Rook+pawns vs two minors

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lkaufman
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Re: Rook+pawns vs two minors

Post by lkaufman » Mon Jan 18, 2010 2:14 am

Thanks for the examples. In the Doch versions out now, the pieces are reasonably "centered", but the pawn is not, so that even though in Doch 1.2 the Exchange was nominally worth two pawns, in reality it was more like 1.75 pawns, which (together with the factors you mention) explains the scores. In our current private version, the pawn is reasonably centered, and so the Exchange gets a higher "real" value, close to the nominal 1.97 pawns on average. This will be one of the improvements in our next release.

BubbaTough
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Re: Rook+pawns vs two minors

Post by BubbaTough » Mon Jan 18, 2010 2:39 am

lkaufman wrote:Thanks for the examples. In the Doch versions out now, the pieces are reasonably "centered", but the pawn is not, so that even though in Doch 1.2 the Exchange was nominally worth two pawns, in reality it was more like 1.75 pawns, which (together with the factors you mention) explains the scores. In our current private version, the pawn is reasonably centered, and so the Exchange gets a higher "real" value, close to the nominal 1.97 pawns on average. This will be one of the improvements in our next release.
I am not sure what you mean when you say the pawn is reasonably centered. It makes some sense to me for piece square tables or something, but for other things like passed pawns or contribution to king shelter, you would have to take a little speed hit to try to balance things like that. For example: you could subtract when the pawn isn't passed as well as adding when it is, but it slows your program down a little. Are you doing things like that?

-Sam

lkaufman
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Re: Rook+pawns vs two minors

Post by lkaufman » Mon Jan 18, 2010 2:58 am

No, I just mean that we make some effort to choose values in the piece location tables and for "average" mobility that will average out to zero over all positions searched. So for example maybe the "average" extra pawn is on the fourth rank and is a candidate for being passed, so we would use values in the pawn table that make this assumption. We're not very precise about it, we just make some effort to keep the nominal values of the chessmen in line with their average "real" value. It's mostly important for changing the eval; for example if we double the bishop mobility it should not affect the average value of a bishop if we have centered it well.

BubbaTough
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Re: Rook+pawns vs two minors

Post by BubbaTough » Mon Jan 18, 2010 3:09 am

lkaufman wrote:No, I just mean that we make some effort to choose values in the piece location tables and for "average" mobility that will average out to zero over all positions searched. So for example maybe the "average" extra pawn is on the fourth rank and is a candidate for being passed, so we would use values in the pawn table that make this assumption. We're not very precise about it, we just make some effort to keep the nominal values of the chessmen in line with their average "real" value. It's mostly important for changing the eval; for example if we double the bishop mobility it should not affect the average value of a bishop if we have centered it well.
OK I understand now. The issue is, the chance of a pawn being passed when you are up 2 pawns is a lot different than the chance of a pawn being passed when you are up 1 pawn. Thus, in some sense the 2nd pawn is on average more valuable than the first pawn. Unless you were really making a lot of effort, I would guess that for material imbalance situations like 2 pawns and a piece for a rook will continue to have, on average, a large positional aspect that off-centers things (in endgame). In middlegames, the same is true, except for when you are down 2 pawns it is often going to be a king safety issue instead of a passed pawn issue unbalancing things. Both factors can often have very large values, so this is not trivial.

If I am right about this then for basically all strong programs, even ones like yours where there has been an effort to balance things, the correct material balance scores with uneven pawns is going to be strongly affected by king safety and passed pawn elements in the evaluation function, and looking at the numbers by themselves without paying attention to these factors will be significantly misleading. It also means the correct material scores are going to be different for different programs, depending on how they handle these evaluation factors. For example, if a good king shelter is added to the score instead of a bad king shelter being subtracted from a score, it will effect the appropriate average value of a pawn in middlegame.

