INTRODUCTION

In this message, move values are calculated for traditional endgame
theory. For modern endgame theory, one would also divide by the tally.

A move value is best calculated as a difference. However, some insist
on calculating a move value as a sum. This tries to use the supposedly
simpler arithmetic operation but causes difficulties explained below.

Even if a move value is calculated as a sum, nevertheless, the
positional value of a settled position is calculated as a difference.
Its calculation is done in different manners, of which I describe two.


POSITIONAL VALUE OF A SETTLED POSITION AS A VALUE FAVOURING A PLAYER

This is the manner preferred by traditionalists. They have never
described the following procedure but apply it.

Determine Black's points. Determine White's points. Determine whether
either points value is larger.

If the points values are equal, the positional value of the settled
position is zero, which favours neither player. It can be stated, for
example, as "the positional value of the settled position is 0" or
"Black has 0 points more than White".

If either points value is larger, from the larger points value
subtract the smaller points value. Call this difference, say, x. This
is the positional value of the settled position favouring the player
having the larger points value. If Black has the larger points value,
the positional value of the settled position is stated, for example,
as "Black has x points more than White". If White has the larger
points value, the positional value of the settled position is stated,
for example, as "White has x points more than Black".

(In diagrams, all outer stones are alive.)

X . X X O O
X X X O O O

Black's points are 1. White's points are 0. Black's points are larger.

The positional value of the settled position is 1 - 0 = 1.

Black has 1 point more than White.

X X O O . O
X X X O O O

Black's points are 0. White's points are 1. White's points are larger.

The positional value of the settled position is 1 - 0 = 1.

White has 1 point more than Black.

X . O . X O O O O O O . O
X X X X X X X X X X O O O

Black's points are 4. White's points are 1. Black's points are larger.

The positional value of the settled position is 4 - 1 = 3.

Black has 3 points more than White.

X . O X O O O O O O O . O
X X X X X X X X X X O O O

Black's points are 3. White's points are 1. Black's points are larger.

The positional value of the settled position is 3 - 1 = 2.

Black has 2 points more than White.

O . X O X X X X X X X . X
O O O O O O O O O O X X X

Black's points are 1. White's points are 3. White's points are larger.

The positional value of the settled position is 3 - 1 = 2.

White has 2 points more than Black.

O . X . O X X X X X X . X
O O O O O O O O O O X X X

Black's points are 1. White's points are 4. White's points are larger.

The positional value of the settled position is 4 - 1 = 3.

White has 3 points more than Black.


POSITIONAL VALUE OF A SETTLED POSITION AS A COUNT

This is the manner preferred by modernists. Except for the general
convention expressing values from Black's value perspective, the
manner has the advantage that one number expresses the positional
value of a position. Informal phrases, such as "Black has more points
than White", are not needed. This is possible because the positional
value of a position can be a negative number, which favours White.
Needless to say, the positional value of a position can also be zero.
The procedure is described in one sentence:

The positional value, the count, of a settled position is Black's
points minus White's points.

X . X X O O
X X X O O O

Black's points are 1. White's points are 0.

The count is 1 - 0 = 1.

X X O O . O
X X X O O O

Black's points are 0. White's points are 1.

The count is 0 - 1 = -1.

X . O . X O O O O O O . O
X X X X X X X X X X O O O

Black's points are 4. White's points are 1.

The count is 4 - 1 = 3.

X . O X O O O O O O O . O
X X X X X X X X X X O O O

Black's points are 3. White's points are 1.

The count is 3 - 1 = 2.

O . X O X X X X X X X . X
O O O O O O O O O O X X X

Black's points are 1. White's points are 3.

The count is 1 - 3 = -2.

O . X . O X X X X X X . X
O O O O O O O O O O X X X

Black's points are 1. White's points are 4.

The count is 1 - 4 = -3.


