Comments/Ratings for a Single Item

You write, '[Divergent] pieces are stronger than either non-divergent piece, but weaker than the unrestricted compound piece.'

This seems exceedingly unlikely.  You seem to be claiming that a piece that moves as a Queen but captures as a Knight is stronger than either; in fact, read strictly, I believe you are claiming that a piece that moves as a Queen but captures as a Bishop is stronger than a Queen, even though it has strictly fewer moves.

I believe conventional wisdom is that such pieces (which George Jeliss calls 'snipers') usually have a value that is somewhere between the strengths of the non-divergent pieces, closer to the capturing component than the non-capturing one.  (Though I'm certain it is possible to craft examples that violate this rule.)

I remain skeptical even of your revised claim.  Even when the components are non-overlapping and 'similar' (though I can only guess what that means), I see no obvious reason that having divergent capturing and non-capturing moves is better than having the same in the general case; only the capture or non-capture will be legal in any given position, so there is no loss due to 'overlap'.  In fact, I've seen the opposite argued, on the grounds that a divergent piece is easier to trap, since it is unable to attack enemies in its way.

When the combination lifts a special disadvantage (such as colorboundness), that is a special case; though it would need to eliminate a disadvantage from EACH of the components in order for that to make it stronger than BOTH, in general.  (And I am unconvinced that lack of triangulation is a disadvantage of any measurable significance.)  If the components were instead Crab and Barc, or Rook and Nightrider, what then?

Comparing a Pawn to a Point is like comparing a Queen/Knight divergent piece to a Knight; that is obviously the weaker parent (Ferz is already stronger than Wazir, and loses much less from the forward-only restriction).  I strongly suspect that the Pawn is weaker than a forward-only Ferz, despite the Ferz being colorbound, because the fF has more possible moves and does not need to capture to change files (the latter advantage being especially important if promotion is allowed).

What is your evidence or reasoning for valuing divergent pieces more highly?

The idea that the Ferz might be stronger thean the Steward is new to me. The Ferz cannot even reach the whole of the board. Is the Ferz stronger the even the Wazir. I would say that on an edge,let alone in a corner, the Ferz is the weakest of the three because of its fewer immediate destinations. The Pawn is a harder call as it will on average have fewer destinations than the Cross, but eight Pawns in a row are certainly stronger than eight Crosses in a row. It is easier to build up chains of non-mutually protected Pawns, and a Pawn on one side can protect while that on the other side progresses.
	I would be interested to know the views of others. Whose view is nearer the general consensus, if there is one?

The only measure to get solid information about the strength of a certain piece is to playtest it (in human play and/or computer play). As far as I know, there are few divergent pieces really tested.

Mike Nelson's Seperate Realms Rook comes out on a value of about 75% Rook plus 25% Dabbabarider, suggesting that capture constitues about 3/4 of the piece value, while moving alone constitutes about 25%. But this particular piece has rather similar move and capture patterns; the more the patterns diverge, the more playtesting is needed, and there is probably no simple formula describing the results.

Note also that the results of playtesting depend on the testbed: What board is choosen, what is the oppenent's army, what pieces are in the same army (sociability effects).

I have done computer measurements of the piece values of the divergent combinations of Q and N and K and N in a FIDE context. The results supported an earlier observation, that a 'Lion' (Betza WFADN) would lose about twice as much value by disabling one of its capture moves (e.g. fW) as it would from disabling the corresponding non-capture. The value of mNcQ was around 7, that of mQcN around 5 (so 2/3 and 1/3 on the way between N and Q, respectively).

The results for the K+N chimera was more surprising. Here we start with pieces thatare already very close in value, so that the interpolation does not do much, and cooperativity effects stand out more visibly. The conclusion was that a pure K (Commoner) is slightly weaker than a pure N (about 0.3 Pawn values), and that mKcN is quite close to pure N, as expected. But the surprise was mNcK, which came out about 0.5 Pawn stronger than pure N!

My speculative explanation is that there are some terms in the piece value that depend on global properties of the move pattern (rather than contributions from individual moves). Like the K value is suppressed because it is a 'slow' piece, (it cannot overtake a moving pawn), but ehanced because of 'concentration' of its capture moves (it can at the same time attack a Pawn, the square in front of it, and the Pawn that protects it). The mNcK has the best of both: concentration of the capture moves, and speed of the Knight.

Thanks once again for your very interesting computer tests, Mr. Muller!

Notice that if this were simply a matter of the K capturing moves being unusually strong and the K non-capturing moves being unusually weak, we should expect mKcN to be weaker than K, but (if I am reading you correctly) the tests say it is stronger.

So if you are correct that K's moves have a higher 'ideal value' but are suppressed by K's overall low speed (which would have been my first guess also), then we must conclude that a piece's 'speed' depends significantly upon its capturing moves, not only its non-capturing moves.  This seems surprising to me, because I would imagine that capturing capabilities are only exercised in a small percentage of the piece's movements.

Alternately, this could be a matter of special synergy rather than a global bonus or penalty--that is, perhaps the combination of speed and concentrated attacks is particularly valuable, but neither component has significant value by itself.  Thus, the mKcN, lacking both, is not significantly worse off than either N or K, which only have one each, but the mNcK stands tall with both.

Curious.

Well, it could be that the end-game value of mKcN is indeed lower than that of a pure Commoner, but that this is masked because the superior forking power makes it very likely you can always trade it for another minor in the midle-game. I measured opening values only.

If that were true, the lower intrinsic value of the mKcN would only be revealed when there is nothing to trade it against, or if all the opponent's minors would be weaker than Commoners.

On the prefixes: given the four "basic" (European, Warhead, Ambush, Nonchalant) patterns of capture/non-capture allowed, we have prefixes for those with zero (Ancient), four (Eurofighter), or two (all six ways) powers. Missing are those with three. Proposals:

anti-European (must capture at least once): HUNGRY

anti-Warhead (may not capture twice): DIETING (no second helpings!)

anti-Ambush (if first is passive, second must be too): ELECTRIC ("It has to warm up... so it can kill you" of Wednesday Addams' electric chair)

Anti-Nonchalant (if first is capture, second must be too: ADDICTED (gotta keep killin'...)

"WARHEAD Pawn: both steps must be capturing (more aggressive than the above)"

- to say that both steps MUST be capturing is not to make the Warhead pawn more aggressive but to allow the Warhead pawn an initial double step only in the extremely unusual event that two pawns are lined up for it to take. Since an opponent is unlikely to allow that to happen, in practice, that means the Warhead pawn is unlikely to be able to make an initial double step at all.

So in practice, makes it more passive, not more aggressive (unless we substitute "can" for "must.")

Edit: I see my comment is insufficient in light of how you use "may" to introduce combinations of initial-step pawns. I will review and comment again later, unless someone can clear all this up in the mean time. I am now thinking perhaps we can create an even more comprehensive list that is also more plausible - unless you already did that, Charles. I wouldn't put anything past you when it comes to creating authoritative lists.

What about the name of Highlander+Lowlander?

What about the name about the symmetry version about pawnbroker and yeomanwarder? I think you have lost these kind of pieces.