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Title: Fixing the rules / eliminating draws Post by Fritzlein on Apr 3rd, 2005, 11:27am If I remember correctly, we had basically reached a consensus that the repetition rule should be changed to include the player to move, i.e. "a third occurance of the same position" should be changed to "a third occurance of the same position with the same side to move". At the same time we make that change, however, I suggest we make an additional rule change, one proposed by Bram Cohen on his blog. http://www.livejournal.com/users/bramcohen/15337.html Bram proposes that a player who loses all eight rabbits should lose the game immediately. That seems so intuitive to me now, I wonder why it didn't occur to me earlier. Wasn't Arimaa intentionally designed not to have any draws? There are very few draws as is, but why not iron out this last little glitch so that there are NO draws? It seems like good PR to be able to say, not that Arimaa rarely has draws, but that it NEVER has draws. I'm trying to think of a disadvantage of this rule change, and not coming up with much. It doesn't seem that someone who could have won under the current rules is suddenly going to lose when draws are eliminated. If I can't keep my last rabit from being captured, it is improbable that I could later immobilize my opponent. So the rule only means that someone may lose who would have drawn the game under the present rules. Why give them a drawing chance when they have no winning chance? There might be situations where a player gives up on trying to score a goal, and instead tries to hunt down his opponent's last rabbit or two. But in what type of situation would someone have an army powerful enough to capture all opposing rabbits, but not powerful enough to force a goal? If the rabbit-hunting player had the stronger army, then s/he would probably have won eventually anyway. If the rabbit-hunting player had the weaker army, then how would she/he force the capture of pieces the opponent would be trying desperately to protect? Besides which, capturing the opponent's last rabbit is already highly desirable, so it isn't clear that gameplay would change much if we made it more desirable. If this rule change alters the feeling of the game at all (and I doubt it will alter it much) it would be to place slightly more emphasis on the rabbits, and maybe create some midgame situations where you wouldn't want to trade a rabbit for a cat, for example. Mostly, though, it would only change things in those rare endgames (see Adanac vs Belbo postal) where both sides have very few rabbits. For these esoteric positions, I think the proposed rule change would improve game dynamics by eliminating possible protracted maneuvering when one side gives up on winning and only plays for a draw. Besides, I think it in general creates more interesting strategic dilemmas to have the weakest piece be the most important piece. When would it be worth giving up a dog for two rabbits? Only in the endgame, or also in the opening? If getting two rabbits for a dog is often a good trade, what about two rabbits for a horse? Slightly increasing the importance of rabbits would natually create more dilemmas in which human judgement outweighs computer calculation in the long run. Because computers rely heavily on material consideration, to muddy the waters of who is materially ahead would seem to favor humans. What do folks think of this proposal? |
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Title: Re: Fixing the rules / eliminating draws Post by jdb on Apr 4th, 2005, 9:15am Quote:
Here is my completely unrealistic contrived example. Gold Pieces: Rc3,Ec4 Silver Pieces: ra2,rb2,rc2,hd2,re2,rf2,rg2,rh2 ea1, mb1, hc1, ,de1,df1,cg1,ch1 Gold to Move However, I agree that it would be a reasonable rule, if a player lost all their rabbits, they lose the game. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 4th, 2005, 1:45pm Cute example, JDB. It never occurred to me that the size of the larger army could work against it. It's true that this example is not realistic, but it reminds me that I didn't report Bram Cohen's suggestion faithfully. He specifically suggested that if a player loses and wins on the same turn, by abandoning the last friendly rabbit but also capturing the last opposing rabbit, then the moving player wins. That could be extended to immobilization just for the sake of completeness. I challenge you to come up with an example where the weaker side can't save its last rabbit, but could win by immobilization not on the same move. Is there even a contrived example that achieves this? Arimanator, it doesn't secure a draw to throw away the last of your own rabbits unless you capture the last of the opposing rabbits too. Suicide is no goal defense! |
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Title: Re: Fixing the rules / eliminating draws Post by jdb on Apr 4th, 2005, 7:40pm Here are a couple of (barely) realistic cases to consider. a) Gold: Ed4,Re4 Silver: ed5,re5 b) Gold: Ed4,Re4 Silver: rd5,ee5 |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 5th, 2005, 8:45am on 04/04/05 at 19:40:51, jdb wrote:
Those are much more realistic than your first case! But are these examples of how the proposed rule change will convert a winner into a loser? It seems more likely that they will convert what would have been a draw into a win for the first player to move. If that's so, I take these positions as evidence that the proposed rule would be a good thing! |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 5th, 2005, 8:49am on 04/05/05 at 01:07:43, Arimanator wrote:
