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Topic: Analysis of Arimaa with reduced material (Read 2461 times) |
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pago
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Analysis of Arimaa with reduced material
« on: Aug 18th, 2010, 8:40am » |
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There have been a few discussions on the forum to know whether there is an advantage to Gold or to Silver in Arimaa. As far as I have understood the discussions, it seemed that strong players felt that there should be a slight advantage to Gold due to the first move but statistics showed that there could be a slight advantage to silver due to their piece placement according to Gold placement.
I would like to open a (hopefully) new topic that I feel related to this issue : Theoretical analysis of Arimaa with reduced material (assuming that we keep all other rules, in particular for piece disposition)
The idea would be to perform perfect analysis with reduced material (for example Elephant + 1 Rabbit versus 1 elephant + 1 rabbit  ER vs er etc…) and to see whether we could reach a conclusion about gold or silver advantage.
One additional interest (or maybe the first interest) could be that nice solutions would sometimes emerge in the spirit of studies of elementary endings in chess.
Computers would be very useful for these systematic tree searches.
I propose Two kinds of analysis.
1) Rabbits alone (although Elephant removal is very artificial)
2) Rabbits + pieces
First analysis show that a few rabbits without piece give a big advantage to gold :
1 gold rabbit wins against 2 silver rabbits
2 gold rabbits win against 4 silver rabbits
3 gold rabbits win against 4 silver rabbits (obviously) but loses against 5 silver rabbits (TBC)
4 gold rabbits win against 5 silver rabbits (TBC)
To be continued…
Would someone be able to analyze 8 gold Rabbits against 8 silver rabbits ?
Although I can’t make an accurate analysis, it seems that the addition of a piece gives an advantage to silver with equal material because he can take into account gold disposition.
For example with 1 Rabbit + 1 Elephant against 1 rabbit + 1 elephant, silver strategy would be to place his elephant on the seventh rank in face of the gold rabbit and to place his rabbit on a7 or h7 on a free column as far as possible from the gold elephant.
The plan would to let silver elephant in front of gold rabbit, advance toward it when possible to take it in hostage in order to free silver rabbit.
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pago
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1) Rabbits alone analysis
« Reply #1 on: Aug 18th, 2010, 8:43am » |
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1.1) 1 gold Rabbit against 1 silver rabbit (1R vs 1r)
Gold win is obvious.
1.2) 1 gold Rabbit against 2 silver rabbits  1R vs 2r (huge material advantage to silver)
There still is a win for gold but it is not so trivial. The only mean to win is to put the gold rabbit back to a trap that could seem at first sight paradoxical.
1.2.1) First wrong gold attempt (gold rabbit on the first column)
1g Ra2
1s ra7 rb7
2g Ra2n Ra3n Ra4n Ra5e (if Ra5n then 2s rb7s and gold can’t move)
2s ra7s ra6s rb7s (only three steps !) with a win at the following move
1.2.2) Second wrong gold attempt (gold rabbit on the first column)
1g Rb2 (with the threat to reach the c5 square)
1s ra7 rc7 (only mean to cover all the threats)
1.2.2.1) 2g Rb2n Rb3n Rb4n
2s ra7s ra6s rc7w rb7s with a win at the following move
1.2.2.2) 2g Rb2n Rb3n Rb4n Rb5w
2s ra7s rc7w rb7s rb6s and gold can’t move
1.2.2.3) 2g Rb2n Rb3n Rb4n Rb5e
2s rc7e ra7e rb7s rb6s with a win at the following move
1.2.3) Third wrong gold attempt (gold rabbit on a central column)
1g Rd2
