Ideal Values and Practical Values (part 5):
By Ralph Betza The "lame" jumper is a piece that cannot leap over an intervening square if that square is blocked. For example, a lame Alfil can move from f1 to d3 only if e2 is empty.
The magic number is clearly applicable here, and it says that the lame Alfil is worth somewhere around 0.66 to 0.7 times as much as a normal (leaping) Alfil.
By the same reasoning, the lame H (the non-leaping version of the (0,3) jumper) is worth 0.49 (or 0.44) standard atoms because it needs both f2 and f3 to be empty in order for it to go from f1 to f4.
A Knight-valued piece can be made combining:
- Lame H with
- vertical Wazir (forwards and backwards but not sideways) with
- narrow Knight (from e4 to d6, f6, d2, or f2 but unable to go from e4 to c3, c5, g3, or g5)
Another type of lame jumper can be seen in the zF2, a Crooked Bishop limited to short moves. In effect, it makes either a Ferz move or a lame Dabbabah move, but instead of the D jumping over the square in front of it (a normal lame D could not go from f1 to f2 if f2 was occupied), it jumps over either of the diagonal squares (the zF2 can go from f1 to f3 if e2 is empty or if g2 is empty). The probability that both of two squares are occupied ranges from 0.09 to 0.16 depending on the magic value, and so the crooked lame Dabbabah is worth 0.91 to 0.84 times as much as a normal D -- not enough difference to make a difference. The same math would apply to an H that was allowed to jump over zero pieces or one piece, but not two.
The Cannon from Xiang Qi is a special case of lame jumper. When it captures, it is allowed to jump over one piece, but not to jump over zero pieces. Simple subtraction shows that if a normal lame H has 0.36 to 0.49 as much mobility as a leaping H, and if an H jumping either zero or one piece has .84 to .91, then the cannon H (gH, jumping one but not zero nor two pieces) must be 0.48 to 0.42 as much as an H; but the trick is that the normal lame H increases in value as the board empties out but the cannon H loses value when there are fewer pieces for it to jump over.
The traditional Xiang Qi Cannon, which moves as R but captures as pR, provides balance and is an excellent combination because as the board empties out it becomes easier to move the piece but harder to capture with it (its R component benefits from the open board but its pR component suffers from the same cause).
Note that in Xiang Qi both Kings are confined to their castles and therefore the Cannon can always be useful, using its own K as a screen and the enemy King's defenders as a target. This means that the imbalance of movement and capture power that would afflict the Xiang Qi Cannon if used in a FIDE setup does not hurt it nearly as much in an Xiang Qi setup.
The simplifications presented in the earlier articles of this series allow us to say that the ability to move like a Rook (but not to capture in that manner) is worth half as much as a Rook, as long as the capturing power of the final piece is above some minimum theshold value. The threshold is unknown, which is unfortunate, but in the late endgame when very few pieces are on the board the power to capture as pR certainly falls below that threshold. See this page for details.
In the initial position, the average mobility of pR is two-thirds that of R, around the magic number it is half, and late in the game it falls as low as one sixth. I'm willing to take "half" as a guess, so that the combination of mRcpR would be theoretically worth 0.75 as much as a Rook, but with the warning that you really need to trade it off before the endgame. I follow Rudolph Spielmann's belief that a minor piece is worth two thirds of a Rook (not the traditional 0.6), and I have won tournament games against strong players based on this belief in a difference of estimated values amounting to a mere 0.0666 of a Rook; because the mRcpR's estimated value is 0.08 of a Rook more than a minor piece (that would be 0.4 Pawns, an advantage large enough to feel), according to this guess at its value having a pair of standard Xiang Qi Cannons against a pair of Knights should be enough to win on an 8x8 board with the standard arrangement of 32 pieces, and all we need is a bit of playtesting to find out if the guess is correct.
Actually, it's not so simple. Take the standard FIDE army, replace the Knights with mRcpR Cannons, and now stare at the board for a while and try to figure out how to develop your pieces! Actually, even replacing the Knights or Bishops with Rooks isn't that easy to win, so doing it with mere Cannons would perhaps cancel completely any material advantage.
An alternative would be to replace B with HFD and R with mRcpR; although this adds an advantage of rapid jumping development to the presumed material edge, I think that one could get a feel for whether or not the Xiang Qi Cannon is really worth that much more than a minor piece by playing several games of this; or of course one could simply replace the R4 of the Remarkable Rookies with mRcpR Cannons...
Part of the problem with cannon values is that the FIDE setup is dense and crowded. In Xiang Qi, when a Cannon aims into the enemy position there is room to interpose or to step aside, but against the FIDE setup, a Cannon attack may be instantly fatal -- for example, if the piece on a1 both moves and captures as cannon-Rook, then 1 pRa1-a3 threatens to capture Ra8, and after 1...Nb8-a6, simply 2 Nb1-c3 e7-e6 3 pRa1-e3+ Bf8-e7 4 pRh1-h3 Ng8-h6 5 pRh3-d3 wins the Queen. Thus, there is always danger with Cannon pieces, and you need to be very careful when adding one to a game.
My new idea that balance makes it possible to get better estimates of piece values tells me that if a piece has sufficient non-Cannon movement, preferably movement that increases in value as the game goes on, it ought to be possible to make a good guess about its value based on its average mobility using the magic number for emptiness of the board.
The idea of balance also tells me that it's not the best idea to make up themed armies like the Clobberers or the Avians; but these armies are perhaps more fun to design and more fun to play than balanced armies would be, and that's a value in its own right.
