Thanks for the info.

End of game procedures may differ for different rule sets, but with some exceptions, correct play by territory scoring is also correct by area scoring.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Problems with my theory. The environment.

As I said, I considered part of the environment part of the ko ensemble. For instance, if the environment consists of a number of simple gote with swing values G0 ≥ G1 ≥ G2 ≥ . . . , in deciding whether to take or win a ko I might compare G0 with K - Gn + Gn+1 - Gn+2 + . . . , where K is the swing value of the ko, and the value of n depended on the ko threat situation. In my mind G0 was the swing value of the largest play on the board besides the ko, and therefore K - Gn + . . . was the swing value of the ko ensemble. Kos did not have values independent of the rest of the board.

There is nothing particularly wrong with this theory. After all, it led me to an understanding of komonster. But it is unwieldy in practice. How deeply do you look into the environment? 8 moves? 10 moves? (This is related to the famous magic number 7, plus or minus 2 that John Fairbairn often brings up. ) As a practical matter, at some point you have to stop. (The fact that the environment does not just consist of simple gote is another matter.)

You can estimate the value of Gn - Gn+1 + Gn+2 - . . . by (Gn)/2. So we can decide whether to take or win the ko or not by comparing G0 with K - (Gn)/2, or with K - Gn + (Gn+1)/2, etc. Which estimate we use is up to us. OC, if the difference between Gn and Gn+1 is significant, then we should not use (Gn)/2, as it will probably be an overestimate.

Another problem I discovered with this framework is that it makes the analysis of approach kos difficult.

Fast forward to 1994, when I attended a talk by Professor Berlekamp in which he presented the idea of komaster (which he had developed in the '90s). The komaster is able to win the ko, but is not komonster. After the talk I sent him a note about komonster effects (without using the term) and suggesting a possible modification of his methods. By using the idea of komaster, however, Berlekamp had been able to evaluate approach kos and 10,000 year kos.

The methodology used by Berlekamp, called thermography, utilizes the concept of temperature, or a tax on making a play. For non-ko positions it gets the same results as traditional go evaluation, and also provides additional information. My theory departed from traditional go evaluation, so that is a plus for thermography.

It turns out that thermography can be adapted to my approach. For instance, if we decide to stop the analysis at Gn and use (Gn)/2, just set the temperature, t, to (Gn)/2 and treat the larger gote as hot plays. Thus, where my theory says to win the ko if K > G0 + (G2)/2, change that to K > G0 + t. This means that G0 is no longer considered part of the environment. The environment consists of G2 and smaller plays. It is a kind of foreground/background distinction.

In 1998 I used the idea of an environment with temperature to redefine thermography and extend it to the evaluation of multiple kos and superkos.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

ONAG is nice and precise. "Winning Ways" is funny and handwaving. "Mathematical Go" is not a pleasure to read for a mathematician. It starts saying too much before giving definitions. I wouldnt mind a current book that is precise.

Your posts are unreadable for me since the background is missing. "Worth 0.75" - in which valuation system?

ONAG describes the disjunctive sum. But the existence of kos means that go positions do not neatly decompose as disjunctive sums. Is there a mathematically precise definition of thermography that applies to go?

Click Here To Show Diagram Code

[go]$$B AGA rules.
$$ -------------------
$$ | . # O C C C O O O |
$$ | X O O C O O O X X |
$$ | X X O C O X X X C |
$$ | C X O O . b a X C |
$$ | C X . . X X X X C |
$$ | C X O O O O O X C |
$$ | X O O B B O X W . |
$$ | X X O C O O X X X |
$$ | C X . . C O O O . |
$$ -------------------[/go]

Because the number of Black stones and White stones on the board are equal, the area count and territory count are the same, even if the value for individual points may differ. For convenience and the familiarity of the readers I will use the territory count. The marked points and circled stones indicate territory. Black has 10 points, White has 11. Using probabilistic semantics we can evaluate point “a” as 0.75 point for Black and “b” as 0.5 point. Adding those to the rest of Black’s territory yields 11.25. The ko stone in the top left corner is usually valued as 1/3 point for White. By komonster analysis its value is 1 point for White, which gives White 12 points for a net value of 0.75 for White.

Why is the ko worth 1 point for White? Again, using probabilistic semantics, half the time Black will fill the ko for 0 points of territory, and half the time White will win the ko, as in the next diagram.

Click Here To Show Diagram Code

[go]$$B AGA rules.
$$ -------------------
$$ | W C O . . . O O O |
$$ | X O O . O O O X X |
$$ | X X O . O X X X . |
$$ | . X O O . . . X . |
$$ | . X . . X X X X . |
$$ | . X O O O O O X . |
$$ | X O O X X O X O . |
$$ | X X O . O O X X X |
$$ | . X . . . O O O . |
$$ -------------------[/go]

Because White is komonster he does not have to fill the ko (before the end of play), so the marked point is one point of territory and White gets one point for the captured stone, for a total of two points. The original ko is worth the average, or 1 point for White.