-Sam

lkaufman
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Re: Rook+pawns vs two minors

Post by lkaufman » Mon Jan 18, 2010 3:45 am

It's true that the second extra pawn is more likely to be passed than the first, but Doch considers candidate passed pawns to be almost as important as passed pawns, so I don't think this factor significantly skews the numbers, at least for us. I did a test on a standard theoretical position from the Gruenfeld where White has a knight and two pawns for a rook, with no passed pawns and no net major positional advantages for either side. r4r1k/p5bp/4q1p1/4Np2/4nB2/4PN1P/PPQ2PP1/5RK1 w - - 0 21. I checked the latest version of every strong program I had. Every single one prefers White by at least a quarter pawn (2 ply searches) except Zappa Mexico II which narrowly prefers Black (-0.03), and almost all want to trade queens by Qc4, which no human master has ever done; a human GM would definitely prefer Black after this move. So in this rather normal case, the computers all value the Exchange at significantly less than two pawns, whereas the humans would rate it at worth over two pawns here! This is the opposite of my general conclusion that humans rate the Exchange at 1 3/4 pawns while computers rate it at 2. I'm rather puzzled by this; if computers normally value the Exchange higher than humans, why do they value it lower in this case? It's strange.

BubbaTough
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Re: Rook+pawns vs two minors

Post by BubbaTough » Mon Jan 18, 2010 5:12 am

lkaufman wrote:It's true that the second extra pawn is more likely to be passed than the first, but Doch considers candidate passed pawns to be almost as important as passed pawns, so I don't think this factor significantly skews the numbers, at least for us. I did a test on a standard theoretical position from the Gruenfeld where White has a knight and two pawns for a rook, with no passed pawns and no net major positional advantages for either side. r4r1k/p5bp/4q1p1/4Np2/4nB2/4PN1P/PPQ2PP1/5RK1 w - - 0 21. I checked the latest version of every strong program I had. Every single one prefers White by at least a quarter pawn (2 ply searches) except Zappa Mexico II which narrowly prefers Black (-0.03), and almost all want to trade queens by Qc4, which no human master has ever done; a human GM would definitely prefer Black after this move. So in this rather normal case, the computers all value the Exchange at significantly less than two pawns, whereas the humans would rate it at worth over two pawns here! This is the opposite of my general conclusion that humans rate the Exchange at 1 3/4 pawns while computers rate it at 2. I'm rather puzzled by this; if computers normally value the Exchange higher than humans, why do they value it lower in this case? It's strange.
OK, this is a good example. I will show it for any audience members :).

[d] r4r1k/p5bp/4q1p1/4Np2/4nB2/4PN1P/PPQ2PP1/5RK1 w - - 0 21.

Doch 1.2 at 2 ply +0.28 1. Qc4 Qxc4
Stockfish1.6 at 2 ply: +0.52 playing 1. Qc4 Rae8
Spark 0.3a at 2 ply: +0.24 playing 1. Qa4 Rab8
Rybka 2.3.2a at 2 ply: +0.23 playing 1. Qa4
My current program at 2 ply: +0.55 1. Qc7 g5

So lets see, no passed pawns, but predictably 2 candidate passed pawns. White's pieces have no obvious advantages, black's king safety is maybe a little worse but no big deal, so I would say this kind of position emphasizes (from a computer program's perspective) the side with the rook because unlike the positions I gave, white has not gotten his knight on a nice outpost like d4 (yet) or gotten a nice protected passed pawn.

Nevertheless, as you say, most computer programs value the exchange at about 1.5 - 1.75 (hmmm, kind of like humans would estimate :)). I think its a little unfair to compare a 2 ply judgement of this position with a human judgement, because I would guess the only reason the humans like the rooks more here is because it is obvious to them they are going to get very active on those open files, but with only a 2 ply search, the computers are no seeing that yet. So lets check scores a little bit deeper.

Doch1.2 at 12 ply: +0.01 Ne5c4
Stockfish1.6 at 12 ply +0.52 Qa4
Spark0.3a at 12 ply +0.31 Qa4
Rybka2.3.2a at 12 ply: +0.10 Qa4
My program at 12 ply: +0.18 Qa4

OK, so programs still like white a little bit...a big correction for Dock and my program, but not that much for the others.

