Life In 19x19

RobertJasiek wrote:Bill, I lost track. What is your current method and why is it simpler than your 1998 paper?

My current method finds the minimax result for each player at the current temperature. So in the diagrammed example White to play moves locally to -23 and Black replies in the environment at temperature, t, so the result is -23 + t.

Black to play moves locally to the mean value, m(B), of the Black follower and White replies locally to -6 or in the environment to m(B) - t. So the result is min(-6, m(B)-t). We find m(B) = -3, which makes the result min(-6, -3-t).

To find the mean value, m, where the result is the same regardless of who plays first we solve the equations:

m = min(-6, -3-t)
m = -23 + t

We can solve for t in the second case

-3 - t = -23 + t
2t = 20
t = 10

Then

m = min(-6,-3-10) = min(-6,-13) = -13

The average value of the position is 13 pts. for White and a play by either player gains 10 pts. on average.

Edited for clarity.

Knotwilg wrote:

c) this is probably what is meant here but the answer isn't different than a) or b) (depending on the question). The worst one can do at move 1 is passing (hypothesis not tested: maybe playing 1-1 is worse). In the course of the game, the impact of a move on the final score can change a lot. This is evoked by the concept "temperature". Making life for a large group or not can make the end result differ by dozens of points (the are is "hot"). After having made life, no such move exists, not in the vicinity of the group (local temperature drops) or anywhere (global temperature drops). At move 1 however, temperature is pretty low. The difference between playing or not playing is about 14 points.

Reading this paragraph above, and your other posts it appears to me that, when the temperature in a area is high, then we are encourage to play urgently in this area. If this is true does that mean that the temperature of an area is simply an estimation of a play in this area ?

The problem with such definition is that it hurts the common sense concerning temperature.

Let's take the two following examples:

diag 1

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . O . . . b . . . . .
$$ | O O . O O a . . . . .
$$ | O O O X X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

diag 2

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . O O O . b c . . . .
$$ | O O O . O a . . d e .
$$ | O O O O X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

If you consider that black can attack the triangle white stone then the temperature is very hot and in the two diagrams everybody understands that black is urged to play "a".
But the two diagrams are very different.
After black "a" in diag1 my feeling is that the temperature has become very low : the attack of the triangle white stone can now take place and locally each player has in mind that black will be able to play "b" in sente (I suppose the white groupe in the corner quite big). In my mind the temperature is very low because there is no uncertainty concerning a black sente move on "b".
In diag2 the situation is really different. After black "a" black can now choose between move on "b" or move on "c" aiming at a black move at ""d" or "e". My common sense tells me that for white point of view the temperature is quite high because white has always to take into account a black move at "b" or "c". In addition as black player and seeing this hot temperature for white I am encourage to avoid choosing between "b" or "c" in order to keep this difficulty for white.

Why I take these examples? To show you that for my "common sense" a hot temperature is not necessarilly the consequence of a big move in an area (a black move at b is not really big, it is a relative small sente move for black) but it could be an uncertainty with very great influence on the game.
In that case we are facing a hot temperature but, despite of this, neither black nor white want to play in this area.

Could you define what you call temperature?

I don't think temperature has anything to do with the amount of uncertainty/choice in a position.

In both of your diagrams, black playing at b after a greatly increase the local temperature : white is more likely to play a move in the area than before black play at b

No doubt for me, temperature has nothing to do with the amount of uncertainty/choice in a position, at least directly (?).
I see perfectly that the intitial position (before a black move at "a") must have a high temperature. But after this black play at "a" I do not know how the temperature take into the existence of such sente move like "b" or "c". The existence of these sente moves are of great importance for future attack on triangle stones (may be not immediatly but certainly in a near future) but I feel that white will not play locally to avoid a gote move here and black will not play either because it is its privilege to play here and because it will find some interest to keep uncertainty here. In these conditions, considering the sente move is not intrinsincly a large move but is of crucial importance (even if not played) it is not clear to see the temperature after black "a".
In addition if black plays "b" it seems that the evaluation of the position greatly change (at least half the value of the corner ?) as well as the temperature (great increase) though for the players it may look only as a "simple" sente move played without the feeling that the temperature grows. On the contrary the feeling is certainly that the temperature drops to zero. In other words it looks like an artificial complication if we say that the temperature becomes very high beetween the black sente move and the white reply. For me the black sente move and the white reply have to be analysed globally. Of course I agree to say that the temperature will increase greatly in the case white does not reply immediatly to black move!

In addition if black plays "b" it seems that the evaluation of the position greatly change (at least half the value of the corner ?) as well as the temperature (great increase) though for the players it may look only as a "simple" sente move played without the feeling that the temperature grows. On the contrary the feeling is certainly that the temperature drops to zero. In other words it looks like an artificial complication if we say that the temperature becomes very high beetween the black sente move and the white reply. For me the black sente move and the white reply have to be analysed globally. Of course I agree to say that the temperature will increase greatly in the case white does not reply immediatly to black move!

Then your concept of temperature is not what Bill call temperature.

When black play b, the local temperature greatly increase : that's why b is (localy) sente. If the temperature didn't increase, it wouldn't be (localy) sente

Brief remarks on temperature and sente

The term, temperature, started out in Combinatorial Game Theory (CGT) as part of thermography. Hence the thermo- in thermograph. Temperature was a parameter associated with a tax on making a play. In 1998, to extend thermography to multiple kos (and superkos), I redefined thermography, not as considering a tax on making a play, but as considering the gain from making a different play in an ideal environment.