MOVE VALUE DERIVED FROM POSITIONAL VALUES OF FOLLOWERS

A follower is a follow-up position. In our simple examples, all
followers are settled positions.

A sequence started by Black creates the black follower. A sequence
started by White creates the white follower.

In an initial position, either Black starts or White starts. Both
cases (or, if you prefer, "possibilities") are considered together
when deriving a move value. A move value is derived from the black
follower and the white follower. More specifically, a move value is
derived from the positional value (or the count) of the black follower
and the positional value (or the count) of the white follower.


MOVE VALUE CALCULATED AS SUM

Some traditionalists want to calculate a move value as a sum. They
prefer to express every positional value of a settled position as a
value favouring a player. Accordingly, they want to derive a move
value as follows:

A move value is the sum of the positional value of the black follower
as a value favouring Black and the positional value of the white
follower as a value favouring White.


X . . . . O
X X X O O O

initial position

X . 3 1 2 O
X X X O O O

sequence after Black's start

X . X X O O
X X X O O O

Black follower: Black has 1 point more than White.

X 2 1 3 . O
X X X O O O

sequence after White's start

X X O O . O
X X X O O O

White follower: White has 1 point more than Black.

The move value is 1 + 1 = 2.

This is correct.


X . O . . O O O O O . . O
X X X X X X X X X X O O O

initial position

X . O . 1 O O O O O 2 . O
X X X X X X X X X X O O O

sequence after Black's start

X . O . X O O O O O O . O
X X X X X X X X X X O O O

Black follower: Black has 3 points more than White.

X . O 2 1 O O O O O 3 . O
X X X X X X X X X X O O O

sequence after White's start

X . O X O O O O O O O . O
X X X X X X X X X X O O O

White follower: Black has 2 points more than White.

This is a problem for the procedure, which expects White to have more
points than Black in the white follower. Therefore, the procedure for calculating the move value is inapplicable.


O . X . . X X X X X . . X
O O O O O O O O O O X X X

initial position

O . X 2 1 X X X X X 3 . X
O O O O O O O O O O X X X

sequence after Black's start

O . X O X X X X X X X . X
O O O O O O O O O O X X X

Black follower: White has 2 points more than Black.

This is a problem for the procedure, which expects Black to have more
points than White in the black follower. Therefore, the procedure for calculating the move value is inapplicable.

O . X . 1 X X X X X 2 . X
O O O O O O O O O O X X X

sequence after White's start

O . X . O X X X X X X . X
O O O O O O O O O O X X X

White follower: White has 3 points more than Black.


As we see, calculation of a move value as a sum can produce problems
of inapplicability if it is derived from the followers' positional
values expressed as values favouring a player. Too much love of
positive numbers and calculations of sums results in desaster. Those traditionalists propagrating calculation of a move value as a sum
without further explanations give wrong advice.


CORRECTION OF MOVE VALUE CALCULATED AS SUM

Since naive calculation of a move value calculated as a sum of always
positive numbers fails, the procedure must be corrected as follows:

The positional value of the black follower is its Black's points minus
its White's points. Call this difference, say, x. Black has x points
more than White. If x is a negative number, this means that actually
White has the absolute of x points more than Black.

The positional value of the white follower is its White's points minus
its Black's points. Call this difference, say, y. White has y points
more than Black. If y is a negative number, this means that actually
Black has the absolute of y points more than White.

A move value is the sum of the positional value of the black follower
as a value favouring Black and the positional value of the white
follower as a value favouring White.


X . . . . O
X X X O O O

initial position

X . 3 1 2 O
X X X O O O

sequence after Black's start

X . X X O O
X X X O O O

Black follower: Black has 1 point more than White.

X 2 1 3 . O
X X X O O O

sequence after White's start

X X O O . O
X X X O O O

White follower: White has 1 point more than Black.

The move value is 1 + 1 = 2.


