Comments/Ratings for a Single Item

Ralph Betza somewhere defined the quantum of advantage (aka one tempo) and
quantified it to 0.33 pawn units.

But: It is not clear at all that the advantage truely exists. For example
look at the game known as Dawson's Chess: Black and White have lines of
chess pawns placed on the 3rd and 5th rank. Winner is whoever manages to
break through the opponent's pawn line.

Whether White or Black wins is intricately dependent on the number of
pawns, there are even mathematical papers on this subject, e.g., 

http://www.math.ucla.edu/~tom/papers/unpublished/DawsonChess.pdf

Dawson himself analysed the game by hand to upto 40 pawns.

In Chess, white has the privilege of choosing his/her favorite, strongest
opening playing offense for the game every time.  By contrast, black must
adapt to whatever opening white uses which is not likely to be his/her
favorite, strongest opening playing defense.  That is only one reason. 
There are others.

Jorg, Derek, thanks for the answers. To clarify completely, I am looking
only at the FIDE game, and the statistics are very roughly: white wins 36%
of the games, black wins 28%, and that leaves 38% draws. So, discarding
draws, which make up 3/8ths of the total games, white wins about 4 games to
black's 3, roughly 30% more wins. This is from the previously-mentioned
Wikipedia article, as is a substantial advantage for white, translating
into an overall points total of 54% vs. blacks 46%. With the very best
players, white's total % goes to 55. Neither tempo nor choosing the
initial direction of the game is, i feel, enough to account for this in
what by ll accounts is a drawn game with perfect play.

HG, what are your numbers again for a one-pawn ad? Ralph Betza's
guesstimate  looks in the ballpark, based on what happens to that first
turn ad by mid-game. It becomes pretty invisible. But then in the final
stats, it shows up with a vengeance. My instincts are that there has to be
more to it than "choosing the initial direction of the game" for such a
diverse outcome between black and white.

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.

I hold the opinion that in Chess, a game with a significant
first-move-of-the-game advantage for white, it is a win for white with
perfect play.  [Unfortunately, Chess will be intractable to computer AI
solutions of this nature for a very long time to come.]  Checkers is a
chess variant (by broad definition) also having a white-black turn order
where it has been proven to be a draw with perfect play.  However, checkers
cannot move more than one space per turn (except when jumping enemy
pieces).  In Chess, a bishop (for example) may move up to seven spaces from
where it rests in one turn if it has a clear path.  This is comparable to
seven consecutive turns in Checkers.  That is why I doubt the same result
will eventually be discovered for both games with perfect play.

Please do not misconstrue the following remark to imply that any move
within a game of Chess is unimportant?  However, the very first move in a
game (by white) is the most important one and all subsequent moves have
slightly, progressively diminishing importance.  This is another clue.

> HG, what are your numbers again for a one-pawn ad?

This corresponded pretty well with Betza's statement. Classical Pawn odds
(averaged over colors) deleting the f-pawn caused a 68% advantage in
Fairy-Max. The first-move advantage there was about 53%. (Probably a bit
smaller than in human games, because Fairy-Max is pretty stupid in the
opening, playing with a pretty strong randomizer there to provide game
diversity.)

So the difference between playing white or black ('1 tempo') is 6% The
difference between pawn ods and standard setup is 18%, i.e. 3 times
larger.

Note that it is quite difficult to be more precise: to determine win rate
with 0.5%-point accuracy already requires some 10,000 games.

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.

> 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."

Obviously, additional explanation of my meaning is needed.

In terms of a chain of events leading to a final outcome ...
the first (a move, in the topic under discussion) is always 
the most important because it has a determinative effect 
upon not just itself (as the last move of the game does) 
but all (moves) that follow.  Even though the very first move
of the game (by white) is not the most exciting,
it (moreso than any other move) determines the course of the
game as defined by its unique move list.

In Chess, where a strict white-black turn order exists, 
all hypothetical talk of non-existent double-move options is
completely irrelevant.

"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."

Technically, you have one point that should be addressed.

No.  White cannot move any piece of unlimited range on the 
first move of the game.  However, by advancing an appropriate 
pawn on the first move, white can then move a queen or bishop 
diagonally on the second move of the game.  [Note:  I don't
recommend actually doing so.]

The important point is the equal burden of development by 
white and black does not diminish the significant, measurable
first-move-of-the-game advantage by white in Chess which 
undeniably exists and is all-but-proven statistically via a 
vast number of reasonably well played games.  After all, 
white has a head start toward this development.

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?

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

First of all, that's a loaded question, but the answer is NO.
Whichever term you prefer, value or importance, is fine with me.

