Arimaa Forum - Print Page


    
      
        Arimaa Forum
        (http://arimaa.com/arimaa/forum/cgi/YaBB.cgi)
      

        Arimaa >> General Discussion >> average captures
        
(Message started by: knarl on Jul 12^th, 2006, 8:26pm)

Title: average captures
Post by knarl on Jul 12^th, 2006, 8:26pm

Gday everyone,

I was just wondering what the average number of captures for a game is? It's probably been discussed already, but I didn't look through past posts because I had another request anyway...

I'd really appreciate it if one or two knowledgable players might look at my game 34660 and add some comments. I played as silver and ended up winning by goal with only one rabbit capture total. I'm sure there's a point when I could have easily put more pressure on the bot, but where?

Cheers,
knarl.

Title: Re: average captures
Post by knarl on Jul 12^th, 2006, 9:43pm

thanks for the comments jbd, I replied to them.

knarl.

Title: Re: average captures
Post by Fritzlein on Jul 12^th, 2006, 9:48pm

I once calculated from the game archive that an average game of Arimaa has 8 captures. I forget how I did it, and what my methodology was. I may try to do it again, limiting my set of games to rated human vs. human encounters.

I also calculated from the database that 3% of Arimaa steps are captures. I wanted to compare the capture density to that of chess, but I couldn't find via Google search what percentage of chess moves are captures. Today, just by coincidence (I mean, before I read your post) I decided to calculate the density of capture in chess. Based on a sample of 60 games, I got that 19% of chess moves are captures, and that an average game of chess has 17 captures.

Now it is true that if 3% of Arimaa steps are captures, then about 12% of Arimaa moves are captures. However, I think the 3% to 19% comparison is more relevant than the 12% to 19% comparison, because most Arimaa bots use step-based search, and not many steps will cause alpha-beta cutoffs based on captures, in comparison to the percentage of chess moves that cause alpha-beta cutoffs.

Title: Re: average captures
Post by Fritzlein on Jul 13^th, 2006, 1:42pm

I just recalculated it. In 1689 rated human vs. human games, there were an average of 7.89 captures. This includes all types of terminations. If we limit it to games ending in goal or immobilization there are 1184 games with an average of 9.06 captures. The games left out of the second calculation are 326 time losses, 176 resignations, 2 repetitions, and 1 abort.

In the Wikipedia article I'm going to stand by the 8 captures per game figure in comparing to the chess average of 17 captures per game, because the majority of chess games aren't played out to conclusion on the board, so it seems like a fairer comparison to include all Arimaa games too.

By the non-tournament rules, it's possible for a game to have 32 captures and therefore end in a draw. In games where each side is playing to win, 31 captures must be the theoretical limit. The record so far in human games is 27, twice achieved.

http://arimaa.com/arimaa/gameroom/replayFlash.cgi?gid=7371

http://arimaa.com/arimaa/gameroom/replayFlash.cgi?gid=22999

In the former game should be considered the true record-holder, because if 99of9 had been thinking about posterity, I'm sure he would have suicided his elephant on the final play to secure a record 28 captures. ;-)

Title: Re: average captures
Post by Fritzlein on Jul 13^th, 2006, 2:05pm

I have no idea what this proves, but here is the average number of captures per game broken down by player. Only rated games against humans and terminating in "g" or "m" are counted.

10.4 blue22
10.3 Adanac
10.3 99of9
10.2 robinson
10.2 Fritzlein
10.0 Belbo
9.8 jdb
9.4 PMertens
9.4 Ryan_Cable
8.9 omar
8.6 chessandgo
8.4 naveed
8.3 Swynndla
8.2 OLTI
8.0 seanick

I think this is a random stat with no bearing whatsoever on anything.

Title: Re: average captures
Post by seanick on Jul 16^th, 2006, 5:08am

I think it shows agressiveness and/or the play strength to act on that aggressiveness. maybe agression is the wrong word. destructiveness? I don't know. but anyway ones tendency to try to kill pieces, as opposed to try to goal.

I think that makes perfect sense, because my aim is usually to try to goal (so therefore my aggressiveness is listed in last place). lately I have learned to be more agressive, especially about rabbits, since PMertens suggested the rabbit-pull exercise against bomb2005p2 which proved both challenging and very helpful. not coincidentally, I have also won a couple more human vs. human games recently.

