  We introduce two min-max problems: the first problem is to minimize the
supremum of finitely many rational functions over a compact basic
semi-algebraic set whereas the second problem is a 2-player zero-sum polynomial
game in randomized strategies and with compact basic semi-algebraic pure
strategy sets. It is proved that their optimal solution can be approximated by
solving a hierarchy of semidefinite relaxations, in the spirit of the moment
approach developed in Lasserre. This provides a unified approach and a class of
algorithms to approximate all Nash equilibria and min-max strategies of many
static and dynamic games. Each semidefinite relaxation can be solved in time
which is polynomial in its input size and practice from global optimization
suggests that very often few relaxations are needed for a good approximation
(and sometimes even finite convergence).