If I told you that appr. 50,000 years ago, the only homo sapiens 
on Earth were a small number in East Africa (probably, black) 
and that some of the things they did which by objective, modern 
standards seem relatively unimportant were actually important 
toward determining the present state of the entire human race, 
would you fail completely to follow my reasoning?

The first event in a cause-effect chain is always supremely important.
Do you know what the butterfly effect is?

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.

"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."

Please don't take my mention of the butterfly effect literally?
I am not seriously asserting that it (and anything similar) explains the
first-move-of-the-game advantage for white.  However, I am asserting that
the advantage for white in having the very first move in Chess carries all
the way thru the midgame and endgame to the last move of the game and is,
in fact, greater than virtually all Chess players have the complex
foresight to appreciate.  After all, Chess is a deterministic game of
perfect information.

You seem to want to argue with established facts and plausible attempts by
others to explain them.  Naysayers typically offer no or few ideas.

Hey, Jeremy. It was designing chess variants that made me realize that the
primary purpose for pawns, in the set-up, is to hem in their own side so
the game doesn't degenerate into a catastrophic shoot-out from the first
move. [Grin, see Texas Two-Step.] Thanks for the Betza reference.

Derek, that white wins is a minority view, and, often as I am in the
minority, I do not join you in this one. However, I do note it is unproven,
so at this point, neither of us is demonstrably right or wrong. I also
think there are very high value moves throughout the game, for both sides,
so unless your argument about move order vs value is statistically borne
out through all possible games, I'd disagree on that. I would note here
there are generally many more bad moves than good in any given turn. But I
understand the concept behind what you are saying, and how 2 follows from
1.

Let me reprise my argument. Find/design a game that does not have the
quality you are examining, but is similar enough otherwise to allow valid
comparison. In other words, argue from contrafactuals. What was changed to
create the 2nd game? I used Chieftain Chess as the 2nd game, because I
understand it well enough to do a decent initial analysis, and I can
demonstrate the game almost certainly cannot have a first turn advantage.
As a verbal demo, I can say any reasonably highly-skilled player can play
black and pass the first turn, giving white 2 first moves, without
detriment to black's game position. 

Chief did 3 basic things that eliminated first turn advantage for white, as
far as I can see. It reduced all movement ranges to no more than 1, 2, or 3
squares/turn, depending on piece. It lengthened the board, putting more
space between the [bulk of] the armies. And it allowed backward movement,
actually 360 degree movement, to all pieces, including the pawn
equivalents. It's also multi-move, allowing 4 pieces/turn to move for each
player, and movement is restricted by the requirement for the moving piece
to be reasonably near one of four leader pieces per side. However, I do not
see how these last two differences with FIDE have any real bearing on the
question.

Thank you HG, for that statistical confirmation of ~ 1/3 pawn. So what
creates that imbalance? 

Chief contrafactual, cont: The first change is pawn reversibility. In
chess, pawns and pawn chains are often hung out to dry when the opponent
breaks through the pawn wall. If pawns could move backward along their
file, they would be more able to defend. Increasing the defense of the
pawns this way should help black slightly more than white, given the
unbalanced won-loss statistics. It does not seem that it would play a major
role in what is a 30% win-loss advantage when draws are thrown out.

The second change is board length. The longer board, all other things being
equal, including the pawn double-step first move, will cause the pawns to
take longer to contact each other. This will act to give black easier
access to adequate maneuvering room for his pieces. One strategy is to
squeeze your opponent to death. Allowing more free space for maneuver,
especially while slowing down pawn contact, will allow black to improve the
defense, thus evening things up. The larger board does little to affect the
infinite sliders, however. Any sort of aggressive play by white, however,
would likely do much to negate this effect, so again, I don't see a major
role for either of these first two changes. Even the combination doesn't
appear to be that potent, pretty certainly providing less that half of the
total effect.

That leaves mobility as the last man standing. The reduction of the average
piece movement range to about 2 squares, on a board that is now 12 squares
deep, and with "pawns" that can retreat, certainly accounts for the
mechanics of the 30% edge white gets in won-lost only. And we've already
seen that the reversibility and the larger board, by themselves and
probably even in concert, don't seem to account for the discrepancy in
scores. 

If Chief is close enough to FIDE in concept, and it should be, because it
is played strictly as chess on an individual piece movement basis, the
analysis should hold. Each individual piece is moved in turn, and each move
must be legal when it is made. A slight restriction on what pieces may move
in any given turn shouldn't have any real effect, nor should the
multi-move aspect. You actually could play it moving 1 piece/turn/side, but
it would be one heck of a boring game.

Therefore, the reason white wins from first turn advantage is primarily and
probably overwhelmingly the mobility of the pieces. Can anybody knock this
argument down?