(Historical note: Counting the marked point for White was a possibility for the Japanese rules before they were codified. Both Honinbo Shusai and Go Seigen favored doing that. )



The "Extended thermography" paper is where I redefine thermography in terms of play in an environment.

Some references:

Berlekamp, “The economist’s view of combinatorial games,” in Games of No Chance, Richard J. Nowakowski (ed.), Cambridge University Press(1996)

Spight, “Extended thermography for multiple kos in go,” in Lecture Notes in Computer Science, 1558: Computers and Games, Van den Herik and Iida (eds.), Springer (1999)

Spight, “Go thermography: The 4/19/98 Jiang-Rui endgame,” in More Games of No Chance, Richard J. Nowakowski (ed.), Cambridge University Press (2002)

Siegel, Aaron, Combinatorial Game Theory, American Mathematical Society (2013)

Edit: I almost forgot.

Berlekamp, "Baduk+coupons," and

Spight, "Evaluating kos: A review of the research," both in

Proceedings: ICOB 2006: The 4th International Conference on Baduk, Myongji University and Korean Society for Baduk Studies (2006)

I don't know how easily available those proceedings are.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thanks! I found the Korean proceedings at
http://www.earticle.net/search/pub/?org=106&jour=252
with the two papers mentioned at
http://www.earticle.net/article.aspx?sn=27269 and
http://www.earticle.net/article.aspx?sn=27268

Click Here To Show Diagram Code

[go]$$ Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | . O O . O X . X . |
$$ | X X O X O X . X . |
$$ | O O O X O X O X . |
$$ | . . O . O X O X X |
$$ | . O . O O O . O X |
$$ | . . . O . O O X X |
$$ | . . . . O X . X . |
$$ ------------------[/go]

Black to play. What result with best play?

White to play. What result with best play?

Enjoy!

(Board edited later.)

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

You show a number of variations, but do not explicitly answer the questions.

For extra credit, evaluate the three gote and the ko.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

My apologies to anyone who tried the 9x10 board. It has its own charm, though. More later.

Edit: I have skipped the discussion of the 9x10 board position, despite its charm.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thanks.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$B Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | 3 O O 1 O X 5 X . |
$$ | X X O X O X 4 X . |
$$ | O O O X O X O X . |
$$ | . . O 7 O X O X X |
$$ | . O . O O O 2 O X |
$$ | . . . O . O O X X |
$$ | . . . . O X 6 X . |
$$ ------------------[/go]

Result: Black +1

Click Here To Show Diagram Code

[go]$$B Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | 4 O O 2 O X . X . |
$$ | X X O X O X 3 X . |
$$ | O O O X O X O X . |
$$ | . . O . O X O X X |
$$ | . O . O O O 1 O X |
$$ | . . . O . O O X X |
$$ | . . . . O X 5 X . |
$$ ------------------[/go]

Result: Black +1

Click Here To Show Diagram Code

[go]$$W Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | 3 O O 1 O X . X . |
$$ | X X O X O X 4 X . |
$$ | O O O X O X O X . |
$$ | . . O . O X O X X |
$$ | . O . O O O 2 O X |
$$ | . . . O . O O X X |
$$ | . . . . O X 5 X . |
$$ ------------------[/go]

Result: White +1

Click Here To Show Diagram Code

[go]$$W Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | 3 O O 2 O X 6 X . |
$$ | X X O X O X 5 X . |
$$ | O O O X O X O X . |
$$ | . . O 7 O X O X X |
$$ | . O . O O O 1 O X |
$$ | . . . O . O O X X |
$$ | . . . . O X 4 X . |
$$ ------------------[/go]

Result: White +1



Is it better to take the largest gote or to play in the ko? It does not matter.

Obviously, I constructed the board that way, but the position is not particularly unusual. On a larger board, with more plays in the environment, it would look even more normal. That was my point, to show a fairly normal position where playing the ko was equivalent to taking the largest gote.

How much does taking the largest gote gain? If Black takes it the local result is 0, if White takes it it is 5 points for White. We may write that as {0|−5}. Black moves to positions to the left of the bar and White moves to positions to the right of the bar. The score is from Black's point of view, so 5 points for White is −5 points for Black. The value of the original gote position is −5/2, or −2.5, and each move in the gote gains 2.5.

The ko is a little tricky to evaluate. First suppose that White wins the ko.

Click Here To Show Diagram Code

[go]$$W Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | . O O . O X 4 X . |
$$ | X X O X O X 3 X . |
$$ | O O O X O X O X . |
$$ | . . O . O X O X X |
$$ | . O . O O O 1 O X |
$$ | . . . O . O O X X |
$$ | . . . . O X . X . |
$$ ------------------[/go]

wins the ko and later - is a sente sequence, which we assume that White will be able to play. The local result is 0.

Now suppose that Black takes and wins the ko.

Click Here To Show Diagram Code

[go]$$B Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | . O O . O X C X . |
$$ | X X O X O X 3 X . |
$$ | O O O X O X W X . |
$$ | . . O . O X W X X |
$$ | . O . O O O 1 W X |
$$ | . . . O . O O X X |
$$ | . . . . O X . X . |
$$ ------------------[/go]

The local result is 7 points for Black, the three stones plus the empty point, .

