Ideal Values and Practical Values (part 4):

Odds and Ends

By Ralph Betza

Mobility and Value

The calculation of average mobility which I invented finds out the mobility of a piece, not its value. However, it is easy to confuse average mobility with value because mobility is such a large part of value.

Sometimes I will write that a piece "is worth" some proportion of another piece when it would be more accurate to say that the piece "has as much average mobility as" the proportion of the other. Sometimes when I do this I may be momentarily confused, but most of the time I will do it because "is worth" is so much shorter to type and makes a much less clumsy text for you to read.

The Magic Number

The problem with average mobility is that many pieces, for example the Rook, Bishop, and Queen, become more mobile as the game goes on because as pieces are exchanged there are fewer occupied squares and more empty ones. More empty squares means that these pieces can happily cruise for long distances without bumping into anything.

As a convenience, I use a single number for the average mobility, based on a single "magic number": the mobility is calculated assuming that 2/3 to 70 percent of all squares are empty (I use different numbers according to the direction of the wind). These percentages of emptiness correspond to having anywhere from 41 to 45 empty squares on the board, that is, they correspond to the situation that applies when from 9 to 13 pieces have been traded off.

This seems logical; it is biased towards the opening and midgame where the game can most often be decided, and yet enough pieces are off the board that endgame values have some weight. Not only that, but the results of using a magic number in this range are usually reasonable, and frequently give values of average mobility that provide useful insights into piece values. Unfortunately, it is highly unscientific.

The magic number was derived by doing the average mobility calculation using many different percentages of emptiness, and then I simply chose the value that gave results that I liked. Bad, bad, bad.

I continue to use the magic number because it works pretty well.

Useful Atoms

The W (Wazir), F (Ferz), D (Dabbabah), A (Alfil), and N (Knight) are billed as the basic geometric units of chess, the atoms from which other pieces can be formed. However, other basic geometric units of chess exist. The (0,3) leaper, designated as H in my funny notation, is actually a useful atom. The (7,7) leaper is possible as well -- it is a basic geometric unit of chess -- however, I do not count it as a useful atom on an 8x8 board even though the "lame" (7,7) movement is part of the Bishop's move.

I have recently come up with two rules which sound very impressive and which might be correct; however I implore you not to place excessive faith in these two experimental rules.

The first rule is that if the sum of the two coordinates of a movement is greater than half the average dimension of the board, the movement is much weaker than the basic atoms because it has too little mobility.

For example, the (3,3) jumper (G in my notation) has 6 as the sum of its coordinates. On an 8x8 board, the average of 8 and 8 is 8, and half that makes 4. According to the rule, a (3,3) jumper is worth less than a (2,2) jumper (the Alfil is the (2,2) jumper), but on a 12x12 board the (3,3) jump should be a useful component to any piece.

The second rule is that a forward leap which is half or more the height of the board is too dangerous. For example, a piece combining the (0,3) and (0,4) leaps would win heavy material in just a few moves from the opening position.

Even a forward leap of 3 is dangerous, but I have found that blitzing attacks (jumping out into the middle of the board on move one for a one-piece attack without benefit of Pawns moves) by such pieces can usually be defended against, and the attackers driven back with loss of time. The thing is, if you make a piece that uses an (0,3), (1,3), (2,3), or (3,3) leap, you need to be careful what you add to the mix lest it win out of hand.

Thresholds and Levelling

The average mobility of the A is a mere 2.25, while the W "is worth" 3.5, so how can it be that they have equal values when added to other pieces?

Of course, one reason is that the A has a longer move and also has more forward moves, but that's not the whole story.

I think that small differences in mobility simply don't matter up to some threshold, simply don't make a difference in piece value; in addition, small differences in piece value (less than the "quantum of advantage") have no appreciable effect in practical play; what's more, if one minor piece is a bit more valuable than another, some of the surplus value is taken away by the "levelling effect" -- if the weaker piece attacks the stronger one, even if it is defended the target feels uncomfortable and wishes to flee; but if the stronger piece attacks the defended weaker piece, the target simply sneers.

The net effect of these three things is what makes it possible to have new pieces that have the same practical values as existing pieces, and so we should be glad that these principles exist.

Weaknesses

The practical value of an Alfil when used as a piece by itself is quite low because of the specific weaknesses of the A, but when it is combined with other atoms such that the combination no longer exhibits those weaknesses, the Alfil contributes its full "ideal value" to the practical value of the new piece.

Here is a list of weaknesses I have noticed:

• If a piece does not have a leaping move, it cannot be developed without an additional Pawn move to clear the way for it.
• A colorbound piece cannot see all the squares on the board. Colorboundness comes in various extremes, so that a piece that sees half the squares on the board is not greatly disadvantaged but a piece that sees but one square in four is very much weaker than its mobility would indicate.

  The Knight must always change color when it moves, which is a very mild kind of colorboundness.

• There is some relationship between the geometric length of a piece's longest move and the size of the board, such that a piece can suffer from having too short a range, from being too slow.

  On the 8x8 board, the Knight is on the cusp. Situations often arise in which the shortness of its move is a disadvantage, but most of the time its move is long enough; in comparison, the WD suffers more notably from shortness (but has other advantages in compensation).

• Unbalanced pieces are weak. See next section.
• If the endgame with King plus piece versus King is a draw, that counts as a strike against the piece. (Usually I describe the opposite of this as the "can-mate" advantage.)
• The ability to make a short move of one square is a special strength, and its absence is a weakness. (Maybe; I'm not certain of this one.)
• If a piece needs other pieces to help it move, it suffers from extreme weakness in the endgame.

Balance

Unbalanced pieces are weak.

If a piece can only move forwards but cannot promote, it is much weaker than its mobility. If a piece has tremendous movement power, but no capture power, or vice versa, it is much weaker than the average of its mobility and capture. If all of its moves are different from its captures, it can be blocked.

If a piece both moves and captures Cannon-style, so that it needs other pieces to help it move, it is unbalanced and suffers from such an extreme weakness that its value is impossible to estimate; however, and this is exciting, a balanced piece that uses some Cannon-style movement or capture can have its value estimated in the normal fashion! (This is like finding the Holy Grail!)

The paragraph above this is the lead-in to the next article.

Click here for the previous article in this series.

Click here for the next article in this series.