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        Arimaa >> General Discussion >> Fairy pieces again
        
(Message started by: IdahoEv on Feb 1^st, 2007, 3:42pm)

Title: Fairy pieces again
Post by IdahoEv on Feb 1^st, 2007, 3:42pm

This is what happens when I play too much Arimaa.

As I was falling asleep last night, I came up with an idea for another potentially interesting variation to the game. Define the following new pieces:

The Hare: Like a rabbit, except that it can move in all four directions and cannot score a goal.

The Puck: The same size as a rabbit (i.e. rabbits and hares cannot push it). It cannot move on its own, but it can be pushed or pulled by either player. It can goal. Like a hockey puck, right?

Now start a game with no rabbits. Before the game, each player secretly decides how to divide the eight "pawn" slots between Hares and Pucks. The remaining officers are unchanged.

The object, of course, is to push one of your own pucks to goal. How you decide to split your forces between mobility/strength (hares) and goaling potential (pucks) would be a potentially exciting strategic choice.

Actually I'm divided as to whether pucks should be the same size as hares (meaning only officers can push them) or one size smaller (meaning hares can push them too). I'm leaning toward the former because I think it would be important to be able to use your pucks on the back line as part of a frame of a hare, otherwise it may be extremely hard to gain an advantage and the game may stalemate. This would have to be determined by playtesting. It also might be possible to require a 4/4 hare/puck split rather than allowing the decision because the best strategy might be to go 7/1 and hide your puck in a corner until the endgame. If so, the game would stalemate for the same reason that caused Omar to require that rabbits cannot retreat: it would degenerate to two opposing defenses, nobody able to gain an advantage.

Title: Re: Fairy pieces again
Post by Fritzlein on Feb 1^st, 2007, 9:40pm

That's an interesting idea, but it is possible that even a 4/4 split between pucks and hares could make the game a defensive stalemate. Right now it looks almost possible to keep six rabbits safe, but the extra two (either back and center, or forward on the flanks) will be vulnerable to pulling. If I could put my only four non-retreatable pieces on a1, b1, g1, and h1, I think I could hunker down and keep everything safe from your lone elephant. Eventually you will get frustrated and launch a multi-piece attack, and if I can punish that, next game you will just accept the infinite-move draw.

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 2^nd, 2007, 1:51pm

Hmmm, that seems a distinct possibility.

Some other possibilities that have occurred to me since the initial post:

(1) Define "The Football" as a puck but which can goal in a deep end zone... the last two rows, not the last one. Then instead of splitting between pucks and hares, split between regular rabbits and footballs. Now you have rabbits which cannot retreat and footballs which cannot retreat on their own, so the non-retreatability aspect of the game is potentially retained. The "deep endzone" feature of footballs is added so that the piece is not strictly inferior to rabbits.

(2) Lots of pucks ... no rabbits at all, just eight pucks per side. especially if it was ruled that pucks do not unfreeze your officers (seems reasonable), this might make a purely defensive game untenable.

Title: Re: Fairy pieces again
Post by DorianGaray on Feb 2^nd, 2007, 3:28pm

Hey, I have an idea!

The mouse : It's weaker than any piece even the rabbit but it scares the hell out of the elephant that has to keep at least one square away from it or else is blocked while it is in contact with it.

What do you think?

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 2^nd, 2007, 5:43pm

on 02/02/07 at 15:28:40, DorianGaray wrote:

I proposed a mouse last summer that was pretty much the same: it was smaller than all other pieces except the elephant, but it was bigger than the elephant. It could push and pull and freeze the elephant in the same way that an elephant does to a camel or any other piece.

Thread: Fairy Pieces (http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=talk;action=display;num=1144695637)

I think though that if the mouse is smaller than the rabbits, it will just get quickly pinned to a rabbit and stuck away for most of the game. I think it would be a more balanced piece if it was the "same size" as rabbits, i.e. it can't push, pull or freeze rabbits and neither can they it. So rabbits can't hunt it. But cats can...

At the time i called that one the "rat" (=big mouse), but I think mouse is funnier because of the urban legend that elephants are scared of mice.

Maybe in the distant future when we have Arimaa Server 2.0 it can support variants. ;-) My guess is that most variants will be entertaining for a little while but turn out not to be as deep as the original game.

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 6^th, 2007, 12:02pm

Here's another possibility. Call it Arimaa-bandy.

Each player gets the usual complement of officers plus six hares (rabbits that can move backwards but can't goal).

In the center of the board are four neutral pucks. Neither silver or gold, either player can push or pull them with any piece. If any puck ends up on row 1 or 8, the player on the other side wins.