This is indeed a somewhat plausible scenario. As I mention above, Bram Cohen anticipated it and suggested that it be a win for the moving player. |
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Title: Re: Fixing the rules / eliminating draws Post by mouse on Apr 5th, 2005, 8:53am Now whatabout this position? 1w Ee2 Rh3 Cf2 1b ra8 eh2 Gold to move. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 5th, 2005, 8:55am on 04/05/05 at 03:57:47, Arimanator wrote:
Yet for some reason computers still can't play the chess endgame (until it gets under seven total pieces) as well as humans. Why are they worse at the endgame, with its lower branching factor, than they are at the middlegame with its higher branching factor? I contend it is because captures are more rare in the endgame, and because the endgame is all about queening a pawn, which means position is more important than material. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 5th, 2005, 9:07am on 04/05/05 at 08:53:22, mouse wrote:
What about this position? Are you saying that Gold wins under one set of rules and Silver wins under the other? I haven't analyzed the position carefully, but my intuition is that it is draw under the curent rules, whereas Gold can win if capturing the last opposing rabbit is a win, by using his elephant to protect his own last rabbit and hunting down Silver's last rabbit with the cat. If my superficial analysis is corect, then it seems like yet one more argument for changing the rules to avoid draws, wouldn't you say? |
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Title: Re: Fixing the rules / eliminating draws Post by mouse on Apr 5th, 2005, 9:31am Sorry the position would win for gold if gold plays the best first move. instead: 1w Ee2 Rh4 Cf2 1b ra7 eh3 As far as I can see. Under the current rules this will probably be a draw. But if we introduce first to lose a rabbit silver would win eventhough gold got more material. |
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Title: Re: Fixing the rules / eliminating draws Post by mouse on Apr 5th, 2005, 11:08am on 04/05/05 at 11:05:40, Arimanator wrote:
You can with this link: http://arimaa.com/arimaa/games/planGame.cgi I don't think Omar has put it up in the gameroom yet. |
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Title: Re: Fixing the rules / eliminating draws Post by mouse on Apr 6th, 2005, 10:25am on 04/06/05 at 00:41:22, Arimanator wrote:
I'm not sure that would be really interesting because a lot of the bots too weak in the end game to play well enough to say anything conclusive. Arimaazilla is deffenitly. Maybe Bomb or Clueless knows enough about the endgame to be usefull. But in many endgames you will have to search atleast 3-4 full moves ahead. Which would be something like 25+ steps. Jdb and/or Footland have tried there bots on that search dept before but I dont think they were convinced about the results. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 6th, 2005, 6:04pm Until the ability to play bots form an arbitrary position comes on-line, you can buy the capability (i.e. you can buy the latest version of Bomb) from Smart Games for $20. http://www.smart-games.com/arimaa.html |
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Title: Re: Fixing the rules / eliminating draws Post by mouse on Apr 7th, 2005, 4:21am on 04/06/05 at 13:22:45, Arimanator wrote:
Actually I think the idea of a sparring partner is pretty good. If you could load a position with the tool Omar has produced in such a way that two or more person could see it simultaniusly it would be a great advantages. In that way you could analyse a position in cooperation with somebody else, which could greatly improve the understanding of specific situations. I just think the sparring partner will have to be a human. Because the bots are just not good enough for that. Given your games against Clueless fast I think if you analyse a position for 5-10 minutes you will most likely come up with a better move than any of the bots. Thats the great thing about Arimaa compared to chess. If you play chess you can check your position against a computer (or grand master) and it will give you a answer that will be better than any move you can come up with. In Arimaa the bots are not able to give you the best answer. The best players like Fritzlein and 99of9 are more likely to give you a better answer but they dont have the grand master level like the top chess players so you will have to think for yourself and you may sometimes come up with a better move than them. ;D |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 8th, 2005, 12:00pm on 04/07/05 at 04:21:43, mouse wrote:
Very true. One of the great things about Arimaa is that there is no authority. None of us understand the game to much depth. For example, Omar is in the process of proving both myself and 99of9 wrong about the high value of the camel in the opening. I was convinced in my postal game against Omar that I had a material advantage, but I think now I am losing, so either I was wrong about what a camel is worth, or I played badly after trading because I didn't understand the position. Either way, I was not just overlooking a tactical idea, which I do all the time, but deeply misunderstanding something. Thanks for complimenting my Wikipedia articles, Arimanator, but it is far too early for anyone to "write the book" on Arimaa. As our understanding of the game evolves, what I have written will outdate very quickly. The game is still very open for new ideas, and new understanding. You could be the next one to see something all the rest of us have failed to see. If I tell you "don't put the camel in the back row", you shouldn't pay me any mind, because next year all of us may be doing it. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 8th, 2005, 12:05pm on 04/05/05 at 13:30:33, Arimanator wrote:
