Details of Computer Results


A is Alfil, a jump to (2,2), two squares diagonally. With no qualifications, it means a piece that can move and capture in all four directions: (2,2), (-2,2), (2,-2), and (-2,-2). On the chessboard, this would be from e4 to g6, c6, g2, and c2.

mA is a piece that moves like an Alfil but cannot capture.

cA is Alfil capture without Alfil movement.

fA is Alfil forwards movement and capture.

bA is the Alfil in retreat.

D is Dabaaba, a jump to (0,2), two squares Rookwise.

rlD ("right+left") is the sideways movement of the Dabaaba.

W is Wazir, (0,1), one square Rookwise.

F is Ferz, (1,1), one square diagonally.

H is (0,3), three squares Rookwise.

G is (3,3), three squares diagonally.

L is (1,3), the "long Knight".

J is (2,3), a piece which is too big for the 8x8 board.

_v_ is the word "versus", and separates the description of one side's army from the description of the other side's army.

And here's a diagram to make it easy to remember:

    G J L H L J G
    J A N D N A J
    L N F W F N L
    H D W . W D H
    L N F W F N L
    J A N D N A J
    G J L H L J G

    fG fJ fL fH fL fJ fG
    rJ fA fN fD fN fA lJ
    rL rN fF fW fF lN lL
    rH rD rW .. lW lD lH
    rL rN bF bW bF lN lL
    rJ bA bN bD bN bA lJ
    bG bJ bL bH bL bJ bG

Experimental Controls

A computer plays both sides of the game; the pieces on a1, b1, and c1 in the opening position are permutated six ways, in different games; in the opening position, the moves 1 d2-d4, d7-d5 have already been played.

The program applies a random factor to its evaluations (so that we don't get thousands of copies of the same game!); the results are scanned to check for multiple instances of the same game.

13842 games of standard chess were played, of which White won 49 per cent. The game is declared drawn at move 95; at any time after 45 moves have been made, if one side retains, for five moves in a row, an advantage equivalent to a Rook or more, the game is adjudicated a win for that side.

In most test runs, one side uses the normal army and the other side has one type of piece replaced by a "strange" piece; the program believes that the "strange" piece has the same value as the piece it replaces.

For example, when playing NA versus N, both of White's Knights are replaced by the NA (Knight plus Alfil) piece, and the computer believes that the NA has the same value as the Knight; but when playing NA_v_R, both White's Rooks are replaced by NA, and the computer believes that the NA has the same value as a Rook.

This belief could be a self-fulfilling prophecy, of course. White could freely trade the NA for a Knight or Bishop, thus dragging the results down towards equality.

However, the results of games where one side's Knights are replaced by Rooks, or where one side's Rooks are replaced by Knights, show the Rooks winning by a consistent 73 to 78 per cent in all scenarios. (Relatively few games, a thousand or fewer, were played in each of four different scenario, and therefore the large variation.)

The results are not linear. You cannot extrapolate the meaning of a 60% win rate based on the 75% win rate of "Rook versus Knight".

One Geometry versus Knight

First comes the basic geometrical series, where every basic geometrical piece is matched against the Knight.

A_v_N 0.172101 2484
D_v_N 0.188380 2556
F_v_N 0.175653 2448
G_v_N 0.128754 2664
H_v_N 0.161508 2520
W_v_N 0.131172 2592
How to read this?

2484 games were played in which White had two Alfils replacing his Knights, and White managed to win seventeen percent of those games.
This isn't a very interesting series of results, but it seemed scientific to have them.

In 1092 games of G_v_N in which the G could only move East (oops!), White's win rate was 0.087.

