Although I believe this: ----- Hmm, strategy seems clear: guess the majority color remaining, if any ----- , I'm a lot less sure of the rarish case when it's a tie: Is it wiser to, say, alternate your guesses when it's a tie (to pick up, you know, a second-order advantage, now that you have maxed out your first-order advantage), or is it entirely irrelevant (to your expected gain) what you pick at those times? Actually, now that I'm writing it, I kind of sort of *think* that it is completely obvious that it's entirely irrelevant. On the other hand, suppose that instead of only considering the *expected* value of this game we view it as a 2-player thang. You are playing against an opponent — each playing in isolation from the other — but with the winner being *not* necessarily the person who is ahead by the end. Instead, what if the winner is defined to be *the one who was *leading* more*. Where a scorekeeper has a running tally of each player's progress through the game. Each player gets one metapoint for each completed t'th stage during the ordinary game, 1 <= t <= 52, at which they are strictly ahead of the other. If at the end of the game *leading more* (i.e., having the most metapoints) were the only thing that mattered, would that affect strategy at ties? Now: Onward, to the third-order effects. —Dan