Bounded-Distance Network Creation Games

A network creation game simulates a decentralized and noncooperative construction of a communication network. Informally, there are n players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, thereby incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about her centrality in the network, we introduce a new setting in which a player who maintains her (maximum or average) distance to the other nodes within a given bound incurs a cost equal to the number of activated edges; otherwise her cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, which we call MaxBD and SumBD, respectively. For both games, we show that when distance bounds associated with players are nonuniform, then equilibria can be arbitrarily bad. On the other hand, for MaxBD, we show that when nodes have a uniform bound D ≥ 3 on the maximum distance, then the price of anarchy (PoA) is lower and upper bounded by 2 and O(n^{1/⌊log₃ D ⌋+1}), respectively (i.e., PoA is constant as soon as D is Ω(n^ϵ), for any ϵ > 0), while for the interesting case D=2, we are able to prove that the PoA is Ω(√n) and O(√n log n). For the uniform SumBD, we obtain similar (asymptotically) results and moreover show that PoA becomes constant as soon as the bound on the average distance is 2^{ω(√log n)}.