On Dynamics in Basic Network Creation Games

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1, 2, . . . , n}, each identified with a vertex of a graph (network), where the strategy of player i, i = 1, . . . , n, is to build some edges adjacent to i. The cost of building an edge is α > 0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SUMGAME variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MAXGAME variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α > 273 · n all equilibria in SUMGAME are trees and thus the price of anarchy is constant, and that for α > 129 all equilibria in MAXGAME are trees and the price of anarchy is constant. For SUMGAME this answers (almost completely) one of the fundamental open problems in the field-is price of anarchy of the network creation game constant for all values of α?-in an affirmative way, up to a tiny range of α.

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“…behaviour of players, in which players alternately change their strategies to best/better responses [12,13].…”

Section: Tablementioning

confidence: 99%

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

Mihaľák

Schlegel

2010

Algorithmic Game Theory

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“…This implies that an improving response dynamic may not converge for the MaxNCG game as well (after relaxing such a liveness property). A deeper discussion on dynamics in NCGs can be found in [8,10].…”

Section: Introductionmentioning

confidence: 99%

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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“…Better response dynamics under this assumption have been studied in [25]. A modified version of this model is introduced in [29], where players can only swap their own edges.…”

Section: Related Literaturementioning

confidence: 99%

“…Indeed, this has been the subject of study in many research papers, e.g. [16,1,6,2,17,15,25], to mention only a few of them. Every star is a strong equilibrium (Theorem 2), existence of non-star strong equilibria (Theorem 3) Table 1: Overview of strong equilibria characterization results and structural results.…”

Section: Introductionmentioning

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 Dynamics in Basic Network Creation Games

Cited by 39 publications

References 10 publications

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

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

The max-distance network creation game on general host graphs

On Strong Equilibria and Improvement Dynamics in Network Creation Games

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