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SOHO Chess. Chess on a 10x10 board with Champions, FADs, Wizards & Cannons.[All Comments] [Add Comment or Rating]
💡📝Kevin Pacey wrote on Thu, Dec 27, 2018 04:26 AM UTC:

Once again I'm not sure how to argue with your most recent post, H.G. For that reason, and for the sake of not risking discussing too many points at once, which may in turn multiply (as it seems has been happening), I'll just mainly confine myself for now to answering your query of me, namely:

If I would make a new piece, by starting with a Bishop and replacing one of its Ferz moves by a Wazir move. Would you now argue that a normal Bishop is worth only half as much as this piece, because displacing that one move to a neigboring square made it color bound? Or would you argue that the piece is worth one Pawn more than a Bishop because it is the combination of 1/4 Wazir with a piece that was only handicapped so little compared to a normal Bishop by missing this Ferz move that it had no effect on the value?

The calculation I'd make for such a piece's value is a bit complex. First, I decide that such a piece is treatable as a compound piece of sorts, then I figure out the value of it's components in stages. However, first I need to assume a given board size, namely 8x8. That allows me to figure out what fraction of a bishop is left when a ferz move is taken away from it. A normal B on 8x8 has 10 moves on average on an empty 8x8 board (i.e. 7 minimum, 13 maximum), so taking away a ferz move gives 9 moves on average (just averaging the new minimum and maximum cases, maybe none too precise a thing to do). Thus I'd work out what 90% of a B is worth as the first component of the compound piece.

The second component of this compound piece would be 1/4 wazir (depending on if it was the forward step it would be 2/5 of a wazir, or if it was a sideways or backward step it would be 1/5 of a wazir, based on how I've implemented your previous discussions about the direction of a piece step, but you specified 1/4 wazir this time, perhaps to make things a little easier for me). In this case it does, because I can easily use the value of 1/8th of a guard instead of 1/4 of a wazir (more or less the same thing) which avoids what seems like a worse slight error I get if I used (wazir-P)/4 rather than using (guard-P)/8 - the latter produces the same value as wazir/4 (the only times fractions of pieces might IMHO clearly work out very 'nicely' for me as it were is when division of a piece by 2 is performed, such as a guard broken into its ferz and wazir components, each having values that I deem to be equal, with a Ps worth of co-operativity first subtracted).

Thus, First Component + Second Component + Pawn = value of the [compound] piece you enquired about, according to the way I'd do the calculation (for now, with my imperfect way of doing things). This becomes:

(B-P)x0.9 + (wazir-P)/4 + P = approx. value of compound piece, or (B-P)x0.9 + (guard-P)/8 + P, which becomes:

(3.5-1)x0.9 + (4-1)/8 + 1 = value of compound piece (note I rate B=3.5 and guard=4 on 8x8),

and thus value of [compound] piece that you asked about = 2.25 + 0.375 + 1 = 3.625 on 8x8, which I'd note is clearly far from twice the value of a B on an 8x8 board.

One of the other problems I have to cope with is that this sort of method wouldn't work (in any sort of fashion, at all) if e.g. a wazir's value was necessary to use in a calculation, and it was worth a pawn or less, since wazir-P would then be worth zero or a negative value. This is indeed the case for the value I give a wazir on 10x10, i.e. I put it at 0.75 on that size board (which you objected to, as well, and I'll more or less pass on that, except to note for now that a wazir crosses the board slower on 10x10 than 8x8, which I count as a tangible consideration all the same, though other considerations pro and con may be possible, especially depending on the armies deployed).

Anyway, for 10x10 figuring out the first component I'd do similarly as before, but for 1/4 of a wazir to be computed I'd now definitely first desire to compute the value of a guard, then hope to use the value of 1/8 of a guard (a similar thing as 1/4 of a wazir), i.e. hoping that a guard is worth more than a pawn on 10x10 (if not, I simply have to use wazir/4 rather than (wazir-P)x0.25, with any difference/error being rather small anyway). It just so happens my home formula for a guard's value puts it at approx. 2.5 on a 10x10 board, so I'd figure out 1/8 of a guard by using (guard-P)/8 = 0.1875 (happily the same as wazir/4, again, as it always would be), which in turn is what I'd use for the second component of the compound, if I were to compute its value for on 10x10. The first component I see as worth 12/13ths of a B, so now the compound's value that you asked about (if on 10x10) would be approx. (with having B=3.5 still, on 10x10):

(B-P)x12/13 + (wazir-P)x0.25 +P, or (3.5-1)x12/13 + (guard-P)/8 + P, or

value of [compound] piece you asked about (if on 10x10) = 2.308 approx. + 0.1875 + 1 = 3.496 approx., which I'd note is again nowhere near twice as much as a B's value on the given board size.

I'll have to admit again that my formulae and methods are not completely perfect, and seem unsound (in particular with fractions of pieces), but the values I get with them to date don't seem ever too far out of the ballpark, to me at least. One interesting thought experiment might be what to make of the value of some sort of 'half of an archbishop'. Doing things my way, (archbishop-p)/2 would be the answer, rather than archbishop/2 (or is half an A worth something different altogether?), and clearly A/2 would give a value greater than a B or N if one uses one, or even two, pawns worth of cooperativity between the bishop and knight components. For now, I still just assume one pawn's worth of cooperativity between those two components, so for me on 8x8 archbishop =N+B+P=3.5(approx.)+3.5+P=8 (noting Q=R+B+P=5.5+3.5+1=10), and thus (archbishop-p)/2=3.5 is the value I get for half an archbishop, i.e. about the value of a N or B. That's opposed to archbishop/2=4, or greater than the value of a minor piece. At any rate, there seems to be some sort of consistency with quite a few of the piece values I've come up with over time, in spite of the lack of perfection, at least it seems to me so far.

In a previous post in this thread I dealt at length with how I estimated the value of an alfil and a dabbabah, including how I factored in such things as my x0.5 binding penalties, plus counteracting bonuses for leaping ability and speediness, if you wish to see instances of how I've handled binding penalties in my personal calculations, when compound pieces are not deemed at issue. I've done calculations for the value of a knight in Alice Chess using a way to take into account the type of binding to it that happens on the two boards there, and I came up with a value for the N (and other pieces) close to what the rules page notes gave for piece values (maybe by ZoG?) in the case of that game, at least. The values are on my 4D Quasi-Alice Chess rules page, in the Notes section (note I used the initial chess base values N=B=3, R=5, Q=9 in that particular case, perhaps to try to match the chess base values I thought were probably initially used in the preliminary calculations made, for Alice Chess, by ZoG - I did all that work long ago).