Bram Cohen (bramcohen) wrote,
Bram Cohen

Arimaa Rules

A rules change I proposed is being considered for Arimaa. Omar's comments are on page 2. Omar says that he likes draws (!) and would like for perfect play in Arimaa to result in a draw. This is puzzling on two accounts. First, the number of draws is small enough, and our understanding of how much advantage the first move gives the first player sketchy enough, that the chances of us being able to tweak the game to be a draw with optimal play are extremely small. Second, when two players reach a draw in Arimaa today it doesn't mean that they played a perfect game any more than when two chess players rated 1500 get a draw. It just means they drew, nothing more, nothing less.

In any case, Omar says that although there are officially draws in Arimaa, there's a tiebreak rule which is used to tell who really won the game in tournaments where games need winners. This is sort of like how some go tournaments officially use japanese scoring but revert to chinese scoring if the outcome is different between chinese and japanese scoring. It makes no sense to me, but it seems to keep the neurotypicals happy.

Omar says the tiebreak rule is that whoever captures the first rabbit wins. So the real question is, which is a better tiebreak rule, the player who captures the first rabbit, or the player to capture all of the opponent's rabbits first? The first rabbit rule sounds sketchy to me, because that first rabbit may have been lost as part of an intentional sacrifice. But even if we assume that there aren't any misleading tactical aspects to the first rabbit capture, it still makes sense to go with the all rabbits condition, because that one more reliably indicates which player is better. The winner of a game is the player who makes on average better moves over the course of the entire game, and the longer the game is, the greater the stastical significance of that result. Since the last rabbit is captured after the first rabbit, it contains strictly more information conveyed by the moves.

Omar claims that this rule would result in players trying to avoid draws in the first place, but that's incorrect. If the game starts to approach a potential draw situation, then with the current rule the players know which side effectively has been given draw odds, and hence one player has an incentive to play towards a draw, while with the last rabbit rule either player still has the potential to have lost if the game reaches a drawn position. Omar's claim that the first rabbit rule encourages players to avoid losing the first rabbit is a little strange, since at that point the very ending is so far off and draws are so rare that it isn't worth worrying about, at least not any more than the damage of losing a rabbit is to begin with.

Another thought I had about the rules of Arimaa: The rule about a rabbit reaching the back rank immediately winning actually favors computers a bit, because it's an extremely sudden position shift, and thus heavily favors computers in positions where there are rabbits near the end of the board. A rule which would favor humans a lot more would be that if a rabbit reaches the back rank it's removed from the board and the player who got it there gets to remove two opposing rabbits from the board, and the first player to lose all of their rabbits loses. This rule changes game play dramatically enough that I'm not proposing it as a serious rules change, but it is an interesting thought experiment.

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