We present a new tool for the study of multiplayer stochastic games, namely
the modified game, which is a normal-form game that depends on the discount
factor, the initial state, and for every player a partition of the set of
states and a vector that assigns a real number to each element of the
partition. We study properties of the modified game, like its equilibria,
min-max value, and max-min value. We then show how this tool can be used to
prove the existence of a uniform equilibrium in a certain class of multiplayer
stochastic games