Comments by jlennert

The increased knowledge of mating potential has raised the apparent value
of Commoner, then?

Arimaa has a special set-up step at the start of the game where white
(gold) arranges all his pieces on the board, and then black (silver) gets
to arrange his own pieces after seeing white's arrangement, and then white
gets the first move.  Seeing your opponent's piece arrangement before
arranging your own pieces can only be an advantage, so in this case each
side receives an asymmetrical advantage in the opening, and it's not
obvious how to compare them.  It may be less a case of the first-move
advantage being small, and more a case of these two advantages canceling
out.

Or not; I'm just speculating wildly.

You might also consider repeating your experiment of adding mating
potential to a bishop.

Another consideration:  does the value of mating potential depend
significantly on how many other pieces already have it?  If you replaced
rooks on both sides with a similarly-valued but non-mating piece, does the
value of commoners relative to knights go up?  If you replaced the bishops
on both sides with a similarly-valued piece with mating potential, does it
go down?  I think Betza suggested once that it was important for a side to
have a piece with mating potential, but not so important how many pieces
had it.

I don't think that's true about the bishops' value, David.  In addition to being color-bound, bishops also suffer a significant loss in mobility near the edges and corners of the board, while rooks suffer much less--due to the fact that the board edges line up with the rooks' moves.  (There are also several other strategic differences of uncertain importance, such as the fact that the rook can interdict the movement of the enemy king, and the differing ways each of them interact with pawns.)

In fact, I would guess that this loss of mobility is far more important than colorboundness, because players of cylindrical chess (where bishops do not lose mobility at the sides of the board) have reported bishops to be roughly the equal of rooks--despite still being colorbound.

It wouldn't surprise me if BK and RK are closer in value than B and R, though at a quick guess I'd expect RK would still be noticeably stronger.

> h . . h . . . . . k . . . . . . . K . . . R R .

I'm not sure I follow this diagram, but I think I can now envision an
arrangement with the properties you describe.

Does the same thing still happen if you have nightriders BUT NOT rooks?  I
suppose you could substitute queens for the rooks, though that would
require a promotion...

I don't have a lot of experience with Ultima or similar variants, but some
thoughts occur to me:

I think you've overlooked an important advantage the Long Leaper has
compared to the Displacer:  the Long Leaper may have a choice of several
squares it can stop on after making a capture, while the Displacer only
ever has one choice.  Not only does this give the piece increased mobility,
but it makes it harder to defend a piece that is threatened by a Long
Leaper.

The Advancer/Displacer is an interesting comparison.  The Advancer has
strictly fewer possible moves than the Displacer, BUT you can "defend" a
piece against a Displacer simply by threatening the square it rests on,
while defending against an Advancer requires threatening different squares
depending on the angle of attack, which seems like it is probably an
advantage for the Advancer.  I suspect that the traditional FIDE army would
still beat an army of equivalent capture-by-approach pieces, but perhaps a
mixed army would be more powerful than either simply because it would make
defending pieces much more complex for the opponent?

I would consider using the average number of possible captures a piece can
make rather than the probability of having at least one capture available. 
For one thing, having a choice of several things to capture sounds useful,
especially if some are defended.  For another, I think you'll find it's
noticeably easier to calculate.  Of course, having a choice of 2 possible
captures is very different from having the ability to capture 2 pieces at
once, which seems to indicate that at least one of those things is going to
require special consideration...

The value of an Immobilizer might be estimated by computing the moves that
your opponent would normally be allowed but that the Immobilizer
prevents...though that suggests you're probably going to need to consider
mobility and not just capture potential.  Also, this might be a case where
assuming random distribution could be very misleading.  Perhaps
immobilizing a piece is really more like a suicide-capture, where you
effectively remove the target piece(s) from the game, but also neutralize
your own Immobilizer, which now cannot move without releasing its captive?

I believe Muller did some experiments suggesting that chess pieces
typically got about 1/3 of their value from non-capturing moves and 2/3
from capturing, if their capturing and non-capturing movement patterns were
similar in overall power.  Is this heuristic likely to hold for Ultima
pieces?

Cannon pawns are hoppers, not locusts.  They capture by displacement.  They would not be hindered in ANY way by the absence of border squares.

