Comments by jlennert

I thought the Rook measurement was off by at least half a pawn, but perhaps my memory is in error.

In any case, we have:

- (Commoner) One situation where your empirical measurement differs from common wisdom, and you think it is probably due to a failure of your engine;

- (Rook) One situation where your empirical measurement differs from common wisdom, but you think it's a failure of definition and you weren't really measuring the same thing;

- (Bishop-Knight) One situation where there's a strong possibility that your empirical tests have shone light on an important lesson for theorists, but we still don't know WHY this particular piece would have such a high value, so we can't extrapolate from it (for example, we can't make reliable guesses about the value of a Bishop-Camel or a Bishop-Nightrider)

I have great respect for your work and I think it's very valuable, but these still strike me as emblematic of how far we have yet to go.

> In my empirical piece-value determinations I never noticed any significant discrepancy with the orthodox values.


Odd, I seem to remember reading about your Joker engine testing material values for FIDE and getting a Rook value that was unexpectedly low.

I also seem to recall the same engine testing the Bishop-Knight compound and finding its value unexpectedly high compared to the Queen and Rook-Knight (closer than the values of their component pieces).  And I do not recall anyone offering a predictive theory capable of explaining that.

Betza also performed computer tests and human playtests on the value of the Commoner (nonroyal King), and was convinced that the computer value was wrong.  I recall that you disagree with him on this point, but that Wikipedia article I linked a couple of posts back cites two sources that seem to at least vaguely support his conclusions (end-game fighting power of the king placed higher than knight or bishop).

Have your formulas for short-range leaper values been verified by anything other than your own chess engine?  It's certainly impressive, and it's plausible, but if it's based entirely on one source, then it's hard to tell how far we should trust it.  Perhaps more importantly, symmetrical short-range leapers are a rather special subclass of pieces; I would be hard-pressed to name a single CV whose pieces all fit into that class.



Incidentally, did you ever finish that new chess engine you were working on that you said you wanted to complete before running more complicated tests?  Spartacus, I think.

The alibaba has similar qualities to a knight (leaping piece, similar range, same number of moves), but its movement pattern confines it to 1/4 of the board (similar to how bishops can reach only 1/2 the board, but moreso).

Betza suggested somewhere in this article on the Crooked Bishop that a non-colorbound piece would be 1.1 to 1.2 times more valuable than its colorbound equivalent, which means a colorbound knight ought to be worth ~87% of a knight. But the alibaba is colorbound "twice", which I would imagine warrants at least applying the penalty a second time, giving ~76% of a knight--a very good match for your estimate. (Though I wouldn't be surprised if "double-colorbound" turned out to warrant a greater penalty than that.)

However, Betza appears to have been talking about pairs of colorbound pieces on opposite colors (like the bishops in FIDE Chess), which are generally believed to be more than the sum of their parts. So that estimate is probably only good if you start with 4 alibabas, one on each "color"--they will be less valuable individually, especially in the endgame (when it becomes harder to find targets on your own "color").

Finally, if you are playing with bishops, rooks, and queens, I would guess that the knight is also benefitting from a "stealth" bonus, due to its ability to threaten these pieces without being threatened in return. The alibaba can only do this if there is another piece in the way (that it can jump over but the other pieces can't), and so I would guess that its true value would be a little bit lower again than the estimate above.

So, in summary, I would guess than 3/4 of a knight is close, but probably a little too high, and would expect the value to fall significantly in the endgame or if you don't get a complete set.

I feel I need to ask again whether you are arguing about the SIZE of the
first-turn advantage, or the EXISTENCE of the first-turn advantage? 
Because you said earlier you were arguing over its existence, but all of
your arguments seem to be about its size.

You could be a thousand moves away from mounting a credible attack, but
that doesn't mean the value of a move is zero.  After you move, you will
only be 999 moves away from a credible attack, which surely must be at
least a tiny bit better than 1000?

Your typical player probably won't notice that advantage.  But then, a lot
of players probably don't notice the first-turn advantage in FIDE, either.
 Small is not the same as zero, and what counts as "small" depends on how
good you are and how many times you're playing.

And zero first-turn advantage isn't even necessarily desirable.  Suppose
we have a game where players are allowed to pass on their turn, the initial
array is symmetrical, and the players know that there is no first-turn
advantage.  Since there is no first-turn advantage, passing is (by
definition) at least as good as anything else you can do on your first
turn, so you might as well pass.  Then the second player is in exactly the
same position as the first player on his first turn, so he might as well
pass.  So not only is the perfect strategy obvious, it's also incredibly
boring.