That's a good point. We don't want to eliminate draws in a way that goes against the spirit of the game. Mouse's examples don't really persuade me, becuase it is already the case that someone with more material could lose due to an opposing goal, so it doesn't bother me that someone with more material could lose due to losing their last rabbit. It's in keeping with the spirit of the game. In fact it extends the "no draws" part of the spirit of the game without costing anything elsewhere. But I see your point about giving the win to one player or the other if both lose their last rabbit on the same move. If two winning conditions are met simultaneously, intuitively it feels like a draw. So I guess the "last rabbit rule" wouldn't necessarily eliminate all draws if you give it in the most intuitive form. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 10th, 2005, 9:25am Very interesting discussion; this and also the one on Bram's blog. You might be suprised to learn what my views are on the topic of draws. First of all, having Arimaa be a game with no draws at all was not one of my goals. My primary goals for Arimaa were: 1 It should be playable with a standard chess set 2 It should be difficult for computers 3 The rules should be simple and intuitive to understand 4 The game should be interesting to play Goal number 4 is very subjective and that's where I spent a lot of time trying out different sets of rules. But eliminating all possibility of draws was not one of my primary goals. If a set of rules resulted in a game that had a high draw percentage, I would have rejected it based on goal 4. In my opinion a game that has a very high frequency of draws is not very interesting. But if a set of rules also had no chance of a draw I would also reject it due to goal 4. In my opinion a game that has no possibility of draws could eventually become non-interesting even though the game itself is interesting in every other way. It would eventually be known that one side or the other will win; in all such games it's always been white or first to play that wins. So as the quality of the game play improved you would begin to notice an increase in the number of games won by white. To make such games more balanced the second player usually has the option to switch sides after the first players move. But with perfect play it doesn't really change anything; the second player should always be able to win. In such games the black player can always argue that he lost not because of a flaw in his play, but rather because of a flaw in the game that black is destened to lose. Anyways my view of draws is that a game should allow some draw positions that can be reached with perfect play by both sides. But the frequency of reaching such positions with non-perfect play should be very low. In such a game, a draw becomes a gem and witnessing one is quite a spectacular thing. For the spectators the possibility of a draw actually adds to the excitement of an end game. I feel quite lucky that Arimaa does have the potential for a draw, but has a very low draw percentage even with very good quality of play. So I guess by now, you can tell that I would not want to eliminate draws from Arimaa; in my opinion they're gems. However, there are many practical situations when we don't want a game to end in a draw and must some how resolve a draw (such as elimination tournaments). You may be suprised to know that Arimaa already does have such a rule to eliminate draws if needed. But it is part of the match rules and not a core game rule. If you look near the bottom of the match rules page there is a rule for breaking draws. It says that if both sides lose all the rabbits the first player who lost the first rabbit loses the game. I can't recall now all the reasons why I chose that, but I do remember also considering a few other options; one being the first player who had lost all the rabbits loses the game and another being the last player to lose all the rabbits wins the game. I think my main reason for chosing this one was because this rules causes the players to avoid a draw type situation to begin with. Initially both players would not want to lose even a sigle rabbit. But once one of the players has lost a rabbit that player would want to protect the remaining rabbits very strongly. So when playing with this rule the players would avoid losing rabbits from the begining and thus there is less chance of having a situation where both players have lost all the rabbits. Also all three of these rules can put one of the players in a situation where they do not want to trap the opponents last rabbit. With the rule I chose which player will be in that situation is known as soon as the first rabbit is trapped; so there is plenty of time to avoid that situation from occuring. Finally I don't think we will see this rule getting invoked so much that it really matters. So my view on draws is that I would not want to eliminate them from Arimaa. However, I can always be persuaded to change my view if someone can give a good argument that a game with any chance of draws is theoretically flawed. In my view a game that has a high frequency of draws with good quality, non-perfect play is flawed and also a game with no chance of draws is theoretically flawed due to the black player arguement I mentioned above; unless somehow both players have an equal chance of winning which I don't think is possible in a deterministic game. A game with any chance of a draw always results in a draw with perfect play, but I think I perfer that over the white player always winning with perfect play. