Tree Nash Equilibria in the Network Creation Game

Mamageishvili, Akaki; Mihaľák, Matúš; Müller, D.

doi:10.1007/978-3-319-03536-9_10

Cited by 28 publications

(50 citation statements)

References 9 publications

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“…In this work, we show that in the max-distance network creation game, for α ≥ 23 all equilibrium graphs are trees and the price of anarchy is constant. Compared to Mamageishvili et al [13], we also improve the constant upper bound of the price of anarchy from 4 to 3 for tree equilibria. The technique of our proof is mainly based on the degree approaches, following the previous work [3,13,14].…”

Section: Related Workmentioning

confidence: 89%

“…Mihalák and Schlegel [3] used a technique based on the average degree of the biconnected component and significantly improved this range from α ≥ 12n⌈log n⌉ to α > 273n, which is asymptotically tight. Based on the same idea, Mamageishvili et al [13] improved this range to α > 65n and Alvarez and Messegué [14] further improved it to α > 17n. Recently, Bilò and Lenzner [9] improved the bound to α > 4n − 13, using a technique based on critical pairs and min cycles, which is orthogonal to the known degree approaches.…”

Section: Related Workmentioning

confidence: 99%

“…and O(n 2/α ) for α < 2 √ log n. Mamageishvili et al [13] showed that the price of anarchy is constant for α > 129 and α = O(n −1/2 ) and also proved that it is 2 O( √ log n) for any α. Specifically, they showed that in the max-distance network creation game, for α > 129 all equilibrium graphs are trees.…”

Section: Related Workmentioning

confidence: 99%

“…Although this tree conjecture was disproved by Albers et al [12], the reformulated tree conjecture is well accepted as that in the sum-distance network creation game, every equilibrium graph is a tree for α > n, where n is the number of agents. A series of works have contributed to this threshold from 12n log n [12] to 273n [3], 65n [13], 17n [14], and to 4n − 13 [9]. Very recently a preprint by Dippel and Vetta [15] claimed an improved bound 3n − 3.…”

Section: Introductionmentioning

confidence: 99%

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On Tree Equilibria in Max-Distance Network Creation Games

Wang

2021

Preprint

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We study the Nash equilibrium and the price of anarchy in the max-distance network creation game. The network creation game, first introduced and studied by Fabrikant et al.[1], is a classic model for real-world networks from a game-theoretic point of view. In a network creation game with n selfish vertex agents, each vertex can build undirected edges incident to a subset of the other vertices. The goal of every agent is to minimize its creation cost plus its usage cost, where the creation cost is the unit edge cost α times the number of edges it builds, and the usage cost is the sum of distances to all other agents in the resulting network. The max-distance network creation game, introduced and studied by Demaine et al. [2], is a key variant of the original game, where the usage cost takes into account the maximum distance instead. The main result of this paper shows that for α ≥ 23 all equilibrium graphs in the max-distance network creation game must be trees, while the best bound in previous work is α > 129 [3]. We also improve the constant upper bound on the price of anarchy to 3 for tree equilibria. Our work brings new insights into the structure of Nash equilibria and takes one step forward in settling the so-called tree conjecture in the max-distance network creation game.

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Section: Related Workmentioning

confidence: 89%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Tree Equilibria in Max-Distance Network Creation Games

Wang

2021

Preprint

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“…This bound was improved in [35], where it was shown that for α ≥ 273n all pure equilibria are trees. Later on, in [32], this was further improved by showing that it even holds for α ≥ 65n. Recently, two preprints have been released that make further progress towards proving that the price of anarchy of network creation games is constant: First, in [3], Àlvarez & Messegué show that every pure Nash equilibrium is a tree already when α > 17n, and that the price of anarchy is bounded by a constant for α > 9n.…”

Section: Related Literaturementioning

confidence: 99%

On Strong Equilibria and Improvement Dynamics in Network Creation Games

Keijzer¹,

Janus²

2019

Internet Mathematics

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We study strong equilibria in network creation games. These form a classical and well-studied class of games where a set of players form a network by buying edges to their neighbors at a cost of a fixed parameter α. The cost of a player is defined to be the cost of the bought edges plus the sum of distances to all the players in the resulting graph. We identify and characterize various structural properties of strong equilibria, which lead to a characterization of the set of strong equilibria for all α in the range (0, 2). For α > 2, Andelman et al. (2009) prove that a star graph in which every leaf buys one edge to the center node is a strong equilibrium, and conjecture that in fact any star is a strong equilibrium. We resolve this conjecture in the affirmative. Additionally, we show that when α is large enough (≥ 2n) there exist non-star trees that are strong equilibria. For the strong price of anarchy, we provide precise expressions when α is in the range (0, 2), and we prove a lower bound of 3/2 when α ≥ 2. Lastly, we aim to characterize under which conditions (coalitional) improvement dynamics may converge to a strong equilibrium. To this end, we study the (coalitional) finite improvement property and (coalitional) weak acyclicity property. We prove various conditions under which these properties do and do not hold. Some of these results also hold for the class of pure Nash equilibria.

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On the Price of Anarchy for High-Price Links

Àlvarez¹,

Messegué²

2019

Lecture Notes in Computer Science

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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.

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Tree Nash Equilibria in the Network Creation Game

Cited by 28 publications

References 9 publications

On Tree Equilibria in Max-Distance Network Creation Games

On Tree Equilibria in Max-Distance Network Creation Games

On Strong Equilibria and Improvement Dynamics in Network Creation Games

On the Price of Anarchy for High-Price Links

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