Notice that the normal "rider" move and the cannon-rider move do not duplicate each other, so you could have a piece combining the Rook with the cannon-Rook. Commonsense says that it would be worth a Queen in general, perhaps a bit more. The paradox is that its Rook power gets stronger as the board gets empty, but its cannon-Rook power gets weaker; somehow, this piece never quite seems to be at its best.
For the cannon type of piece, we can simply take the chart we made for the Grasshoppers and multiply the "LEGAL" number for line N by N-minus-one. This is because, for example, when the grasshopper wants to go from a1 to a5, it must find both a2 and a3 empty and also there must be a piece on a4; but the cannon works in this case and also in two others (the screening piece can be on any square).
This way to the numbers.
This gets interesting. The calculations suggest that a cannon-Queen is worth more than a Knight, but experience has shown in CCCC Chess that a WD outvalues the cannon-Queen! In fact, the opening position of that game was thought to be practically a win for White, because his first move could fork the two WD with his cannon-Queen.
According to the calculations, this involved a sacrifice of current value in order to get an advantage in endgame value, and in the opening, the cannon-Queen's value is probably even higher than shown (we used 0.7 as the probability of a square being empty, which corresponds to having 19 pieces on the board).
The value of cannon pieces does not quite get to zero as the endgame approaches, because you always have your own King to act as a friendly screen and help you move; see the endgame study in the discussion of CCCC Chess.
In CCCC Chess, the cannon pieces are perhaps worth less than they might be in other games, because the amazing power of the cannon pieces at the start of the game means that Pawns are quickly slaughtered, and the board becomes empty more quickly (so the cannon pieces become immobile more quickly). The WD is the only piece in CCCC Chess that holds its value as the board empties out.
Also in CCCC Chess, the WD is the only piece that can mate the enemy King (in a Pawnless endgame with nothing but two Kings and one piece). This must add to its value in that game.
Thus, the sacrifice of cannon-Queen for WD is a winning idea because of the specific set of pieces on the board in that specific game; in general, however, (and to the extent that pieces have a value apart from their current values in specific positions), I'd have to say that the cannon-Queen outvalues the WD, and the numbers we calculated sound reasonable.
In fact, I'd have to say that the cannon-Queen outvalues the WD by a rather large amount, because my common sense tells me that its immense mobility in the midgame means that it can always do a lot of work, and can "always" trade itself off for a N or some other minor piece before the cannon-Queen gets too weak.
Xiangqi has a fairly empty board, so the lame Knight is not worth much less than a jumping Knight would be, and the Cannon has more trouble finding screens for its jumping than it would on a more crowded board.
On an 8x8 board with 32 pieces, the Cannon's calculated value would be about 1.1 or 1.2 jumping Knights; that's one or two Point Count Chess "points" more than a Knight. (And assuming that you can just average the values of the capturing power and the moving power.)
In my Funny Notation, the xiangqi Cannon would be "mRcpR".
Useful? Well, you can use them to design pieces like the NpB or the NpR4, with expected values in the Rook range. If you call that useful.
I sort of like the FpB, which ought to be worth almost as much as a Bishop.
Actually, during a game the probability that a randomly-chosen square will be empty ranges from 0.5 at the start of the game to nearly 0.97 when just two Kings are left on the board.
When I run this same calculation using different values as the probability for "empty", I get a lot of numbers.
These 29 values for "empty" correspond to having 32 pieces on the board, then 31, and so on down to just 4 pieces.
Notice how the Rook nearly doubles in value, while the cannon-Rook drops two-thirds of its value. Meanwhile, if we take the average of all 29 values we get 8.65 for the Rook and 3.51 for the cannon-Rook.
For reference, remember that there are two estimates of the value of a Rook relative to a Knight, those being 1.5 and 1.6666; and the Knight's average mobility is 5.25; and 1.5 times 5.25 comes out to 7.875, while 1.6 times 5.25 gives us 8.75; so 8.65 is within the expected range.
Of course, mobility isn't everything, but it is the most important single factor, and so we expect a good calculation of mobility for the Rook to be somewhere within shouting distance of the expected range (from 7.875 to 8.75, or near their average, which is 8.3).
The Rook's calculated value falls into this range when there are from 44 to 47 empty squares, and also when we average all 29 cases. Its lowest value is when the board is most crowded, and it steadily improves until its best value is double its worst.
