It is well known that the rock-paper-scissors game has no pure saddle point.
We show that this holds more generally: A symmetric two-player zero-sum game
has a pure saddle point if and only if it is not a generalized
rock-paper-scissors game. Moreover, we show that every finite symmetric
quasiconcave two-player zero-sum game has a pure saddle point. Further
sufficient conditions for existence are provided. We apply our theory to a rich
collection of examples by noting that the class of symmetric two-player
zero-sum games coincides with the class of relative payoff games associated
with symmetric two-player games. This allows us to derive results on the
existence of a finite population evolutionary stable strategies.