This might be an interesting game even if captures were eliminated, though one might want to reduce the total piece count a bit in that case.

Title: Re: Fairy pieces again
Post by DorianGaray on Feb 8^th, 2007, 6:15am

How about this? You keep all the rules of the classic Arimaa and add this one:

You can push (or pull) your own pieces (E.G. your own rabbit with your own cat).

I've always found it unfair that you can't push your own pieces even when it's convenient.

Title: Re: Fairy pieces again
Post by PMertens on Feb 8^th, 2007, 2:14pm

I am clearly for the ruleset with the highest tactical and strategical depth.
Arimaa seems to be very complex and hard to master since it is not yet clear what kind of play is superior.
Most changes could tipp the balance into one or the other direction.
Imho the rabbit pulling concept is great, because it actually forces action into an otherwise very defensive game.
I can defend any retreatable piece with my phant - no matter if it moves on its own power or has to be pushed..
In addition pushing pieces would have strange side-effects when i.e. the phant is protecting a rabbit on an opponent trap from the side:
It could push itself on the trap and then either push the opposing blocker away or even worse push the now protecting rabbit away (and being declared dead midmove) ...

Title: Re: Fairy pieces again
Post by Fritzlein on Feb 8^th, 2007, 10:32pm

I agree that allowing one to pull and push one's own pieces would make the game more defensive, and probably tip Arimaa over the edge to unplayability. Similarly having hares and the neutral hockey pucks in the middle would probably make for too much defense, although it depends on how the pucks affect trap control.

Arimaa strikes a remarkable balance in my mind, of being so defensive that only slow strategies are sound in the opening, but not so defensive that it stalemates. That's no easy trick. Furthermore Arimaa has a remarkable way of allowing tension to build until it explodes when one player refuses to concede a small disadvantage and ups the ante instead.

Omar clearly got lucky when he designed Arimaa, because he finalized the rules without any notion of camel hostage or elephant blockade, the most basic underpinnings of strategy. He certainly couldn't have anticipated the advanced strategies that have emerged. I'll give Omar credit for play-testing enough to know that rabbits had to be forbidden from moving backwards, but there's no way he could have known how much depth there would be.

I'm apt to be extremely skeptical of any rule change that makes Arimaa more defensive, as allowing rabbits to move backward or be pushed/pulled backward would do. As for rules that make Arimaa less defensive, the litmus test in my mind is whether they could introduce new strategic (as opposed to tactical) features, and what old strategic features they would disrupt. Given that it is always easier to see what would be destroyed than it is to see what would be created, I guess I'll just be skeptical all around. :P

The one kind of rule change that would attract me is one where we don't have to give anything, but might possibly gain. I keep coming back to my idea of replacing one horse with a lion. That would probably make the game a tiny bit less defensive, and it doesn't seem to disrupt any current strategies very much, and possibly would introduce new strategies. But then again, if Arimaa ain't broke, why fix it?

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 9^th, 2007, 12:13pm

Remind me how you define a lion. More powerful than a camel but smaller than an elephant?

Title: Re: Fairy pieces again
Post by Fritzlein on Feb 9^th, 2007, 12:27pm

on 02/09/07 at 12:13:37, IdahoEv wrote:

Remind me how you define a lion. More powerful than a camel but smaller than an elephant?

Yeah, that. The idea is to have fewer ties. Right now the four horses seem to automatically sit on b3, b6, g3, g6. The facing horses can't really attack each other, which deadens the position.

If each side had elephant, lion, camel, horse, then the equivalent of the "camel hostage" would be a "lion hostage". The equivalent of the "elephant-horse attack" would be the "elephant-camel attack". The board would be just as congested, so elephant blockades would be just as possible. The inability of rabbits to move backward would be just as much of an issue.

It's hard to guess how strategic the game would be with a lion, but it is a conservative possibility I would throw out if we ever reached consensus that Arimaa is too defensive as is. A lion would liven up the game a little bit while (hopefully) retaining most strategic features.

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 9^th, 2007, 2:48pm

Well, hopefully at some point in the future we will have a more generalized server that can support variants using a rules plug-in architecture.

Until then - try it on a physical board sometime.

Title: Re: Fairy pieces again
Post by DorianGaray on Feb 9^th, 2007, 2:54pm

on 02/09/07 at 12:27:47, Fritzlein wrote:

I would replace your lion by a rat with the following order:

Elephant, Camel, horse, dog, cat, rat, rabbit

That way it would be easier to adapt from classic Arimaa without having to rethink the relative value of each piece to the Elephant.