1s rc7 re7
1.2.3.1) 2g Rd2n Rd3n Rd4n
2s rc7e rd7s re7s re6s with a win at the following move
1.2.3.2) 2g Rd2n Rd3n Rd4n rd5w
2s re7w r c7w rb7s rb6s with a win at the following move
1.2.3.3) 2g Rd2n Rd3n Rd4n rd5e
2s re7s rc7e rd7s rd6s with a win at the following move
1.2.4) Solution : Gold wins thank to its rabbit back to a trap
1g Rc2 (with two threats : Reach b4 square and d4 square)
1s Rb2 Rd2 (not sufficient attempt to cover the 2 threats)
1.2.4.1) 2g Rc2w Rb2n Rb3n Rb4n
2s rb7w rd7w rc7w
3g Rb5e Rc5e Rd5n Rd6n with a win at the following move
1.2.4.2) 2g Rc2e Rd2n Rd3n Rd4n
2s rd7e rb7e rc7e
3g Rd5w Rc5w Rb5n Rb6n with a win at the following move
1.3) 2 gold Rabbits against 4 silver rabbits  2R vs 4r
This time Rabbits shall not be put back to the traps (if 1g Rc2 Rf2 then 1s Ra7 Rc7 Rf7 Rh7 and silver wins easily)
Gold shall place its rabbits on the second column :
1g Rb2 Rg2
The most effective defense for silver is :
1s ra7 rc7 rf7 rh7 (on other symmetric position gold wins by playing 2g Rb2n Rb3n Rb4n)
1.3.1) First gold wrong attempt : one rabbit straight move
2g Rb2n Rb3n Rb4n (move effective with other silver placements)
2s rc7w rb7s ra7s ra6s with a win at the following move
1.3.2) Second gold wrong attempt : 2 threats on c5 square
2g Rb2n Rb3n Rb4n Rb5e
2s rc7e ra7e rb7s rb6s with a win at the following move
1.3.3) Solution : Gold wins with 2 rabbits advance
2g Rb2n Rb3n Rg2n Rg3n
1.3.3.1) 2s rc7w rf7e ra7s ra6s
3g Rb4w Rg4n Rg5w Rf5w
3s rb7e rc7e rg7w rf7w
4g Re5w Rd5w Rc5w Rb5n with a win at the following move
1.3.3.2) 2s rc7w rf7e ra7s rh7s
3g Rb4e Rc4e Rd4n
3s rb7e rc7e rg7w rf7w
4g Rg4n Rg5n Rg6n Rg7n
1.3.3.3) 2s ra7e rh7w
3g Rb4n Rb5n Rb6w Ra6n
1.4) 3 gold rabbits against 5 silver rabbits
I think that silver wins but I am not totally sure of my analysis. I think that the best gold attempt should be :
1g Rb2 Rd2 Rg2
… and the best silver defence would be :
1s ra7 rc7 rd7rf7 rh7 (on other defence gold wins)
1.4.1) Gold first attempt : 2 threats on f5 square
2g Rg2n Rg3n Rg4n Rg5w
2s rd7e rf7e rh6s rh5s with an easy win
1.4.2) Gold second attempt : 2 rabbit move
2g Rb2n Rb3n Rg2n Rg3n
2s rc7w rf7e rh7s rh6s
3g Rg4e Rd2n Rd3n Rd4n
3s rg7w rf7w ra7s ra6s
4g Rb4w Rd5e Re5e Rf5e
4s re7e rf7e rg7s rd7e
5g Rg5w Rf5w Re5w Rd5n (if Rd5w then 5s re7w rg6s rg5s rg4s)
5s rb7e rc7e rg6s rg5s with a win at the following move
1.5) 4 gold rabbits against 5 silver rabbits
Gold should win
1g Rb2 Rd2 Re2 Rg2 (gold position from the previous section with an additional rabbit on e column)
1s ra7 rc7 rd7rf7 rh7 (same defence than in the previous section
2g Rb2n Rb3n Rg2n Rg3n
2s rc7w rf7e rh7s rh6s
3g Rg4e Re2n Re3n Re4n
3s rg7w rf7w ra7s ra6s
4g Rb4w Re5e Rf5e Rg5n with a win at the following move
1.6) More rabbits
It becomes too difficult for me 
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chessandgo
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Re: Analysis of Arimaa with reduced material
« Reply #2 on: Aug 18th, 2010, 11:00am » |
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There is a thread by jbd where he systematically analyses all *positions* with few pieces, you definitely want to read it. Some of the results you are looking for should directly follow from what he has.
Oh, and by the way, I'm not sure strong players think gold has an advantage. I used to, but now I'm absolutely clueless. I think Fritzl might prefer silver?
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| « Last Edit: Aug 18th, 2010, 11:01am by chessandgo » |
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pago
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Re: Analysis of Arimaa with reduced material
« Reply #3 on: Aug 19th, 2010, 3:33am » |
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Thank you for your reply chessandgo.
