Ok, Nash equilibrium, finally the correct approach. But here's an interesting variation, seemingly very similar, but a bit harder: same probabilities, but introduce a third side with a predefined algorithm and infinite budget. The game still looks the same to you as the player: 50% rock, 50% human RPS player, but in fact suddenly the game depends on so many details...
The setting: you play RPS against a human opponent, but do not communicate with them directly - the interface is operated by the third side and only shows the choices (no chatting, etc). Your opponent is free to choose as he wishes. The operator intervenes with 50% chance (fair coin toss) by replacing your opponent's choice with rock (replacing rock with rock still counts as intervention). Of course the fact of intervention is not communicated to players.
Scoring rules: On "normal" rounds you and your opponent score against each other as in normal RPS. On "intervention" rounds you both score against the operator's budget. Your score depends on your choice vs. operator's rock. Your opponent's score depends on his original choice vs. your choice (so, yes, in this game you can both win/lose simultanously).
1. What is your best strategy in this game?
2. More interestingly: is the best strategy different if your opponent does not know about the existence of interventions (they change nothing visible on his side after all)? If you could tell him about it before the start of the game, would you?
That's my main question, but for the really bored, some more options. Consider a symmetric variant, where your choices as seen by your opponent also get replaced with rock randomly. Does the strategy depend on the operator's algorithm: option 1 - single throw decides no replacement vs. replacement in both directions, option 2 - separate throws for each direction, option 3 - always replace, single throw determines direction?
And what if your choices get replaced with something different than your opponents'? He 50% rock vs. you 50% paper, or he 50% rock vs. you 50% scissors?
And the most difficult one: review all these variants under "no ties" rule - in case of tie you replay the round, but the coin is not thrown again - if there was an intervention it will happen again, if there wasn't, it won't. This is a difficult variant - the same game might be a tie-resolving round for one player and a start of a normal round for the other, so you're never really sure whether your next round will get a throw. You will no longer see 50% rocks, the actual proportion depends on strategies of both players.
Man, coming up with such variations is fun. And details sometimes matter. Adding a side channel for communication between players also might change everything, as you can then cooperate against the operator... And from the player's point of view all these variants technically still fit within the bounds of the same simplified description: random choice of 50% rock, 50% human RPS player...