The original ko position is worth −2 1/3, and each move in the ko gains 2 1/3 points.

2 1/3 is a little less than 2.5, yet the two plays are in this case equivalent. As I mentioned earlier, a small komonster effect is normal.

Gotta run. More later.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

OK. The above study illustrates the equation,



where K is the swing value of the ko, G is the swing value of the largest gote, and t is the (ambient) temperature of the environment. The ko and gote are the only plays hotter than the environment. That is, K > 3t and G > 2t. K = 7 and G = 5, so t = 2. If the ambient temperature is 2, then the player to move is indifferent between taking the gote and making a ko move. If t > 2 then the player takes the gote; if t < 2 then the player makes a ko play.

It may not be obvious why we may consider the temperature to be 2 on this board. After all, we can read the whole play out at temperature 0. Well, we can do that, but it would have been tedious and error-prone for me to provide a realistic environment at temperature 2 that it would have been impractical to read out. So I took a short cut.

Click Here To Show Diagram Code

[go]$$ Japanese rules. No komi.
$$ ------------------
$$ | . . . O X X . . . |
$$ | X X X X O X O X . |
$$ | . O O . O X . X . |
$$ | X X O X O X . X . |
$$ | O O O X O X O X . |
$$ | . . O . O X O X X |
$$ | . O . O O O . O X |
$$ | . . . O . O O X X |
$$ | . . . . O X . X . |
$$ ------------------[/go]

The swing values for the second and third hottest gote are 4 and 2, respectively. If we takes the second largest gote as the hottest play in the environment, then the temperature is 2, which is how much a play in that gote gains.

First, let us look at the case where Black to play can take the ko. As we can see from the sequences in the previous note, all of the plays change hands. At temperature 0 we have this equation:



Check.

Now let us look at the case where White to play can win the ko. In each sequence White gets the second hottest gote, so it does not figure into the calculations. At temperature 0 we have this equation:



Check.

The trick is to have the third largest gote be half the size of the second largest gote. Doing so yields an effective temperature in our equation of 2. (This is the trick behind button go, BTW. ) On a larger board I could have had a more normal environment with plays having swing values of 4, 3, 2, and 1, to the same effect, but the space on the 9x9 is cramped, so I used that trick.

Aside from the equation, what is the takeaway from all this? Note that the player to move makes a ko play when t is small enough (and K > G). To put it another way, when G is significantly hotter than 2t. OC, we may see that G is significantly hotter than 2t by inspection, but there is a time when we expect G to be hotter than 2t, when the opponent has just played a ko threat and we have to decide whether to answer it or not. If neither player has any other threats at that point, then the above comparison may be our guide.

Now, when I was learning go the rule of thumb was this. Answer a ko threat when G > 2K/3, that is, when a play in the threat gains more than a play in the ko, on average. But often we should ignore such a threat. G > K − t is a better guide. (And it is even better to take other threats into account if possible, as we shall see. )

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hey, Bill. I don't mean to detract from the discussion, but I was wondering if you ever considered writing a book on this material. The topic seems deep enough that you'd have enough material.

_________________
be immersed

Thanks, Kirby. I appreciate the encouragement.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

No problem... I'll take that as a "Yes"

_________________
be immersed

To continue the discussion of ko and the environment I need to move to larger boards. So today let me introduce a new topic:

Sente and gote

In a way, I was fortunate that the second go book I bought was Korschelt's The Theory and Practice of Go. (The first was Edward Laker's, Go and Go-Moku.) For some strange reason through the double translation from Japanese to German to English, sente became Upper Hand. However, Korschelt was clear that good players tried to take and keep Upper Hand. So I avoided the beginner's trap of following White around the board. OC, that meant that a lot of my groups died, but ç'est la vie! Or ç'est la guerre.

Later on, when I was around 3 kyu, I learned that good players also tried to get the Last Play (tedomari) not just at the end of the game, but at earlier stages of the game, as well, such as the opening. OC, if you get the last play of the opening, you give up sente, so those two principles are apparently at odds. There is a go proverb that attempts to reconcile them by saying "Tedomari is worth sente." Sorry folks, but, unlike most go proverbs, that one is just misleading. I'll explain why in a later note.

Now, the idea of taking and keeping sente hearkens to the meaning of sente as the initiative. (I think that that would have been a better translation, even though one meaning of te is hand. )

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I've never come across this in any language. Is it perchance a western invention, Bill?



I await the promised explanation, but I don't mind admitting that I for one do not yet see why "OC". Imagine an alleyway with Batman at one end and the Joker in the middle. If Robin gets the last big point by blocking the other end of the alley, Batman and Robin surely have the initiative (i.e. sente). The Joker has to respond. But if the Joker's accomplice got to the end point of the alley before Robin, the pressure's off. In general it seems that a tedomari in the opening does imply a threat or a follow up. Because the board is so open at this stage, the opponent can naturally decide to ignore the tedomari's threats, but that doesn't seem automatically to confer the initiative on him. At best he may get a local initiative, but the whole-board initiative surely still rests with the guy who got the tedomari.