Sometimes some topics are well hidden in the middle of the forum : it reminds me the final scene of the first Indiana Jones 
I had'nt read your opinion about the advantage for gold or silver until today but I thought that Fritzlein's opinion was slightly for Gold. I have misunderstood his point of view (As Assimil says : "My talor is rich... but my english is poor")
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Fritzlein
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Re: Analysis of Arimaa with reduced material
« Reply #4 on: Aug 19th, 2010, 8:38am » |
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I don't know which color has the advantage. I play devil's advocate against anyone who thinks it is an obvious question. These days I think most top players slightly favor Silver. That is why I argue that Gold is in fact advantaged. 
Indeed, I can be doubly contrarian, because the essential reason given for supposed Silver advantage is that Silver can use EHH setup but Gold can't. I respond that neither Silver nor Gold can use EHH setup, only centralized EM with split horses is sound, which means second setup is worthless and the first move is the only advantage. However, if this stance becomes conventional wisdom and everyone reverts to the 99of9 setup even with Silver, I will make the opposite argument to prove that Silver is better off after making an unbalanced setup.
The important thing is to stop anybody from believing that he knows the answer. It's like Stephen Hawking placing a bet that black holes don't exist because he wanted to remind everyone of the difference between consensus opinion and fact. Nobody knows if Gold or Silver has the advantage; that is my only point. 
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omar
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Re: Analysis of Arimaa with reduced material
« Reply #5 on: Aug 20th, 2010, 8:16am » |
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Very interesting analysis. If we notice a pattern emerging then maybe we can at least theoretically say if Arimaa is ultimately a win for the first or second player. In Go I think it has been solved up to 5x5 and found that even size boards result in a draw and odd size boards result in a win for the first player. Based on this pattern one can expect that 19x19 Go is a win for the first player.
Does this correctly summarize what you've found so far?
1R>1r
1R>2r
1R<3r
2R>2r
2R>3r
2R>4r
2R<5r
3R>3r
3R>4r
3R<5r ?
4R>4r
4R>5r
4R?6r not yet tested
So for games with only rabbits it seems that the first move advantage is worth about 1.5 rabbits. 2R>4r seems to break the rule though.
To analyse with more pieces I think you will need to use a computer. Maybe use one of the bots on the Downloads page and run tests with that.
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speek
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Re: Analysis of Arimaa with reduced material
« Reply #6 on: Aug 20th, 2010, 8:45am » |
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I'm still wondering how silver could have the advantage. If gold sets up EHH on the left and M on the right, how does silver capitalize on that? It seems to me, whatever silver does, there are match-ups across the board that favor one player or the other, but I don't see how all the match-ups can favor silver, in which case, gold has the first move, and at least one favorable match-up to exploit.
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jdb
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Re: Analysis of Arimaa with reduced material
« Reply #7 on: Aug 20th, 2010, 9:06am » |
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Just so I understand the conditions of the study.
1) Gold and Silver must set up on the first two ranks, as normal.
2) Only rabbits and no other pieces.
Is this correct?
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pago
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Re: Analysis of Arimaa with reduced material
« Reply #8 on: Aug 20th, 2010, 9:20am » |
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To Omar : You have a talent to be very friendly on your comments even when the posts are not transcendent
Your summarize is correct (assuming that my analysis are correct !)
To jdb :
My idea was to make studies repecting all Arimaa rules (in particular the 2 rank initial setup) excepted that we would use only a few pieces to be able to make perfect analysis.
The easiest analysis would be rabbit without other pieces.
After that I hoped to introduce other pieces (ER vs er ; ERR vs err ; EMR vs emr)
As far as I understand chessandgo reply, your already had the same idea.