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 10th, 2005, 11:47am Very interesting, Omar! I didn't realize that you would prefer the game to be drawn with perfect play. Now that I think about it, I probably agree with you, as long as draws are quite rare. Ideally both sides would have an equal chance of winning, but no one could lose unless they made a mistake. If I had to lay a bet right now, I would bet that Arimaa is NOT drawn with perfect play on both sides, but in fact Gold has enough of an initiative to win. Setting up the pieces second doesn't seem to quite compensate for the first move. I believe the near 50-50 win percentage for the two colors in the past was partly due bots playing more games as Gold than as Silver (because humans were seeking a greater challenge). One notices that when the bots were broken and only playing Silver, the win percentage for Gold shot way up. But this poses a new question: If it is a design flaw for the first player to have a forced win with perfect play on both sides, and if I'm right about my hunch, how could the game be rebalanced? One idea would be to allow Silver to take one step after setting up the pieces, or even two steps with two different pieces. This would go some distance to disrupting any Gold initiative, if only because it would allow Silver to block an initial "elephant forward 4" opening. Perhaps one could allow Silver an unrestricted two steps, but then I suspect Silver would just play "elephant forward two" and claim a miniscule advantage. Anyway, this is all hypothetical discussion with minimal practical impact, because the game seems very well balanced as is, with even marginally better play likely to tip the balance towards a win from an even position. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 13th, 2005, 6:37pm It is possible for a game to be drawn with perfect play and still show a significant advantage for the white (first to move) player with non-perfect play. The advantage is greatest at random play and diminishes slowly as the quality of play improves and more and more games become drawn. It eventually disappears at perfect play where all games are drawn. I noticed that when I was experimenting with tic-tac-toe. See the discussion about rating of a perfect chess player for some numbers. Im not sure what happens in games where perfect play is a win for white. It may well be that Arimaa is a win for white with perfect play even though the potential for draws is there with non-perfect play. Connect4 turned out to be this way. It is possible for a game to be drawn in Connect4, but with perfect play it is always a win for the first player. Experiments with connect4 might show what happens to the white advantage and frequency of draws for games that are win for white with perfect play. Does the white player advantage continue to increase as the quality of play increases or does it first decrease from the random play level before increasing again? The idea proposed by Karl of giving the first player only 2 steps on the first turn is an interesting one. It was also proposed by R. Wayne Schmittberger (editor of GAMES magazine). I am open to modifying the core rules of Arimaa to add this if we find that there is a significant advantage of about 4 or 5 percent for the first player in high quality games. |
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Title: Re: Fixing the rules / eliminating draws Post by 99of9 on Apr 14th, 2005, 8:52am on 04/13/05 at 18:37:57, omar wrote:
Maybe I'm just an over-cautious scientist, but I think the first sentence in the quote above is unjustified. It may well have happened this way for tic-tac-toe, but I don't think it's necessarily generally true. For example in arimaa, if high quality human games turn out to have a bigger advantage than the 0.3% that random steppers show, that doesn't necessarily mean that arimaa is a theoretical win for gold. It may still be a theoretical draw - we just aren't playing it well enough yet. The reason the split opens up may just be some peculiarity about who gets to choose the path of the game. About changing the rules. I think the 2 step idea is a good one, and if you were inventing arimaa now, I'd say you should go with it. But I agree with Arimanator that it's very desirable to have a continuity of rules, and not to change them on a suspicion of an advantage! Even 20-30% would be acceptable as long as tournaments mix up the gold and silver or use double rounds (eg chess). Which reminds me of one of the gripes I had this year in the WC. I really don't think it is good to make the top rated players play all silver. If nothing else it should be randomized. (Not that I'm blaming my poor performance on that of course - Naveed played a spectacular game to knock me out in round 1.) I do have another gripe, but I'll save it for another day and another posting topic. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Apr 14th, 2005, 9:11am I, too, am suspicious of the claim that the advantage of going first is greatest with random play and diminishes as the quality of play improves. I think I recall reading that in chess the 55%-45% advantage for white holds more or less steady throughout the spectrum, even as the percentage of draws increases. Of course, if chess is theoretically drawn then this advantage for white will go away with perfect play, and diminish with near-perfect play, but on the scale at which humans play, I believe that 2500-rated players and 1000-rated players show about the same advantage for white in games between equally rated players. The data to verify this should be out there somewhere. In fact, Omar, didn't you acquire a huge database of chess results for a different purpose? I'm curious to see if my recollection is correct. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 14th, 2005, 12:53pm on 04/14/05 at 01:00:17, Arimanator wrote:
