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        Arimaa >> General Discussion >> Correlation of winning to first capture
        
(Message started by: Fritzlein on Apr 24^th, 2006, 10:26pm)

Title: Correlation of winning to first capture
Post by Fritzlein on Apr 24^th, 2006, 10:26pm

This post is in response to a request from 99of9 as to how winning correlates with making the first capture. In my database there are

28786 games, of which
20673 are rated games, of which
19601 had at least one capture, of which
19592 were decisive, of which
19526 didn't have a suicide on 2w or 2b.

I eliminated the suicide captures because I speculated that they were handicap games, but maybe I should have left them in because they involved a totally stupid bot. Anyway 13092 of the 19526 remaining were won by the side making the first capture, which is 67%.

To eliminate bot silliness, I further pared the remaining games to the 1340 which were between two humans. Of those, 936 were won by the player making the first capture, or 70%.

By the way, I defined the side to make the first capture as the other side from the side first losing a piece. Thus if you frame my rabbit on a trap, and I abandon it later, I credit you with making the first capture even though the piece died on my move. I hope that is what you wanted, 99of9. If this doesn't answer your question, I'll be happy to rejigger the query.

Title: Re: Correlation of winning to first capture
Post by frostlad on Apr 24^th, 2006, 11:40pm

Is there any trend as to what was lost when people were able to come back and win?
I would imagine that games when a rabbit was captured first that the win / loss ratio would be a lot closer.
It is interesting to see though that the odds of winning are so lopsided to the player making the first capture though.

Title: Re: Correlation of winning to first capture
Post by 99of9 on Apr 25^th, 2006, 12:08am

on 04/24/06 at 22:26:17, Fritzlein wrote:

I defined the side to make the first capture as the other side from the side first losing a piece. Thus if you frame my rabbit on a trap, and I abandon it later, I credit you with making the first capture even though the piece died on my move. I hope that is what you wanted, 99of9.

Yes, that's exactly what I was interested in. Thanks heaps for analysing this.

A 7:3 advantage wasn't quite as conclusive as I expected. But I think it is an important piece of data to give us hints about the value of the opening, and the value of material (relative to positional influence).

This could be dominated by totally uneven contests. Would it be possible to run the same query limited to humans rated within 200 rating points of one another.

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 25^th, 2006, 12:20am

on 04/25/06 at 00:08:19, 99of9 wrote:

Yes, that's exactly what I was interested in. Thanks heaps for analysing this.

My pleasure.

Quote:

This could be dominated by totally uneven contests. Would it be possible to run the same query limited to humans rated within 200 rating points of one another.

That cuts it to 674 games, of which 454 were won by the player making the first capture. That's 67%.

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 25^th, 2006, 12:40am

on 04/24/06 at 23:40:06, frostlad wrote:

Is there any trend as to what was lost when people were able to come back and win?

In order to have enough games to get a reasonable percentage, I dropped back to allowing bot involvement. The winning percentage as a function of what the first capture was is:

R 67%
C 63%
D 65%
H 68%
M 78%
E 85%

I know you're all dying to know about the games where the first capture was an elephant, and the other side won. The game numbers are

798
1253
1926
4124
7593
8107
11022
11957
13697
16451
16942
17126
17485
23621

Title: Re: Correlation of winning to first capture
Post by frostlad on Apr 25^th, 2006, 12:49am

wait, so those players lost an elephant first and still managed to win? Is that possible?

I spose we could save it for a new challenge eh evan? ( yes I'm from minnesota I do say eh )

Title: Re: Correlation of winning to first capture
Post by 99of9 on Apr 25^th, 2006, 4:50am

on 04/25/06 at 00:40:16, Fritzlein wrote:

R 67%
C 63%
D 65%
H 68%
M 78%
E 85%

So the first rabbit is more valuable than a dog? :-)

Title: Re: Correlation of winning to first capture
Post by mouse on Apr 25^th, 2006, 6:01am

on 04/25/06 at 04:50:45, 99of9 wrote:

So the first rabbit is more valuable than a dog? :-)

This is interesting.