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jdb
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Re: 1) Rabbits alone analysis
« Reply #9 on: Aug 20th, 2010, 9:35am » |
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on Aug 18th, 2010, 8:43am, pago wrote:
1.3) 2 gold Rabbits against 4 silver rabbits  2R vs 4r
This time Rabbits shall not be put back to the traps (if 1g Rc2 Rf2 then 1s Ra7 Rc7 Rf7 Rh7 and silver wins easily)
Gold shall place its rabbits on the second column :
1g Rb2 Rg2
The most effective defense for silver is :
1s ra7 rc7 rf7 rh7 (on other symmetric position gold wins by playing 2g Rb2n Rb3n Rb4n)
1.3.1) First gold wrong attempt : one rabbit straight move
2g Rb2n Rb3n Rb4n (move effective with other silver placements)
2s rc7w rb7s ra7s ra6s with a win at the following move
1.3.2) Second gold wrong attempt : 2 threats on c5 square
2g Rb2n Rb3n Rb4n Rb5e
2s rc7e ra7e rb7s rb6s with a win at the following move
1.3.3) Solution : Gold wins with 2 rabbits advance
2g Rb2n Rb3n Rg2n Rg3n
1.3.3.1) 2s rc7w rf7e ra7s ra6s
3g Rb4w Rg4n Rg5w Rf5w
3s rb7e rc7e rg7w rf7w
4g Re5w Rd5w Rc5w Rb5n with a win at the following move
1.3.3.2) 2s rc7w rf7e ra7s rh7s
3g Rb4e Rc4e Rd4n
3s rb7e rc7e rg7w rf7w
4g Rg4n Rg5n Rg6n Rg7n
1.3.3.3) 2s ra7e rh7w
3g Rb4n Rb5n Rb6w Ra6n
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For 1.3.3:
1g Rb2 Rg2
1s ra7 rc7 rf7 rh7
2g Rb2n Rb3n Rg3n Rg4n
2s ra7s ra6s ra5e rf7e
Then some kind of repetition fight starts.
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pago
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Re: Analysis of Arimaa with reduced material
« Reply #10 on: Aug 20th, 2010, 10:29am » |
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I didn't see this defense.
If I continue this repetition fight we would have :
1g Rb2 Rg2
1s ra7 rc7 rf7 rh7
2g Rb2n Rb3n Rg2n Rg3n
2s ra7s ra6s ra5e rf7e
3g Rg4n Rg5w Rf5w
3s rc7e rg7w rf7w rh7w
4g Re5e Rf5e
4s rg7e re7e rf7e rd7e
5g Rg5w Rf5w
5s re7w rg7w rf7w rh7w
6g Re5e Rf5e
etc…
and silver would lose the repetition fight because he would repeat 3 times the position issued from 3s.
Am I right ? May I still conclude that 2R wins against 4r ?
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Fritzlein
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Re: Analysis of Arimaa with reduced material
« Reply #11 on: Aug 20th, 2010, 12:26pm » |
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on Aug 20th, 2010, 8:45am, speek wrote:| I'm still wondering how silver could have the advantage. If gold sets up EHH on the left and M on the right, how does silver capitalize on that? It seems to me, whatever silver does, there are match-ups across the board that favor one player or the other, but I don't see how all the match-ups can favor silver, in which case, gold has the first move, and at least one favorable match-up to exploit. |
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If you think of the board as having only two theaters, east and west, then Silver can choose between being equal in both or having advantage in one and disadvantage in the other, with Gold's first move being telling in either case. However, if you go to the other extreme of thinking the board has eight theaters, one for each column, then Silver can easily get more favorable oppositions than unfavorable ones. For example (assuming rabbits set up in back):
E > c
M > c
H < e
H < m
D < h
D < h
C < d
C < d
in which case Silver is winning six of the eight oppositions!
Even if you only consider three theaters (east, west, and center) Silver can win two out of three against Gold's unbalanced setup by pitting "m" against EHH, "edd" against M, leaving "hh" to win the middle. But maybe winning the middle is worthless.
It's quite muddled. It isn't clear to me that Gold's EHH setup is unsound. We are far from having explored all of Silver's options, never mind refuting them all. My goal is just to bring doubt into the mind of anyone who thinks it is obvious which side has the advantage.
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Fritzlein
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Re: Analysis of Arimaa with reduced material
« Reply #12 on: Aug 20th, 2010, 12:38pm » |
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on Aug 20th, 2010, 8:16am, omar wrote:If we notice a pattern emerging then maybe we can at least theoretically say if Arimaa is ultimately a win for the first or second player.
[...]
So for games with only rabbits it seems that the first move advantage is worth about 1.5 rabbits. |
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I'm quite confident that one pattern we will observe is that the advantage of the first move decreases as the the number of pieces increases, since we know from practical experience that endgames are sharper than openings. Supposing we even get up to having enough material to block all columns, do you still expect such a value on the first move? Are you forecasting seven rabbits for Gold would beat eight rabbits for Silver? It might, but it's far from clear to me.
Furthermore, the current thinking that Silver has an advantage relies on favorable piece alignments (e.g. M opposite HH), which are a non-factor when all pieces are rabbits. It's hard to generalize from cases without freezing to cases with freezing.
