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Topic: Draws? (Read 5896 times) |
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arimaaking
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To those here with a better grasp on game theory than myself: As far as I know, there is no way to draw in Arimaa. If that is true, how can the game not be broken, with a forced win for one player?
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Arimabuff
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Re: Draws?
« Reply #1 on: Mar 30th, 2009, 1:39pm » |
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Well I guess there is a remote possibility of a draw if a game lasts beyond its assigned limit. If it is still decided on scores and the scores happen to be equal for both colors then I guess it would be a draw.
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arimaaking
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Re: Draws?
« Reply #2 on: Mar 30th, 2009, 1:41pm » |
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So, then, is the game broken when played "casually", with no time control, or in games with no total time control, just per-move and reserve?
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Simon
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Re: Draws?
« Reply #3 on: Mar 30th, 2009, 2:11pm » |
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Yes, technically there must be a forced win for one player. However, in practice the game is not broken as long as the forced win is far beyond anyone's reach, as it seems to be.
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Fritzlein
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Re: Draws?
« Reply #4 on: Mar 30th, 2009, 2:19pm » |
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First, although Arimaa can't be drawn according to the rules, the three-fold repetition rule might not ensure that a game ends before the sun burns out. There are positions from which it is impossible for either player to make progress. For example, if you put all eight rabbits of each player on his fourth rank, with his other pieces behind, both players would have nothing to do except shuffle their own pieces on their home three ranks and try to avoid repetition. In a tournament such a game will end after the game's allotted clock time, and a winner will be declared based on who has more pieces, but that resolution hardly matters. If it happened in practice that games of Arimaa devolved into pointless piece-shuffling, Arimaa would be a broken game whether those positions were called draws or wins and losses. We are fortunate that no such position has arisen in practice in the 100,000 games of Arimaa we have played so far.
From a broader perspective, every finite two-player game of perfect information is "broken" because it is a forced draw, or a forced win for the first player, or a forced win for the second player. I think the implication of your question is that a game "should" be a draw with best play on both sides, but I don't see how that would save a solved game from being broken. If players get good enough so that every game is drawn, the game is just as broken as if the players get good enough for the same side to win every game. Tic-tac-toe is broken. Quarto, Abalone, and Teeko are all broken. What is the fun in playing a game that is drawn every time?
The relevant question for Arimaa (given the absence of draws) is whether the results are intolerably lopsided in favor of Gold or Silver. So far the difference in winning percentage between Gold and Silver is statistically insignificant. Thus we can say that at our current skill level Arimaa is not broken. Of course it is entirely possible that the more we improve in skill at Arimaa, the more our results will skew toward the perfect result. That's what happened to Pente. Once the players started getting really good at Pente, whoever moved first started to win most of the games. It's hard to play a Pente tournament, because each player must get a chance to go first to make it fair, but then what do you do when each player wins the game in which he moved first?
I happen to think it will be a long time before we get that good at Arimaa. Pente doesn't have a lot of features which allow a player to come back from a worse position; Arimaa does. Indeed, I predict it will be at least another five years before we can be confident we know whether Gold or Silver has the advantage, or whether well-played Arimaa has tendencies to stalemate. As long as we aren't even good enough to make a reliable guess as to the result of perfect play, we certainly don't have to worry that our results will be too close to the result of perfect play.
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Fritzlein
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Re: Draws?
« Reply #5 on: Mar 30th, 2009, 2:27pm » |
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on Mar 30th, 2009, 1:39pm, Arimabuff wrote:| If it is still decided on scores and the scores happen to be equal for both colors then I guess it would be a draw. |
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The score function at present is the number of pieces left. If both players have the same number of pieces left, the player who was reduced to that size army first is the loser. If no captures happened at all, Silver wins.
This is all irrelevant, because games are never decided on score. I really, really, really hope that the score stays irrelevant. If the score starts coming into play in actual games, then we will realize that the score can't save Arimaa from being broken.
