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        Arimaa >> General Discussion >> variant start to even out theoretical advantage
        
(Message started by: johncf1018 on May 27^th, 2011, 5:22pm)

Title: variant start to even out theoretical advantage
Post by johncf1018 on May 27^th, 2011, 5:22pm

Hi Everyone - this is my first post in these forums. Anyways, I've been slowly working my way up the bot ladder and thoroughly enjoying myself. I know there isn't much of an advantage one way or the other regarding 1st or second player, but after playing a few games of connect6 I thought about the following modification to the arimaa ruleset in order minimize any future advantage. What if the first player only had 2 moves for his first move, and then play proceeded as usual (4 moves each)? Has this been suggested before? Any thoughts for or against the idea? I know I'm new to the game but I couldn't help thinking this might be a worthwhile modification.

Title: Re: variant start to even out theoretical advantag
Post by Fritzlein on May 27^th, 2011, 6:31pm

Welcome to the Arimaa Forum. Thanks for posting!

Yes, it has been proposed to take two steps away from Gold's first turn. There is widespread suspicion, however, that Silver already has the advantage as the rules stand. If that is true, then taking two steps away from Gold will make the game less balanced, not more.

The theory behind Silver having an advantage is that the Silver setup is often used to create a favorable alignment of the heavy pieces. For example, if Gold puts his camel in the east, Silver can put both of her horses in the west, away from the camel. Gold may thus feel compelled to start with a centralized camel. Silver, on the other hand, is under no such constraint. If she wants to start with her camel in the east, it is too late for Gold to set up both of his horses in the west.

The suspicion, far from proved in practice, is that Silver gains more than two steps of time from setting up her pieces after knowing where Gold's pieces have been set up. In the early days of Arimaa, when everyone played defensively, the second setup didn't seem very useful, but now that everyone is coming to believe that attack is best, it seems that the second setup can give a leg up on attacking while leaving enough defense to hinder the opposing attack.

If we want to re-balance Arimaa, we can't start with the first move; we have to start with the setup itself. It might be that the problem we need to cure is not a Gold advantage, but a Silver one. Therefore Omar dialed your proposal back to the very beginning as follows:

1g Gold places ER
1s Silver places EMRR
2g Gold places MHRR
2s Silver places HHRR
3g Gold places HDRR
3s Silver places DDRR
4g Gold places DCRR
4s Silver places CCRR
5g Gold places CR and makes two steps
5s Silver makes four steps
etc.

I opposed this rule change. To me it seemed like adding a paragraph to the rules and three turns to every game for no purpose. It is not obvious that Arimaa is imbalanced at present. (If anyone can think of good terms of a bet, I will defy the conventional wisdom that Silver has an inherent advantage.) There might not be any problem that needs solving. But even if there is a problem, there is no guarantee that a staggered setup would solve it. That would itself take much study to prove.

I say that it is soon enough to fix the balance of Arimaa when we have evidence that it is broken.

Title: Re: variant start to even out theoretical advantag
Post by johncf1018 on May 27^th, 2011, 6:44pm

Interesting thoughts about setup and possible silver advantage. While I don't think it is appropriate or necessary for casual games, it does seem like an interesting idea for tournaments etc. I can appreciate the "if it ain't broke don't fix it" sentiment but still think it is worth investigating the correction of even theoretical imbalances as long as such measures are not overly cumbersome.

It looks like there are currently two possible sources of imbalance: setup order and turn order. Currently it sounds like they might more or less balance each other out and that any measures to fine tune balance must be address both.

Title: Re: variant start to even out theoretical advantag
Post by robinz on May 28^th, 2011, 5:32am

As Fritzlein says, it is not clear that either side should have an advantage, even theoretically. (My personal view is that Gold does - while it is true that Silver setting up second can get an advantage if Gold chooses a strange or imbalanced set up, I don't really see what Silver can do to gain an advantage if Gold starts with a sane and balanced setup such as the 99of9 one.) I'd rather look at the statistics from practice to see which side has won most games, and if this is statistically significant. I'm not sure where the thread is, but I do recall reading recently that, based on statistics from all games from this site, silver does win more often, but that this may be due to the fact that most often in casual games it is the stronger player who takes silver. (Certainly, having started playing autopostal games recently, I seem to be always given the gold pieces when my opponent has a higher rating, and vice versa.)