The reason can be the first capute of a dog is followed by a capture by the other side in the next move. If this is the case the first capture is not really a capture but part of an exchange. Which could explain the relative low correlation.

Title: Re: Correlation of winning to first capture
Post by NIC1138 on Apr 25^th, 2006, 8:38pm

Following a bad influence from Swynndla, I´m developping the habit of suiciding my elephants in the winning move. I hope you are excluding these games from the statistics... 8) (of course this is not the FIRST kill/capture, but the last...)

Title: Re: Correlation of winning to first capture
Post by Janzert on Apr 25^th, 2006, 8:39pm

Very interesting results, Fritzlein.

As mouse sort of alluded to it would be interesting to see the numbers split out for outright captures and piece exchanges. I wonder how many of the 35% losses after first capture of a dog were when the player was already about to lose say a horse and took the dog for compensation. The tricky part of course would be defining an exchange, maybe the two pieces captured within 3(?) moves of each other.

Janzert

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 25^th, 2006, 9:15pm

on 04/25/06 at 20:39:14, Janzert wrote:

As mouse sort of alluded to it would be interesting to see the numbers split out for outright captures and piece exchanges.

That's beyond my ability to do with an MS Access query, but what about all the FAME values you calculated? For each game you could calculate the eventual winner's worst FAME score at any point. That would tell us how often the winner led materially wire-to-wire, and by how much they came back if not.

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 25^th, 2006, 9:21pm

on 04/25/06 at 06:01:11, mouse wrote:

The reason can be the first capute of a dog is followed by a capture by the other side in the next move. If this is the case the first capture is not really a capture but part of an exchange. Which could explain the relative low correlation.

An alternative explanation would be this: Often the first capture of a rabbit goes against the side that has an additional strategic disadvantage such as one of their pieces held hostage or framed. Really the rabbit is the second blow against that player, although it is the first capture.

A dog capture, meanwhile, might be more likely part of an open position, which could involve a simple exchange as already suggested, but also could be part of a larger race. In either case, the advantage for the capturing player isn't as likely to be a dog plus something as it is to be a rabbit plus something.

In particular, suppose I'm playing a dumb, materialistic bot. (Is there another kind?) If I frame a dog and then threaten a rabbit in the other trap, it will let me take the rabbit rather than abandoning the dog, so I have a rabbit capture plus a framed dog. ALSO if I frame a rabbit in one trap and threaten a dog in the other, the bot will give up its rabbit, in which case I again have a rabbit capture and a hostage dog. I actually get to take the dog only when (A) the position is open or (B) I have a yet larger threat to horse or camel, and the bot can't throw away a rabbit or cat to get the dog out of trouble.

Clearly this is all speculation, but it is speculation of the best kind, namely the kind that is extremely difficult to refute or verify. I can talk and talk, and never fear being proven wrong, because the truth is buried deep within the movelists generated according to the strategies of players past. ;D

Title: Re: Correlation of winning to first capture
Post by chessandgo on Apr 25^th, 2006, 10:25pm

that's poetry ! (I try to resist saying "amen") ::)

Title: Re: Correlation of winning to first capture
Post by Janzert on Apr 26^th, 2006, 2:19am

While not what Fritzlein asked about here's some related fame score information until I can look at that question directly.

First any game that had a non-zero fame score in move 2 was excluded. All other games were included. This left 29046 games that were looked at.

There were 9754 games (33%) that had a fame score that favored one side then the other at some point in the game, of those 4232 or 43% were won by the side initially favored by fame (i.e. making the first capture).

1963 (6.7%) had a neutral fame score throughout the game.

17429 games (60%) had a fame score that only favored one side. This is split into 13982 games (80% or 48% of the total) where fame correctly favored the winning side and 3447 games (20% or 12% total) where fame favored the side that lost.

Narrowing it to games involving at least one human and both players within 200 points it leaves 12030 games. With these results:

A mixed FAME score in 4447 games (36.97%) with correct first capture in 1947 (43
.78% of mixed)
FAME neutral throughout game in 725 games (6.03%)
FAME favored one side only in 6858 games (57%)
FAME was correct 5372 games (78.33% or 44.66% of the total)
FAME favored the wrong side in 1486 games (21.67% or 12.35% of the total)

Which are probably not a significant difference.