No I would not do it without a lot of play testing and only if it made a significant improvement. Our current estimate of the first player advantage is less than 1% so I don't think it would make much of a difference. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 14th, 2005, 1:05pm What if it turns out that for every gold setup there is a counter silver setup that can defeat it. Since gold has to go first and silver can see golds setup, it may even be possible that Arimaa is a win for silver with perfect play. But I hope that is not the case and that Arimaa is a draw with perfect play. So it is necassary to not eliminate draws if we hope to have a chance that the game is a draw with perfect play. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 14th, 2005, 1:27pm on 04/14/05 at 09:11:32, Fritzlein wrote:
I posted a reply to this, but in the 'How great an advantage is the first move?' thread. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Apr 14th, 2005, 1:38pm on 04/14/05 at 09:11:32, Fritzlein wrote:
Yes, it was for the 'Rating of a perfect chess player' thread. See the notes in this file for where to get the data. http://arimaa.com/arimaa/rating/humanChessDrawPerc.xls |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Jun 7th, 2005, 9:40pm Bram's comments regarding the tie break rule have caused me to re-evaluate the situation. The way I was thinking about it before is that only after a game has ended in a draw (i.e. both players have lost all the rabbits) the tie break rules kicks in. However, I now think that in games where draws are not acceptable, it might be better to just use the rule Bram suggested and declare the first player who captures all of the opponents rabbits as the winner right away and the game does not progress any further. Though this introduces an additional way to win the game, I don't think it will impact the main objective of the game. I've modified the 'Match Rules' page to reflect this. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Jun 8th, 2005, 12:24am I also was persuaded by Bram's arguments. This is quite a separate issue from whether draws are a bad thing that should be totally eliminated from Arimaa. I personally would favor wiping out the possibility of any draws ever, but I respect your position, Omar, that draws need not be eliminated and can be an interesting side show if they ever happen. But because there are cases (e.g. a single-elimination tournament) where we all agree that draws are not acceptable, it makes it an important question how to eliminate draws in those cases. The "first to capture a rabbit" rule seems less desirable than the "first to capture all opposing rabbits" rule for reasons Bram outlined and other reasons as well. Since losing all rabbits happens very deep into the game, it is almost surely indicative of inferior skill, whereas losing one rabbit early could happen for all kinds of reasons, for example sacrificing a rabbit to gain a camel hostage. Also Bram's rule gives less advantage to Gold. Also Bram's rule won't kick in except in situations where both players know very well what is on the line. And finally, if rulebook draws occurred with regularity, and if they were decided by the first rabbit capture, then opening play would be weirdly influenced by a reluctance to lose the first rabbit, even for material gain, i.e. winning a dog for a rabbit in the opening. Bram's rule, in contrast, won't affect opening play at all, which makes it a much better rule for that reason alone. So, in brief, I heartily applaud the change you just made, Omar. :-) |
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Title: Re: Fixing the rules / eliminating draws Post by jdb on Jun 23rd, 2005, 3:45pm I just played a game against shallowblue, where I lost all my rabbits. The game was not over. Has the server been updated to reflect this new rule? If so, I have mistunderstood the rules. Is capturing all your opponents rabbits an automatic win, or does it only win under certain circumstances? (I certainly hope the rule applies at all times!) Thanks |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Jun 23rd, 2005, 9:29pm on 06/23/05 at 15:45:32, jdb wrote:
If I understand Omar's intent, it is that the rule only applies in situations (e.g. single-elimination tournaments) where draws are unacceptable. Otherwise, you can still play on hoping for a draw after you lose all your rabbits. |
