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Ideal Values and Practical Values (part 3):

The Rider Problem

By Ralph Betza

The Rook is technically known as a Wazir-Rider: it makes a Wazir move, and if it lands on an empty square it may make another in the same direction, repeating this process any number of times. Of course, the Bishop is a Ferz-Rider.

The ideal values of Ferz and Wazir are the same, but the well-known practical values of the Rook and Bishop are quite different. Why?

The ideal values of the R and B are presumed to be very close to their practical values, and so the Rider Puzzle is very much in the scope of a discussion of ideal values.

I have mentioned more than once that there is a formula for the probability that a given jump is on the board:

Consider a one-step move of displacement x and y
(for example a Knight move has x = 1 and y = 2 (or x = 2 and y = -1 and so on for all combinations, but we call that a (1,2) jump))
being made on a board of dimensions w and h
(the normal chessboard has w = 8 and h = 8)
the answer is ((w - x) times (h - y)) divided by (w times h)
And for a Knight the result is (7 * 6) / 64, but there are 8 directions in which a Knight can move, and so we multiply the result by 8, and it comes out to 5.25 which is exactly the average number of moves a Knight has when you put it on every square and count them up. So there.

A Rook makes an (0,1) jump, then if it landed on an empty square it may continue to (0,2) and so on. Therefore its average mobility is the probability that (0,1) is on the board plus the product of the probability that a square is empty and the probability that (0,2) is on the board, and so on. The probability that a square is empty varies (gets larger as the game goes on), so there isn't one perfect number for the average mobility of a rider; and although average mobility is a very important part of piece values, I can't find a reliable way to calculate one from the other.

It would be nice to know both the ideal and the practical values of rider pieces. Even the Rook and Bishop, whose practical values are fairly well-known, have unknown ideal values. I assume that the ideal value of the R is roughly equal to its practical value, and that the ideal value of the Bishop is a bit larger than its practical value; one possible clue is that the Queen is worth a notable amount more than the separate R and B, but this seems to be mostly because pieces that concentrate great value are as a general rule worth more than their separate component pieces (more forking power).

The Chancellor is roughly equivalent to the Queen even though the ideal value of N is presumably less than Bishop: the Bishop is colorbound and its practical value is ever so slightly more than a Knight, combining it with R removes the colorboundness, and therefore is a classical case of "combining pieces to mask their weaknesses and thus allow their practical values to be fully expressed"; and therefore one might expect the Q to be worth notably more than the Chancellor.

One hypothesis about why the Chancellor does so well is that the R has a weakness that is masked when N is added to form Chancellor. This weakness would be its relative slowness and difficulty of development, and perhaps its lack of forwardness (it has only one forwards direction).

Digression: the endgame with WKa1, BKg7, BPf7 is drawn if W has a Q, but won if W has a Chancellor. You should work it out for yourself because it's interesting.
The Nightrider is another special case. It moves in twice as many directions as the Rook, but covers two squares with each step and therefore cannot take many steps in one direction before being stopped by the edge of the board. Despite this, its practical value is very much equal to the Rook -- but one might expect the ideal value of the R to be larger than the NN's.

The Dabbabahrider moves in (0,2) increments as opposed to the Rook's (0,1) steps. As a piece by itself, it is much weaker than a Knight, the main reason being that it is colorbound times colorbound -- it can visit only one fourth of all squares on the board. The Alfilrider is even worse, and can see only one eighth of all squares. Because of this extreme limitation, we have the interesting case where the AD (Alfil plus Dabbabah) has the same ideal value as the Knight but is much weaker in practice, while the AADD (Alfilrider plus Dabbabahrider) has an ideal value which is unknown but which must be appreciably larger than Knight -- but the practical value of AADD seems to be a bit less than a Knight.

In order to use these atomic movements in combination with others and thus define new pieces, one would like to know their ideal values.

One way of looking at the rider-value is that the Wazir-rider (the Rook) seems to be worth 3 times the Wazir, while the Ferzrider (the Bishop) is only worth twice as much as a Ferz; and remember, the ideal values of Wazir and Ferz are the same. (I really want to find a formula using the inverse of the geometrical distance of one step [1], but can't find any convincing reason.) The Nightrider is worth a mere 1.5 times as much as a Knight. The multiplier for Dabbabahrider should be somewhere between 1.5 and 2, while the multiplier for the Alfilrider is probably less than 1.5 (and by definition must be greater than 1.0).

If the multiplier for the DD is 1.75, exactly between the two proposed limits, the difference between D and DD is worth a Pawn; and if the AA is worth 1.25 times the A, the difference between the two pieces is just one the "quantum of advantage", the smallest difference in values that you can notice.

The above two paragraphs give a better idea of the values of AA and DD than we ever had before, but aren't very exact nor probably very accurate.

I can't solve the Rider Puzzle. This is depressing. Let's have some fun instead.

The Avian Airforce

Consider, for example, the Faalcon, FAA, Ferz plus Alfilrider. The FA by itself has an ideal value equivalent to N or B, and in practice is worth perhaps one quantum of advantage less than N or B. Making the A into an AA should add the missing quantum, and (because of concentrated power is worth more) maybe even more. Therefore I expect the FAA to be fully equal or even slightly better than N or B.

Consider also the Wader, WDD, Wazir plus Dabbabahrider. The WD by itself is a good match for the N, weaker in the opening but stronger in the endgame. The WDD must clearly be slightly weaker than the R, but it's definitely in the "major piece" range so that the levelling effect should help the WDD by pulling its value up towards R.

Combined, we get the Flying Kingfisher, WFAADD, which must be worth about as much as a Queen, how much more or less who can guess?

That's nearly a whole army. What to use for the Knight? Wazir plus long crab is tempting as "more of the same", but development can get too awkward. Commoner would provide balance -- this army is a little bit weak but has so much early mobility that it should be playable; using Commoners as Knights would give it some endgame punch -- but so common. I want something brand new, something subatomic in fact...

Thus, the Darter: moves forward as narrow Knight, all directions as Wazir, and retreats as -- Alfilrider!! The Alfilrider retreat increases its value so it's a tiny bit stronger than a N, but the army as a whole needs extra strength and so it's okay. The riding retreat is not nearly so strong as a riding advance, and since long diagonal retreats are the moves that Grandmasters most frequently overlook, there's a certain charm to the choice.

The Avian Air Force would be the army's name, and of course it's classed as an experimental army.

You can see how research into values leads directly to creation of new chess variants. On the other hand, using the Avians can be construed as research because practical experience with the Faalcon and Wader and Kingfisher will make it possible to refine the blind guesses about the ideal values of the AA and DD!

In the next article, I will ferociously attempt to establish ideal values for some other types of pieces, but will probably fail.

[1]: An interesting point is that the practical values of R and B in Cylindrical Chess are thought to be roughly equal by players who have much experience in the game.

Click here for the previous article in this series.

Click here for the next article in this series.

Written by Ralph Betza.
WWW page created: October 15th, 2001.