In Monster Chess, one side has only a King and four Pawns, but they get to move twice per turn (including the possibility of moving two different pieces one move each, which a doublemove army can do but a doublemove piece cannot). The other side has the normal setup of peces, making the normal one move per turn.
If we assume that this is an equal game, simple arithmetic indicates that a doublemove piece must be worth 6 times as much as a normal piece (or a bit less, depending on what value you give to the King).
Experience with the game indicates that it is far from equal. The player with the full set of pieces will lose some games at first, but will gradually learn how to win every time.
SPOILER: the following rot13 text takes all the fun out of this game.
ROT13: Va beqre gb jva jvgu gur shyy nezl, fvzcyl nqinapr lbhe Dhrra'f Ebbx'f Cnja, gur rarzl Xvat zhfg pbzr bire gb pncgher vg naq gura lbh chfu gur bgure Ebbx'f Cnja n juvyr, hagvy ng ynfg lbh pna fnsryl oevat n Ebbx gb gur sbhegu enax naq sbez n oneevre gung gur qbhoyrzbir Xvat pna arire crargengr. Bs pbhefr, va gur zrnagvzr lbh zhfg or pnershy gb pbhagre gur rarzl Cnjaf, ohg fvapr lbh unccvyl tvir n Xavtug be Ovfubc sbe n Cnja, vg fubhyq or rnfl.
Part of the problem is that Pawns cannot retreat, and there are so few doublemove Pawns on the board that you cannot afford to trade them. In fact, trading the four minor pieces for the four 2move Pawns produces an easy win even if a few of the full-army Pawns get lost along the way.
Part of the problem with the theoretical side of this is that the 2move King in Monster Chess is the King, the royal King subject to check and mate. Giving checkmate to this monster requires a lot of force. Does the fact that the 2move power is available for escaping check make the Monster King worth more than an ordinary 2move piece would be?
Although Monster Chess does not solve the theoretical problem of how much a 2move piece is worth, it does give us a few helpful clues.
For example, we know that the value of this particular doublemove army is less than 6 times the value of the same 1move army; and it's helpful to have an upper limit.
The basic scheme I have come up with is that White gets 8 normal Pawns and a normal King plus a few doublemove pieces, and Black gets a complete normal setup. Sometimes Black moves first; for consistency, WHite is always the one with the small army of 2move pieces.
In addition, I have adopted the common rule of multimove chess, "check stops the move": therefore, for example, if White starts with no Pawns and a 2move Bishop on f1, the move 1.Bf1-b5 is not checkmate, and in fact it isn't even check. Because of this rule, the doublemove pieces in my variant are not as strong as the doublemove pieces in Monster Chess.
The reason for the "check stops" is that the rampaging checks against the undeveloped Black position make it harder to see the real value of the 2move pieces.
Furthermore, I allow the 2move piece to make a single move if desired: 1. Ng1-f3 is legal, you do not have to make the second move of this 2move Knight.
Last of all, these are two-move pieces, instead of the 2move army used in Monster Chess: you cannot make 1 move with one piece and 1 move with another.
1. b3 h5 2. Bb2 Rh6 3. g3 d6!
After 3...a5? 4. Bg2 Ra8-a6 5. Bb2-e5-f4!, Black loses a piece.
1. b3 h5 2. Bb2 Rh6 3. g3 d6! 4. Bg2 c6 5. Bb2:g7-c3 a7-a5
Black will gladly give up the Queen for half WHite's army; besides which, I think that the 2move B is worth more than the Q in any case.
1. b3 h5 2. Bb2 Rh6 3. g3 d6! 4. Bg2 c6 5. Bb2:g7-c3 a7-a5 6. a2-a4 h5-h4 7. g3-g4 h4-h3 8. Bf3 Rg6 9. g5 Bh6! 10. e3 R:g5
That's enough; probably both players have made inferior moves, and we have enough of a position to see that Black can reasonably hope to develop his pieces and make gradual progress, while White can reasonably hope to gradually pick off materaial, a Pawn at a time, and reach an position where Black hasn't got enough stuff to block all of the threats and then start picking off pieces.
If the material is even, simple arithmetic shows that a 2move Bishop-pair must be worth about 5 times as much as a normal Bishop-pair. We can see that the material in this game is at least close enough to show that the 2move Bishop-pair is worth at least 4 times as much, and not more than 6 times as much. as the normal Bishop-pair.
In addition, we have a new chess variant, Two Bishop Muenster Cheese, that seems to be better balanced than Monster Chess and at least as interesting to play.
Here is another sample game, showing that a blitzkrieg strategy will not work:
1. e4 e5 2. Bb5 Nf6 3. d3 h6 4. B:h6-g5 Rh5! 5. B:f6:d8? K:d8
Shooting pieces tend to be unbalancing because chess just isn't geared to it; of course, doublemove pieces are even worse, but it's quite possible that "shooting" is worth almost half as much as doublemove.
It looks as if the value of 2move pieces may be determined more by their "empty-board maximum mobility" than by their "crowded-board average mobility". If this is so, we would expect to find that Two Knight Muenster Cheese greatly favors Black, because 2move Knights must be much weasker than 2move Bishops.
