Ideal Values and Practical Values (part 1)
By Ralph Betza My research into the values of chess pieces was conducted with the specific end in view of making it possible to construct the game of Chess With Different Armies, although of course the research has also contributed to many other interesting chess variants.
In this research, I found that the basic pieces formed from a single type of movement, that is, the Alfil, Dabbabah, Ferz, and Wazir, had very different values when considered as pieces in their own right -- but when one of these pieces was added to a Knight, the combination of the two was equal for all practical purposes. (The NA is as strong as the ND which is as strong as the NW or the NF, even though the F by itself is worth nearly twice the A by itself.)
In fact, it seems that each piece has two values. One value is the practical value of that particular piece, considering all its specific weaknesses and strengths; and the other value is the ideal value of that piece, the abstract value that the piece has when combining the moves of two pieces masks the weaknesses of both.
I devoted a great many words to the question of practical values, and came to the conclusion that although I could not really solve the problem I could at least develop some useful guidelines; however, I largely ignored the question of ideal values.
As it turns out, ideal values of simple one-step pieces are absurdly simple: however many different moves it can make, that's its ideal value. An Alfil moves to 4 different squares, so it is ideally worth half as much as a Knight, which moves to 8. A piece that combines the moves of Alfil and Dabaaba moves to 8 different squares, so it has the same ideal value as the Knight.
Although this is absurdly simple, I believe it to be a Truth; and equally, the ideal value of a piece combining many moves is the most important component of its practical value.
For example, the practical value of the AD is much less than the practical value of a Knight, and it is obvious that the main reason for the weakness of the AD is caused by its extreme colorboundness (it can only go to squares of one color, and what's worse, it can only go to half the squares of one color). If we mask that weakness by adding a W, we get the WAD, which is as strong as a NW for all practical purposes.
"For all practical purposes" is vague because A piece is as strong as the hand that holds it. A grandmaster who is thoroughly familiar with both pieces might find that one or the other has a winning advantage; but a mere master who has only a nodding acquaintance with them does in fact find them to be roughly equal.
In fact, experience shows that a NW or a WAD is in practice roughly as strong as a Rook, and by extension the NWAD or NWB (adding something the value of a minor piece to something the value of a Rook) must be roughly as strong as a Queen.
When I realized that this simplification could be made to work, I was pleased and excited -- think about how regular it is! Take a pile of piece components each of which is ideally worth half a Knight (the W, F, A, and D of course, but also the Crab, the Barc, the narrow half Knight, and the wide half Knight), and if you combine any two of them you have something ideally worth as much as a Knight or Bishop; combine any three of them, you get a Rook's value; combine five, and you've got a Queen.
How simple, how regular, how symmetrical!
Not only that, but if you combine four of these units, it's an Archbishop (Bishop plus Knight, a piece seen in many chess variants); combine seven (forget that the choices I listed only go up to 6) and it's an Amazon; combine six, and it's a piece of a value that's rarely been used (worth as much as a piece combining the Rook and the Knightrider).
How beautiful, simple, regular, and symmetrical! The several common values of chess pieces are reduced to a simple quantity, take two, take three, take five.
The ideal values at least are thus reduced. The practical values can still raise complicated questions. On the other hand, it is easy enough to choose combinations of choices that mask the weaknesses of the components and bring a piece approximately to its ideal value in practice -- for example, adding W or Crab to a colorbound piece nearly always does the trick.
"Nearly always does the trick" should be exciting to all the chess variant inventors out there. It means that you can design chess variants in which the players have different armies, using pieces that have never been seen before, and although you will still want to playtest it a bit you can expect to have a good chance of getting it right the first time.
For example, suppose you invent a new piece called the Crabbish, which combines the powers of Bishop and Crab. You know from the start that its ideal value is exactly a Rook, and you also know that its practical value is sure to be very close to that of a Rook. You can give one player a Rook, and give the other player a Crabbish, and you can be fairly confident that you haven't created a major problem with your game's balance. (In practice, K plus R versus K is an easy win and K + Crabbish versus K is certainly difficult and perhaps impossible; but if the player with the Crabbish has this "can-checkmate" advantage with some other pair of pieces, for example WD versus N, it should even out.)
There is more to be said on this subject, but this seems a good place to stop, and allow both the author and the reader to think about what has been said so far.
Click here for the next article in this series.
Written by Ralph Betza.
WWW page created: October 10th, 2001.