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.