So basically, as you say, the computers prefer piece and 2 pawns to a rook. Well, that is not surprising to me at all, since that is what I think as a human, and what I programmed as a chess programmer :). I never realized that other people thought computers did this differently until I saw this thread. In fact, when playing on ICC, I occasionally run into folks that tell me that analyzing with Rybka has taught them to value an exchange as worth less than a piece and 2 pawns. I would guess that if you put up a bunch of positions you will find that regardless of what you may read in the piece value code, when you factor the positional factors in you will find that in most normal positions strong programs will value the exchange as less than 2 pawns in practice. Which makes sense, given most strong chess players believe it is worth less than 2 pawns in practice.

-Sam

BubbaTough
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Re: Rook+pawns vs two minors

Post by BubbaTough » Mon Jan 18, 2010 5:22 am

A minor tweak of the position for my amusement:

[d]r4r1k/p5bp/4q1p1/5p2/Q2NnB2/2P2N1P/PP3PP1/5RK1 w - - 1 21
Doch1.2 at 2 ply: +.61 (1...Qb6 2 Qc2)

-Sam

Michael Sherwin
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Re: Rook+pawns vs two minors

Post by Michael Sherwin » Mon Jan 18, 2010 5:35 am

metax wrote:How many pawns does the side with the rook need to make the material configuration rook+pawns vs. two minors about equal, excluding bonuses and penalties based on bishop pair and/or rook pair redundancy issues in your opinion?
RomiChess gets in a lot of these type games on both sides of the issue and wins far more than she looses. Romi's weaknesses seem to be elsewhere. The number of pawns does not seem to matter nearly as much as other factors. Romi usually wins with a more active king and/or more dangerous pawns, even if they are fewer in number. If I can find a couple of examples I'll post them.
I hate if statements. Pawns demand if statements. Therefore I hate pawns.

Michael Sherwin
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Re: Rook+pawns vs two minors

Post by Michael Sherwin » Mon Jan 18, 2010 6:45 am

Michael Sherwin wrote:
metax wrote:How many pawns does the side with the rook need to make the material configuration rook+pawns vs. two minors about equal, excluding bonuses and penalties based on bishop pair and/or rook pair redundancy issues in your opinion?
RomiChess gets in a lot of these type games on both sides of the issue and wins far more than she looses. Romi's weaknesses seem to be elsewhere. The number of pawns does not seem to matter nearly as much as other factors. Romi usually wins with a more active king and/or more dangerous pawns, even if they are fewer in number. If I can find a couple of examples I'll post them.
This is all I found on short notice, an almost example, in that Romi sacs an exchange for one pawn. Romi then wins more pawns.

Diablo does not start to evaluate serious trouble until about move 36 while Romi evaluates a large advantage from the beginning and sticks with it.

Even Stockfish has to think awhile before locking into 20 Rf7.

Although, I am not claiming this to be a really good example it nevertheless illustrates my above points (number of pawns in compensation not that important, active king important and better piece(s) important, implied).

[D]r2bk2r/1p1n1pp1/p2B4/7p/4PRbP/1NN5/PPP3K1/5R2 w kq - 2 20

Last Move: 19...Nd7

[Event "WBEC17 3rd Division"]
[Site "AMD-PHENOM-3100"]
[Date "2010.01.16"]
[Round "6.3"]
[White "RomiChess P3k-x64"]
[Black "Diablo 0.5.1"]
[Result "1-0"]
[Number "863"]
[TimeControl "40/1200"]
[Annotator "2. -0.39 11... -0.62"]

