Great point - it's not quite as trivial as it appears. But I think you could just use the same approach: "Every turn, keep suggesting a strategy until you rece

Great point - it's not quite as trivial as it appears. But I think you could just use the same approach: "Every turn, keep suggesting a strategy until you receive the same strategy you suggest". Once that happens, then you both execute that strategy. Both players are trying to collaborate here, so they'll naturally subordinate themselves on the first opportunity to win the game, they'll implement randomness or backoff naturally.