The only factor that complicates the possible statistical significance of that number, is that it does not take into account the style and skill level of the opponent that each game was against. obviously that would have an effect on the number of captures.

Title: Re: average captures
Post by 99of9 on Jul 16^th, 2006, 7:35am

Just so the threads are linked, here is the reference for the previous analysis (much more detailed):

http://arimaa.com/arimaa/forum/cgi/YaBB.cgi?board=talk;action=display;num=1101770263;start=

Title: Re: average captures
Post by DorianGaray on Jul 16^th, 2006, 9:23am

on 07/13/06 at 13:42:40, Fritzlein wrote:

...By the non-tournament rules, it's possible for a game to have 32 captures and therefore end in a draw...

I am sorry but I don't think it should be a draw but a win by no move as it is the case when you capture all of an opponent's pieces. The first player who's turn it is and has no pieces loses by no move.

Title: Re: average captures
Post by Fritzlein on Jul 16^th, 2006, 9:43am

on 07/16/06 at 09:23:51, DorianGaray wrote:

Well, we might have to ask Omar for clarification. He has said in the past that a player who suicides his last piece doesn't lose instantly. That player loses only on the next turn, because it isn't until next turn that that player has no move. The opponent gets one more move in the mean time. See game 22999 for example.

Meanwhile both sides running out of rabbits seems to kick in instantly. So after the first player suicides his last piece, the second player has a turn to suicide his last piece (a rabbit, or it would have been drawn before all 32 captures), and the server declares it a draw before checking on the next turn for a loss by immobilization.

Note that to me it makes more sense for a player to lose the game by immobilization as soon as soon as he loses his last piece. The next move should not be necessary. But I believe Omar has set it up differently than my intuition would expect, so I think a 32-capture draw is indeed theoretically possible according to the official rules.

Title: Re: average captures
Post by Fritzlein on Jul 16^th, 2006, 9:53am

Thanks for the link to previous discussion, 99of9. I had totally forgotten about it. Re-reading what we said then reminds me that capture density (i.e. captures per move or captures per step) is more interesting to me than captures per game, although both are relevant.

Title: Re: average captures
Post by Fritzlein on Jul 17^th, 2006, 12:37pm

OK, I've run the numbers for capture density. The stats include only rated HvH games, but I include time losses and resignations along with goals and immobilization. Since this is captures per 100 steps, it takes the game length out of the equation.

Note that seanick is now high on the list. I think he was low before simply due to shorter games. Given the duration of his games, they tend to be bloodier than most. Also it sort of makes sense that jdb tops the list: his style is to accept trades, even disadvantageous ones, rather than sweat it out in a positional bind.

3.50 jdb
3.21 blue22
3.16 seanick
3.15 clauchau
3.11 Adanac
3.06 Swynndla
3.05 99of9
3.03 Fritzlein
3.01 Ryan_Cable
2.98 OLTI
2.96 megamau
2.90 robinson
2.89 chessandgo
2.84 Belbo
2.84 PMertens
2.83 naveed
2.78 Paul
2.78 Arimanator
2.66 omar
2.44 kamikazeking

Still, I'm not sure what this proves. It doesn't make sense to me that defensive players like Ryan and me are in the middle of the pack. And PMertens, with his wild flooding attacks, is low in capture density. Perhaps it is because he floods to gain control? But these numbers may be mostly just random and/or unenlightening.

Title: Re: average captures
Post by chessandgo on Jul 17^th, 2006, 1:23pm

one might be as well surprised by Kami's last position ... maybe kamikazing one's pieces is not the best way to capture much ? :)

Title: Re: average captures
Post by seanick on Jul 18^th, 2006, 12:05am

I think it makes perfect sense that Fritzlein at least is closer to the middle or high end of the capture density- for being a defensive player, it sure seems like he subscribes to the "the best defense is a good offense" school of thought. Kind of like some so-called "defensive" or reactive martial arts.... use the opponents attack to deliver their own strength plus your own into a crushing counterattack. like the second game I played against him, about move 10w: http://arimaa.com/arimaa/gameroom/replayFlash.cgi?gid=31540&s=b&client=1.

that was a learning experience. and didn't strike me as all that defensive, though I realize now that itwas just reactive.

anyway. I didn't realize I was so bloody, I thought I had trouble killing pieces. or are those mostly my pieces that are getting killed?

