Finding open-loop Nash equilibrium for variational games

It is well known that in static games the existence of Nash equilibria is often established through the use of a fixed point theorem applied to the ”best reply mapping”. As most fixed point theorems are non-constructive these important theorems provide almost no clues for determining the equilibria. In a dynamic setting the notion of Nash equilibria must also be further qualified as open-loop or closedloop. Here we restrict our attention to the open-loop concept. Recently, based on an optimization result from Leitmann [1], a joint paper by Leitmann and the author has explored a new method for determining open-loop equilibria for a large class of N-player differential games. This ”direct method” transforms the original game into an equivalent one which, hopefully, has a solution which is easier to identify. This method implicitly involves a fixed-point map. In this paper we explore these ideas providing a “constructive method” for finding a fixed point of the best reply map. An example will illustrate the utility of our approach.