;)

Title: Re: Fairy pieces again
Post by Fritzlein on Feb 9^th, 2007, 5:16pm

on 02/09/07 at 14:54:20, DorianGaray wrote:

Good point, but then that rat would be lonely down there by itself, and it might look smaller than a rabbit. How about instead demoting one horse to a donkey, so your pieces become:
1 Elephant
1 Camel
1 Horse
1 Donkey
2 Dogs
2 Cats
8 Rabbits

on 02/09/07 at 14:48:51, IdahoEv wrote:

Well, hopefully at some point in the future we will have a more generalized server that can support variants using a rules plug-in architecture.

Or maybe when ddyer adds Arimaa to his game site, he will decide to add a variant rather than the original.

Quote:

Until then - try it on a physical board sometime.

New pieces are not trivial to represent with a standard chess set. I guess the easiest solution would be to represent the horse by a rook and the donkey by an inverted rook.

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 9^th, 2007, 7:02pm

on 02/09/07 at 17:16:01, Fritzlein wrote:

represent the horse by a rook

I just can't countenance that. The knight piece is a <i>sculpture of a horse</i> fer krissake. Of COURSE the knight needs to be used to represent the horse. Otherwise you're thinking "I'll move my horse here..." and you reach for the piece that <i>doesn't</i> look like a horse. How confusing is that? Insane, I tell you. Mass hysteria!

Yes, in chess the rook is stronger than the knight or bishop. But in chess the movement and capture rules are all screwed up anyway. Why look to chess as a guide for anything?

Title: Re: Fairy pieces again
Post by IdahoEv on Feb 9^th, 2007, 7:06pm

I also can't quite get over the idea of a donkey. All I can see is Mike Meyers portraying Fritzlein in the first Arimaa movie, yelling at his donkey in a scots accent when it gets into trouble.

"Donkeh! Ah tohld yeh not te guh near thaht horse!"

Title: Re: Fairy pieces again
Post by DorianGaray on Feb 10^th, 2007, 8:09am

on 02/09/07 at 17:16:01, Fritzlein wrote:

Good thinking! I went a little too far with my rat. Silly me forgot about the pairs of dogs and cats.

I'd propose though to replace the donkey by a pony. I've seen donkeys that looked quite strong and I am not sure if I'd instinctively consider them weaker than horses.

Title: Re: Fairy pieces again
Post by aaaa on May 6^th, 2007, 9:02pm

on 02/09/07 at 19:02:31, IdahoEv wrote:

I agree. In fact, did you know that the knight chess piece is actually called a "horse" in many languages (http://www.geocities.com/TimesSquare/Metro/9154/nap-pieces.htm)? I can understand that for pedagogical reasons one would like to make the pieces roughly correspond depending on their respective strengths in both games, but there are two things here:
One is the complication that the relative value of a knight versus a bishop is not so clear-cut; it's mostly having the bishop pair that is an advantage, whereas the value of the knight for its part depends significantly on the phase of a game (it decreases over time). See this (http://mywebpages.comcast.net/danheisman/Articles/evaluation_of_material_imbalance.htm) for interesting statistical insight into the relative values of chess pieces (BTW, I'm planning to do something similar with respect to Arimaa).
More importantly however, if you want to present Arimaa as a fun game for children to learn, you'll just confuse them with something as blatantly counterintuitive like this.
Despite the "official" representation rule as mandated by Omar Syed, I wonder whether in real-life games the horse has ever not been represented by the piece that is actually shaped like one.

This is just another leftover from the chess origins of Arimaa that should be done away with.

Title: Re: Fairy pieces again
Post by Fritzlein on May 6^th, 2007, 10:17pm

Thanks for the very interesting link to Kaufman's article. One similarity between chess and Arimaa is the significance of redundant pieces. Winning a camel for a horse in the opening is a substantial advantage, but this advantage is much reduced if a pair of horses are traded. Why? The side with HH has a redundant horse, while the side with MH doesn't. Trading the redundant horse for the non-redundant horse is a win for the weaker side.

I reckon M for HRR is about even, and so is MHR for HHRR, so I guess I'm saying the difference between the redundant and non-redundant horse is about a full rabbit in that situation.

Title: Re: Fairy pieces again
Post by Fritzlein on May 6^th, 2007, 10:24pm

on 05/06/07 at 21:02:14, aaaa wrote:

(BTW, I'm planning to do something similar with respect to Arimaa).