Title: Re: average captures
Post by Fritzlein on Jul 18^th, 2006, 2:21pm

Now that I think about it, seanick, it's not too weird that my capture density is middle of the road. I'm a defensive player, true, but I'm very capture-oriented. When I have an advantage, I don't try to use it to force goal, I use it to try to take pieces. Unlike goal-oriented people, I wait until I have accumulated a large material advantage before I try to force a win. And even when I'm behind material, it takes a lot to induce me to launch a desperation goal attack. Instead I usually try to catch up on material in any way possible. So my general reluctance to launch a goal attack ever probably drives up my capture numbers overall.

I guess the aggressive/defensive distinction is not the same as the goal/capture distinction. For example, I refer to Adanac and PMertens as aggressive players. Both are eager to take the fight to the opposing half of the board, rather than having it on their own side. But both are materialistic, compared to Robinson and Blue22. I wonder if the following categorizations are fair:

aggressive/goal: robinson, blue22
aggressive/capture: Adanac, PMertens
defensive/goal: Belbo
defensive/capture: Fritzlein, Ryan_Cable

It's a bit weird to think of a defensive player being focused on goal, since you can't threaten a goal without taking the fight to the other side of the board in some way, but I think it reasonably describes Belbo's penchant for worming through a rabbit without necessarily making a bunch of captures on the way.

Sadly, this doesn't help much to explain capture density. For that maybe other axes are more useful, along the lines of "comfortable with elephants apart" vs. "comfortable with elephants together". JDB is at the former extreme of the scale, whereas I tend to gravitate more towards the latter.

Title: Re: average captures
Post by Adanac on Jul 19^th, 2006, 8:29am

on 07/17/06 at 12:37:51, Fritzlein wrote:

3.50 jdb
3.21 blue22
3.16 seanick
3.15 clauchau
3.11 Adanac
3.06 Swynndla
3.05 99of9
3.03 Fritzlein
3.01 Ryan_Cable
2.98 OLTI
2.96 megamau
2.90 robinson
2.89 chessandgo
2.84 Belbo
2.84 PMertens
2.83 naveed
2.78 Paul
2.78 Arimanator
2.66 omar
2.44 kamikazeking

Most of my HvH games have been against players in the middle of the list, but comparing top vs. bottom I've definitely played more games against the high-capture players rather than low-capture players. I'm sure that's one of several reasons that I have a higher than average score.

Title: Re: average captures
Post by Fritzlein on Jul 21^st, 2006, 3:50pm

on 07/19/06 at 08:29:19, Adanac wrote:

I've definitely played more games against the high-capture players rather than low-capture players. I'm sure that's one of several reasons that I have a higher than average score.

Oh, dear. You had to go and complicate the problem by saying your capture density is due to who your opponents have been.

To account for who your opponents happen to have been, I set up the problem differently. I put a list of all the rated HvH games into a spreadsheet, and then made a capture density variable for each player. For each game, I calculated the expected number of captures as the average capture density of the two players times the number of steps in the game. Then I let the Excel solver tool adjust everyone's capture density variable to minimize the sum of the square error between actual captures and expected captures.

You will notice from the new list, Adanac, that the high capture density in your games is your fault, and not to be blamed on your opponents. :P I can't interpret these numbers any better than the previous list, but this method should correct for anything that is due to the style of your opponent rather than due to your own style.

4.06 jdb
3.71 clauchau
3.64 blue22
3.33 Adanac
3.29 99of9
3.17 Fritzlein
3.09 megamau
2.98 Ryan_Cable
2.96 seanick
2.93 chessandgo
2.82 robinson
2.80 Paul
2.77 OLTI
2.68 PMertens
2.65 Belbo
2.54 Arimanator
2.46 naveed
2.39 Swynndla
2.32 omar
2.03 kamikazeking

P.S. Based on our last game, Adanac, we are both to blame for a high number of captures. When I got the material advantage, I didn't go for a direct goal, but rather tried to retain control of the game. Meanwhile you tried to prolong the game to 60 moves for Player of the Month. Thus I was forced to take a bunch of your pieces before winning.

