I wonder how OP calculated the Nash equilibrium for each of those game variants.

The only secret strategy I know only works against my good friends son who literally starts every new round of rock scissors paper with scissors. Play him 10 minutes later he starts with scissors.

We we noticed this so we began to say let's play a round and throw paper and lose. 5 minutes later repeat and lose again. THEN, once our drink was almost finished we would say come on one more game and if I win you have to go get me a can from the fridge. He would agree and we would this time throw rock beating him and winning. It works every single time. Lol