The Price of Anarchy in Network Creation Games Is (Mostly) Constant

Mihaľák, Matúš; Schlegel, Jan Christoph

doi:10.1007/s00224-013-9459-y

Cited by 39 publications

(61 citation statements)

References 14 publications

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“…First, we prove that computing a best response for a player is NP-hard for any 0 < α = o(n), thus extending a similar result given in [13] for MaxNCG when α = 2/n. Then, we prove that MaxNCG(H) is not a potential game, by showing that an improving response dynamic does not guarantee to converge to an equilibrium, even if we assume a minimal liveness property that no player is prevented from moving for arbitrarily many steps.…”

Section: Introductionsupporting

confidence: 61%

“…Consequently, the Nash Equilibria 1 (NE) space of the game is a function of it. Actually, if we characterize such a space in terms of the Price of Anarchy (PoA), then this has been shown to be constant for all values of α except for n 1−ε ≤ α ≤ 65 n, for any ε ≥ 1/ log n (see [12,13]). …”

Section: Introductionmentioning

confidence: 99%

“…This variant, named MaxNCG, received further attention in [13], where the authors improved the PoA of the game on the whole range of values of α, obtaining in this case that the PoA is constant for all values of α except for 129 > α = ω(1/ √ n). Besides these two basic models, many variations on the theme have been defined.…”

Section: Introductionmentioning

confidence: 99%

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The max-distance network creation game on general host graphs

Bilò

Gualà

Leucci

et al. 2015

Theoretical Computer Science

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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 counter-intuitive 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 k-regular (for any constant k ≥ 3), or a 2-dimensional grid, the PoA is still Ω(1 + min{α, n α }), 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.

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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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The max-distance network creation game on general host graphs

Bilò

Gualà

Leucci

et al. 2015

Theoretical Computer Science

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“…Moreover, the authors show that the price of anarchy is constant for α ≤ √ n and for α ≥ 12n log n. In the latter case they prove that any pure equilibrium is a tree. This bound was improved in [30], where it was shown that for α ≥ 273n all pure equilibria are trees. Later on, in [27], this was further improved by showing that it even holds for α ≥ 65n.…”

Section: Related Literaturementioning

confidence: 99%

“…The paper [13] introduces a version of the game where the distance cost of a player is defined the maximum distance from i to any other player (instead of the sum of distances), and studies the price of anarchy for these games. Further results on those games can be found in [30]. Another natural variant of a cost sharing game is one where both endpoints of an edge can contribute to its creation, as proposed in [29], or must share its creation cost equally as proposed in [12] and further investigated in [13].…”

Section: Related Literaturementioning

confidence: 99%

On Strong Equilibria and Improvement Dynamics in Network Creation Games

Janus

Keijzer

2017

Web and Internet Economics

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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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The Price of Anarchy in Network Creation Games Is (Mostly) Constant

Abstract: Abstract. We study the price of anarchy and the structure of equilibria in network creation games.

Cited by 39 publications

References 14 publications

The max-distance network creation game on general host graphs

The max-distance network creation game on general host graphs

On Strong Equilibria and Improvement Dynamics in Network Creation Games

On the Price of Anarchy for High-Price Links

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