But even if passing isn't allowed, the first player either has a move that
is EXACTLY AS GOOD as passing--which I'm not sure is possible, and I
don't think it changes the outcome compared to allowing passing--or else
the best possible move is WORSE than no move at all, which means we've
simply traded a first-turn advantage for a SECOND-turn advantage.

All else being equal, I think we want the first-turn advantage to be
"small".  We might even want people to be uncertain whether the advantage
lies with the first player or the second player, perhaps by using an
asymmetric starting array or placing special restrictions on the first move
(such as moving half as many pieces as normal).  But if you could somehow
prove that the first-turn advantage was exactly zero, I think that would
probably end up being bad (not so much because the advantage was zero, but
because you were able to prove it).

Joe, notice that all the theories you have advanced to explain the lack of
a first-turn advantage are general properties of the game, NOT unique to
the opening array.  The reversible pieces don't suddenly become
irreversible in the late game; the short-range pieces don't turn into
long-range ones; etc.  If those properties were sufficient to prevent a
move from having value, they would prevent ANY move from having value, not
just the first or second one.

But as Muller points out, it seems pretty obvious that you will quickly
lose if you pass ALL of your moves, which means moves must have some value
at some point.  IF there truly is no first-turn advantage whatsoever, the
reason needs to be something special about the opening array, NOT the
general properties of the game.  The things you cited MIGHT make each move
less valuable, but they cannot possibly reduce the value all the way to
zero.

And while it is conceivable that there is something special about the
opening array that puts the first player in a position of zugzwang, it is
intrinsically unlikely.  Most possible positions in most Chess-like games
are NOT instances of zugzwang.  And the facts that the opening array
appears to be a "calm" position, and that the pieces are reversible, both
make it substantially LESS likely to be a position of zugzwang--after all,
if my second move can be to reverse my first, and my opponent cannot do
anything to hurt me in the meantime, it is difficult to see how the first
move could have harmed me.

Asking us to verify the non-existence of a first-move advantage by pushing
a few pieces around is silly.  Based on this conversation so far, the
first-move advantage in FIDE is barely large enough to be noticed by
masters (it's estimated at approximately one "quantum of advantage"). 
Perhaps you understand Chieftain Chess as well as a master understands
FIDE, but the rest of us certainly do not.  Hypothetically, Chieftain could
have a first-turn advantage that is substantially larger than FIDE and it
would still be all but impossible for us to demonstrate it to you.

We "proved" the existence of a first-turn advantage in FIDE only by
recourse to a statistically-significant sample of high-level games.  Unless
you have a similar statistical collection for Chieftain, then none of us
have any real evidence one way or the other, so we are reduced to arguing
generalities--and IN GENERAL it is safe to assume that a randomly-selected
Chess variant has a first-turn advantage.

I wouldn't expect that the addition of noise would EVER completely
eliminate the first-turn advantage, just make it less significant.

Assuming that players strictly alternate full turns, and that nothing other
than the positions of pieces affects the game (e.g. there's no time
limit), and general chesslike properties such as perfect information, the
only way I can see to have NO first-turn advantage is if the first player
has literally nothing useful to do with his turn.  No possible piece
development, no moving forward to claim extra territory, no starting to
launch an attack or race for a promotion, NOTHING.  And I question whether
that would even be desirable.

If white gets to take an entire turn before black starts, then I think
it's appropriate to measure game length in turns, not in moves (if
different).

Though I've seen several double-move Chess variants that restrict white to
a single move on the first turn in an attempt to counteract the first-turn
advantage; have you considered a Chieftain variant that only allows white
to make half the usual number of allowed moves on the first turn?

If the race metaphor is accurate, I would expect the head-start to be
compared not to the length of the runner's stride, but to the length of
the entire race.  I would guess that a 10m head-start would be much more
likely to be decisive in a 100m sprint than in a marathon.

Perhaps we should not be looking at the mobility of individual pieces, but
the length of the overall game?  Piece mobility would likely be a factor in
game length, but board size, number of pieces, and several other factors
could also be highly significant.

What do you mean by "demonstrated"?  You have some proof of the absence
of first-turn advantage in Chieftain Chess?  Could you perhaps share this
proof?