Two Geometries versus Knight

This series of results is more interesting. The strongest three combinations of two geometries are very nearly equal in strength to the Knight.
Although the HF ((1,1) plus (0,3)) may seem to be nearly as strong as the Knight in these results which were obtained with a permuted opening position, it is obvious that if the HF were used in place of the Knight in the standard opening position, there would be no good way to develop it!.
HF_v_N 0.473977 6552
FA_v_N 0.457447 5076
WA_v_N 0.457002 5256
WD_v_N 0.448696 5292
HD_v_N 0.438271 6480
HW_v_N 0.433047 4548
DF_v_N 0.432960 3744
WG_v_N 0.405278 5400
HA_v_N 0.403836 3780
GA_v_N 0.396096 3816
GF_v_N 0.386905 2520
DA_v_N 0.386414 3636
WF_v_N 0.367452 5598
DG_v_N 0.366081 3924
HG_v_N 0.260858 3960
In the "erroneous G" series, where the G (3,3) could only move East, about a thousand games each were played, with results GA 0.24, GF 0.26, DG 0.29, WG 0.24, and HG 0.20

Three Geometries versus Rook and Knight

HDF_v_N 0.695008 5328			HDF_v_R 0.615054 5580
HWF_v_N 0.686151 5004			HWF_v_R 0.586334 5832
WFA_v_N 0.678819 4896			WFA_v_R 0.537138 4968
HDA_v_N 0.676965 5292			HDA_v_R 0.487915 5544
WFD_v_N 0.669547 4860			WFD_v_R 0.582434 5004
WGA_v_N 0.669236 5256			WGA_v_R 0.45613  5220
HWD_v_N 0.664403 5742			HWD_v_R 0.558867 5436
WDA_v_N 0.658119 5148			WDA_v_R 0.508669 5652
WFG_v_N 0.656308 6516			WFG_v_R 0.499629 5400
WHA_v_N 0.653186 3672			HWA_v_R 0.532819 4860
WDG_v_N 0.652496 5328			WDG_v_R 0.516703 5508
HFA_v_N 0.651921 5256			HFA_v_R 0.529551 5076
HWG_v_N 0.643595 4536			HWG_v_R 0.422709 5544
DGA_v_N 0.634412 5472			DGA_v_R 0.412431 5076
DGF_v_N 0.625267 5472			DGF_v_R 0.480374 4968
FDA_v_N 0.624143 5256			FDA_v_R 0.448184 6552
HGF_v_N 0.617871 5400			HGF_v_R 0.433107 7056
GFA_v_N 0.61768  5328			GFA_v_R 0.409285 5148
HDG_v_N 0.579952 5472			HDG_v_R 0.396019 6732
HGA_v_N 0.558753 5472			HGA_v_R 0.397436 7020

N_v_WFD 0.302987 5256			R_v_WFD 0.411959 6372
B_v_WFD 0.319538 5328			WFD_v_B 0.663783 6588
This group of results is interesting. Especially fascinating is the fact that the results versus Rook do not track the results versus Knight.

Some time ago, when not as many games had been played, and the results were very slightly different from those above, I took the above group of results, got the total of all 10 occurrences of W, F, D, A, H, and G, (there were no "versus Rook" results yet), found the average, and found the difference from the average for each basic geometrical unit:

Piece	Diff from average
=====	=================
W		.1877233334
F		.0714133334
D		.0267833334
A		-.0186366666
H		-.0506366666
G		-.2166466666
This table implies that in some sense the W is stronger than the F, at least as a geometry to be added to other pieces. Take it with a grain of salt.

This table also simply ranks the basic geometries in the order of their average mobility!

Knight plus One Geometry

This series of results gave rise to the game of Different Augmented Knights.

NA_v_N 0.685323  4824		NA_v_R 0.563368 5184
NH_v_N 0.684156  4860		NH_v_R 0.537905 5184
NW_v_N 0.667370  5688		NW_v_R 0.557363 5256
NF_v_N 0.6644965 4608		NF_v_R 0.564314 3996
ND_v_N 0.6615269 4572		ND_v_R 0.547787 4248
NG_v_N 0.6246245 3996		NG_v_R 0.407658 3996
LD_v_N 0.638888 5292        LD_v_R 0.434907 5400
LW_v_N 0.641333 4500        LW_v_R 0.535755 5328
LF_v_N 0.660648 4320        LF_v_R 0.480463 5400
You'll notice that there are some results here with the Long Knight instead of the Knight; there are no results with LH, because it wins with an early fork in the permuted opening position: LH-h5 with h7 undefended wins.

Different Augmented Knights

Both sides have their Knights replaced by new pieces. The computer believed the new pieces have the same value, which is a bit more than the value of a Rook.