[EDIT] I should say that score has been relevant in a few bot games. Those few cases, however, showed that the bots involved were stupid, not that there is a problem with Arimaa.
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| « Last Edit: Mar 30th, 2009, 2:34pm by Fritzlein » |
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The_Jeh
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Re: Draws?
« Reply #6 on: Mar 30th, 2009, 3:18pm » |
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Quote:| I think the implication of your question is that a game "should" be a draw with best play on both sides, but I don't see how that would save a solved game from being broken. |
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It may not solve the game from being broken, but you might say a game is only "fair" if theoretically neither player can force a win from the opening positon. What makes a "fair" game broken or not, provided its game tree is sufficiently large, seems to be how sensitive it is to the precision of play.
Let's say that in a certain game, player A plays an amount x better than player B. Then player A will win the game if and only if x>=y, where y is the game's sensitivity. Otherwise, a draw will occur. Now you might say that x can only be measured in terms of the game tree itself. I do not refer to the game tree in measuring x, but rather the intent imbedded in the game's concept to make some moves better than others.
For chess, you might say that y is a little larger than would be liked.
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| « Last Edit: Mar 30th, 2009, 3:20pm by The_Jeh » |
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Fritzlein
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Re: Draws?
« Reply #7 on: Mar 30th, 2009, 3:41pm » |
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on Mar 30th, 2009, 3:18pm, The_Jeh wrote:| For chess, you might say that y is a little larger than would be liked. |
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You are assuming that chess is a theoretical draw and not a first player win. You should add "z" to your equation to represent the first-player advantage. Then the first player wins if x+z > y. Probably z<y, but we don't know for sure. It's possible that the first player could win with x=0.
What we do know is that chess suffers from a first-player advantage large enough to be an annoyance at all levels of play, and that chess suffers from a proportion of draws that threaten to break the game at the highest level of play.
One fairly reliable indicator of a broken game is when experts start suggesting alternate rules. We have
Chess: Capablanca wants to add more pieces, Fischer wants to randomize the opening, and organizers try to ban agreed draws.
Abalaone: Top players agree to change the starting position to make pure defense infeasible.
Quarto: Top players suggest adding a winning condition (four-in-a-square as well as four-in-a-line) because otherwise every game would be drawn.
Teeko: The inventor changes the rules so that the players still move their own pieces, but set up the opponent's pieces. Alas, it proves to be a forced draw anyway.
English Checkers: Tournament organizers make the first three moves on behalf of the players to force them out of known-drawn lines.
All of these examples are evidence of the same thing: too many draws breaks a game. You don't need new rules unless something is badly wrong.
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| « Last Edit: Mar 30th, 2009, 3:44pm by Fritzlein » |
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arimaaking
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Re: Draws?
« Reply #8 on: Mar 30th, 2009, 3:43pm » |
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I suppose you're right. As long as perfect play is not practically achievable on a consistent basis, the question is irrelevant, and a drawish game really is not much more fun than a goldish or silverish game. However, if perfect play ever became achievable even a small percentage of the time, then the absence of draws would bother me.
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Fritzlein
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Re: Draws?
« Reply #9 on: Mar 30th, 2009, 3:53pm » |
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on Mar 30th, 2009, 3:43pm, arimaaking wrote:| However, if perfect play ever became achievable even a small percentage of the time, then the absence of draws would bother me. |
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If we get near perfect play, I will be bothered. I can't tell you in advance how I will be bothered, but the fun will definitely be ending. Arimaa will be played out. I think it is just a matter of personal preference whether one would rather a game flare up into a slugfest where one player has too much advantage, or cool down into a stalemate where neither player can force a win. It puts me in mind of the poem by Robert Frost:
Some say the world will end in fire;
Some say in ice.
From what I've tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice.
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| « Last Edit: Mar 30th, 2009, 3:53pm by Fritzlein » |
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arimaaking
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Re: Draws?