I am so far unconvinced.  You tell me that you pretty much always win your
Chieftain games, which suggests you have not played against any opponents
that seriously tax you, which in turn suggests that you have no sample
games with high-level play on both sides to use for reference.  But you
also say that you win with an aggressive strategy, which suggests that you
think you can get some advantage by doing something rather than idly
maintaining your position, which suggests that tempos have value (at least,
you are playing as if they do).

Unless you have solved the game or you have a large, high-quality
statistical sample showing that black wins at least as often as white, I'm
not sure how you could demonstrate the absence of a first-turn advantage,
nor does such an absence seem inherently likely to me based either on your
testimony or from reading the rules.

Keep in mind that the first-move advantage in FIDE Chess is thought (at least by Betza) to be approximately the minimum advantage that MASTER level players will notice in practice; I would hazard that no one currently alive is as good at Chieftain Chess as a master-level player is at FIDE, and so it seems plausible to me that you might not easily notice the first-move advantage even if it were LARGER than the one in FIDE.

So, is your personal standard strategy in a game of Chieftain Chess to
maintain a holding pattern and let the enemy come to you?  If both players
do that, it seems like the game would be awfully boring.

Joe, just to be clear:  are you saying you believe that white has NO
first-turn advantage in Chieftain Chess, or are you saying that you believe
white has a first-turn advantage, but that it is SMALLER than the
first-turn advantage in FIDE Chess?

I think it is entirely plausible that weaker pieces will lead to a smaller
first-turn advantage, since the weaker your pieces, the less you can
accomplish each turn, and therefore the smaller the value of a turn.

But saying that there is NO first-turn advantage is equivalent to saying
that the null move, if it were allowed, would be the best possible opening
move.  Do you recommend moving as few pieces as possible in the early part
of a Chieftain Chess game?  Do you think that the best possible second move
is to reverse the move(s) made during your first turn?  If not, it seems
unlikely that the null move is really optimal.  Zugzwang certainly exists
in Chess, but it's pretty rare.



Also, your proposed "Moving 1 Square Chess" rules for knights appear to
boil down to "knights move as wazirs". You mandate changing parity every
move, but a wazir move changes color, so it will always change parity,
while a ferz move preserves color, so it will never change parity.

The first event in a causal chain can be important. I completely fail to follow the "always" part. Perhaps you can find a hurricane that wouldn't have formed if a particular butterfly hadn't flapped its wings, but not every flap of a butterfly's wing causes a hurricane.

But you seem to have missed the thrust of my last post, which was that, even if you were right, that would contradict your earlier suggestion that we can learn something about Chess based on the importance of its first move. If the first move is always the most important, then we cannot learn anything at all from the fact that the first move is most important in this particular case. (See also: Bayes' Theorem.)

But you are also wrong about the first move being the most important, for reasons I have already explained. If we ask how important X is to the outcome of some system, we are comparing two hypothetical situations, one where X obtains and one where it does not, and exploring the difference in the evolution of these two hypothetical systems. So if we ask how valuable a move is in a Chess game, hypothetical examples where we break the normal turn sequence are not only relevant, they're mandatory.

Or, here's a completely unrelated point: ever heard of zugzwang? It's important in, among other situations, the KRvK endgame. The fact that zugzwang exists proves that a move can have negative value, and from that it seems fairly safe to assume that some move after that point has a value higher than it. So that alone shoots down your theory that move-value is strictly decreasing.

In other words, it is possible (fairly easy, in fact) to devise a chesslike board game where black (that is, player two) has a forced win. Therefore, the first move cannot always be the most valuable.

You suggest that we should take the prime importance of the first move as a "clue", and then you justify your belief in the importance of the first move by saying that the first move is always and by definition the most important in all games?

Um.

If I told you we were discussing "value" rather than "importance", would that short-circuit this loop and get us back on topic?

> So the difference between playing white or black ('1 tempo')

Shouldn't the difference between white and black be half a tempo?  Giving
black a free tempo at the beginning of the game doesn't cancel out white's advantage, it transfers it to black, so the tempo must be
twice that advantage.

The first move in a game of Chess isn't even CLOSE to the most important
one in a typical game.  If you look through the log of a decisive game, I
bet you will easily find at least one point where allowing the player who
eventually lost to take 2 moves in a row would EASILY have turned that loss
into a win (for example, maybe around the time the queens were
exchanged?).