I'm quite interested in the result, although you face a difficulty Kaufman didn't. He had 300,000 master-level games to work from. You have a few hundred :-/

I really liked IdahoEv's numerical analysis of the relative values of the pieces, which showed that we severely overvalue the heavy pieces relative to rabbits. However, I wonder how much those numbers are affected by the emptiness of the board. The emptier the board, the less stronger pieces matter and the more numerical superiority matters. Maybe our more traditional values are about right for a full board, and totally false for a depleted board?

Title: Re: Fairy pieces again
Post by aaaa on May 14^th, 2007, 12:30pm

This post is sort of off-topic for this thread, but I am responding to the previous post:

on 05/06/07 at 22:24:56, Fritzlein wrote:

I'm quite interested in the result, although you face a difficulty Kaufman didn't. He had 300,000 master-level games to work from. You have a few hundred :-/

Unfortunately that has turned out to be an insurmountable obstacle to empirically arriving at a material evaluation function. Since I don't want to contaminate the data with (suboptimally playing) bots, I feel I should restrict myself to rated human vs human games, but there are so few of them that I can hardly get a dog to be seen to be worth more than a cat.

on 05/06/07 at 22:24:56, Fritzlein wrote:

I really liked IdahoEv's numerical analysis of the relative values of the pieces, which showed that we severely overvalue the heavy pieces relative to rabbits. However, I wonder how much those numbers are affected by the emptiness of the board. The emptier the board, the less stronger pieces matter and the more numerical superiority matters. Maybe our more traditional values are about right for a full board, and totally false for a depleted board?

IdahoEv's research is itself laudable, but there is just too little data to work with and the problem of overfitting is casting its shadow over any possible conclusion. A better source would come from Arimaa experts systematically following the scientific method by repeatedly agreeing on values for various material imbalances, coming up with a hypothesis in the form of a evaluation function that fits those existing imbalances, seeing how it works out in new situations and repeating the whole process. Perhaps this can be automatized to a certain degree.

Title: Re: Fairy pieces again
Post by chessandgo on May 15^th, 2007, 10:03am

on 05/14/07 at 12:30:40, aaaa wrote:

but there are so few of them that I can hardly get a dog to be seen to be worth more than a cat.

But is a dog worth significantly more than a cat (in the opening) ? I would say no, so please don't take this for a proof of failure. Exchanging a dog for a cat as first trade still gives 50-50 chances fr the outcome, and winning a cat or winning a dog should be almost exactly the same advantage.

Title: Re: Fairy pieces again
Post by IdahoEv on Jun 11^th, 2007, 1:04pm

on 05/14/07 at 12:30:40, aaaa wrote:

IdahoEv's research is itself laudable, but there is just too little data to work with and the problem of overfitting is casting its shadow over any possible conclusion.

Actually, I'm going to disagree with you here. While there may not be as much data as chess, there was more than sufficient data when I did that project to avoid any serious overfitting.

The models I fit to the data had anywhere from 2 (LinearABC) to 7 (optimized DAPE) adjustable coefficients. These formulas were empirically fit to a dataset of ~18000 states from ~3000 different games. When the number of data points is three orders of magnitude larger than the number of adjustable parameters, overfitting is not a huge problem.

In the cases of high numbers of adjustable parameters (like oDape), there was evidence of a very minor degree of overfitting - some solutions would grab one small cluster of states (10s of states, not 100s) while others would grab a different cluster. But the OVERALL resulting function was altered only very little, and the general conclusions reached by the optimized evaluators were highly consistent regardless of the form of the evaluation function.

If you ask oDAPE (optimized DAPE) and LinearAB and LinearABC about who is winning in any particular case, they will agree almost all of the time (~99%), and they will be correct far more often than the hand-fit versions.

There was "overfitting" but it only affected an extremely small number of corner cases. For the bulk of real-world arimaa material states, I'm quite confident in the results I posted last year.

Perhaps soon I'll re-run those analyses to see if anything has changed with an additional year of data. I'm pretty confident predicting that the coefficients will change very little, perhaps only in the fourth or fifth significant figure in the significant cases.

Using coefficients from that research is the only significant change between Zombie and Faerie ... and that was enough to put Zombie two steps higher in the CC rankings.

Title: Re: Fairy pieces again
Post by roule on Jun 17^th, 2007, 7:33pm

I asked myself this question while reading the topic.

Does all rabbits need to be prevented to move backwards? Why not replace camel by a extremely weak piece, and prevent this piece from moving backwards? Or maybe only one of the rabbits is not allowed... can the opponent still force an opening? Anyone has ideas on this?