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 also 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 [KB+10], 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 bound on convergence time obtained for that dynamics was exponential in network size for the unequal division case. This bound is tight, in the sense that there exists instances on which the dynamics of [KB+10] converges only after exponential time. 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 \eps-UD solution in time poly(|V|,1/\eps). We also provide a fast local algorithm for finding \eps-UD solution, providing further justification that a market can find such a solution.Comment: 18 pages; Amin Saberi (Ed.): Internet and Network Economics - 6th International Workshop, WINE 2010, Stanford, CA, USA, December 13-17, 2010. Proceedings

A theory on power in networks

The eigenvector centrality equation $\lambda x = A \, x$ is a successful compromise between simplicity and expressivity. It claims that central actors are those connected with central others. For at least 70 years, this equation has been explored in disparate contexts, including econometrics, sociometry, bibliometrics, Web information retrieval, and network science. We propose an equally elegant counterpart: the power equation $x = A x^{\div}$, where $x^{\div}$ is the vector whose entries are the reciprocal of those of $x$. It asserts that power is in the hands of those connected with powerless others. It is meaningful, for instance, in bargaining situations, where it is advantageous to be connected to those who have few options. We tell the parallel, mostly unexplored story of this intriguing equation

Joint Head Selection and Airtime Allocation for Data Dissemination in Mobile Social Networks

Mobile social networks (MSNs) enable people with similar interests to interact without Internet access. By forming a temporary group, users can disseminate their data to other interested users in proximity with short-range communication technologies. However, due to user mobility, airtime available for users in the same group to disseminate data is limited. In addition, for practical consideration, a star network topology among users in the group is expected. For the former, unfair airtime allocation among the users will undermine their willingness to participate in MSNs. For the latter, a group head is required to connect other users. These two problems have to be properly addressed to enable real implementation and adoption of MSNs. To this aim, we propose a Nash bargaining-based joint head selection and airtime allocation scheme for data dissemination within the group. Specifically, the bargaining game of joint head selection and airtime allocation is first formulated. Then, Nash bargaining solution (NBS) based optimization problems are proposed for a homogeneous case and a more general heterogeneous case. For both cases, the existence of solution to the optimization problem is proved, which guarantees Pareto optimality and proportional fairness. Next, an algorithm, allowing distributed implementation, for join head selection and airtime allocation is introduced. Finally, numerical results are presented to evaluate the performance, validate intuitions and derive insights of the proposed scheme

The Theory of Relational Cohesion: Review of a Research Program

In this paper we analyze and review the theory of relational cohesion and attendant program of research. Since the early 1990s, the theory has evolved to answer a number of basic questions regarding cohesion and commitment in social exchange relations. Drawing from the sociology of emotion and modem theories of social identity, the theory asserts that joint activity in the form of frequent exchange unleashes positive emotions and perceptions of relational cohesion. In turn, relational cohesion is predicted to be the primary cause of commitment behavior in a range of situations. Here we outline the theory of relational cohesion, tracing its development through the present day, and summarize the corpus of empirical evidence for the theory’s claims. We conclude by looking ahead to future projects and discussing some of the more general issues informed by our work