On the PoA Conjecture: Trees versus Biconnected Components

Abstract

In the classical model of network creation games introduced by Fabrikant et al. [On a network creation game, in Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing (PODC‘03), 2003, pp. 347–351], players correspond to the nodes of a network buying links of price and to the other players with the goal of being well-connected to the resulting network. Still as an open problem, the constant PoA conjecture states that the Price of Anarchy (PoA) is constant for any . When tackling this problem distinct behaviors must be taken into the account depending on whether has either large or low value. It is known that for every ne is a tree and for with the diameter of networks that are in equilibrium when restricting to deviations that consist only in buying links (buying equilibria) is at most a constant. These results imply that the PoA is constant for the disjoint union of the two ranges, and thus the constant PoA conjecture seems to be true for most of all the possible values . In this paper we study the PoA for the remaining range of and we show the following: (i) For the PoA is constant by proving that the size of any biconnected component of an equilibrium graph is constant. (ii) For we have that , where is the maximum diameter of an equilibrium graph for the same range of . Therefore if the constant PoA conjecture was false, it would suffice to construct equilibria of nonconstant diameter. Towards this direction we find nontrivial buying equilibria of nonconstant diameter when and , exploring new intimate relationships between distance-uniform graphs and buying equilibria.