On Stability and Convergence of Multi-Commodity Networks and Services

The rise of distributed services and user-driven networking concepts in recent years poses the critical question of stability. Can a system operating under non-cooperation and self-interest converge to a stable state? and how fast? The answers to these questions readily lend themselves to game theory analysis, and to the study of congestion games in particular. In the past, much work have been done on establishing the existence of pure Nash equilibria in congestion games, and has shown that finding a pure Nash equilibrium is PLS-complete [1] and hence convergence to a pure Nash equilibrium is very difficult (exponential time in worst case). Furthermore, much of the convergence analysis have been carried out on simple single-commodity game models. In this paper, we attempt to construct a more realistic multi-commodity congestion game model suited for distributed service and user-driven networking scenarios. We introduce the desirability of equilibrium concept that is helpful in determining whether a system state meets the quality requirements of the users and services. Desirability is an alternative concept to price of anarchy. In fact we show the desirability ratio is a special case of price of anarchy. We then define the α-threshold congestion game whose minimum potential state corresponds to a desirable equilibrium (if the system permits one) and we bound its convergence to polynomial time through game transformation. Finally, we present a mechanism for partial simultaneous moves. To the best of our knowledge, there has been no prior establishment of the desirability concept and no bound given on the convergence of asymmetric multi-commodity congestion games with exponential cost function.