Jeremy, Derek, I think you are arguing somewhat at cross-purposes here. You
are closer together on some positions than you believe, and you are
discussing 2 slightly different things. Further, Derek, you seem to not
like contrafactuals. 

Derek is merely maintaining that the mere fact of moving first gives white
a 1/3 pawn advantage in the game. Jeremy is maintaining that it is
demonstrable in individual games there are higher-value moves than first
move. For example, checkmate is the highest value move outcome you can
have, and that never happens on the first turn. ;-) 

Still, for the question of what the mechanism is by which white gains a
first turn advantage, can anyone come up with a better answer than
primarily mobility, and maybe some add-on from board size and possibly some
from pawn irreversibility?

I think there is a likely chain of events in Chess whereby ...

Having the very first move in the game along with control of a white-black
turn order tempo gives white a head start toward development.  This, in
turn, gives white an irrefutible advantage in mobility throughout the
opening game and results in a small positional advantage.  A small
positional advantage should be built into a large positional advantage.  A
large positional advantage should be built into a small material advantage.
 A small material advantage should be built into a large material
advantage.  A large material advantage will probably, eventually enable
white to checkmate its opponent (black).

If all of the links in this chain of events (plus any I have overlooked)
are solid, they may account for the observed win-loss discrepancy between
white & black without resorting to any mysterious theories.

Http://en.wikipedia.org/wiki/First-move_advantage_in_chess.

> So what creates that imbalance? 

Chess is a game of balance, where creating a local majority of force is
usually the key to achieving strategic objectives.So you are constantly
manoeuvring to concentrate your forces against a weak spot of the opponent,
or to strengthen your weak spots that are in danger of collapsing under the
pressure he musters.

So typically any constellation of opponent pieces can be cracked if you are
given the time to organize your pieces in a constellation needed to crack
it. So a game of chess is a constant race between concentrating your attack
force, and the opponent strengthening the spot against which you direct
that attack. Being allowed to do two quiet moves in a row (which is what
happens when the opponent loses 1 tempo) makes it more likely you will win
that race.

Only in positions where nothing can be achieved no matter what (i.e. a
static defense exists that has no weak spots weak enough to succumb to even
total concentrated attack of all enemy material), a tempo loses its value.
Such fortresses are quite rare. An example would be the 'Charge of the
light brigade' position (3 Queens vs 7 Knights, in the presence of King
and Pawns). Once the Knights sufficiently protect each other, the Queens
are totally powerless to inflict any damage to Knight army, and basically
have to stand by idly, watching the Knights gang up slowly but surely on
their King.

George, thanks for the reference. That's where I took my numbers from.
I'd run across it when I was searching for the answer, and that's the
best I did in that search. I was quite surprised to see nothing there,
other than it might be this or it might be that, and everybody likes tempo.
But given the specific rules of chess, with move order and general
structure built in, the thing that most gives white that 30% win-lose
advantage is mobility primarily, as far as I can tell. No one, anyhow, has
shot down the contrafactual argument. If you accept the premises, the
result seems to follow inexorably. 

Derek, you may be right with that chain of "if A, then B; if B, then C, if
C...", but like the Drake equation, it is expandable, and at each stop
along the way [a, b, c... etc] you lose some of what you're looking for. I
think you present too long a chain of events, giving too many opportunities
to go wrong. I like short, sharp and simple here. Occam's Razor. Mobility,
with or without board size and pawn reversibility, seemingly can be tested
by using pieces that are all short range. Check the stats of great
Shatranj, where all the pieces move only one or two squares [but all jump],
and run FIDE-type games with B2, R2, and Q2, and/or B3, R3, and Q3
repalcing the infinite sliders, and see what happens to the stats. Even HG
Muller's piece value estimation method, properly carried out, will give an
indication, because it does show white with a 53-47 ad.

Drake Eq Calculator
http://www.symmetryperfect.com/SETI

Just an aside.

Hi, HG, thanks for the responses. I grabbed this part of your last comment:
"Being allowed to do two quiet moves in a row (which is what happens when
the opponent loses 1 tempo) makes it more likely you will win that race."
I would argue this presupposes that the pieces are mobile enough to need
only 1 turn to make the key attack. Hmm, guess that's the same argument I
made last comment in a different guise, or at least from the other
direction. The small board certainly plays into it. 

Nice aside, Derek!

Via causality, the small advantage white holds at the beginning of the game
(in Chess), given appr. equal quality play for white & black, gets
amplified into a large advantage by the end of the game roughly consistent
with known win-loss stats for white & black.

[Bravo to Occam's Razor.]