Fast Convergence of Natural Bargaining Dynamics in Exchange Networks

Bargaining networks model the behavior of a set of players who need to reach pairwise agreements for making profits. Nash bargaining solutions in this context correspond to solutions which are stable and balanced. Kleinberg and Tardos [19] proved that, if such solutions exist, then they can by calculated in polynomial time. This left open the question: Are there dynamics which can describe the bargaining process of real-world players, and which converge quickly to a Nash bargaining solution? This paper provides an affirmative answer to that question. The contribution of this paper is threefold: (1) We introduce a single-stage local dynamics which models the way in which actual players could bargain. We show that (approximate) fixed points of our dynamics are in one-to-one correspondence with (approximate) Nash bargaining solutions. (2) We prove that our dynamics converges to an ǫ-fixed point in O(1/ǫ2) iterations independent of the network size when the potential earnings (weights) are uniformly bounded. We use this to prove that an approximate Nash bargaining solution is reached in time polynomial in 1/ǫ, the network size and 1/g. Here g is the difference between the weights of the two corners of the matching polytope having largest weights, and controls the behavior of fast message passing algorithms for maximum weight matching (matching naturally arises as a subproblem of Nash bargaining). (3) Our proof introduces a new powerful technique from functional analysis to this set of problems. The technique allows us to extend our results in various directions. We believe the tools introduced here will be useful in many related problems. As a corollary, for bipartite graphs we prove polynomial time convergence to an approximate Nash bargaining solution, with probability close to one under small perturbations.