ON THE EXISTENCE OF EQUILIBRIA IN NONCOOPERATIVE OPTIMAL FLOW-CONTROL

The existence of Nash equilibria in noncooperative flow control in a g eneral product-form network shared by K users is investigated. The per formance objective of each user is to maximize its average throughput subject to an upper bound on its average time-delay. Previous attempts to study existence of equilibria for this flow control model were not successful, partly because the time-delay constraints couple the stra tegy spaces of the individual users in a way that does not allow the a pplication of standard equilibrium existence theorems from the game th eory literature. To overcome this difficulty, a more general approach to study the existence of Nash equilibria for decentralized control sc hemes is introduced. This approach is based on directly proving the ex istence of a fixed point of the best reply correspondence of the under lying game. For the investigated flow control model, the best reply co rrespondence is shown to be a function, implicitly defined by means of K interdependent linear programs. Employing an appropriate definition for continuity of the set of optimal solutions of parametrized linear programs, it is shown that, under appropriate conditions, the best re ply function is continuous. Brouwer's theorem implies, then, that the best reply function has a fixed point.