Janzert

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 26^th, 2006, 2:08pm

on 04/25/06 at 21:21:47, Fritzlein wrote:

Clearly this is all speculation, but it is speculation of the best kind, namely the kind that is extremely difficult to refute or verify.

Actually, I thought of a query I could do, namely correlating the first capture with the second capture. In 17398 rated games in which there were at least two piece deaths (and the first wasn't apparent suicide on move 2), I found that 58% of the time both of the first two piece deaths were from the same army. Breaking this down by the type of the first piece captured gives:

R 66%
C 59%
D 55%
H 48%
M 51%
E 87%

So when the first capture is a rabbit, it is usually a rabbit plus something, but when the first capture is a horse, it is usually a horse for something. In the latter case it is interesting that the player capturing the horse still wins 68% of the time despite having given up something in return 52% of the time.

One final interesting stat from the same dataset: There were 1929 cases of the first two deaths being rabbits on the same side, and only 575 cases of the first two deaths being rabbits from opposite sides. Starting the game with a rabbit trade is far less common than I would have expected.

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 26^th, 2006, 2:31pm

on 04/26/06 at 02:19:38, Janzert wrote:

FAME favored the wrong side in 1486 games (21.67%)

Wow, this really shows what a mediocre indicator material advantage is. I wonder what percentage of chess games are won by a player that never leads in material at any point in the game. I'll bet it is way less than 20%.

Title: Re: Correlation of winning to first capture
Post by chessandgo on Apr 26^th, 2006, 3:01pm

on 04/26/06 at 14:31:51, Fritzlein wrote:

Quite sure. But I don't understand, does the 20 % figure given by Clauchau correspond to that ??

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 26^th, 2006, 8:08pm

on 04/26/06 at 15:01:46, chessandgo wrote:

Quite sure. But I don't understand, does the 20 % figure given by Clauchau correspond to that ??

Hmm... I should be more precise. Janzert seems to be saying that in about 60% of Arimaa games, one side is materially ahead at some point while the other side is never materially ahead. It would be interesting to know how often this occurs in chess, as opposed to games in which material is even the whole way, or the material lead switches at some point.

However many such games there are in chess, it would be interesting to know further: Of those games, how often did the player that led but never trailed materially end up losing the game? For example, in the Muzio Gambit, Black my lead materially the whole way and yet be checkmated in short order.

I guess that in chess, if one side leads in material at some point and also never trails, they are more than 80% likely to have won the game, Muzio notwithstanding.

Title: Re: Correlation of winning to first capture
Post by mouse on Apr 27^th, 2006, 5:25am

on 04/26/06 at 20:08:54, Fritzlein wrote:

However many such games there are in chess, it would be interesting to know further: Of those games, how often did the player that led but never trailed materially end up losing the game? For example, in the Muzio Gambit, Black my lead materially the whole way and yet be checkmated in short order.

I guess that in chess, if one side leads in material at some point and also never trails, they are more than 80% likely to have won the game, Muzio notwithstanding.

I think you are thinking about top level chess between GM's. In that case you are probably correct in assuming very few winners never being ahead materially.

But I think in lower level of chess (which is probably more comparable to the level in arimaa) it is much more common to have winners never being ahead on material. As a result of blunders leading to a quick mate by the opponent. And because different types of gambits are a lot more common at lower levels of chess.

Title: Re: Correlation of winning to first capture
Post by mouse on Apr 27^th, 2006, 5:56am

on 04/26/06 at 14:08:20, Fritzlein wrote:

I wonder if there is a difference between h vs h games and h vs b games. I tend to believe there would be.

Title: Re: Correlation of winning to first capture
Post by Janzert on Apr 27^th, 2006, 12:26pm

Still not really answering Fritzlein's question, but something I wanted to look at for a while and I think it will answer the spirit of his question. ;) If not let me know Fritzlein and I can take a look at your actual question as well.

