An FPTAS for Bargaining Networks with Unequal Bargaining Powers

Bargaining networks model social or economic situations in which agents seek to form the most lucrative partnership with another agent from among several alternatives. There has been a flurry of recent research studying Nash bargaining solutions (also called 'balanced outcomes') in bargaining networks, so that we now know when such solutions exist, and that they can be computed efficiently, even by market agents behaving in a natural manner. In this work we study a generalization of Nash bargaining, that models the possibility of unequal 'bargaining powers'. This generalization was introduced in [12], where it was shown that the corresponding 'unequal division' (UD) solutions exist if and only if Nash bargaining solutions exist, and also that a certain local dynamics converges to UD solutions when they exist. However, the convergence time for that dynamics was exponential in network size for the unequal division case. Other approaches, such as the one of Kleinberg and Tardos, do not generalize to the unsymmetrical case. Thus, the question of computational tractability of UD solutions has remained open. In this paper, we provide an FPTAS for the computation of UD solutions, when such solutions exist. On a graph G = (V,E) with weights (i.e. pairwise profit opportunities) uniformly bounded above by 1, our FPTAS finds an e-UD solution in time polynomial in the input and 1/e. We also provide a fast local algorithm for finding e-UD solution.