« Reply #10 on: Mar 31st, 2009, 11:46am » |
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If we had perfect play just a little bit of the time, I would prefer a theoretical draw. That way, games would only produce a winner when someone made a mistake, which would happen inevitably in matches or over a long period of time. If, however, we got perfect play a lot of the time, I think I wouldn't care anymore, regardless of the theoretical outcome, and would probably find another game, one that was not a "solved puzzle" yet.
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| « Last Edit: Mar 31st, 2009, 11:48am by arimaaking » |
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omar
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Re: Draws?
« Reply #11 on: Mar 31st, 2009, 11:48am » |
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on Mar 30th, 2009, 1:26pm, arimaaking wrote:| To those here with a better grasp on game theory than myself: As far as I know, there is no way to draw in Arimaa. If that is true, how can the game not be broken, with a forced win for one player? |
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This is exactly what I used to think. Ideally I wanted a game to allow for draws but with a very low margin. Arimaa fit this pretty well with the rule that if both sides lose all the rabbits the game is a draw. Draws were possible but pretty rare. However, in practice I was finding that for elimination type tournaments (like the finals of our world championship tournament) I did not want there to be any possibility of draws, because another game has to be played to break the draw and that makes the tournament schedule less predictable. So for tournaments we were already using the rabbit elimination rules to not allow draws.
I was pretty content with this situation because I knew that ideally Arimaa would be a draw, but for tournaments we could make it drawless. Then one day Karl suggested to me that even though Arimaa ideally allows for draws a perfectly played game of Arimaa might always be win for one player or the other. A solved game that fits this is Connect4. It allows for draws and well played games often result in draws, but a perfectly played game is always a win for the first player. So just because a game allows for draws doesn't mean that a perfectly played game will be a draw. I'd never be able to prove that Arimaa is not like Connect4 (without solving it  ), so why bother carrying two sets of rules when in practice you want the game to be drawless. That convinced me to make Arimaa a drawless game.
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| « Last Edit: Mar 31st, 2009, 11:49am by omar » |
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JoeHead
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Re: Draws?
« Reply #12 on: Feb 18th, 2010, 3:12am » |
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And what would happen if the game is decided by score and both players have the exactly same score?
The game would be draw? Why? Or cancel the results and count zero point for both players? If we want the game to be perfect just wipe draws out. I don´t want to imagine terrible logistic complications during live offline tournament.
Game of go got rid of draws by getting 0,5 point to the white. So the same we can apply for arimaa.
Gold score = A
Silver score = A + 0,5
thus silver wins, the same way as white wins in the game of go if both players have same number of points at the end.
The formula for arimaa score is exelent in my opinion. I was thinking about it before I knew it was already there. So my thoughts were: if the game is not decided in a reasonable time, so who has more rabbits?, more advanced rabbits?, more pieces? more valuable pieces? hmmm... but how to connect these values?
Then I read about scoring system... and you take the rank on which your rabbit is, then take cube power of it and sum it with another rabbit´s rank cubed, then pieces, nubmer of rabbits plus one etc. It was much greater then I thought, simple and valueing rabbits so much as it should be...
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Fritzlein
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Re: Draws?
« Reply #13 on: Feb 18th, 2010, 7:46am » |
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on Feb 18th, 2010, 3:12am, JoeHead wrote:| And what would happen if the game is decided by score and both players have the exactly same score? |
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Omar already anticipated this, and decided that Silver wins.
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mattj256
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Re: Draws?
« Reply #14 on: Apr 22nd, 2013, 12:08am » |
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on Mar 31st, 2009, 11:48am, omar wrote:| [F]or elimination type tournaments ... I did not want there to be any possibility of draws, because another game has to be played to break the draw and that makes the tournament schedule less predictable. |
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(I'm bumping this up because of the current suggestion to replace the scoring function.)
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