Nash equilibria: Gradient and decomposition algorithms

We show that static Nash equilibrium problems are amenable to several variational formulations. To each of these formulations corresponds a 'natural' class of iterative computational algorithms (including simple gradient-like algorithms). We discuss and compare the assumptions required to prove convergence of these algorithms. We also study how each class of algorithms lends itself to decomposition purposes, that is whether each iteration may be viewed as the task of solving a collection of independent minimization problems, one for each player, or even, when each player is actually made up of a 'team' of players, whether decomposition can match this finer structure. By the way, the question of convergence of iterative algorithms raises the more fundamental issue of 'stability' of Nash equilibria: the property implied by the definition itself (which may justify the term 'equilibrium') is indeed, from this point of view, very weak.