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Title: Re: Fixing the rules / eliminating draws Post by omar on Jun 26th, 2005, 9:36am Capturing all of the opponents rabbits does not equal a win. By default the game continues. However in tournaments where draws are not acceptable then this rule can be used to prevent a draw situation from occuring. |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Jul 9th, 2005, 8:31am I haven't been following the games closely for a few weeks, but I'm guessing that if there had been a draw, someone would have mentioned it. Thus we have again gone several thousand games with no draws. When was the last near-draw? The closest I can think of is Belbo vs. Adanac postal, and that wasn't actually very close to drawing. My guess is that draws are most likely when the players start exchanging rabbits before pieces. If there are lots of pieces left at the end and only a few rabbits, then there may be more possibility to sacrifice in exchange for the opponent's last rabbit or two. On the other hand, if games start out with piece exchanges, which empties the board but leaves lots of rabbits on, then the game becomes even less drawish than at the start. About a year ago 99of9 was prediciting a draw soon between top-level players as knowledge of strategy got more refined. It seemed reasonable at the time, because top-level games were dominated by lone-elephant openings resulting in rabbit pulls. Nowadays, not only is it more common to see pieces in addition to the elephant committed to an attack, but also it seems that lone-elephant attacks are more focused on the opposing heavy pieces than on rabbits. Therefore it seems that, unless the style of game involving opposing rabbit pulls returns to prevalence, we may not see a top-level draw (or any draw at all) for years to come. |
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Title: Re: Fixing the rules / eliminating draws Post by 99of9 on Jul 10th, 2005, 7:48am Yes, I agree with that analysis Fritz. The chance of a draw has definitely decresed since the era of rabbit-pulling has gone out of favour. I of course am still of that mindset - so as soon as someone similar comes along we can have that draw! |
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Title: Re: Fixing the rules / eliminating draws Post by Fritzlein on Jul 10th, 2005, 2:22pm on 07/10/05 at 07:48:38, 99of9 wrote:
I have seen you punish the opening elephant-horse attack mercilessly on defense. Somehow you always get the attacking horse framed (which shouldn't be as easy as you make it look) and somehow you convert the framed horse into a captured horse (which also shouldn't be as easy as you make it look). Because of watching your games, I'm not comfortable with anything other than lone-elephant attacks in the opening, except against bots, of course. The more multi-piece attacks you refute, the more converts you will make to your lone-elephant-rabbit-pulling style. Interestingly, though, I noticed in my current postal game against RonWeasley that as soon as we traded a pair of horses, I was suddenly eager to launch an E+H attack and afraid he would be able to stage an E+H attack on me first. I conclude that the balance between offense and defense in the opening position is so fine that a single trade can violently tip the scales. You have opined that a camel trade stabilizes the opening position, which seems reasonable in my judgement, but given this it is counter-intuitive how much a horse trade destablizes the opening position. To stray off-topic, let me add that the more I learn about Arimaa strategy the more mystical it becomes to me. I keep groping for easy rules of thumb, but the divergent effect between a camel trade and a horse trade as the first trade of the game doesn't yet fit into a larger picture for me. I know what I feel about a position before I can explain why, which didn't happen to me much in chess. Maybe that's just the difference between learning chess strategy from books and learning Arimaa strategy by playing, but I don't think that's the whole picture. I think Arimaa is simply a more strategic game than chess. Back to the topic of draws, I think that sentiment might well swing back in the direction of lone-elephant rabbit-pulling contests, because I've seen nothing yet to discredit this strategy. Another possibility is that opening theory might lead to early camel trades, which would then force the game into rabbit-pulling mode. Moreover, I'm not even convinced that any progress at all can be made against the perfect blend of defense and counter attack. Perhaps all multi-piece attacks from the opening position are unsound. Perhaps in dual-lone-elephant openings piece trades can always be avoided, and futher it might be that for one player to guarantee himself a rabbit pull, he has to give up a horse hostage, which isn't worth it. Arimaa might then devolve into endless thrust and parry, a practical draw albeit not a rulebook draw. On the other hand, the recent awareness of more attacking possibilities makes the game seem so rich and complicated that I doubt we can "solve" Arimaa any time soon. For example, it might be that when a rabbit is being pulled on one wing, the E+H (or even E+M!) attack gains strength on that wing and can become sound. Recall how aggressively Omar used his camel in the postal tournament to good effect, not to mention his recently beating BombP3 with a cat and two rabbits for a lost camel. Our ideas of "lone-elephant rabbit pulling opening" and "multi-piece attack opening" don't stay conveniently separate from one another in practice. In fact, I don't yet discount the E+M+H opening attack: BlackKnight almost beat me with such an attack in last year's World Championship match, and I still don't know what I should have done differently. Arimaa is so far too mind-boggling to guess what will happen next in opening theory. Rather than predicting the frequency/infrequency of draws in the future, I should instead predict only this: Arimaa theory will remain unpredictable for a long time. |
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