The Spacious CannoneersCommon sense also tells me that making up an army whose power is based mostly on the combination of a new and untested estimate of the values of Cannon pieces plus a new and untested kind of piece is risky; but what the heck, if the Spacious Cannoneers turn out to be too weak or too strong when used in different-army games, at least one can always enjoy games where both sides use the same army, so let's do it.
The Spacious Cannoneers R == the Mortar and the Howitzer
The Spacious Cannoneer Rook comes in two flavors, and I like the idea of having both in the same game.
Both kinds of Rook move and capture as Wazir or as Spacious Rook. (The Wazir power is useful only when the Spacious Rook power cannot make a one-square move.)
To this, the Mortar adds the power of capturing (but not moving) as a Rookwise Cannon, while the Howitzer adds the power of moving (but not capturing) as a Rookwise Cannon.
It is recommended to start with Mortar on a1 or h8, Howitzer on h1 or a8; Black gets the one that attacks the enemy Kingside position as compensation for White's first-move advantage. In games with different armies, perhaps only the Howitzer should be used.
Valuewise, the Spacious Rook should be worth two thirds of a R, half the power of a Cannon Rook should be worth one quarter of a R, and the small amount added by the Wazir move should be worth one ninth of a R. This adds up to a bit more than a Rook, not enough to be significant (the error in the estimate of value is likely to be much larger than that!).
The Spacious Cannoneers B == the Carronade
The Spacious Cannoneer Bishop is named the Carronade and moves and captures as Spacious Bishop or as Cannon Bishop.
Valuewise, two thirds plus one third, nothing could be more simple.
Note that 1. Carronade f1-b5+ is legal but not a good move.
The Spacious Cannoneers Q == Big Bertha
The Spacious Cannoneer Queen is named Big Bertha and combines the powers of Howitzer and Carronade.
Note that combining Mortar with Carronade would allow 1. BigB d1-h5+ or 1. BigB d1-a4+ not only winning a R, but also with unstoppable 2. BigbB-e4 (or to e5) checkmate.
The Spacious Cannoneers N == the Napoleon
The Napoleon is an equine piece that is neither Spacious nor Cannonistic, but is named the Napoleon after the horse-drawn field artillery of Napoleonic times.
I have chosen the fbNW -- narrow Knight plus Wazir -- simply because it is a good Knight substitute that I have often mentioned but have never used in any game.
The Spacious Cannoneers == Summary
This army is highly experimental and could be much too strong or much too weak to use against other standard armies.
This army is quite exotic and takes some getting used to. Players who haven't played enough Xiang Qi to get accustomed to cannon moves may well be baffled by a game that combines two exotic powers at once. Players who know the Cannon well will still be startled by the Spacious pieces.
In the late endgame, the long-distance pieces lose very little by being Spacious and gain very little by being Cannonate; but the small tactical details can be interesting, when the difference between a Bishop and a Carronade suddenly makes itself felt.
In the early game, Spaciousness should frustrate many moves you want to make, while Cannonization should give you pleasant choices to make up for that.
The Spacious Cannoneers == Sample Game
Let us suppose W has the Fabulous FIDE army, and Black has the Spacious Cannoneers with Mortar on h8, Howitzer a8.
1. e4 e5 2. Nf3 d7-d5
The fbNW is a good piece in general, but can't defend e5.
2...Carronade c8-g4 might be good. It pins the Nf3 with Spacious power and attacks the Qd1 with Cannon power, and after the forced Bf1-e2, both e2 and f3 are pinned by the Cannon power, though the Nf3 is not attacked.
2...d7-d6 does not block the Carronade at f8 from developing to c5, however it does block it from developing to e7!
The Pe5 is halfway defended, because 3 Nxe5 BigB d8-a5+ 4 d2-d4 (f8 is occupied, so Spacious capture on e1 is illegal) looks mighty risky.
But wait a minute! 1. e4?, BigB d8-h4+ and wins the Pe4.
1.e2-e4? is just a bad move, because 1. d2-d4 BigBd8-h4+? 2. g2-g3 BigBh4-e4 3. Bf1-g2! defends, develops, and drives back the attack with loss of time. On the other hand 1 d2-d4 allows Carronade f8-b4 checkmate.
3.Bf1-g2 is picturesque, don't you think? But in fact the Rh1 wasn't attacked because a8 is occupied.
Hmmm, 1 e2-e4 BigBd8-h4+ 2 g2-g3 BigBh4xe4 (not check!) 3 Nb1-c3 Carronade f8-b4 (pin) 4 Bf1-g2 BigB e4-g6! attacks g2, White doesn't have enough compensation for the Pawn. Instead 4.a2-a3 Carr b4-a5 (now b2-b4 allows Carr a5xc3), 5 Qd1-e2 BigB e4xc2, not enough compensation.
Therefore 1. Ng1-f3 is going to be a popular opening move. 1 Nc3 is also good, or 1.g2-g3, and enough other choices that the opening is playable.
If White has the Spacious Cannoneers with two Howitzers and Black has the Clobberers, 1. BigB d1-h5+? g7-g6, e5 is defended by BDh8, h7 can't be captured because h8 occupied, W must retreat in shame; or if Black had the FIDEs, 1. BigB-h5+ g6 2. BigB-e5 makes no threat, is simply a bad move.
As I said, Cannon powers can be dangerous. It appears as if there is danger here, but that the opposing army can, with a bit of care, always survive and even profit from attempts at early raids.
In addition, this little bit of analysis gives me more faith in the proposition that the weakness of Spaciousness balances the strengths of artillery.
I've always wanted to make up a variant where one side had the FIDE army but the other side had an army with Cannon pieces. Well, not really always, only since 1977 or so, close enough.
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Written by Ralph Betza.
WWW page created: October 22nd, 2001.