I recall reading about a variant on this site where each player begins the
game with the right to make a double-move at one point of their choice
during the game.  The author suggested that forcing your opponent to use up
this ability was critical, having an equivalent material value of AT LEAST
queen + rook--almost two orders of magnitude above the 1/3 of a pawn we've
been assigning to the first-move advantage in this thread.

I also see no particular reason to think that a Bishop moving 7 squares has
equivalent value to taking 7 consecutive moves in a game of
checkers--but if it were true, that would seem to severely undermine your
theory that the first move in Chess is the most important one, since no
piece can move farther than 2 squares on the first turn.

I'm no expert on Chess, but I think about it this way:

It seems to me that in the opening position of FIDE Chess, most pieces are in quite poor positions, with their mobility greatly restricted by overcrowding and no immediate opportunities to threaten enemy pieces. An extra move can be used to develop your pieces into more advantageous positions, which translates into a higher probability of winning. As a corollary, the value of an extra move changes throughout the game based on how rapidly your position can currently be improved.

In On Numbers and Games, John Conway develops a theory of combinatorial games (Chess doesn't quite fit in this set, but it's close) as being a superset of numbers. Certain games (or sub-positions within larger games) have an exact numerical value because their existence gives a direct advantage to one player (positive) or the other (negative). But other positions have a "fuzzy" value; it's not clear how much advantage they give because it depends on how many moves each player takes (and in what order) trying to improve that particular sub-position rather than some other sub-position elsewhere in the game. He has a concept of "heat" that corresponds to the volatility of a position; the "hotter" a game is, the more advantage can be gained by whoever makes the next move.

(Note: I don't quite follow all of the math in this book and it's possible my description of his theory isn't entirely accurate.)

As for Betza's "quantum of advantage", I think his position was a little more complicated than that. He said that a third of a pawn was roughly the smallest advantage that a master player would notice in practice, and therefore that it makes a good approximation for lots of different minor advantages that are NOT actually equal but all roughly on that scale, and lists a tempo as one of several examples. I believe he talks about this in part 3 of About the Values of Chess Pieces.

Carlos, that sounds pretty weird to me. I think the spirit of the rules of Chess is that you really are trying to capture the enemy king, we just forbid moving into check to prevent games ending prematurely due to a dumb mistake; in variants, unless the intent is clearly otherwise, I think we ought to follow that spirit, which means "check" must be construed to mean "your opponent could capture your king next move". Playing on because white doesn't "perceive" the danger, and then forbidding black from actually making the capture because "the goal is to checkmate", seems to me to fly directly in the face of that spirit.

There's no reason you can't make a game that works like that, but inferring that as another designer's intent seems extremely implausible.

...

I think I see a potential for paradox in these rules. Suppose white pawn d3, black pawn d5. At first glance, both pawns seem to threaten e4. But that means that e4 doesn't exist for either player, which means neither pawn can move there. But that means that it isn't threatened, so it does exist...

Or you look at one side at a time, and say, for example, that white is threatening d4, so black can't go there, so black doesn't threaten it. Then there's no contradiction...except that you could equally well rule that black is threatening it and so white isn't! How do you decide between them?

Such an app would basically just be an interface to a giant table listing pieces for every possible combination of moves. Assuming you could create that table, the app would be straightforward. I question its usefulness, though.

Firstly, most of the table would either be empty or taken up with obscure pieces that few would be interested in. For every combination of squares you could light up that corresponds to a common compound, there's a ton of combinations like "forward and left as a knight, forward and right as an alfil, or backwards as a ferz or rook" that are just weird. I'm sure you can dig up names and variants for several pieces like that, but it could take a lot of work if you're really being thorough, and the odds of a user stumbling upon any particular one of them are pretty low.

Secondly, for popular pieces, there would be an unwieldy amount of information to display. Take a look at the Piececlopedia page for the Bishop-Knight Compound, for example; it's got a bunch of different names and has been used in a ton of variants (the list on that page can't possibly be complete, but it's already more than one screen of info).

Thirdly, your database would have to leave out (or misrepresent) a lot of important and interesting pieces, because pieces aren't defined only by which squares they move to. The Knight, the Mao, and the Moa all move to the same squares, but they take different paths to get there. Some pieces, like the Cannon, require a hurdle to move. Some pieces capture differently than they move; Ultima is a variant based around giving every piece a different method of capture. Some pieces can make special moves under specific circumstances, like the pawn's initial double-step or en passant capture in FIDE. Some pieces just have weird special rules, like the ability to capture multiple times during their movement, or copy another piece's movements, or move other pieces with them.