So basically I'm very much in the "it ain't broke, let's not try to fix it" camp, as is Fritz :) Certainly I wouldn't want to introduce any complicated rules to try to fix any perceived imbalance. One small modification that has occurred to me would be that one could have gold starting by just placing his elephant, then silver places their elephant, then gold places their camel, and so on. This would add a more dynamic aspect to the setup phase (both players get to react to the other), and I think could be fun to try, but I wouldn't be sure it was theoretically better. (It would seem to be that this proposal would be more favourable to gold than the current rules, by also allowing gold some room to react to silver's placement, so whether it would make the game more or less balanced probably depends on whether gold or silver has the advantage in the standard rules, which, as we've seen, is an open question.)

As a final thought, note that chess most undoubtedly does have a first-player advantage, yet this hasn't stopped people playing chess tournaments, up to world championship level, without suggesting alternative rules for the first few moves to balance this. Chess copes fine with it because, firstly, the first-player advantage isn't large even at the top level (and all but nonexistent for non-experts), and secondly, in a match or tournament one can try to give each player an equal number of games with each colour. Something similar could easily be done in arimaa as well - I don't believe anyone thinks that any imbalance in arimaa is large enough for this to be unsatisfactory.

Title: Re: variant start to even out theoretical advantag
Post by Hippo on May 28^th, 2011, 10:04am

Yes, it could be interesting to limit white pawns to 3rd rank on first turn.
I must say I was happy with color assignment during 2011 WC as I got silver against strong players in the preliminary and they started symmetric (except Harren). That allowed me to play my agressie setup in all the games. I expect I could be punished for that by silver opponent, but when my silver opponent chosen symmetric, I got both advantages :).

Title: Re: variant start to even out theoretical advantag
Post by Nazgand on May 31^st, 2011, 1:07pm

Gold's # of setup choices is 16!/(8!*2!*2!*2!*1!*1!)/2=32432400 (because of symmetry)
Silver's number of responses is 16!/(8!*2!*2!*2!*1!*1!)=64864800

Because no ties exist, one of them must have the advantage. Gold(whom I suspect has it) has it iff for any of the 32432400 setups no response allows silver to force a win.

Title: Re: variant start to even out theoretical advantag
Post by Fritzlein on May 31^st, 2011, 1:51pm

It's a forced loss for Gold, because Silver can force Gold to repeat position on move 7,413,547,680, but if Gold chooses the second best setup, Silver can win in only 247,118,256 moves. ;D

Title: Re: variant start to even out theoretical advantag
Post by Nazgand on May 31^st, 2011, 4:19pm

I really would like to see your source, because I have a strong suspicion that you made it up.

Title: Re: variant start to even out theoretical advantag
Post by Fritzlein on May 31^st, 2011, 4:59pm

on 05/31/11 at 16:19:13, Nazgand wrote:

I really would like to see your source, because I have a strong suspicion that you made it up.

You don't trust my result? :'( If I had known I would meet with such skepticism, I would have used my 3900 yottaflops computer for some other purpose than solving Arimaa. Geez, next time I'll just cure cancer instead. :P

Title: Re: variant start to even out theoretical advantag
Post by Nazgand on May 31^st, 2011, 5:45pm

Nice computer! Maybe you should let play online. You could not only win the Computer Challenge, but also tweak it it go for immobilizations whenever forcible. You will obviously never lose, cause when you are gold, someone's bound to make a mistake within the first 7,413,547,680 moves.

Are you sure it's only a yottaflop computer? Searching 7,660,665,936 ply deep an a yottaflop computer would (assuming a low average of 10000 unique moves per ply) would take(10^30642663744-1)/9999=10^3064266370 evaluations; if we assume an evaluation is 1 float operation, then it would take 10^3064266346 seconds=10^3064266338.5 years

Unless, of course... Time travel!

Title: Re: variant start to even out theoretical advantag
Post by lightvector on May 31^st, 2011, 7:05pm

Except that the state space of Arimaa is only around 10^43, meaning tons of repeated positions in all that computation. Maybe Fritz did some sort of clever retrograde analysis, or position caching. With a machine that can do 3900 yottaflops, I wouldn't be surprised. ;D

Title: Re: variant start to even out theoretical advantag
Post by Fritzlein on May 31^st, 2011, 7:36pm

Heh, I forgot to explain about the time travel bit. I thought it was implicit from megajester's post (http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=other;action=display;num=1306786202).

Actually the point of my first post (insofar as I had a point other than being a wise guy) was that Arimaa is not theoretically drawless in a practical sense of what a draw is. Yes, mathematically either Gold or Silver has a forced win, but that forced win may be so long as to be a draw in essence. It might be impossible to watch a perfect game of Arimaa being played out. I keep "forced win by repetition after a long, long game" separate in my mind as a third option along with "Gold has a forced win in a reasonable number of moves" and "Silver has a forced win in a reasonable number of moves".