To add something, and to clarify (though I don't think you've made any mistake here, but many people misunderstand this subject): The 33%-each Nash Equilibrium

To add something, and to clarify (though I don't think you've made any mistake here, but many people misunderstand this subject): The 33%-each Nash Equilibrium is the mixed strategy Nash Equilibrium of the micro game (i.e. a single round of RPS, averaged over all possible randomizations). This is in no way the Nash strategy of the tournament game, which is "win the tournament given a pool of unknown participants and a set of rules for whom you face when". You have to add additional assumptions (e.g. that everyone else is going to play the uniform random strategy) in order for uniform random to be the Nash strategy for the tournament game. If the pool includes players who deviate from the single-round nash equilibrium strategy, there is opportunity to exploit them (and in doing so, open yourself to possible exploitation). This is why pure random play can often perform very poorly at the tournament game.