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
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".
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 2592How 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.
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 3960In 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
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 6588This 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 -.2166466666This 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!
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 5400You'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.
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 6228Doing 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 40752In 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.
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 directionsIn 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.
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 0.705Yes, a piece that moves as Rook plus Knight is as strong as a Queen!