# American Institute of Mathematical Sciences

2005, 2005(Special): 153-163. doi: 10.3934/proc.2005.2005.153

## Finding open-loop Nash equilibrium for variational games

 1 Mathematical Reviews, 416 Fourth St, P.O. Box 8604, Ann Arbor, MI 48107-2604, United States

Received  September 2004 Revised  March 2005 Published  September 2005

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.
Citation: Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
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