"The enemy's key point is my own" is often invoked, for example, as a
reason to occupy the central point of a _nakade_ shape, or to play a
double sente point, or to make an extension that would also be an
extension for the opponent.
I would like now to talk about it in the context of the potential
modification of program behavior, or "style", via opponent modeling.
There's another oft-repeated maxim which states that "there are as many
styles of play as there are go-players." If true, this adage implies a
continuum along which each player may find himself or herself.
Or itself.
One measure of playing style is the degree to which the bulk of a given
player's selected behaviors display a concern with one's own position,
or a concern with the position of the enemy.
That is to say, or rather ask, of the moves in which a player most
often chooses to indulge, whether they exhibit what might be called
a "defensive" character, seeking to shore up or expand one's own
position, or do they instead more often exhibit a "combative"
character, seeking to destroy or diminish that of the other?
Of course all players indulge in both types of behavior, but we all
know that there are some players who prefer to build, and some who
prefer to destroy, when they are given a choice.
Now, for each potential behavior on the board -- even those acts in
which we would never choose to indulge -- suppose that we already have
a pair of numbers, in this case percentages, where one represents the
likelihood (or our belief about the probability) that such a point is
advantageous for us to play, and where another such number represents,
conversely, the likelihood (or our belief) that the same point (were we
to pass and it be selected by the foe) would be advantageous for the foe.
How we have arrived at these "probabilities" may be important,
certainly, but it's beyond the scope of this message. Let's
just say that we already have them.
Remembering that frequently-quoted proverb, namely: "The enemy's key
point is my own", we may wish to consider giving some weight to the
notion of playing on those points that we believe would be best for our
opponent to place a stone.
For example, suppose that it is our belief that a specific move has a
15% chance of being our best move, and say, a 35% chance of being best
for the foe.
If we give equal weight to these percentages, we may simply average
them, or "split the difference", arriving at a value of 25%, if we wish
to take advantage of the notion embodied in the proverb, as well the
notion of merely looking for plays that will be advantageous to our own
position. Note that this average is given by:
( 0.5 * 0.15 ) + ( 0.5 * 0.35 ) = 0.25
where the weights are equal, that is, fifty-fifty.
However, IRL ("in real life") each go player in the world is not only
likely to -- but in practice _does_ -- give a different weight, either
more or less, to each notion, as his character, or even his whim, dictates.
At the one extreme we may have a player who is defensive, pacifistic,
self-obsessed, overly-concerned about his own stones, to the point of
ignoring the other entirely. Such a player is oblivious to the foe's
plans and position, trying always to take one's own best point. Such
a player completely disregards the proverb: "The enemy's key point is
my own."
At the other extreme we have the player who is aggressive, combative,
entirely obsessed with the other, overly-concerned about the opponent's
stones, to the point of ignoring his own best interests entirely. Such
a player is oblivious to his own plans and postion, trying always to
take the foe's best point. This player has _pathologically_ taken to
heart the proverb: "_Only_ the enemy's key point is my own."
Both styles are pathological, and somewhere between these two extremes,
lies _every_ go-player, it seems.
Thus, a moderately-defensive player might be likely, instead of as in
the above example, to give more weight to the 15% chance (his own) and
less to the the 35% chance (the foe's).
I use the term "chance" loosely here, as we all know that go, being a
game of perfect information, does not depend on chance. [Whatever that is!]
Yet if a certain player were known to be, in general, say, about 63%
likely to be aggressive (and 37% defensive), we might then calculate
that his chance of selecting the above specific point is given by:
( 0.63 * 0.15 ) + ( 0.37 * 0.35 ) = 0.224
because such a player gives more weight (63%) to playing _our_ moves,
and less (37%) to playing his own.
In this particular example, there is not so much difference between our
original 50/50 estimate of the chance that this is a good move, and the
new, 63/37 estimate.
However, as the degree of aggressiveness of the player approaches one
or the other of the extremes, and also as our _a_priori_ beliefs vary
about the chances of a particular point being a good one for one or the
other of us, there are cases where the deviation from the 50/50 case
will be much greater. (Consider the case where "our" move is 95%
likely, for example.)
This all reminds me of a differential equation, although I'm not wholly
sure why, because although I passed "diff E.Q.", as we called it, I
don't know how I passed. However, in any case, it's trivial to define
a "slider", a variable setting wherein the sum of these two weights is
always 100%, but where the "degree of bellicosity", if you will, ranges
from 0% (self-obsessed, defensive) to 100% (other-obsessed, combative).
While this duality is almost certainly not the only measure of "style"
-- whatever that is -- it may be considered at least one component
thereof. [And I further assert that it is one of the key components.]
As for modifying our program's play, we can of course immediately
provide several distinct playing styles, simply by arbitrarily
setting the slider to different value-pairs. That in itself may be
of interest; perhaps there is an optimal ratio to discover, at least
for some given program, by holding tournaments among various styles.
But we need not stop there.
Beyond that, if we know our opponent, which is to say that we have
observed his games and concluded that he is, say, a 63/37 player,
might it not make some sense to counter that by adjusting our own
style, becoming a 37/63 player for the duration of our contest?
That is, to play in such a way that we would give the same weight
to the foe's "best" choices as this foe gives, and indeed to give
the same weight to our own "best" choices as this foe gives, thereby,
it is to be hoped, frequently occupying _exactly_ the point on which
this foe -- this particular foe, mind you -- wishes to play?
Not only is the enemy's key point my own, but if I know _a_priori_ what
sorts of behavior in which my _specific_ enemy is likely to indulge,
perhaps I may be able to play, sometimes anyway, on the very points
where he or she (or it) is likely to play!
I personally find it quite annoying when a good player takes all of
"my" points, and these are the very games I often have trouble winning!
The hypothesis outlined above, as of yet, has neither been confirmed
nor falsified by experimentation, but it kinda gives a whole new
meaning to the proverb, don't it?
Your thoughts and comments are solicited, and appreciated.
In particular, if there is someone who can explain to me why I have the
nagging suspicion that a differential equation is involved here, or who
can otherwise somehow formalize the above with P's and d's and such, I
will be grateful.
--
The best thing about being an amateur mathematician is that I have
nothing to prove.
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