Title: Re: average captures
Post by 99of9 on Jul 21^st, 2006, 6:20pm

Is there an easy way to split this figure up:

*captures of my pieces when I won
* captures of my pieces when I lost
* captures of his pieces when I won
* captures of his pieces when I lost

That might help us analyse the results more easily.

Title: Re: average captures
Post by Fritzlein on Jul 31^st, 2006, 5:37pm

Ok, 99of9, you had to ask. For this calculation, I did not take into account the opponents in each game. I merely calculated the average captures/losses for each player in games they won/lost.

The first column is how many pieces per game the player took.
The second column is how many pieces per game the player lost.
The third column is how many pieces per game the winner captured more than the loser. (A positive number for everyone, i.e. nobody wins while capturing less than their opponent, or loses while capturing more, at least on average.)

I find extremes in the third column interesting. Among losers, omar and Fritzlein capture nearly as many pieces as their opponents, which suggests that they frequently lose by goal when up on material. Seanick is at the other extreme of losers, suggesting that when he loses, he's way behind on material. Swynndla is a distant second in losing seanick-style.

Among winners, kamikazeking, clauchau, and blue22 capture barely more than their opponents, suggesting they win by goal more than by material domination. The other extreme of winners is again seanick, suggesting that he accumulates a large material advantage before winning. Others who win with material advantage seem to be Swynndla, 99of9, and Fritzlein.

This breakdown seems more enlightening than previous ones, because there can be different axes for winning and losing. It seems Fritzlein wins materially and loses by goal, seanick wins materially and loses materially, jdb wins by goal and loses materially, and Adanac tends to win by goal and lose by goal. Interesting stuff...

6.09 2.87 3.22 99of9 wins
4.15 6.20 2.04 99of9 losses
5.90 3.66 2.24 Adanac wins
3.94 5.71 1.77 Adanac losses
5.09 1.95 3.14 Arimanator wins
1.61 4.00 2.39 Arimanator losses
5.83 3.19 2.64 Belbo wins
3.83 6.00 2.17 Belbo losses
5.00 2.88 2.12 blue22 wins
3.18 5.34 2.16 blue22 losses
5.05 2.24 2.81 chessandgo wins
2.89 4.80 1.91 chessandgo losses
3.55 1.50 2.05 clauchau wins
2.67 5.43 2.76 clauchau losses
6.32 3.15 3.17 Fritzlein wins
4.85 5.52 0.67 Fritzlein losses
5.15 2.78 2.37 jdb wins
3.36 7.00 3.64 jdb losses
4.03 2.53 1.50 kamikazeking wins
1.59 4.51 2.92 kamikazeking losses
5.40 3.07 2.33 megamau wins
2.97 4.97 2.00 megamau losses
5.32 2.25 3.07 naveed wins
2.61 3.93 1.32 naveed losses
5.25 2.57 2.68 OLTI wins
1.65 5.07 3.42 OLTI losses
4.83 1.77 3.06 omar wins
3.24 3.98 0.75 omar losses
5.27 2.77 2.50 Paul wins
2.92 5.69 2.77 Paul losses
5.49 2.59 2.90 PMertens wins
3.06 5.93 2.88 PMertens losses
5.28 2.89 2.38 robinson wins
4.36 7.34 2.99 robinson losses
5.46 2.76 2.69 Ryan_Cable wins
3.03 5.63 2.60 Ryan_Cable losses
6.38 2.88 3.50 seanick wins
1.24 6.35 5.11 seanick losses
4.97 1.60 3.37 Swynndla wins
2.54 6.40 3.86 Swynndla losses

Title: Re: average captures
Post by seanick on Jul 31^st, 2006, 11:24pm

Indeed, that is the most enlightening that this class of statistic has ever been.

hmm... how to increase my win by goal, and decrease my loss of material...

solution: assimilate the style of other goal-winner low-material-loser player(s) - look for recent games by
omar and fritz to analyze.

also, realize that my games are the bloodiest. but... how else to clear the path for a rabbit? otherwise, the other pieces keep getting in the way :)