On the Tree Conjecture for the Network Creation Game

Bilò, Davide; Lenzner, Pascal

doi:10.1007/s00224-019-09945-9

Cited by 22 publications

(35 citation statements)

References 42 publications

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“…For large values of α it has been shown constant PoA for the intervals α > n 3/2 [17], α > 12n log n [3], α > 273n [19], α > 65n [20], α > 17n [1] and α > 4n − 13 [7], by proving that every ne for each of these ranges is a tree, that is, proving that the tree conjecture holds for the corresponding range of α.…”

Section: Historical Overviewmentioning

confidence: 99%

“…In contrast, Bilò and Lenzner in [7] consider a different approach. Instead of using the technique of bounding the average degree, they introduce, for any non-trivial biconnected component H of a graph G, the concepts of critical pair, strong critical pair, and then, show that every minimal cycle for the corresponding range of α is directed.…”

Section: Historical Overviewmentioning

confidence: 99%

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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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Section: Historical Overviewmentioning

confidence: 99%

Section: Historical Overviewmentioning

confidence: 99%

On the Price of Anarchy for High-Price Links

Àlvarez¹,

Messegué²

2019

Lecture Notes in Computer Science

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“…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. Shortly after that, Bilò & Lenzner [12] proved that all Nash equilibria are trees for α > 4n − 13 and additionally provide an improved upper bound on the price of anarchy of tree equilibria. In a very recent preprint [4], the bound on α has been further improved, as the authors show that the price of anarchy is constant if α > n(1 + ), for all > 0.…”

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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“…It has been proved that if all equilibrium graphs are trees, the price of anarchy will be constant in either sum-distance setting or max-distance setting [1,3,9]. This result will help to justify the famous small-world phenomenon [10,11], which suggests selfishly built networks are indeed very efficient, at least not unacceptably bad.…”

Section: Introductionmentioning

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%

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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On the Tree Conjecture for the Network Creation Game

Cited by 22 publications

References 42 publications

On the Price of Anarchy for High-Price Links

On the Price of Anarchy for High-Price Links

On Strong Equilibria and Improvement Dynamics in Network Creation Games

On Tree Equilibria in Max-Distance Network Creation Games

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