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

Abstract

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.

On Tree Equilibria in Max-Distance Network Creation Games

“…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.…”

“…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.…”

On Strong Equilibria and Improvement Dynamics in Network Creation Games

“…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 [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.…”

“…The paper [16] 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 [35]. Another natural variant of a cost sharing game is one where both endpoints of an edge can contribute to its creation, as proposed in [34], or must share its creation cost equally as proposed in [15] and further investigated in [16].…”

The Price of Anarchy for Network Formation in an Adversary Model

“…Upper bounds were shown for ULF and tight bounds for BLF. For ULF, improved bounds were recently shown in [MS10].…”