1.e4 c5 2.Nf3
{-0.39/8 0.2}
2...d6 3.d4
{-0.89/8 0.1}
3...cxd4 4.Nxd4
{-0.97/8 0.2}
4...Nf6 5.Nc3
{-0.69/9 0.2}
5...a6 6.Be2
{-0.73/10 0.2}
6...e5 7.Nb3
{-1.02/10 0.3}
7...Be7 8.O-O
{-1.02/9 0.1}
8...Be6 9.f4
{-0.72/9 0.1}
9...Qc7 10.g4
{-0.25/8 0.1}
10...exf4 11.Bxf4
{-0.14/8 0.2}
11...h6
{-0.62/11 32}
12.h4
{+0.43/15 37}
12...Qb6+
{-0.22/11 32}
13.Qd4
{+0.12/17 48}
13...Bd8
{-0.51/11}
14.Kg2
{+0.34/16 48}
14...Nxg4
{-0.55/11 34}
15.Qxb6
{+0.26/16 48}
15...Bxb6
{-0.25/13 34}
16.Bxg4
{-0.17/17 49}
16...Bxg4
{-0.10/14}
17.Bxd6
{+0.17/17 46}
17...Bd8
{-0.12/13 36}
18.Rf4
{+0.54/17 46}
18...h5
{+0.07/13 36}
19.Raf1
{+0.72/17 41}
19...Nd7
{-0.38/13 5}
20.Rxf7
{+1.39/18 46}
20...Bf6
{-0.49/14 38}
21.R1xf6
{+2.20/19 46}
21...gxf6
{-0.53/15 0.8}
22.Rg7
{+1.94/18 40}
22...b6
{-0.52/14 9}
23.Nd5
{+1.67/18 47}
23...O-O-O
{-0.49/13 4}
24.Nd4
{+1.67/17 32}
24...Kb7
{-0.60/15 21}
25.Nxf6
{+1.80/18 41}
25...Ka8
{-0.41/15 14}
26.Nd5
{+1.53/19 45}
26...Rhg8
{-0.17/14 48}
27.Rxg8
{+1.92/19 45}
27...Rxg8
{-0.21/15 12}
28.Kf2
{+1.77/18 45}
28...Kb7
{-0.21/14 15}
29.Ke3
{+1.56/18 45}
29...Bh3
{-0.19/14 55}
30.Nf4
{+1.50/18 45}
30...Bg4
{-0.22/15 19}
31.c4
{+1.56/18 30}
31...Re8
{-0.17/14 1:00}
32.b4
{+1.59/19 41}
32...Ne5
{-0.27/15 1:02}
33.Bxe5
{+1.70/20 29}
33...Rxe5
{-0.29/17 40}
34.Nd5
{+1.79/20 32}
34...Re8
{-0.33/16 41}
35.c5
{+1.99/21 26}
35...bxc5
{-0.50/17 52}
36.bxc5
{+1.94/21 25}
36...Rf8
{-0.76/16 59}
37.c6+
{+2.30/18 20}
37...Kc8
{-0.96/17 1:12}
38.Nf5
{+3.23/18 15}
38...Bxf5
{-1.96/17 1:25}
39.exf5
{+4.01/21 14}
39...Kb8
{-3.88/19 1:41}
40.f6
{+5.39/22 16}
40...Kc8
{-4.88/18 2:10}
41.Kf4
{+6.00/21 37}
41...a5
{-5.05/18}
42.Kf5
{+6.56/21 27}
42...Rd8
{-5.43/18 11}
43.Nf4
{+6.85/21 38}
43...a4
{-6.84/16 0.4}
44.f7
{+12.18/19 38}
44...Rf8
{-11.44/15 27}
45.Kf6
{+13.49/21 38}
45...Rxf7+
{-15.00/20 1.1}
46.Kxf7
{+16.20/24 28}
46...Kc7
{Xboard adjudication}
1-0



