💡📝Kevin Pacey wrote on Tue, Dec 4, 2018 10:36 PM UTC:
On 8x8 I value a N not at 3, but at Euwe's 3.5 (or I'd say unofficially 3.49, just to make it a hair less than a B, but 3.5 is a rounder number), and also I value R=5.5 (Euwe), A=D=(N-P)/4(approx.)=0.625 and W=(Man-P)/2(approx.)=1.5 (though I'd suppose it's actually a little less since a ferz is thought worth a little more than a W). Cooperativity (your own idea?) I'm not sure about the meaning of. One thing that somewhat influenced me to believe my value for a WAD (or Wizard) on 10x10 was about okay is that the strategy page for Omega Chess' commercial website advises that B=4, like for a Champion(WAD) or Wizard, but it was advised not to trade a B for a Champion, in that 10x10 game (plus its 4 extra cells), I seem to recall. Not sure if the advice was meant just for the opening phase of a game. Nor do I know who wrote the strategy page, either. I don't rate a B as worth 4 pawns, quite (prefering never to use more than e.g. 3.99 for any board size with at least some breathing room on it - otherwise the thought that in an endgame a B can sometimes restrain 3 passed pawns, but seldom 4 of them, affects my reasoning, correctly or not).
Regarding the above, an A has half as many targets as a N (which moves in a somewhat similar way, at least in terms of range and being a leaper), and is thrice 'binded' too (which I take as calling for a [further] halving, only, since as I see it there is a x2 leaper bonus offsetting a x0.5 penalty I would give for one of the bindings, and I also gave a x2 bonus for an A being able to move faster across a board to a certain square at times than a N, as a [possibly generous] way to offset one other binding x0.5 penalty). A D has a similar story, except it's only twice 'binded', but I also don't think of it as quite as often being speedier than a N to get to a given cell that both might want to reach eventually (aside from the square reached via a one move leap by the D - though that is not so bad for the N to get to as for it getting to a cell that an A moves to in just 1 turn). It may seem the above reckoning is fishy somewhere, but the value I get for A (and maybe also for a D - note it can be slower at times than an A to get to a given cell) seem about right to me. The wikis for As and Ds rate either as worth a bit more than a P on 8x8, which I find hard to believe in the case of an A especially. Anyway, a Man is a compound of a W and a ferz, the latter two being of roughly equal value IMHO (though your own results disagree with Man=4), and I value Man=W+ferz+P, similar to Q=R+B+P in chess (the latter is an equation I often depend on heavily as an analogy when I calculate/estimate values, quite possibly incorrectly at times, but it helps keep my life simpler).
Fwiw, for the WAD on 10x10, first I rate a (lone) Man there worth 2.5 approx. (using my rather unproven formula for Man value that doesn't apply to all possible board sizes - I have an even more complex such formula for N value), thus rating a W or ferz there worth 0.75, and an A or a D there as worth 0.5 each. Assuming these values aren't too far off, since Q=R+B+P in chess, I hazard to rate a WAD=(W+A+P)+D+P where P=1, to get WAD=3.75 (for 8x8 I get WAD=4.75 with such a calculation). Thus 2Ps worth of value is being added in by this calculation (I neglected to mention this detail earlier). I figure a camel is still worth 2 on 10x10 (as on 8x8) due to its considerable range, so a wizard=camel+ferz+pawn=3.75 I'd say similarly (for 8x8 I get wizard=4.5 with such a calculation). Unfortunately, for 10x10 at least, my results don't agree with the effects of the notion of cooperativity, as you described them. I have trouble understanding the effects described, too. For example, on 8x8 a camel is worth 2 and has 8 targets (max.). A wizard has 1.5 more targets, so I'm guessing by your example calculation that only camelx1.5=3 must be exceeded for the value of a wizard on 8x8, which isn't saying too much yet. Unless describing the effects of cooperativity depends on using a (normally more valuable) N rather than (e.g.) a camel for one's example calculated estimate of a 12-target leaper's approx. value (on any size board).
