# Comments/Ratings for a Single Item

There seems to be a bug on the recent comments page: Mark Bates' comment, which is on another game, immediately folllows H.G. Muller's comment (below) with no intermediate heading row, as if they were on the same game, and yet the two comments after that, which are both on Buypoint Chess, each have their own separate heading. Not sure where to report that.

On the topic of Rook development, it seems possible to me that an R4 or R5 might have its development hindered less than an R7 would, since by advancing past the pawn wall the piece will also become somewhat more centralized, which seems of some benefit to a short rook but probably not to a full one. But I certainly could be wrong.

On the topic of software, For the Crown includes some sufficiently exotic pieces that it seems unlikely I will find any software that can represent all of them without modification (I'm a programmer and could potentially perform some modifications myself)--but testing only a subset of them would still have some value. You can download the rulebook from here (near the bottom of the page) if you're interested in the details, but some highlights include:

- Asymmetrical pieces
- Long-range leapers (bison)
- Bent riders
- Nightriders
- Pieces that can promote as if they were pawns
- A long-range rider with a
*minimum*move distance - A piece that can exchange places with another friendly piece as a move
- A piece that stays in its original square when making a capture
- A piece that can capture mid-move and continue moving
- A piece that, when captured, goes into it's owner's hand and can be dropped back into play

The ideal software would also handle multiple royal pieces on a side (win by capturing all of them, no restriction on moving into check; corollary: no stalemate) and the option to start with pieces in hand that can be dropped into an empty square on the first rank in place of a move (some pieces can be dropped on the first or second rank).

There's an expansion coming out in a month with yet more weird pieces (example: a piece that moves once "for free" each turn, before you move a different piece).

Well, you somehow decided that the values would follow the formulas:

R2 = R1 * (1 + p)

R3 = R1 * (1 + p + p^2)

R4 = R1 * (1 + p + p^2 + p^3)

and so forth.

That looks like a mobility calculation to me, but whether you choose to call it that or not, the fact remains that your "interpolation" is following a curve that you derived based on ONE of Ralph Betza's ideas regarding piece value while ignoring many other important ideas that he also had about piece value.

I'm saying that I don't think the intermediate piece values are actually going to fall along that particular curve.

JÃ¶rg, I'm not sure you've given due consideration to board geometry. Betza's mobility calculations attempt to account for both the probability that a move will be blocked by another piece AND the probability that a move will be blocked by the edge of the board. If you assume that pieces are distributed randomly, then the odds of being blocked by a piece are p^(distance-1), as your formula suggests, but the odds of being blocked by the edge of the board follow a completely different pattern (e.g. for a rook, it's 1 - (distance/8)). That means it can't be accurately represented simply by plugging a different value for p into the polynomial shown in your post.

You also seem to be assuming that piece value is directly proportional to mobility. Most people believe that value has a super-linear relationship to mobility; the evidence being that compound pieces tend to be worth more than their component parts. There are also many things other than mobility that might affect a piece's value; Betza attempted to compile a list here.

Finally, you might want to consider that Betza believed a full Rook had a value closer to 3/2 of a Knight than to 5/3 of a Knight. If that's the case, then it's unclear whether a piece priced at 4 points should aim to be worth 4/5 of a Rook or 4/3 of a Knight--a difference of perhaps 10%, or roughly the difference in your calculated mobility between an R4 and a R5.

Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.

This gives the following formula for the value of a full rook:

R = R1 * (1 + p + p^{2}+ p^{3}+ p^{3}+ p^{4}+ p^{5}+ p^{6})

Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.

The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is â‰ˆ0.715.

The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.

The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.

Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.

X1 | X2 | X3 | X4 | X5 | X6 | X7 | magic number | |

Rook | 150 | 260 | 339 | 398 | 440 | 471 | 494 | 0.73 |

Bishop | 150 | 225 | 262 | 282 | 291 | 296 | 298 | 0.5 |

Queen | 300 | 515 | 668 | 777 | 855 | 910 | 950 | 0.715 |

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## Extra pieces:

Q6 8

zB 6

NN 5

DA 2

WzB 8

qN 5

t[FR] 8

Rt[FR] 14

QN 11

QNN 15

t[WB] 7

Bt[WB] 10

Qt[FR]t[WB] 18

NLJ 12 (10 but has SERIOUS checkmate power)

NNB 9

R2 2

AA 2

WAAmD 4

t[WB]t[FR] 11

HHWD 7

## This will be updated regularly