Stockfish 1.6s JA 64bit:
1 00:00 224 0 +2.38 Rf4xf7
2 00:00 260 0 +2.38 Rf4xf7 Bd8xh4 Rf7xg7
3 00:00 848 0 +0.44 Rf4xf7 Bd8f6 Rf1xf6 g7xf6
4 00:00 1.750 0 +0.44 Rf4xf7 Bd8f6 Rf1xf6 g7xf6
4 00:00 2.960 0 +0.52 e4e5 Bg4e6 Nb3d4 Be6c4
5 00:00 4.211 0 +0.48 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1
6 00:00 7.607 475.437 +0.32 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1 Bd8b6
7 00:00 24.234 1.514.625 +0.44 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1 Rh8h6 b2b3 Bd8a5 Nc3e4 Rh6g6+ Ne4g5 Bc4d5+ Kg2g3
8 00:00 90.659 1.928.914 +0.52 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1 Rh8h6 Kg2h3 Ra8c8 Rd1g1 g7g6 Kh3g3 Bd8b6
9 00:00 155.045 2.461.031 +0.28 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1 Rh8h6 Nd4f5 Rh6g6+ Kg2h2 Bc4e6 Nc3d5 Be6xd5 Rd1xd5 Ra8c8
10 00:00 218.413 2.323.542 +0.28 e4e5 Bg4e6 Nb3d4 Be6c4 Rf1d1 Rh8h6 Nd4f5 Rh6g6+ Kg2h2 Bc4e6 Nc3d5 Be6xd5 Rd1xd5 Ra8c8
10 00:00 309.442 2.475.536 +0.48 Rf4xf7 Bd8f6 Rf1xf6 g7xf6 Rf7g7 b7b6 Nc3d5 000 Nb3d4 Rd8e8 Bd6e7
11 00:00 680.481 2.908.038 +1.05 Rf4xf7 Bd8f6 Rf1xf6 g7xf6 Rf7g7 Ra8c8 Rg7e7+ Ke8d8 Nb3a5 Rc8xc3 b2xc3 Kd8c8 Kg2f2 Rh8g8 Re7f7
12 00:00 939.533 3.011.323 +0.88 Rf4xf7 Bd8f6 Rf1xf6 g7xf6 Rf7g7 Ra8c8 Nb3a5 b7b6 Rg7e7+ Ke8f8 Re7xd7+ Kf8g8 Rd7a7 b6xa5 Ra7xa6 Rc8d8 Nc3d5
13 00:01 3.446.105 3.619.858 +1.25 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h2 Ra8c8 Nb3d4 Rc8c4 Rf7f4 Bd8xh4 Nc3d5 Bh4d8 c2c3 Rc4a4 a2a3 Ra4c4
14 00:01 4.874.370 3.589.374 +1.37 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h2 Ra8c8 Nb3d4 Rc8c4 Rf7f4 Bd8xh4 Nc3d5 Bh4d8 c2c3 Rc4a4 a2a3 Ra4c4 Rf4f2 Rg6g5 b2b3 Rc4c8
15 00:02 6.917.448 3.574.908 +1.05 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h2 Ra8c8 Nb3d4 Rc8c4 Rf7f4 Bd8xh4 Nc3d5 Bh4d8 c2c3 Rc4a4 a2a3 Ra4a5 Rf4f8+ Nd7xf8 Rf1xf8+ Ke8d7 Rf8xd8+ Kd7xd8 Bd6c7+ Kd8d7 Bc7xa5 Bg4e6
16 00:03 12.570.111 3.628.784 +1.29 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h2 Ra8c8 Nb3d4 Rc8c4 Rf7f4 Bd8xh4 Nc3d5 Bh4g3+ Kh2xg3 Bg4e6+ Kg3h4 Be6xd5 Kh4xh5 Rg6g2 Rf1d1 Ke8d8 b2b3 Rc4c3 Rf4g4 Rc3h3+ Kh5g5 Rg2f2
17 00:07 27.878.320 3.746.079 +0.80 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h2 Ra8c8 Nb3d4 Rc8c4 Rf7f4 Bd8xh4 Nc3d5 Bh4g3+ Kh2xg3 Bg4e6+ Kg3h4 Be6xd5 c2c3 Nd7c5 Kh4xh5 Rg6h6+ Kh5g4 Nc5e6 Nd4xe6 Bd5xe6+ Kg4g5 Rc4xf4 Rf1xf4 Rh6h2 Rf4b4 Rh2g2+ Kg5h5 b7b5
18 00:20 76.034.599 3.843.819 +1.01 Rf4xf7 Rh8h6 e4e5 Rh6g6 Kg2h1 Bg4h3 Rf1f2 Ra8c8 Rf7f4 Bh3g4 Nc3d5 Bg4e6 Nd5e3 Rg6g3 Nb3d4 Rg3xe3 Nd4xe6 Nd7xe5 Ne6xg7+ Ke8d7 Bd6xe5 Re3xe5 Rf2d2+ Kd7e7 Ng7f5+ Ke7e6 Nf5d6 Bd8c7 Rf4d4



