Count of Basic Ko and Dame

RobertJasiek · #1

This discusses the count and local temperature of a ko ensemble consisting of one basic endgame ko and one dame under territory scoring and area scoring. The start is about only one basic endgame ko.

ABBREVIATIONS

C = count of the initial position's locale
B = count of the black follower
W = count of the white follower
T = local temperature in the locale = size per play = miai value
b = excess number of black plays to the black follower
w = excess number of white plays to the black follower
N = b + w = tally
P = prisoner difference = white prisoner stones minus black prisoner stones

Positive counts favour Black, negative counts favour White.

All counts apply to the locales only.

ONE BASIC KO OPEN FOR BLACK UNDER TERRITORY SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X . X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X C B O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 1

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 0

Click Here To Show Diagram Code: [go]$$W White first, 2 = pass $$ ------------- $$ | . X 1 3 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 2

Click Here To Show Diagram Code: [go]$$B White follower, P = -1 $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -1

N = b + w = 1 + 2 = 3

T = (B - W) / N = (0 - (-1)) / 3 = 1/3

C = B - b*T = 0 - 1 * 1/3 = -1/3

Interpretation:
- One black move increases the initial count -1/3 by 1/3 to create the black follower's count 0.
- Two white moves decrease the initial count -1/3 by 2*(-1/3) to create the white follower's count -1.

ONE BASIC KO OPEN FOR BLACK UNDER AREA SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X . X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X C B O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 1

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 2

Click Here To Show Diagram Code: [go]$$W White first, 2 = pass $$ ------------- $$ | . X 1 3 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 2

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -2

N = b + w = 1 + 2 = 3

T = (B - W) / N = (2 - (-2)) / 3 = 4/3

C = B - b*T = 2 - 1 * 4/3 = 2/3

Interpretation:
- One black move increases the initial count 2/3 by 4/3 to create the black follower's count 2.
- Two white moves decrease the initial count 2/3 by 2*(-4/3) to create the white follower's count -2.

ONE BASIC KO OPEN FOR WHITE UNDER TERRITORY SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X O . O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X W C O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first, 2 = pass $$ ------------- $$ | . X 3 1 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 2

Click Here To Show Diagram Code: [go]$$B Black follower, P = 1 $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 1

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X O 1 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 1

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = 0

N = b + w = 2 + 1 = 3

T = (B - W) / N = (1 - 0) / 3 = 1/3

C = B - b*T = 1 - 2 * 1/3 = 1/3

Interpretation:
- Two black moves increase the initial count 1/3 by 2*1/3 to create the black follower's count 1.
- One white move decreases the initial count 1/3 by -1/3 to create the white follower's count 0.

ONE BASIC KO OPEN FOR WHITE UNDER AREA SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X O . O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X W C O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first, 2 = pass $$ ------------- $$ | . X 3 1 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 2

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 2

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X O 1 O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 1

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -2

N = b + w = 2 + 1 = 3

T = (B - W) / N = (2 - (-2)) / 3 = 4/3

C = B - b*T = 2 - 2 * 4/3 = 6/3 - 8/3 = -2/3

Interpretation:
- Two black moves increase the initial count -2/3 by 2*4/3 to create the black follower's count 2.
- One white move decreases the initial count -2/3 by -4/3 to create the white follower's count -2.

After these warming up exercises, now we come to the more interesting examples.

ONE BASIC KO OPEN FOR BLACK AND ONE DAME UNDER TERRITORY SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X . X O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X C B O | $$ | . X X O O | $$ | . X C O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 1 // only the play in the ko is counted

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 0

Click Here To Show Diagram Code: [go]$$W White first, P = -1 $$ ------------- $$ | . X 1 3 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 2 // only the plays in the ko are counted

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X X O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -1

N = b + w = 1 + 2 = 3

T = (B - W) / N = (0 - (-1)) / 3 = 1/3

C = B - b*T = 0 - 1 * 1/3 = -1/3

Interpretation:
- One black move increases the initial count -1/3 by 1/3 to create the black follower's count 0.
- Two white moves decrease the initial count -1/3 by 2*(-1/3) to create the white follower's count -1.

ONE BASIC KO OPEN FOR BLACK AND ONE DAME UNDER AREA SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X . X O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X C B O | $$ | . X X O O | $$ | . X C O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 1 // only the play in the ko is counted

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 1

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X 1 3 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 2 // only the plays in the ko are counted

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X X O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -1

N = b + w = 1 + 2 = 3

T = (B - W) / N = (1 - (-1)) / 3 = 2/3

C = B - b*T = 1 - 1 * 2/3 = 1/3

Interpretation:
- One black move increases the initial count 1/3 by 2/3 to create the black follower's count 1.
- Two white moves decrease the initial count 1/3 by 2*(-2/3) to create the white follower's count -1.