X . O . . O O O O O . . O
X X X X X X X X X X O O O

initial position

X . O . 1 O O O O O 2 . O
X X X X X X X X X X O O O

sequence after Black's start

X . O . X O O O O O O . O
X X X X X X X X X X O O O

Black follower: Black has 3 points more than White.

X . O 2 1 O O O O O 3 . O
X X X X X X X X X X O O O

sequence after White's start

X . O X O O O O O O O . O
X X X X X X X X X X O O O

White follower: White has -2 points more than Black.

The move value is 3 + (-2) = 1.


O . X . . X X X X X . . X
O O O O O O O O O O X X X

initial position

O . X 2 1 X X X X X 3 . X
O O O O O O O O O O X X X

sequence after Black's start

O . X O X X X X X X X . X
O O O O O O O O O O X X X

Black follower: Black has -2 points more than White.

O . X . 1 X X X X X 2 . X
O O O O O O O O O O X X X

sequence after White's start

O . X . O X X X X X X . X
O O O O O O O O O O X X X

White follower: White has 3 points more than Black.

The move value is -2 + 3 = 1.


In conclusion, a move value can be calculated correctly as a sum if
negative numbers are used as necessary.

However, I have never seen anybody propagating calculation of a move
value as a sum explaining this correctly because the tradtionalists
preferring this want to avoid negative numbers. They tolerate
calculation of differences, like we need for the positional value of a
settled position, but they reject negative numbers, as if negative
temperatures did not exist.

So what do they actually do when calculating a correct move value?
They apply a case analysis, which they have never described in
general. Consider the last example: the black follower's positional
value is that White has 2 points more than Black and the white
follower's positional value is that White has 3 points more than
Black. The move value is the white follower's positional value 3 minus
the black follower's positional value 2, that is, 3 - 2 = 1. Wait,
what? In some cases, they need to calculate a move value as a
difference so that they can avoid negative numbers! Hence, their claim
that they would always calculate the move value as a sum is false!


MOVE VALUE CALCULATED AS DIFFERENCE

If a move value is derived from counts, its calculation as a
difference in the following procedure is easy, except for a possible
double negation when subtracting a negative number:

The move value is the black follower's count minus the white
follower's count.


X . . . . O
X X X O O O

initial position

X . 3 1 2 O
X X X O O O

sequence after Black's start

X . X X O O
X X X O O O

Black follower: The count is 1.

X 2 1 3 . O
X X X O O O

sequence after White's start

X X O O . O
X X X O O O

White follower: The count is -1.

The move value is 1 - (-1) = 2.


X . O . . O O O O O . . O
X X X X X X X X X X O O O

initial position

X . O . 1 O O O O O 2 . O
X X X X X X X X X X O O O

sequence after Black's start

X . O . X O O O O O O . O
X X X X X X X X X X O O O

Black follower: The count is 3.

X . O 2 1 O O O O O 3 . O
X X X X X X X X X X O O O

sequence after White's start

X . O X O O O O O O O . O
X X X X X X X X X X O O O

White follower: The count is 2.

The move value is 3 - 2 = 1.


O . X . . X X X X X . . X
O O O O O O O O O O X X X

initial position

O . X 2 1 X X X X X 3 . X
O O O O O O O O O O X X X

sequence after Black's start

O . X O X X X X X X X . X
O O O O O O O O O O X X X

Black follower: The count is -2.

O . X . 1 X X X X X 2 . X
O O O O O O O O O O X X X

sequence after White's start

O . X . O X X X X X X . X
O O O O O O O O O O X X X

White follower: The count is -3.

The move value is -2 - (-3) = 1.


Now, some might still want to resurrect calculation as a sum by hiding
the first calculation step of 1 - (-1) = 1 + 1 = 2 and only writing 1
+ 1 = 2. They must accept that 3 - 2 = 1 cannot be simplified as a sum
and that -2 - (-3) = -2 + 3 = 1 still contains a negative number.


In conclusion, a move value is best calculated as a difference!

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