Rock beats paper 1% of the time?

The nash equilibrium can be written as:

Play rock with probability r.

Play scissors with probability s.

Play paper with probability 1-r-s.

Against a player with this strategy, it does not matter which strategy you choose. Therefore, playing 100% rock or 100% scissors or 100% paper is equivalent. This gives you 2 simple equations which you can use to solve for r and s.