Convergence to and quality of equilibria in distributed systems

• Authors: Damon Mosk-Aoyama
• Venue: Convergence to and quality of equilibria in distributed systems
• Year: 2009

A natural property of a distributed system to study is its set of stable points, or equilibria. In a system comprising individual agents where the communication between the agents is local, determining whether the system converges over time to an equilibrium is important for understanding whether a global consensus is reached by the agents. A proof that such a system converges to a fixed point shows that the dynamic process executed by the agents can be considered a local computation that produces a global output. In a distributed system with selfish users, an equilibrium in which no user can benefit via a unilateral deviation is an expected outcome of the system. From the point of view of a system designer, a comparison between the aggregate welfare of all the users at an equilibrium and that at an optimal state provides a useful measure of system performance.In this work, we study the equilibria of several distributed systems. One setting that we consider is a network in which nodes obtain global information by repeatedly communicating with their neighbors. We study a gossip algorithm of Deb and Médard for information dissemination in the network. For a general network topology, we provide an upper bound on the time required for the network to converge to a state in which every node has every message to be disseminated. We also show that if each node begins with a positive number and executes a simple gossip algorithm, the system converges to a state in which every node has an accurate estimate of the sum of the numbers. Using this approximate summation algorithm as a subroutine, we develop a simple distributed algorithm for a class of convex optimization problems. In this algorithm, the nodes repeatedly execute a gradient ascent procedure that converges to an approximate solution to the convex optimization problem.Another setting that we consider involves the production of a good that is to be consumed by multiple users. Each user requests a quantity of the good. The system produces enough of the good to satisfy all the requests, and recovers the cost of producing the total amount requested from the users by assigning a cost share to each user. The utility of a user is a function of the user's requested quantity and assigned cost share. When users act to maximize their individual utilities, a strategic game is a natural model of the system. Under standard assumptions on the utility functions and a quadratic cost function, and two well-known cost sharing methods, average cost pricing and serial cost sharing, this game is guaranteed to have a Nash equilibrium in which no user can benefit by requesting a different quantity. We show how to determine the worst-case ratio between the aggregate welfare of the users under a Nash equilibrium and the optimal aggregate welfare for each cost sharing method in a class that interpolates between these two methods.