system
Given the following post, respond as if you're commenting on Reddit:
human
How much does it increase your odds of winning in Rock, Paper, Scissors to say “best two out of three” after you lose a game?

That game would have been one out of one. You have to win the next two games which is 25% odds from there but I’m thinking about the advantage you would gain from having the second chance to win. Is it just 50% instant win plus the 25% second chance giving you a 75% win rate using this trick? So your odds of winning go up 25%, right? I just thought of this ‘cheat’ and keep over thinking and second guessing myself. Thanks!
gpt
It depends how you formulate it. If only you call out "best two out of three" to initiate more rounds after you lose, then we can break the cases down as follows:

You win half the time, so you don't play any more games

You lose the first game the other half, and of those times you only win best two out of three a quarter of the time.

Adding the two you get a 1/2(1) + 1/2(1/4) = 5/8, which means your odds of winning do increase if you are a crybaby.

On the other hand, if your opponent is also going to call "best two out of three" to initiate more games if they lose, then the odds go back to being 50/50.