ONE BASIC KO OPEN FOR WHITE AND ONE DAME UNDER TERRITORY SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X O . O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X W C O | $$ | . X X O O | $$ | . X C O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first, P = 1 $$ ------------- $$ | . X 3 1 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 2 // only the plays in the ko are counted

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 1

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X O 1 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 1 // only the play in the ko is counted

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X X O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = 0

N = b + w = 2 + 1 = 3

T = (B - W) / N = (1 - 0) / 3 = 1/3

C = B - b*T = 1 - 2 * 1/3 = 1/3

Interpretation:
- Two black moves increase the initial count 1/3 by 2*1/3 to create the black follower's count 1.
- One white move decreases the initial count 1/3 by -1/3 to create the white follower's count 0.

ONE BASIC KO OPEN FOR WHITE AND ONE DAME UNDER AREA SCORING

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X O . O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Click Here To Show Diagram Code: [go]$$B locale $$ ------------- $$ | . X W C O | $$ | . X X O O | $$ | . X C O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The marked intersections are the considered locale, for which count and local temperature shall be determined.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 3 1 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

b = 2 // only the plays in the ko are counted

Click Here To Show Diagram Code: [go]$$B Black follower $$ ------------- $$ | . X X X O | $$ | . X X O O | $$ | . X O O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

B = 1

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X O 1 O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

w = 1 // only the play in the ko is counted

Click Here To Show Diagram Code: [go]$$B White follower $$ ------------- $$ | . X O O O | $$ | . X X O O | $$ | . X X O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

W = -1

N = b + w = 2 + 1 = 3

T = (B - W) / N = (1 - (-1)) / 3 = 2/3

C = B - b*T = 1 - 2 * 2/3 = -1/3

Interpretation:
- Two black moves increase the initial count -1/3 by 2*2/3 to create the black follower's count 1.
- One white move decreases the initial count -1/3 by -2/3 to create the white follower's count -1.

QUESTIONS

1) Have I made any mistakes and which?

2) Why do the numbers of ko threat plays not count for the tally?

emeraldemon · #2

Are you using count and temperature here in the Conway/Berlekamp sense?

RobertJasiek · #3

I think basically yes, but they use some fine details, of which I am not aware. However, note that I also use the area count variant, which of course is derived from area scoring. For temperature, I use the LOCAL temperature.

http://senseis.xmp.net/?Count
http://senseis.xmp.net/?Temperature

Bill Spight · #4

Quick discussion of ko and dame combination.

First, there is no one way of evaluation that is the only right one. There are a number that are consistent and produce the same game outcome. What I will discuss is the traditional way. It appears in the Mueller, Berlekamp, Spight paper of 1996.

Click Here To Show Diagram Code: [go]$$B initial position $$ ------------- $$ | . X . X O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

The ko and dame are, under area scoring, not strictly independent, but treating them as independent works. The count of the ko is ⅔ (for Black) and each play in the ko gains 1⅓ points. The count of the dame is 0 and each play in the dame gains 1 point. The count of the combination is the sum of the independent counts, or ⅔.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

Filling the ko gains 1⅓ points, then filling the dame gains 1 point, for a local result of ⅔ + 1⅓ - 1 = 1. That is correct.

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X 1 B O | $$ | . X X O O | $$ | . X 2 O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

fills the ko at :bc:

Taking the ko gains the ko gains 1⅓ points, then filling the dame gains 1 point, then filling the ko gains 1⅓ points, for a local result of ⅔ - 1⅓ + 1 - 1⅓ = -1. That is correct.

Traditional theory, treating the ko and dame as independent, gives a consistent account.

----

Evaluating the combination as a whole is trickier. When Black plays first the result is +1 in an even number of plays. When White plays first the result is -1 in an odd number of plays, one more for White than for Black. You could tell a consistent story by evaluating the combination as a 2 point sente for Black with the starting position worth +1 and each play gaining 2 points. The main reason that we do not do that is that we want to treat the ko and dame as independent to the extent that we can. That is what lets us use evaluation to guide play. It is not an infallible guide, since the best play is not always the one that gains the most, but it is in the vast majority of cases.

To evaluate a play or combination we make use of the idea of an environment. This idea is new with modern mathematical go, but has been lurking informally for a long time. When we explain a sente by saying that the reply is the largest play on the board, we are evoking the notion of an environment (the rest of the board). For evaluation we abstract the idea of the environment as a stack of coupons of the same value, such that taking a coupon gains the value of a coupon, which is called the temperature. Taking a coupon also lifts any ko or superko ban. There are indefinitely many coupons, as many as needed. Play stops when each player prefers to take a coupon instead of playing on the board, and the second player takes the last coupon. (I have modified the way that play stops to take care of some special situations, but this simple procedure works for nearly all positions.)

When Black plays first, we may draw lines to indicate the results up to temperature 1, the temperature of a dame. When Black plays first, that line is vertical at a local score of +1. When White plays first, that line goes from -1 at temperature 0 to 0 at temperature 1. Plainly the lines do not intersect. Above temperature 1 neither player will fill the dame. Here is how play will go at temperature t when t > 1 and less than the temperature of the ko.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------- $$ | . X 1 X O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

,

,

take a coupon

Black fills the ko, then the players take coupons to stop play. The local result is 2 - t, 2 points for Black on the board, while White has one extra coupon worth t.

