Asynchronous Algorithms in Non-cooperative Games*

An equilibrium solution of an N-person game is said to be globally stable (or, simply, stable) with respect to an a priori fixed ordering of computation of policies if, regardless of what initial policies the players start with, the generated algorithm converges to that equilibrium. At each step of such an iteration, each player determines his policy (or action, if the underlying game is static) by optimizing his individual cost function, using the most recently available policies of the other players as given. Clearly, any deviation from such an equilibrium followed by an optimal adjustment process that respects the given fixed ordering for computation of policies, would lead back to the same equilibrium, and furthermore a globally stable equilibrium is necessarily unique [Bagar and Olsder (1982), Bagar (1986)]. An equilibrium solution is said to be locally stable, if it is globally stable in a domain which is an open neighborhood of the equilibrium solution, and naturally a locally stable equilibrium need not be unique. If the order in which the players respond to a non-equilibrium point and the information available to them at the time of their computation are not predetermined, then the notion of a stable equilibrium becomes more restrictive. Existence of such equilibria can best be studied by treating the underlying algorithm as an asynchronous algorithm [Bertsekas (1983) Tsitsiklis et al. (1986), Li and Bagar (1987a)] and we will refer to such equilibria as asynchronously stable equilibria. Stable equilibria (be they synchronous or asynchronous) are appealing since they allow for real-time iterative computation of non-cooperative equilibria. The only information carried along the line is the most current policy information (or action information, in the case of static games), and each