  Bargaining networks model the behavior of a set of players that need to reach
pairwise agreements for making profits. Nash bargaining solutions are special
outcomes of such games that are both stable and balanced. Kleinberg and Tardos
proved a sharp algorithmic characterization of such outcomes, but left open the
problem of how the actual bargaining process converges to them. A partial
answer was provided by Azar et al. who proposed a distributed algorithm for
constructing Nash bargaining solutions, but without polynomial bounds on its
convergence rate. In this paper, we introduce a simple and natural model for
this process, and study its convergence rate to Nash bargaining solutions. At
each time step, each player proposes a deal to each of her neighbors. The
proposal consists of a share of the potential profit in case of agreement. The
share is chosen to be balanced in Nash's sense as far as this is feasible (with
respect to the current best alternatives for both players). We prove that,
whenever the Nash bargaining solution is unique (and satisfies a positive gap
condition) this dynamics converges to it in polynomial time. Our analysis is
based on an approximate decoupling phenomenon between the dynamics on different
substructures of the network. This approach may be of general interest for the
analysis of local algorithms on networks.
