|
||||
|
Title: Analysis of Arimaa with reduced material Post by pago on Aug 18th, 2010, 8:40am 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. |
||||
|
Title: 1) Rabbits alone analysis Post by pago on Aug 18th, 2010, 8:43am 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 ??? |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by chessandgo on Aug 18th, 2010, 11:00am 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? |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by pago on Aug 19th, 2010, 3:33am 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") |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by Fritzlein on Aug 19th, 2010, 8:38am 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. :P |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by omar on Aug 20th, 2010, 8:16am 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. |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by speek on Aug 20th, 2010, 8:45am 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. |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by jdb on Aug 20th, 2010, 9:06am 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? |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by pago on Aug 20th, 2010, 9:20am 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. |
||||
|
Title: Re: 1) Rabbits alone analysis Post by jdb on Aug 20th, 2010, 9:35am on 08/18/10 at 08:43:23, pago wrote:
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. |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by pago on Aug 20th, 2010, 10:29am ??? 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 ? |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by Fritzlein on Aug 20th, 2010, 12:26pm on 08/20/10 at 08:45:32, speek wrote:
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. ;) |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by Fritzlein on Aug 20th, 2010, 12:38pm on 08/20/10 at 08:16:02, omar wrote:
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. ;D |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by speek on Aug 20th, 2010, 1:24pm on 08/20/10 at 12:26:33, Fritzlein wrote:
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. |
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by jdb on Aug 21st, 2010, 9:11am on 08/20/10 at 09:20:06, pago wrote:
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:
|
||||
|
Title: Re: Analysis of Arimaa with reduced material Post by chessandgo on Aug 25th, 2010, 7:50am Cool thread! :) I've not had time to go through the analysis, but I hope I will when I'm back in France. on 08/20/10 at 12:38:00, Fritzlein wrote:
Cool question! That's kind of mind blowing. My best guess would be that yes, the first move should allow gold to paralyze several silver rabbits by advancing directly to the 6th line (like leaving the c-file empty, and advancing to e6 on move 1g). I'm looking forward to the opportunity of playing it out a bit :) on 08/21/10 at 09:11:04, jdb wrote:
Ah, thanks for the link. And wow, all combos of EDCRs???? That must be a freakingly huge number of positions? And argh, I don't read the bot dev forum, so I have completely missed most of the linked thread... I have some catching up to do. I'm pretty excited by the glimpses I have taken on it :) |
||||
|
Arimaa Forum » Powered by YaBB 1 Gold - SP 1.3.1! YaBB © 2000-2003. All Rights Reserved. |