Kevin Pacey wrote on Sat, Nov 7, 2020 04:29 AM UTC:
@ H.G:
Leaving aside the leaper formula, which is rather mysterious, I may have misunderstood (due to insufficient clarity?) the non-capture penalty of 2, which you wrote of somewhere long ago, besides just now. Apparently you meant Capturing-able piece has 2x1+1x1=3 and Non-Capturing piece has 2x0+1x1=1, in terms of their factors.
Yet, looking at things afresh, I believe I read somewhere (Betza?) to the effect that the chance of any given square on the average path of squares (traversed by a piece) being unoccupied by any other piece, on a board with some pieces per side, during a typical CV game (if there is such a thing), notably with setup piece density of roughly 50% pieces to squares on the board, is =0.7. That is, there is an obstruction rate of 0.3 for each of the squares on the paths e.g. sliders take, if not for the destinations of leapers as well (in a way), I might add. If we assume on average half of non-empty squares leapers might arrive at in one move are occupied by an enemy piece, rather than a friendly one, then that's 0.15 for that.
The average chance of a non-capture on a given square is 0.7, as observed above. So, a capturing-able leaper gets 0.15x2+0.7x1, or 1.0, and a non-capturing leaper gets only 0.7x1 (noting it cannot take an 'obstruction'), in terms of a crude value multiplier compared to a capturing-able piece. Then 0.7 divided by 1.0 =0.7 would then now be my preferred non-capture multiplier penalty (applied to the value of a piece that would otherwise normally be a capture-able one), which is admittedly also somewhat far from the original 0.5 non-capture multiplier penalty that I had used (in your view erroneously) when looking at things differently than yourself.
However, 0.7 does seem rather an over-generous piece value multiplier penalty, so for the moment I could try a piece value multiplier penalty of 0.33 (in my quest for simpler and ideally unified formulae, however [slightly?] inaccurate the results), which still wouldn't affect my previous calculation for the Alpaca that much, as unsound as it seemed to you (and still would if I only used 0.33 instead of 0.5, and changed nothing else). At any rate, I still suspect the Alpaca as being worth more than 2, since at the least, as you mentioned, it helps in the odd mating attack.
@ H.G:
Leaving aside the leaper formula, which is rather mysterious, I may have misunderstood (due to insufficient clarity?) the non-capture penalty of 2, which you wrote of somewhere long ago, besides just now. Apparently you meant Capturing-able piece has 2x1+1x1=3 and Non-Capturing piece has 2x0+1x1=1, in terms of their factors.
Yet, looking at things afresh, I believe I read somewhere (Betza?) to the effect that the chance of any given square on the average path of squares (traversed by a piece) being unoccupied by any other piece, on a board with some pieces per side, during a typical CV game (if there is such a thing), notably with setup piece density of roughly 50% pieces to squares on the board, is =0.7. That is, there is an obstruction rate of 0.3 for each of the squares on the paths e.g. sliders take, if not for the destinations of leapers as well (in a way), I might add. If we assume on average half of non-empty squares leapers might arrive at in one move are occupied by an enemy piece, rather than a friendly one, then that's 0.15 for that.
The average chance of a non-capture on a given square is 0.7, as observed above. So, a capturing-able leaper gets 0.15x2+0.7x1, or 1.0, and a non-capturing leaper gets only 0.7x1 (noting it cannot take an 'obstruction'), in terms of a crude value multiplier compared to a capturing-able piece. Then 0.7 divided by 1.0 =0.7 would then now be my preferred non-capture multiplier penalty (applied to the value of a piece that would otherwise normally be a capture-able one), which is admittedly also somewhat far from the original 0.5 non-capture multiplier penalty that I had used (in your view erroneously) when looking at things differently than yourself.
However, 0.7 does seem rather an over-generous piece value multiplier penalty, so for the moment I could try a piece value multiplier penalty of 0.33 (in my quest for simpler and ideally unified formulae, however [slightly?] inaccurate the results), which still wouldn't affect my previous calculation for the Alpaca that much, as unsound as it seemed to you (and still would if I only used 0.33 instead of 0.5, and changed nothing else). At any rate, I still suspect the Alpaca as being worth more than 2, since at the least, as you mentioned, it helps in the odd mating attack.