My attempt at a solution: if you see two flags of the same color, guess the opposite color, otherwise don’t guess. This wins 75% of the time.
Lemma 1: it’s impossible that everyone chooses not to guess. Proof: some two people have the same color, because there are three people and only two colors.
Lemma 2: the chance of losing is 25%. Proof: by lemma 1, the team can only lose if someone guessed wrong, which implies all three colors are the same, which is 2 out of 8 possible assignments.
This leaves open the question of whether this strategy is optimal. I highly suspect it is, but don’t have a proof yet.
My attempt at a solution: if you see two flags of the same color, guess the opposite color, otherwise don’t guess. This wins 75% of the time.
Lemma 1: it’s impossible that everyone chooses not to guess. Proof: some two people have the same color, because there are three people and only two colors.
Lemma 2: the chance of losing is 25%. Proof: by lemma 1, the team can only lose if someone guessed wrong, which implies all three colors are the same, which is 2 out of 8 possible assignments.
This leaves open the question of whether this strategy is optimal. I highly suspect it is, but don’t have a proof yet.
UPDATE: here’s a proof I just found on the Internet, it’s elegant but not easy to come up with. I wonder if there’s a simpler one.