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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...
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Motivated by applications in peer-to-peer and overlay networks we define and study the \emph{Bounded Degree Network Formation} (BDNF) game. In an (n,k)(n,k)(n,k)-BDNF game, we are given nnn nodes, a bound kkk on the out-degree of each node, and a weight wvuw_{vu}wvu for each ordered pair (v,u)(v,u)(v,u) representing the traffic rate from node vvv to node uuu. Each node vvv uses up to kkk directed links to connect to other nodes with an objective to minimize its average distance, using weights wvuw_{vu}wvu, to all other destinations. We study the existence of pure Nash equilibria for (n,k)(n,k)(n,k)-BDNF games. We show that if the weights are arbitrary, then a pure Nash wiring may not exist. Furthermore, it is NP-hard to determine whether a pure Nash wiring exists for a given (n,k)(n,k)(n,k)-BDNF instance. A major focus of this paper is on uniform (n,k)(n,k)(n,k)-BDNF games, in which all weights are 1. We describe how to construct a pure Nash equilibrium wiring given any nnn and kkk, and establish that in all pure Nash wirings the cost of individual nodes cannot differ by more than a factor of nearly 2, whereas the diameter cannot exceed O(sqrtnlogkn)O(\sqrt{n \log_k n})O(sqrtnlogkn). We also analyze best-response walks on the configuration space defined by the uniform game, and show that starting from any initial configuration, strong connectivity is reached within Theta(n2)\Theta(n^2)Theta(n2) rounds. Convergence to a pure Nash equilibrium, however, is not guaranteed. We present simulation results that suggest that loop-free best-response walks always exist, but may not be polynomially bounded. We also study a special family of \emph{regular} wirings, the class of Abelian Cayley graphs, in which all nodes imitate the same wiring pattern, and show that if nnn is sufficiently large no such regular wiring can be a pure Nash equilibrium.
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Geometric Network Creation Games
The 31st ACM Symposium on Parallelism in Algorithms and Architectures
Network Creation Games are a well-known approach for explaining and analyzing the structure, quality and dynamics of real-world networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unit-weight edges. In practice, e.g. when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al. [PODC'03] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state-of-the-art for the unit-weight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight non-constant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the non-metric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of a variant of the classical Network Design Problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.
``A Survey of Models of Network Formation: Stability and Efficiency,''
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. JEL Classification Numbers: A14, C71, C72
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 ...
A game-theoretic network formation model
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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.
Definitions of equilibrium in network formation games
International Journal of Game Theory, 2006
We examine a variety of stability and equilibrium de…nitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of de…nitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible. JEL Classi…cation Numbers: A14, C71, C72
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.