This is not to say investigation of rabbits-only positions is uninteresting, just that it is unrealistic to hope we can infer anything about the full-set evaluation from them. For starters, the first move is not worth 1.5 rabbits, and I'll trade the first move for just one rabbit to anyone who thinks it is. 
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| « Last Edit: Aug 20th, 2010, 12:39pm by Fritzlein » |
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speek
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Re: Analysis of Arimaa with reduced material
« Reply #13 on: Aug 20th, 2010, 1:24pm » |
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on Aug 20th, 2010, 12:26pm, Fritzlein wrote:
Even if you only consider three theaters (east, west, and center) Silver can win two out of three against Gold's unbalanced setup by pitting "m" against EHH, "edd" against M, leaving "hh" to win the middle. But maybe winning the middle is worthless. |
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And, of course, if you have HHE, your elephant is half in the middle anyway.
From what I've seen so far, gold has an initiative in the game that sometimes takes many moves to dissipate.
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jdb
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Re: Analysis of Arimaa with reduced material
« Reply #14 on: Aug 21st, 2010, 9:11am » |
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on Aug 20th, 2010, 9:20am, pago wrote:
To jdb :
My idea was to make studies repecting all Arimaa rules (in particular the 2 rank initial setup) excepted that we would use only a few pieces to be able to make perfect analysis.
The easiest analysis would be rabbit without other pieces.
After that I hoped to introduce other pieces (ER vs er ; ERR vs err ; EMR vs emr)
As far as I understand chessandgo reply, your already had the same idea. |
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The thread where I initially posted some results is here:
http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=devTalk;action=display ;num=1062013358;start=90
Everything I did included elephants, since I want to use the results for a material evaluator. Currently, I'm running games with all combinations of EDCR. In other words, no horses or camels. This makes 72 different material combinations. There is roughly a 3000 elo point spread from E1R to EDDCC8R.
For example, here are the head to head results for E4R taken from bayeselo.
Code:
63 Clueless_E4R -779 332.0 (143.0 : 189.0)
4.0 ( 0.0 : 4.0) Clueless_ECC8R 691
4.0 ( 0.0 : 4.0) Clueless_ECC7R 527
4.0 ( 0.0 : 4.0) Clueless_ECC6R 397
4.0 ( 0.0 : 4.0) Clueless_EC8R 374
4.0 ( 0.0 : 4.0) Clueless_ECC5R 241
4.0 ( 0.0 : 4.0) Clueless_EC7R 201
4.0 ( 0.0 : 4.0) Clueless_ECC4R 56
4.0 ( 0.0 : 4.0) Clueless_EC6R 31
4.0 ( 0.0 : 4.0) Clueless_E8R -26
4.0 ( 0.0 : 4.0) Clueless_EC5R -156
4.0 ( 0.0 : 4.0) Clueless_ECC3R -174
21.0 ( 0.0 : 21.0) Clueless_E7R -225
22.0 ( 2.0 : 20.0) Clueless_EC4R -358
24.0 ( 2.0 : 22.0) Clueless_E6R -404
1.0 ( 1.0 : 0.0) Clueless_EDC2R -410
2.0 ( 0.0 : 2.0) Clueless_EDCC1R -434
24.0 ( 4.0 : 20.0) Clueless_ECC2R -462
2.0 ( 1.0 : 1.0) Clueless_ED3R -516
24.0 ( 6.0 : 18.0) Clueless_EC3R -566
24.0 ( 1.0 : 23.0) Clueless_E5R -580
2.0 ( 0.0 : 2.0) Clueless_ED2R -770
2.0 ( 2.0 : 0.0) Clueless_EDC1R -809
1.0 ( 1.0 : 0.0) Clueless_EDD1R -912
24.0 ( 16.0 : 8.0) Clueless_ECC1R -925
24.0 ( 17.0 : 7.0) Clueless_EC2R -959
24.0 ( 24.0 : 0.0) Clueless_E3R -1007
1.0 ( 1.0 : 0.0) Clueless_ED1R -1249
22.0 ( 22.0 : 0.0) Clueless_E2R -1332
22.0 ( 21.0 : 1.0) Clueless_EC1R -1469
22.0 ( 22.0 : 0.0) Clueless_E1R -1832
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