LW_v_ND 0.486111 4680		ND_v_LW 0.502576 4464
LW_v_NF 0.462998 4716		NF_v_LW 0.504015 4608
LW_v_NW 0.459430 4104		NW_v_LW 0.518880 4608
LW_v_NA 0.482002 6084		NA_v_LW 0.471600 4824
ND_v_NA 0.493175 6300		NA_v_ND 0.475233 5148
NF_v_NA 0.511767 5184		NA_v_NF 0.444060 6516
NA_v_NW 0.467242 6624		NW_v_NA 0.504876 8100
NF_v_NW 0.514918 7776		NW_v_NF 0.471451 7776
ND_v_NW 0.487137 6336		NW_v_ND 0.503979 6408
NF_v_ND 0.509424 6048		ND_v_NF 0.486044 7452

ND_v_WFA 0.480184 5652		WFA_v_ND 0.482616 5580
NA_v_WFA 0.478676 6120		WFA_v_NA 0.497338 6012
NF_v_WFA 0.487180 6084		WFA_v_NF 0.483418 5940

HWD_v_NA 0.502119 6372		NA_v_HWD 0.504708 6372
HWD_v_NF 0.467084 6228		NF_v_HWD 0.49719 6228
Doing the averaging bit again, and with fewer results than the above, I got:

	NA     0.483100 47484
	ND     0.495616 35244
	NW     0.499913 40572
	LW     0.503746  9612
	NF     0.522686 40752
In the real game of Augmented Knights, the NA and ND have the advantage of easy and rapid development, which makes them at least as strong as the others.

The Value of Forward Movement and Capture

I don't have a complete and perfect series yet, but here are a few partials:

WFD_v_N		0.669547	4860		All Directions, move and capture
FDfW_v_N	0.554771	4464		Forward move and capture
FDrlmW_v_N	0.505922	4644 		Sideways move but not capture
FDfmW_v_N	0.499783	4608		Forward move but not capture
FDfbmW_v_N	0.496713	4716		Forward and Back, move but not capture
FDbW_v_N	0.493734	4788		Backwards move and capture
FDbmW_v_N	0.455757	4464		Backwards move but not capture
DF_v_N		0.432960 	3744		No directions

FDA_v_N 	0.624143 	5256		All Directions, move and capture
AFffD_v_N 	0.558073 	6156		Forward, move and capture
AFbbD_v_N 	0.479372 	6084		Backwards, move and capture
FA_v_N 		0.457447 	5076		No directions, move and capture

WFD_v_N 	0.669547 	4860		All Directions, move and capture
WFmD_v_N 	0.520356 	4716		All Directions, move but not capture
WFfrlmD_v_N 0.5262 		4752		Forward and Sideways, move but not cap
WFfbmD_v_N 	0.480297 	4644		Forward and Back, move but not cap
WFrlmD_v_N 	0.446741	4572		Sideways, move but not cap
WF_v_N 		0.367452 	5598		No directions

NW_v_N 		0.667370  	5688		All Directions, move and capture
NcW_v_N 	0.621476 	4824		All Directions, capture but not move
NmW_v_N 	0.556379 	4860		All Directions, move but not capture
Norm		0.49227		13842		No directions
In the above, remember that 0.505922 is really the same as 0.496713, but that 0.493734 is very likely smaller than 0.505922 (statistically, that is).

You can see that the backwards move and capture aren't worth much, and that the forwards move/capture adds fully half as much to the win rate as does the all-direction move/capture. (But remember, the results aren't necessarily linear!)

You can also see that capture is worth much more than movement.

The End

I have presented only the most interesting results which are also up-to-date (run with the latest major version of the program, since 1990 or so), and have enough games played to make the results reliable.

Just for fun, here's a sample of more:

		  Q		  R		  N
RN		0.503	0.827	0.808
FDN		0.469	0.820	0.758
FAN		0.458	0.808	0.770
KN      0.448   0.793   0.748
KAD		0.435	0.768	0.757
NWD		0.439	0.766	0.740
NB		0.401	0.729	0.705
Yes, a piece that moves as Rook plus Knight is as strong as a Queen!

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