You have failed to define the term "piece", which seems to be pretty
important.  Under your proposed definitions, I think one could make a
reasonable argument that StarCraft or baseball are "board games" (in that
they are entertainments with rules and goals that contain things that could
reasonably be described as "pieces" that move among a predefined set of
possible "locations"), which is probably not what you intended.  Your
definition of "battle game" also seems very overbroad; it includes many
games that most people would say have nothing to do with "battle", such
as Klondike (card solitaire), Pandemic, etc.

You've defined "piece type" as "a group of pieces that are identical to
each other".  That definition seems to imply that it is a set of specific
pieces, whereas I think most people think of a "piece type" as a
taxonomical grouping.  Furthermore, your definition seems to imply that
pieces owned by different players are never the same type, which I think
also conflicts with common usage.

Your count of "over 2,000 documented board games", while technically
accurate, is far short of the true number.  BoardGameGeek's game database
currently contains 61,611 board games, and if your definition of "unique
game" is generous enough to include all the variants on chessvariants.org,
then I'd wager the true number is at least 100 times that.

You say that "at some point everyone will need to pick a game to play." 
If you mean that everyone will need to agree on one game to be the only
game ever played by any person ever again, I think the idea that this is
necessary, desirable, or even feasible is extremely naive.  Conversely, if
you only mean that a single person typically will not want to play 2,000
games SIMULTANEOUSLY, and therefore any specific individual will need to
settle on one game to play on any given lunch break, then the segue to
talking about the "best game" seems unjustified.

Furthermore, your suggested criteria of "originality" seems to directly
undermine the concept that such a thing as a "best game" is possible,
since it implies a yearning for novelty.  No single game will remain novel
forever.

Jörg, I'm not sure you've given due consideration to board geometry. Betza's mobility calculations attempt to account for both the probability that a move will be blocked by another piece AND the probability that a move will be blocked by the edge of the board. If you assume that pieces are distributed randomly, then the odds of being blocked by a piece are p^(distance-1), as your formula suggests, but the odds of being blocked by the edge of the board follow a completely different pattern (e.g. for a rook, it's 1 - (distance/8)). That means it can't be accurately represented simply by plugging a different value for p into the polynomial shown in your post.

You also seem to be assuming that piece value is directly proportional to mobility. Most people believe that value has a super-linear relationship to mobility; the evidence being that compound pieces tend to be worth more than their component parts. There are also many things other than mobility that might affect a piece's value; Betza attempted to compile a list here.

Finally, you might want to consider that Betza believed a full Rook had a value closer to 3/2 of a Knight than to 5/3 of a Knight. If that's the case, then it's unclear whether a piece priced at 4 points should aim to be worth 4/5 of a Rook or 4/3 of a Knight--a difference of perhaps 10%, or roughly the difference in your calculated mobility between an R4 and a R5.

Seems pretty complex for a single piece.  The combination of double-move
AND leap, and the nuanced anti-trading restrictions, jumps out at me as
being an evolutionary design and not very elegant.

Kind of like FIDE pawns (with initial double-move, en passant,
promotion...).  Pawns have been heavily tweaked over the years.  That
probably makes FIDE a better game, but I would be reluctant to import all
of those quirks into a new game--especially one where pawns weren't a
major focus and/or where the target audience wasn't already familiar with
them.  (And in fact, many casual chess players don't even know about en passant.)

Building an unbreachable defense is probably a part of any winning strategy for white, but surviving long enough to build it looks like the hard part.  After what opening is it possibly safe to castle?

Currently, a stalemate in Chess is widely recognized as a draw.

Why?

My understanding is that, originally, Chess had no prohibition against moving into check, so "stalemate" didn't exist. The rule against moving into check was added to prevent interesting games from ending early due to a dumb mistake. (I personally think this is a dubious justification--there are many blunders that could lead to the swift and unexpected end of an interesting game, and my gut feeling is that an opponent should be allowed to retract any of them in a casual game and none of them in a tournament or other serious game. I don't see why this specific blunder should be enshrined in rule.)

That changed the win condition from "capturing the king" to "checkmate", and as a side effect created the possibility of "stalemate". But the situations that we now call "stalemate" would have been wins for the side delivering the stalemate before the above rule modification, and the above rule modification was not (so far as I know) specifically targeted at such situations, so it's not clear why it ought to change how the game is resolved in those situations.

Wikipedia has a brief history of the stalemate rule, and points out various people who have argued for or against changing the current rule. But I'm looking for a game design reason, rather than a historical or political reason--is the game BETTER because stalemate is a draw rather than a win? Why or why not?