Empty-Board Average Random Mobility

Assuming that a simple single-step piece moves and captures the same way, and that its movement is symmetrical (forward equals rearward, right equals left), and that the board is square, its Empty-Board Average Random Mobility can be calculated quite simply.

First we represent the piece's move as an (x,y) co-ordinate pair.

For example, the Knight could be (1,2) or (2,1); it doesn't matter for this simplified case. This is really a shorthand for the 8 different displacements from the original square that the Knight can make: (-1, -2), (-1, 2), (1, -2), (1, 2), (-2, -1), and so on.

The board's width and height are separately represented as "w" and "h" even though they are the same value in this simplification.

"N" is the number of directions. If x is equal to y, or if either x or y are equal to zero, N is 4; otherwise, N is 8.

The formula for Empty-Board Average Random Mobility is

    E = ((w-x)*(h-y)/(w*h))*N

If you care about the arithmetic, you can easily adapt the formula for 8x10 chessboards and such; and you can use this formula to find the empty-board mobility of a Rook by finding the sum of all the solutions for (x=0,y=1) through (x=0,y=w); and I have shown how to use this to get the crowded-board mobility of the Rook just as well.

However, I think it's important not to get too deeply into these calculations without first understanding their limitations.

Limitations of Empty-Board Average Random Mobility

This calculation of mobility is imperfect, and mobility isn't the whole answer.
  1. "Random" is not Chess. Chessplayers move their pieces to good squares. If you look at 64,000 randomly-generated positions containing exactly one White Knight per position, you will find the White Knight on a1 in 1,000 of them; but if you look at 64,000 randomly-chosen positions from actual games, you will find a White Knight on a1 in many fewer cases.

    On the other hand, random is meaningful. Often when you optimize the placement of one piece, you block or impede another piece. (You will most likely optimize your strongest pieces, therefore random is more important for weaker pieces.)

  2. "Average Mobility" is not nearly as important in Chess as "Attacking or defending the right square". This is why more than one person has thought of using the Center Control measurement to modify mobility.

  3. Generalizing to pieces that have different moves than captures, or different advances than retreats, is dangerous.

Average Random Crowded-Board Mobility

"Runners" (such as Rooks and Queens and Bishops) don't get to use their Empty-Board Mobility until the endgame. I invented the idea of adjusting the Empty-Board Mobility of the Rook by allowing for the fact that the board may be crowded: simply multiply each square's value by the probability of getting there.

This is an important calculation, and contains a lot of truth; however, the other factors that contribute to a piece's value are important enough that you cannot simply run this number and get the value of a piece. The answer you get might be wrong by as much as one-third of the value of the piece in question.

An important step is picking the base probability value, that is, the likelihood that any given square will be empty. Earlier, I suggested that one could arbitrarily choose 0.69 in order to make the values seem to come out right, but also pointed out that this would just be lying with numbers.

I have often run this calculation using 0.7, just to get a rough idea, and I have usually found that the values of Rooks were underestimated. "Random" is not Chess. I don't know what the "correct" value ought to be.

Mobility versus Capturing Power

Most pieces move the same way that they capture, and most other pieces that have been examined have capturing powers roughly equivalent in strength to their movement powers.

When the powers of movement and capture are very different, this factor might have strange values...

But does it? Or is the value really just an average of the move-power and capture-power? You see, you really need the ability to move without capturing in order to aim your piece at different targets; but the more capture-power it has, the less you need to move it in order to hit something. I tried a piece that moves a Ferz but captures as Rook, and its value was close to the Knight; in this case, the average of the two values was close to the real value.

On the other hand, a Bishop or a Knight are each stronger than either a piece that moves as Knight but captures as Bishop, or one that moves as Bishop and captures as Knight. (As far as I can tell, anyway.)

Somebody brave should try playing some games with a piece that moves as Queen but captures a Ferz (mQcF), or a piece that moves as Ferz but captures as Q (mFcQ). Perhaps these pieces are equal in value, and both are generally equal to a Rook.

Forwards and Backwards Mobility

My quick and dirty rule of thumb:

The forward move of the Rook or Wazir or Dabaaba is worth about as much as both its sideways moves; the forward move is worth three or four times as much as its retreating move, but perhaps in the endgame the retreating move is more valuable than that.

The two forward moves of the Bishop are worth two or three times as much as its two retreats. Again, in the endgame retreats may be more nearly equal to advances.

The four forward moves of the Knight, about like the Bishop; but the two forwardmost (from e4 to d6 or f6) are worth a bunch more than the two slightly forward ones (c5 or f5).

Of course, a piece that has no way to retreat is an exception to the rules of piece value.

Cannons and Grasshoppers

The Cannon in xiangqi moves as a Rook, and captures almost like a Rook; in order to capture, it must be screened by exactly one piece. The Crowded-Board Mobility for this rule is fairly easy to calculate; in order for it to capture at a distance of N squares, there must be exactly one piece in the intervening "N minus one" squares.

Grasshopper capture and movement, which are similar to cannon capture and movement, have often been used in problems. It's a pretty weak power, as it requires the screening piece to be adjacent to the destination: to move a Grasshopper Rook from a1 to a8, there must be a piece on a7, and none on a2,a3,...,a6.

Many other powers are susceptible to calculation, and the results of the calculations are likely to provide reasonable estimates of the worth of those powers. The probability that a square is empty is the key to it all.

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