The cannon-Rook has no "expected value", because it's an unknown piece. Its best values are when there are from 32 to 42 empty squares on the board, after which it begins to lose value at an accelerating pace, falling almost to one-third of its best value. (Looks like a bell curve.)
The RpR, combining the powers of both, gets slightly stronger as the game goes on, but its best and worst values are rather close to each other. Remember that the Cannon, from xiangqi, moves as Rook, captures as cannon-Rook, should therefore have a value exactly half of the RpR.
Remember that in real life the Q should get a bonus for moving in 8 directions, which is shown in real life by the fact that Q is worth R, B, *and* Pawn.
That was pretty good, so let's look at Bishops. Now the average for all 29 is 6.179386 Bishop, 1.881799 pB.
The Bishop, of course, is colorbound and moves only 4 directions, so it needs a greater mobility to be equal in value to the Knight.
Well, if we only had one number for the cannon-pieces, we wouldn't be able to get the same feel for the way their strength drops off; and putting the Rook side by side with the cannon-Rook helps contrast their trends; and the sum of the two (the QpQ column) shows that the increasing trend of the Bishop is always stronger than the decrease of the cannon-B; and doing it for all three types (Rook, Queen, Bishop) lets us see the differences.
We see that the R, N, and B all hit their average values with 47 empty squares, when the probability of an empty square is 0.73; but the pR, pB, and pQ hit their average at 50 empty squares, when the probability of an empty square is 0.77; and can we figure out what that means? Maybe you can, but I can't.
The poorer endgame piece hits its average value when the board is less full than the better endgame piece. Seems counterintuitive.
Earlier, we may notice that the cannon-Rook operates even better at a distance of 3 squares than at a distance of 2; and does pretty well at 4 squares.
And when you think about it, it's interesting that the cannon-piece always has to go two squares or further. Elsewhere, I concluded that a Rook may get as much as one-third of its value from its ability to move just one square. Makes you want to consider the value of the WpR, doesn't it? (WpR is not as good as Rook, by about a Pawn. According to the numbers, anyway.)
That was quick. Does this mean that the pQ is really worth a lot more? Look, it also wins if Black is the one with the pQ, and it wins against the Clobberers army or the Nutty Knights, but maybe not against the Rookies.
Or is it just a tactical accident in the specific position that starts the game? If we go back to the first try (both sides use the FIDE army, but White has pQ instead of N), and simply require the opening moves 1.e4 2. Bc4, the problem disappears. No more instant win.
1. e4 e5 2. Bc4 Nf6 3. d3 Nc6 4. (pQ)b1-b3
White attacks f7, threatens (pQ)b3-h3 to prevent Black from Castling, reserves (pQ)g1:g7 to inhibit Black's development. Black's mechanical play has allowed White to gain a small advantage.
We required 2. Bc4, otherwise 2. (pQ)b1-b5+ and 3. (pQ)b5-h5+ would be strong; and 1...e5 was as good a defense to the numerous overwhelming threats as any other move. But 2. Bc4 is a decent move, a logical move, a developing move. It throws away the accidental tactical win, and lets us examine the values of the pieces better.
1. e4 e5 2. Bc4 Nf6! 3. d3 c6!
That was hard to find. The pQ is expected to be a bit stronger than the Knight over the entire course of the game, but here in the opening, it's really dangerous. Naturally, all Black needs to do is survive awhile and trade pieces but avoid trading the pQ, and he'll have an advantage; but it isn't easy.
1. e4 e5 2. Bc4 Nf6 3. (pQ)b1-b5+ c6? 4. (pQ)b5:b8
White will win the Pa7 with 5. (pQ)g1:a7
1. e4 e5 2. Bc4 Nf6 3. (pQ)b1-b5+ Nc6! 4. (pQ)b5-h5+ N:h5 5. Qd1:h5 Qe7 (or 5...g6)
I'm not impressed by White's position.
1. e4 e5 2. Bc4 Nf6 3. (pQ)b1-b3
And wins the Pa7? Maybe not. Of course, ...N:e4 is dangerous because of (pQ)g1-e3 with a pin, but
1. e4 e5 2. Bc4 Nf6 3. (pQ)b1-b3 Nc6! 4. (pQ)b3:f7 Na5 5. Qd1-e2 N:c4 (Not 5...b5 6. (pQ)f7-f3!) 6. Q:c4 d5