Click Here To Show Diagram Code: [go]$$W White first $$ ------------- $$ | . X 1 B O | $$ | . X X O O | $$ | . X . O . | $$ | X X X O O | $$ | . . X O . | $$ -------------[/go]

takes a coupon, :w3:

fills ko at :bc:

,

,

,

take a coupon

The local result is -2 + 2 * t. White gets 2 points on the board, while Black has two extra coupons.

Note that when t = 1, the result when Black plays first is 2 - 1 = 1, and the result when White plays first is -2 + 2 = 0, as advertised.

Suppose that the temperature is such that the players are indifferent about who plays first and whether to play on the board or not. Then 2 - t = -2 + 2 * t. Then t = 1⅓ and the value of the initial position is ⅔ . Not surprisingly, that is the same value as we get for the ko alone.

¿Es claro?

RobertJasiek · #5

RobertJasiek wrote:

ONE BASIC KO OPEN FOR BLACK AND ONE DAME UNDER AREA SCORING

C = B - b*T = 1 - 1 * 2/3 = 1/3

Interpretation:
- One black move increases the initial count 1/3 by 2/3 to create the black follower's count 1.
- Two white moves decrease the initial count 1/3 by 2*(-2/3) to create the white follower's count -1.

ONE BASIC KO OPEN FOR WHITE AND ONE DAME UNDER AREA SCORING

C = B - b*T = 1 - 2 * 2/3 = -1/3

Interpretation:
- Two black moves increase the initial count -1/3 by 2*2/3 to create the black follower's count 1.
- One white move decreases the initial count -1/3 by -2/3 to create the white follower's count -1.

So the cause of my mistakes here was the false assumption that each play, whether in the ko or on the dame, would have the same size? Since the size is not constant for all the plays, a simple division by the tally is wrong. A more complicated model is needed, and there coupon environments and the like come in.

RobertJasiek · #6

Bill, many thanks!

Bill Spight wrote:

there is no one way of evaluation that is the only right one. There are a number that are consistent and produce the same game outcome.

Which would be a list of the known, possible, consistent ways?

Quote:

It appears in the Mueller, Berlekamp, Spight paper of 1996.

Unfortunately, the results of that and similar papers are still well hidden for ordinary players because an easily applicable translation to go players' thinking is still missing. (Hint, hint, I wish you could write a go book on that topic! :) )

Quote:

treating them as independent works.

Which are the conditions to decide, for an arbitrary ko ensemble,
- whether it consists of independent parts (is this possible at all for any ko?),
- whether such a treatment as independent parts works correctly?

I think now I understand your calculation as independent parts for the particular area scoring example: The ko has count 2/3 and local temperature 4/3, the dame has count 0 and local temperature 1. Black first means calculating 2/3 + 0 + 4/3 - 1 = 1. White first means calculating 2/3 + 0 + 1 - 2*4/3 = -1. The general idea here is to

form the sum of the initial counts of the parts of the ko ensemble, to add the value of all black plays and to subtract the value of white plays made until reaching a follower.

Quote:

consistent

What do you mean by consistent?

Quote:

Evaluating the combination as a whole is trickier. When Black plays first the result is +1 in an even number of plays. When White plays first the result is -1 in an odd number of plays, one more for White than for Black. You could tell a consistent story by evaluating the combination as a 2 point sente for Black with the starting position worth +1 and each play gaining 2 points.

I am lost here.

Quote:

The main reason that we do not do that is that we want to treat the ko and dame as independent to the extent that we can.

Why do we want that?

Quote:

That is what lets us use evaluation to guide play.

What does it mean to "let us use evaluation to guide play"?

Quote:

It is not an infallible guide, since the best play is not always the one that gains the most, but it is in the vast majority of cases.

Which are the minority cases?

Quote:

To evaluate a play or combination we make use of the idea of an environment.

Although adding token values is easy enough, I do not understand why / exactly under which conditions we may do that. We have a ko position's ensemble. Then we add the imagined environment's values. Then we get some conclusion for the local(?) temperature's value. Then we imagine to remove all the environment's values (because what we have in reality is only the position and not the helping coupons). Then why is the determined temperature value still correct?

Does such an environment model serve only the function of a probe? We use it to find a candidate value of the local temperature and ko ensemble's count? If the determined values work when adding them up for the plays to the followers, fine. If the determined values do not work, then the environment model was insufficient and we need to invent another, possible still unknown method?

Quote:

For evaluation we abstract the idea of the environment as a stack of coupons of the same value, such that taking a coupon gains the value of a coupon, which is called the temperature.

IIRC, there are are also other, alternative environment models such as NTE or other finite or infinite decreasing value stacks?

Quote:

Taking a coupon also lifts any ko or superko ban.

By definition?

Quote:

the second player takes the last coupon.

Why? Why is that a good model? Why does it sometimes allow to equate temperature terms, as you do under certain assumptions in 2 - t = -2 + 2 * t?

Quote:

we may draw lines

All those temperature graphs have never made me happy. Isn't algebra and test temperatures within useful ranges (here: 0..1, 1..4/3, 4/3..oo) easier?

Quote:

Not surprisingly, that is the same value as we get for the ko alone.

Frankly, it still does surprise me:)

Count of Basic Ko and Dame

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