Here is a graph showing the percentage of times a side wins after getting that FAME score in a game.

Once again games with captures in move 2 are excluded. "All" includes all other games. HvH are human vs. human and BvB are bot vs. bot respectively. The "200" lines are limited to players within 200 rating points of each other. Each set is stopped when the number of samples falls below 100.

http://arimaa.janzert.com/othergraphs/fameacc.png

and one showing fame scores up to 15.

http://arimaa.janzert.com/othergraphs/fameacclow.png

There are several things that surprised me here. But possibly the most striking was that a large material advantage is much more important in HvH games than it is in BvB. Also related, note that a very small material advantage (1 rabbit or less) is most important in overall BvB but least important in closely rated BvB.

Janzert

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 27^th, 2006, 1:28pm

Janzert, that is a totally awesome graph. Thanks. I'm not sure what my question was, but I love your answer.

Would you mind spelling out the methodology in a little more detail? In particular, did you take all positions in your database, and for each positoin put it in a bin corresponding to its FAME score, and then calculate a win percentage for each bin? In hindsight, I wish I had thought to ask for that graph, because it is exactly what I want to know, and a much more interesting indicator than the crude notion of "first capture". You rock.

One thing that jumps out at me from the graph is that if FAME is computing any advantage whatsoever, even 0.1 rabbits, the side with the material lead has almost a 2/3 chance of winning.

Another interesting feature is the little jumps that occur at values of 1, 1.5, 2, 2.5, 3.1, and 5.6. These correspond to an initial capture of a rabbit, cat, dog, (something), horse, and camel. I can't figure out what the common material difference would be that results in an advantage of 2.5. Maybe a cat and a rabbit for nothing is a common advantage to have?

Title: Re: Correlation of winning to first capture
Post by Fritzlein on Apr 27^th, 2006, 3:31pm

on 04/27/06 at 05:56:52, mouse wrote:

I wonder if there is a difference between h vs h games and h vs b games. I tend to believe there would be.

Wow, you are totally right. I should have checked for this myself, but thanks for reminding me since I didn't.

Across all games there is a 58% percent chance that the first two captures will be of the same type, but in human games there is only a 48% chance the first two captures will be of the same type. If you restrict it to humans within 200 points of each other, it drops further to 45%.

However, that is still quite a high probability of consecutive captures rather than exchange. This is another number I'd like to compare to chess. My subjective guess is that only 20% of the time in chess the first two captures come from the same army rather than being an exchange.

Title: Re: Correlation of winning to first capture
Post by Janzert on Apr 27^th, 2006, 9:45pm

on 04/27/06 at 13:28:33, Fritzlein wrote:

Would you mind spelling out the methodology in a little more detail? In particular, did you take all positions in your database, and for each positoin put it in a bin corresponding to its FAME score, and then calculate a win percentage for each bin?

Sure, in trying to fit the description into one sentence I think I lost some important details.

I did recently refactor my database to seperate positions from moves, but I didn't use that for this graph at all.

The method you describe above would be a similiar idea but not quite the same as what I actually did here. In particular your method would show the percentage of wins after a game had been exactly at a FAME score. What I looked at was percentage of wins after achieving at least a certain FAME score. Unfortunately my rather bad description above does imply what you describe instead of what I actually did.

The actual steps were to examine each game and find the largest FAME score in favor of the winner and of the loser (or opposing the winner, same thing just different ways to look at it).

That's the easy part to describe. With the concept and the implementation of it matching. For the second part the abstract idea I'm trying to accomplish is not very straight forward in the implementation. So let me first describe the concept, then I'll give the actual implementation steps so someone can double check that I got it right.

Secondly, for any FAME score the percentage of sides (players) that made it at least to that score are noted.

Which of course is only conceptually what I'm doing. Here's the details of what I'm actually doing.

Take both lists of scores reached by the winning and losing sides in a game. While both lists have values, pop the value off the list with the smallest value. If both lists are the same pop it off both. If this score is different (larger) than the last iteration, note the score and percentage of winning sides remaining (number of winning scores divided by number of the total (winning and losing) scores remaining).

Janzert