Following up on my other comment about the power of coordination to eliminate certain failure modes: If everyone guesses randomly they stand a (1/2)^100 chance

Following up on my other comment about the power of coordination to eliminate certain failure modes: If everyone guesses randomly they stand a (1/2)^100 chance. If they all guess the majority, with, say, blue as the tie-breaker just to have a deterministic algorithm, then they win twice as often because they capture the all-blues and all-reds cases, each with probability (1/2)^100. I don't know if that's optimal but I always find these "improve random outcomes even with really stupid coordination mechanisms" situations amusing. If there's something better to be done I'd be curious to know :) [Edit: raldi's clever answer that partitions by parity eliminates all but the we're-all-right and we're-all-wrong outcomes, and is clearly superior :)]