Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

Abstract

We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply sampling techniques which allow us to achieve polynomial running time. The algorithm builds on the algorithmic ideas of [7], however, to the best of our knowledge, this is the first algorithmic result on computation of approximate equilibria using other than proportional shares as player costs in this setting. We present a novel relation that approximates the Shapley value of a player by her proportional share and vice versa. As side results, we upper bound the approximate price of anarchy of such games and significantly improve the best known factor for computing approximate pure Nash equilibria in weighted congestion games of [7].

Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

“…For any γ ∈ (0, 1] define the game Gγ , where we rescale the player weights and resource cost coefficients in G as wi = γw i , ãe = γ d+1−ke a e , ce (x) = ãe x ke . (13) This changes the quantities in (12) for Gγ as ãmin ≥ γ d a min , W = γW, cmax = γ d+1 c max .…”

Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

“…This enabled them to bound from above the approximation factor that the algorithmic framework of [7] yields when applied to unweighted games with general cost functions. Most recently, Feldotto et al [19] designed a algorithm similar to [8] which computes d d+o(d) -approximate equilibria in weighted congestion games. However, it is important to note that their algorithm is randomized and finds d d+o(d) -approximate equilibria only with high probability.…”

“…However, with respect to the analysis of the approximation guarantee of the algorithm (see the following Section 4.4), it will actually be treated as a parameter: the approximation factor will be given as a function of d. We want to emphasize here that all these assumptions are standard in the literature of algorithms for approximate equilibria in congestion games (see e.g. [7,8,20,19]). 11 One can think of this second type of moves as "preparatory" for the next phase.…”

“…which indeed holds, due to Lemma 14. We now move to proving (19), which can be equivalently rewritten as…”

“…As far as deterministic solutions are concerned. See below for a discussion of the recent randomized algorithm by Feldotto et al[19] 3. See Footnote 8 for a formal definition.…”