FEN: r2bk2r/1p1n1pp1/p2B4/7p/4PRbP/1NN5/PPP3K1/5R2 w kq - 0 1

RomiChess96:
1 00:00 63 63 +2.06 f4f7
2 00:00 195 195 +2.92 f4f7 d8h4
3 00:00 542 542 +2.60 f4f7 d8h4 f7g7 a8d8
4 00:00 2.596 2.596 +1.34 f4f7 d8f6 f1f6
5 00:00 18.231 18.231 +2.15 f4f7 d8f6 f1f6 g7f6 f7e7 e8d8 b3d4
6 00:00 20.065 20.065 +2.15 f4f7 d8f6 f1f6 g7f6 f7e7 e8d8 b3d4
7 00:00 41.109 4.110.900 +2.16 f4f7 d8f6 f1f6 g7f6 f7e7 e8d8 b3d4 h8f8 c3d5
8 00:00 347.112 3.856.800 +1.07 f4f7 h8h6 e4e5 h6g6 g2h2 d8f6 f1f6
9 00:00 452.259 3.768.825 +1.57 f4f7 h8h6
10 00:00 643.310 3.573.944 +1.57 f4f7 h8h6 e4e5
11 00:00 932.990 3.588.423 +1.57 f4f7 h8h6 e4e5 h6g6
12 00:01 3.915.635 3.659.471 +1.07 f4f7
13 00:01 4.856.743 3.679.350 +1.07 f4f7 h8h6 e4e5 h6g6
14 00:03 10.874.066 3.698.661 +1.07 f4f7 h8h6 e4e5
15 00:07 25.887.544 3.698.220 +1.61 f4f7 h8h6 e4e5 h6g6 g2h2 g6f6 f7f6 g7f6 e5f6 d7f6 b3c5 b7b6 c5d3 g4e6 d3f4 f6g4 h2g1
16 00:08 31.005.628 3.875.703 +1.61 f4f7 h8h6 e4e5 h6g6 g2h2 g6f6 f7f6 g7f6 e5f6 d7f6 b3c5 b7b6 c5d3 g4e6 d3f4
17 00:16 62.365.844 3.897.865 +1.41 f4f7 h8h6 e4e5 h6g6 g2h2 a8c8 b3d4 c8c4 f7f4 d8h4 c3d5 h4d8 d5e3 c4a4 e3g4 g6g4 f4g4
18 00:28 109.814.008 3.921.928 +1.41 f4f7 h8h6 e4e5 h6g6 g2h2 a8c8 b3d4 c8c4 f7f4 d8h4 c3d5 h4d8 d5e3 c4a4 e3g4
I hate if statements. Pawns demand if statements. Therefore I hate pawns.

Uri Blass
Posts: 8594
Joined: Wed Mar 08, 2006 11:37 pm
Location: Tel-Aviv Israel

Re: Rook+pawns vs two minors

Post by Uri Blass » Mon Jan 18, 2010 7:21 am

diep wrote:
metax wrote:How many pawns does the side with the rook need to make the material configuration rook+pawns vs. two minors about equal, excluding bonuses and penalties based on bishop pair and/or rook pair redundancy issues in your opinion?

Traditional count implies a value of 1 pawn, which is definitely too little.
Kaufman values imply 1.5 pawns.
The following material values

{ P=1; N=3.75; B=3.75; R=5.50; Q=10.75 }

are basically increased Kaufman values, minor pieces and rook increased by 0.5 pawns and queen increased by 1 pawn. These values keep the value of the Exchange and the value of Q vs 2R, increasing the values of minor piece vs pawns, rook vs two minor pieces and queen vs three minor pieces. The first and the last are clearly favorable because the minor pieces are often undervalued in these cases. However, the values would imply a value of 2.5 pawns to equal two minors with the rook. Including a bishop pair advantage of 0.5 pawns and a rook pair penalty of 0.25 pawns and assuming a rather closed position, there may be cases where the rook needs over 3.5 pawns to equal the two minors, which seems almost certainly exaggerated.

So what do you think about all this?
typically you see that most top engines use rook is 2.5 pawns better than a piece. A few use 2.2 pawns, only amateur engines use less than 2.2 pawns.

Again just material we speak about nothing else. Some rooks can get a mighty amount of bonus when put on 7th rank in todays software.

Thanks,
Vincent
This is simply not correct and all the examples that people posted in this thread show that programs evaluate the difference between rook and bishop as not more than 2 pawns.

Maybe you take basic value too seriously and if you give knight and bishop higher mobility score than you give to rooks then you need to compensate the rook.

Uri
Uri

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