Anarchy Is Free in Network Creation

Abstract

The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by Fabrikant et al. In it, each of n agents corresponds to a vertex, which can create edges to other vertices at a cost of α each, for some parameter α. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent's cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices.Previous research proved that for many regimes of the (α, n) parameter space, the total social cost (sum of all agents' costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game theoretic nomenclature, this approximation ratio is called the price of anarchy. In our paper, we significantly sharpen some of those results, proving that for all constant non-integral α > 2, the price of anarchy is in fact 1 + o(1), i.e., not only is it bounded by a constant, but it tends to 1 as n → ∞. For constant integral α ≥ 2, we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.

Locality-based network creation games

“…If we characterize the space of NE in terms of the Price of Anarchy (PoA), i.e., the ratio between the social cost of the costlier NE to the optimal (centralized) social cost, then it has been shown this is constant for all values of α except for n 1−ε ≤ α < 65 n, for any ε ≥ 1/ log n (see [Mamageishvili et al 2013;Mihal ák and Schlegel 2013]), while an upper bound of 2 O( √ log n) is known for the remaining values of α [Demaine et al 2012]. Moreover, very recently, in [Graham et al 2013] it was proven that for all constant non-integral α ≥ 2, the PoA is bounded by 1 + o(1).…”

Selfish Network Creation with Non-uniform Edge Cost

“…the structure of the equilibrium networks changes from a clique for very low α to trees for high α. For different regimes of α different proof techniques yield a constant PoA [15,1,13,24,22,16] but it is still open whether the PoA is constant for all α. In particular, constant upper bounds on the PoA are known for α < n 1−ε , for any fixed ε > 1 log n [13], and if α > 65n [22].…”

Quality of Service in Network Creation Games

“…Recently, by Mihalák and Schlegel [15] and improved by [14], it was shown that for α ≥ 65n all equilibria are trees (and thus the PoA is constant). For non-integral constant values of α > 2, Graham et al [10] showed that the PoA tends to 1 as n → ∞.…”