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Halflings: Corrected calculations

I could swear I wrote a page that followed the calculation of a Rook's value, and a Short Rook's value, in great detail.

I can't find it, and I want to refer to it. So I'll write it again.

So let's run through the calculation.

If we assume that the board is 0.69 empty, then the chance of a Rook being able to go from a1 to a3 without having its motion blocked by something on a2 is 0.69; but when it crosses two squares, the chance is 0.4761, that is, 0.69 times 0.69; and so the chart

Distance 1 2 3 4 5 6 7
Probability 1.0 0.69 0.4761 0.3285 0.2267 0.1564 0.1079

However, in addition, in order for the Rook to move seven squares, there must be a square on the board that is that far away; and of course a Rook on b2 cannot move seven squares in any direction.

If a piece wants to move x squares laterally and y squares vertically on a board that is w squares wide and h squares high, the probability that the desired square exists on the board (in other words, the probability that the piece is not trying to move off the edge of the board) is (((w - x)/w) * ((h - y)/h)). Chart:

Move 0,1 0,2 0,3 0,4 0,5 0,6 0,7
Probability 0.875 0.75 0.625 0.5 0.375 0.25 0.125

Thus the average value contributed to the Rook by its ability to move 7 squares is a mere 0.0134897703, that is, 0.1079 times 0.125, as in the following chart:

Distance 1 2 3 4 5 6 7
Value 0.875 0.5175 0.2976 0.1643 0.0850 0.0391 0.0135
Times 4 3.5 2.07 1.19 0.657 0.34 0.1564 0.054

The "Times 4" is because the Rook makes an identical move in each of 4 directions.

Thus, a one-move Rook, also known as the Wazir, has a calculated mobility of 3.5, while a Rook which may never go more than two moves has a calculated mobility of 3.5 plus 2.07, or 5.57; and already we see reason to distrust the calculations because the R2 is not as valuable as the Knight (whose mobility is 5.25).

Even so, fill out the chart:

Piece R1 R2 R3 R4 R5 R6 R7
Mobility 3.5 5.57 6.76 7.42 7.76 7.91 7.97

The betterway page said that the R was worth 7.88, but it used 0.68 rather than 0.69; both 0.68 and 0.69 are mere guesses, empty estimates that are used because I can find no convincing reason to choose a different number.

The short rook page says that a R2 has the mobility of 0.566 Rooks, and it says it was using 0.69 as the empty square probability. However, the chart seems to be completely wrong, and instead I now get

             RATIO OF PIECE TO
              =================
    PIECE      R7          R1         R(N-1)     WIN
    =====     =====       =====       =====      ====
    R1        0.439
    R2        0.699
    R3        0.848
    R4        0.931
    R5        0.974
    R6        0.992
    R7        1.000
	

I think my new numbers are right, and my old chart was a mess. That's too bad, because I tried to interpolate the value of a Halfling Rook into the old chart, and thought it was between R3 and R4, when actually it's smack dab in the middle between R2 and R3; so, instead of being worth a whole Knight, the Halfling Rook is worth half a Rook!

The Halfling Rook is a bit harder to calculate. I looked at its moves in 4 directions on 16 squares, and found it can move a distance of 4 squares in 8 cases, exactly 3 squares in 16 cases (and therefore "3 or more squares" in 24 cases), 2 in 16, 1 in 16, and zero in 8 cases. That works out to a mobility of 6.1, and so the Halfling Rook is worth Half a Rook.

Remember, the calculation is not a calculation of the value of the piece, but merely of its mobility. There are other factors that enter into the value. The R1 contributes 44 per cent of a Rook's mobility, but experience shows that it is responsible for only one third of the Rook's value.

The approximate values of the R, R2, R3, and R4 are known from experience, and it seems logical that the mobility of the Halfling Rook can be interpolated into the list of mobilities of pieces with similar charecteristics and known values to determine an approximate value for the halfling.

Halflings are worth half.

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Written by Ralph Betza
WWW page created: March 8, 2001.