On the other hand, Knights can develop without waiting for Pawn moves, and therefore can take advantage of Black's lack of development in a manner that distorts the values of the pieces.
1. Ne5 d6 2. N:f7:h8 Nc6! 3. Nd5 Rb8! 4. N:e7:g8? Be6!
The Knight on g8 cannot escape because f6 gives check, and so Black has an easy win. Better might be 2. N-g6:h8, but in any case it seems that a blitzkrieg strategy won't win outright. However, at least White could grab one Rook...
1. Nd5 e6!? 2. Nd5-b6:a8 Qf6!? 3. d4 Ne7 4. N:c7+ Kd8 5. Nc7-b5:a7 Ne7-c6 6. Na7:c8-b6 Q:d4?! 7. Ng1:d4 Nc6:d4 8. Nb6:d7:b8 Bf8-d6 9. Nb8-d7-b6
This is all very nice, and it feels as though 2move Knights were even stronger than 2move Bishops; but I suggest you look at the same game where Black gets to start with Pawns on d6 and e6:
1. ... d7-d6, e7-e6 2. Nc4 b5! 3. N:a8 Bd7! 4. Nf4 Qf6!
What an amazing difference! This shows that the game of Two Knight Muenster Cheese is too unbalanced to be playable, but if we force both sides to make a few calm opening moves perhaps it's still possible to use this game to get a feel for the strength of these Knights:
1. Nc3 e6 2. a3 Nf6 3. Nf3 Nc6 4. e4 h6 5. d4 d6
Another amazing difference. White has no credible attack, and can only wait and try to catch Black in an error; Black need merely advance carefully, and it is all over. After some moves like 6. d5 ed5 7. ed5 Ne7 8. Nd4 Bd7, or even 8...Qd7!, the 2move Knights don't seem to have been worth much at all.
Perhaps the problem is not so much the mobility as the distance the piece can go? Let's try it with Long Knights, but because the L is so clumsy let's use "Long Fibnif"s instead.
Of course, White has doublemove fbLF!
1. (fbLF)b1-c4 [threatens mate at f5 or d5] e7-e6 2. (fbLF)c4-b5:a8
It's the old blitzkrieg story. Let's force a "calm" position:
1. e4 e6 2. d4 d6 3. (fbLF)e2 Nf6 4. (fbLF)d2 Nc6 5. d5 ed5 6. ed5 Ne7 7. (fbLF)d2-c3-d4 Rg8 8. (fbLF)e2-b3-c4
The Ra8 is hanging, and if it goes to b8 it can be captured. Incorrect is 8...Bd7?? 9. (fbLF)c4:b7 mate, and without examining the position further we can say that the 2move fbLF is very obviously a lot stronger than the 2move Knight.
I think we can safely say that doublemove pieces (with the "check stops the move" rule) have a value that depends on the single-move value, the max-empty-board-mobility, and the distance of the move.
I think it is safe to say that a 2move Knight is worth 3 or 4 normal Knights PLUS a large blitzkrieg bonus, and a 2move Bishop-pair is worth 4 or 5 normal Bishop-pairs (but I didn't show you any examples of how a single 2move Bishop is much weaker than you'd expect, so that its colorboundness is worse than that of a normal Bishop).
It is a reasonable guess that a 2move Commoner is scarcely worth 3 normal ones, and a 2move Ferz or a 2move Wazir would be worth only a bit more than a Knight, but that a 2move Rook would be worth 5 or 6 Rooks and a 2move Queen would be worth 7 or 10 normal Queens.
This is just fine, except that the nominal value of a Pawn is 1.0, and we know that a 2move Pawn is worth more than a 1move Pawn, so there is a problem: what number could this "N" possibly be?
I think that a doublemove Knight will be too expensive, not worth what you pay for it, a 2move Rook will have a fair price, and a 2move Queen will be a bargain.
If you can survive on the board while saving up 54 zorkmids or so to buy a doublemove Queen, you deserve to win.
Shooting is also a new modifier. I think it would be too confusing to have a piece with some shooting captures and some normal captures; therefore, I made this a whole-piece modifier.
Just as for doublemove, the modifier is probably too expensive when used with cheap pieces.
When a shooting piece captures, it does not move into the square of the piece it captured. For example, a "shooting Knight" would capture a Pawn on e5 by shooting it: at the end of the move, e5 is empty and the "shooting Knight" is still on f3.
A 2move Ferz combines the powers of F2, zF2, and "shooting Ferz". Actually, it's even a bit better than that because it can capture 2 pieces in one move. Unlike the FAD, the 2move F in the opening position cannot jump out from f1 to d3, ignoring the Pe2 which is in its way: the 2move F, in a normal setup, has no legal move until a Pawn makes way for it.
I think a FAD is worth about 4 Pawns, and I think that a 2move F should be close in value to a FAD. Is this "only a bit more than a Knight"? Close enough, I guess.
The army that uses the FAD is, of course, the Colorbound Clobberers. A variant version of this army, using 2move F instead of FAD, sounds interesting.