On 8x8 I value a N not at 3, but at Euwe's 3.5 (or I'd say unofficially 3.49, just to make it a hair less than a B, but 3.5 is a rounder number), and also I value R=5.5 (Euwe), A=D=(N-P)/4(approx.)=0.625 and W=(Man-P)/2(approx.)=1.5 (though I'd suppose it's actually a little less since a ferz is thought worth a little more than a W). Cooperativity (your own idea?) I'm not sure about the meaning of. One thing that somewhat influenced me to believe my value for a WAD (or Wizard) on 10x10 was about okay is that the strategy page for Omega Chess' commercial website advises that B=4, like for a Champion(WAD) or Wizard, but it was advised not to trade a B for a Champion, in that 10x10 game (plus its 4 extra cells), I seem to recall. Not sure if the advice was meant just for the opening phase of a game. Nor do I know who wrote the strategy page, either. I don't rate a B as worth 4 pawns, quite (prefering never to use more than e.g. 3.99 for any board size with at least some breathing room on it - otherwise the thought that in an endgame a B can sometimes restrain 3 passed pawns, but seldom 4 of them, affects my reasoning, correctly or not).
Regarding the above, an A has half as many targets as a N (which moves in a somewhat similar way, at least in terms of range and being a leaper), and is thrice 'binded' too (which I take as calling for a [further] halving, only, since as I see it there is a x2 leaper bonus offsetting a x0.5 penalty I would give for one of the bindings, and I also gave a x2 bonus for an A being able to move faster across a board to a certain square at times than a N, as a [possibly generous] way to offset one other binding x0.5 penalty). A D has a similar story, except it's only twice 'binded', but I also don't think of it as quite as often being speedier than a N to get to a given cell that both might want to reach eventually (aside from the square reached via a one move leap by the D - though that is not so bad for the N to get to as for it getting to a cell that an A moves to in just 1 turn). It may seem the above reckoning is fishy somewhere, but the value I get for A (and maybe also for a D - note it can be slower at times than an A to get to a given cell) seem about right to me. The wikis for As and Ds rate either as worth a bit more than a P on 8x8, which I find hard to believe in the case of an A especially. Anyway, a Man is a compound of a W and a ferz, the latter two being of roughly equal value IMHO (though your own results disagree with Man=4), and I value Man=W+ferz+P, similar to Q=R+B+P in chess (the latter is an equation I often depend on heavily as an analogy when I calculate/estimate values, quite possibly incorrectly at times, but it helps keep my life simpler).
Fwiw, for the WAD on 10x10, first I rate a (lone) Man there worth 2.5 approx. (using my rather unproven formula for Man value that doesn't apply to all possible board sizes - I have an even more complex such formula for N value), thus rating a W or ferz there worth 0.75, and an A or a D there as worth 0.5 each. Assuming these values aren't too far off, since Q=R+B+P in chess, I hazard to rate a WAD=(W+A+P)+D+P where P=1, to get WAD=3.75 (for 8x8 I get WAD=4.75 with such a calculation). Thus 2Ps worth of value is being added in by this calculation (I neglected to mention this detail earlier). I figure a camel is still worth 2 on 10x10 (as on 8x8) due to its considerable range, so a wizard=camel+ferz+pawn=3.75 I'd say similarly (for 8x8 I get wizard=4.5 with such a calculation). Unfortunately, for 10x10 at least, my results don't agree with the effects of the notion of cooperativity, as you described them. I have trouble understanding the effects described, too. For example, on 8x8 a camel is worth 2 and has 8 targets (max.). A wizard has 1.5 more targets, so I'm guessing by your example calculation that only camelx1.5=3 must be exceeded for the value of a wizard on 8x8, which isn't saying too much yet. Unless describing the effects of cooperativity depends on using a (normally more valuable) N rather than (e.g.) a camel for one's example calculated estimate of a 12-target leaper's approx. value (on any size board).
https://omegachess.com/strategy.htm