Factors that Contribute to Piece Values

There are quite a few factors that could be part of the value of a chess piece. The following list is an attempt to mention every possible one, including some I could never think of how to measure, much less how to use such a measurement.

In the pages that follow, I will look more closely at most of these attributes.

Some of the items on this list are merely different ways of thinking about the same attribute.

Empty-Board Mobility
Put a piece on every square of an empty board, count how many squares it can move to, divide by the number of squares.

I forgot to mention that for a jumping piece whose move is given as (x,y) (this would be (1,2) in the case of the Knight), there is a formula for a square board of width w and height h:

where N is the number of directions. 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).

The square-board formula is a simplification of the general formula for rectangular boards. I discovered both myself, but doubt I was the first to do so.

Empty-Board Mobility was calculated in the 19th century, and probably earlier.

Crowded-Board Mobility
"Runners" (such as Rooks and Queens and Bishops) don't get to use their Empty-Board Mobility until the endgame; before then, their moves are limited by the other pieces on the board, because the other pieces get in their way and impede the free movement of these "Runners".

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.

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...

Cannons and Grasshoppers
Some pieces use slightly unusual rules of movement and capture. The mobility factor for some of these rules is easy to calculate.

Jump or Walk or Hop or Run or Step
The Knight, Alfil, and Dabaaba are true jumpers because they jump to a place that is more than one square distant from their origin; the King takes a single step, while the Queen is a "runner"; the Knight from xiangqi is a "lame jumper" because it cannot jump over obstacles; and the Knightrider is a runner-jumper -- it runs in the same directions as the Knight.

There is something special about each of these categories. I'll devote a chapter to this later.

Forward Mobility versus Rearward
Most pieces move forward the same way they move backwards, but there are some asymmetrical pieces.

Colorboundness, Colorblindness, or "Phase" refers to the fact that some pieces see only part of the board. Bishops, for example, see only half the board.

How far a jumper jumps, or how far a runner runs, is important for more than one reason.

In the language of endgame studies, a Bishop on c5 is said to "dominate" a Knight on c8: the Knight is not attacked, but the squares to which it can move are all attacked by the Bishop.

Number of Directions
A Bishop or Rook moves the same way it captures, and can move in 4 directions. When it moves, it can attack two pieces that it did not previously attack, and at most one of those attacks will be in the "forward" direction.

The Chancellor (Rook plus Knight) can attack ten new squares at once, and five of them will be "forwards".

The reason the Queen is worth more than the separate Rook and Bishop is that she gets a bonus from having 8 directions of movement.

Forwardness is a combined measure of number of directions of forward capture, number of directions of forward movement, distance of the movement and capture, and average mobility attributed to that movement.

It was my idea, and is interesting.

When the King wants to go from e6 to e8, it has three different roads it can take. Other pieces are inflexible and awkward.

Are flexible pieces more valuable?

The masters of Shatranj thought that a Knight was worth 5 Pawns; today we think it is worth 3. The only thing that has changed is that in Shatranj the Pawn could only be promoted to a Ferz, and today it can be promoted to a Queen.

It seems logical to conclude that one-third of a Pawn's value comes from its ability to be promoted to a powerful piece (which usually decides the game).

Special Powers
Castling, capturing en passant, relaying your power to the next piece, stepping through the looking glass, co-powers, rifle capture, and so on: how can the worth of these powers be evaluated?

The Knight in Xiangqi cannot jump over obstacles, and the Gryphon in Tamerlane's Great Chess cannot make the one-step diagonal move that is part of its path (or anyway, one source says so). How much does this hurt them?

Center Control
Place a piece on each square of the board and see how many center squares it controls.

Carpet Bombing
The Amazon Queen (Rook plus Knight plus Bishop) attacks everything nearby; moves into position, blankets a whole area with attacks, can mate the King with no help at all from other pieces.

In contrast, the Gryphon/Griffion has most of the power of two Rooks, but it attacks a more disperesed pattern of squares.

The WD is a lot weaker than the HFD, but the Pawnless endgame of K+WD versus K is a win, and the endgame of K+HFD versus K is only a draw.

Pawn Cooperation
There is a certain synergy between the way the Rook moves and the way that Pawns move. Does this make the Rook more valuable?
Your strongest piece often "fights" the enemy's strongest piece.

Is it better to have a strong strong piece or a weak strong piece (which allows you to buy more expensive weak pieces)?

Depending on the other pieces in the same army, what is the main job of this piece and how well does it do its job?

Depending on this piece's rank and role, does it start the game in the best place or the worst place?

Restrictions on Being Captured
You're not allowed to let your King be captured. In the game "Knight Relay II", the Knights cannot be captured. In one historic game, the Nemesis could not be captured, and could capture nothing except the enemy King.

How does this sort of thing affect piece values?

Every piece is worth something just because it occupies a square.

When piece A can attack piece B without in turn being attacked by piece B, then piece A is stealthy with respect to piece B.

This is an important new idea, and will have its own chapter.

Briefly, it is a combination of other factors that shows whether a piece does better in endgames or in middlegames.

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