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.)

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . X X O O
$$|X X X O O O[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X X O O . O
$$|X X X O O O[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . X O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O X O O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X O X X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . O X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . X X O O
$$|X X X O O O[/go]

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

The count is 1 - 0 = 1.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X X O O . O
$$|X X X O O O[/go]

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

The count is 0 - 1 = -1.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . X O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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

The count is 4 - 1 = 3.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O X O O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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

The count is 3 - 1 = 2.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X O X X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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

The count is 1 - 3 = -2.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . O X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . . . . O
$$|X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . 3 1 2 O
$$|X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . X X O O
$$|X X X O O O[/go]

Black follower: Black has 1 point more than White.

Click Here To Show Diagram Code

[go]$$W
$$------------
$$|X 2 1 3 . O
$$|X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X X O O . O
$$|X X X O O O[/go]

White follower: White has 1 point more than Black.

The move value is 1 + 1 = 2.

This is correct.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . . O O O O O . . O
$$|X X X X X X X X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . 1 O O O O O 2 . O
$$|X X X X X X X X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . X O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

Black follower: Black has 3 points more than White.

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|X . O 2 1 O O O O O 3 . O
$$|X X X X X X X X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O X O O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . . X X X X X . . X
$$|O O O O O O O O O O X X X[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X 2 1 X X X X X 3 . X
$$|O O O O O O O O O O X X X[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X O X X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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.

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|O . X . 1 X X X X X 2 . X
$$|O O O O O O O O O O X X X[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . O X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . . . . O
$$|X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . 3 1 2 O
$$|X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . X X O O
$$|X X X O O O[/go]

Black follower: Black has 1 point more than White.

Click Here To Show Diagram Code

[go]$$W
$$------------
$$|X 2 1 3 . O
$$|X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X X O O . O
$$|X X X O O O[/go]

White follower: White has 1 point more than Black.

The move value is 1 + 1 = 2.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . . O O O O O . . O
$$|X X X X X X X X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . 1 O O O O O 2 . O
$$|X X X X X X X X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . X O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

Black follower: Black has 3 points more than White.

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|X . O 2 1 O O O O O 3 . O
$$|X X X X X X X X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O X O O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

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

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

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . . X X X X X . . X
$$|O O O O O O O O O O X X X[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X 2 1 X X X X X 3 . X
$$|O O O O O O O O O O X X X[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X O X X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|O . X . 1 X X X X X 2 . X
$$|O O O O O O O O O O X X X[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . O X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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 xplaining 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, what 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.

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . . . . O
$$|X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . 3 1 2 O
$$|X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X . X X O O
$$|X X X O O O[/go]

Black follower: The count is 1.

Click Here To Show Diagram Code

[go]$$W
$$------------
$$|X 2 1 3 . O
$$|X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$------------
$$|X X O O . O
$$|X X X O O O[/go]

White follower: The count is -1.

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

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . . O O O O O . . O
$$|X X X X X X X X X X O O O[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . 1 O O O O O 2 . O
$$|X X X X X X X X X X O O O[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O . X O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

Black follower: The count is 3.

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|X . O 2 1 O O O O O 3 . O
$$|X X X X X X X X X X O O O[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|X . O X O O O O O O O . O
$$|X X X X X X X X X X O O O[/go]

White follower: The count is 2.

The move value is 3 - 2 = 1.

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . . X X X X X . . X
$$|O O O O O O O O O O X X X[/go]

initial position

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X 2 1 X X X X X 3 . X
$$|O O O O O O O O O O X X X[/go]

sequence after Black's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X O X X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

Black follower: The count is -2.

Click Here To Show Diagram Code

[go]$$W
$$--------------------------
$$|O . X . 1 X X X X X 2 . X
$$|O O O O O O O O O O X X X[/go]

sequence after White's start

Click Here To Show Diagram Code

[go]$$B
$$--------------------------
$$|O . X . O X X X X X X . X
$$|O O O O O O O O O O X X X[/go]

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!