Network design with weighted players

We consider a model of game-theoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. [2] proved that pure-strategy Nash equilibria always exist and that the price of stability--the ratio in costs of a minimumcost Nash equilibrium and an optimal solution--is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight wi ≥ 1, and its cost share of an edge in its path equals wi times the edge cost, divided by the total weight of the players using the edge.This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria--outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log wmax)-approximate Nash equilibria exist in all weighted Shapley network design games, where wmax is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α = Ω(log wmax), the price of stability with respect to O(α)- approximate Nash equilibria is O((log W)/α), where W is the sum of the players' weights. In particular, there is always an O(logW)-approximate Nash equilibrium with cost within a constant factor of optimal.Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log wmax/ log log wmax)-approximate Nash equilibria, and show that for all α = Ω(logwmax/ log log wmax), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.