  We consider Gillette's two-person zero-sum stochastic games with perfect
information. For each $k \in \ZZ_+$ we introduce an effective reward function,
called $k$-total. For $k = 0$ and $1$ this function is known as {\it mean
payoff} and {\it total reward}, respectively. We restrict our attention to the
deterministic case. For all $k$, we prove the existence of a saddle point which
can be realized by uniformly optimal pure stationary strategies. We also
demonstrate that $k$-total reward games can be embedded into $(k+1)$-total
reward games.