At some point on rec.games.go, go players adopted the term, temperature, to refer to the value of the largest play on the whole board. As adopted, temperature was not a technical term, but an informal term, with the usual ambiguity of everyday speech. See https://senseis.xmp.net/?Temperature . At some point, I and others started using this informal notion of temperature to refer not just to the whole board, but to local regions of the board. But it is still an informal term, not a technical one.

In the discussion here, when talking about the parameter, temperature, I am using it in the technical sense of my CGT paper. When you solve for temperature in finding the average value of a position, the solution indicates how much one or more plays from that position gains. Here is a sente example, a modification of your earlier gote example.

Click Here To Show Diagram Code

[go]$$Bc Sente
$$ -------------------
$$ | X X X . X a X . .
$$ | X X X O X O X . .
$$ | X X O O X O X . .
$$ | O O O O O O X X X
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Assuming that the White stones and the Black stones to the right are immortal, and taking Black's point of view, if White plays first the local score in the corner is -23. If Black plays first and White replies, the local score is -16, and if Black gets to play twice, the local score is 0.

If White plays locally and Black plays in the environment at temperature, t, in either order, the result at that temperature is -23 + t. If Black plays locally and White replies locally the result is -16. If instead White plays in the environment the result is m(B) - t, where m(B) is the average value of local position after Black connects at a. To find the average value, m, of the original position we solve these equations, as above.

m = -23 + t
m = min(-16, m(B)-t)

Solving for m(B), we find that m(B) = -8. Which yields the equation,

m = min(-16, -8-t)

If m = -8 - t, then

t = (23-8)/2 = 7.5

and

m = min(-16, -8 - 7.5) = min(-16, -15.5) = -16.

Also,

-16 = -23 + t
t = 7

Which indicates that White's local play gains, on average, 7 pts. White's reply from the position after Black a gains 8 pts., as you may verify. So Black's initial play at a also gains 8 pts., since the exchange gains zero points on average. In an ideal environment when 8 > t > 7 Black will be able to play locally with sente.

Will Black play with sente in a real game? Who knows? Black's threat is only 1 pt. larger, on average, than White's reverse sente. The window of opportunity is narrow, so maybe not. But we still classify this position as a Black sente and evaluate it as 16 pts. for White.

Edited for correctness and I hope, clarity.

Edit: Note that in usual go parlance Black's local play would be called a 7 pt. sente, which could be confusing, as it actually gains 8 pts. It is White's reverse sente that gains 7 pts.

Gérard TAILLE wrote:No doubt for me, temperature has nothing to do with the amount of uncertainty/choice in a position, at least directly (?).

If you don't mind, since Tryss referenced me I'll reply briefly to this note.

I think you get the idea of the informal sense of temperature.

Earlier you said this:

Gérard TAILLE wrote:diag 1

$$W
$$ -----------------------
$$ | . O . . . b . . . . .
$$ | O O . O O a . . . . .
$$ | O O O X X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . O . . . b . . . . .
$$ | O O . O O a . . . . .
$$ | O O O X X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

diag 2

$$W
$$ -----------------------
$$ | . O O O . b c . . . .
$$ | O O O . O a . . d e .
$$ | O O O O X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . O O O . b c . . . .
$$ | O O O . O a . . d e .
$$ | O O O O X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

If you consider that black can attack the triangle white stone then the temperature is very hot and in the two diagrams everybody understands that black is urged to play "a".
But the two diagrams are very different.

I think there's a typo in Diagram 1, that you meant it to be this.

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . . O . . b . . . . .
$$ | O O . O O a . . . . .
$$ | O O O X X X Q Q . . .
$$ | O O O O X . Q . . . .
$$ | X X X X X . Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

This way Black b is not sente to kill the corner.

Also, the diagrams do not show all the pertinent information, so I don't think we can say for sure that Black should play at a to attack the marked White stones. Maybe Black should play somewhere else to force White to play at a. Who knows?

Gérard TAILLE wrote:I see perfectly that the intitial position (before a black move at "a") must have a high temperature.

That's a big maybe. But it is certainly higher than just taking away a few points.

Gérard TAILLE wrote:But after this black play at "a" I do not know how the temperature take into the existence of such sente move like "b" or "c".

Well, if the temperature of the empty board is around 14, then killing the large corner group gains more than that. Making a ko for the group may gain less, but if White wins the ko he threatens to cut off the Black stone at b, and White can even cut during the ko fight to make it larger. But if Black b or c is sente, given the global context, we assume that Black plays it with sente.

Gérard TAILLE wrote:The existence of these sente moves are of great importance for future attack on triangle stones (may be not immediatly but certainly in a near future) but I feel that white will not play locally to avoid a gote move here and black will not play either because it is its privilege to play here and because it will find some interest to keep uncertainty here.

A matter of judgement.

Gérard TAILLE wrote:In these conditions, considering the sente move is not intrinsincly a large move but is of crucial importance (even if not played) it is not clear to see the temperature after black "a".

We don't have enough information.

Gérard TAILLE wrote:In addition if black plays "b" it seems that the evaluation of the position greatly change (at least half the value of the corner ?) as well as the temperature (great increase) though for the players it may look only as a "simple" sente move played without the feeling that the temperature grows.

If Black plays at b in the second diagram, threatening to kill the corner, then certainly the evaluation of the position changes temporarily before White protects and the local temperature probably increases — we don't have enough information to know what it is. If White correctly protects the corner, then the evaluation of the position indeed would not change, because the original evaluation would take that sente exchange into account. And the temperature would drop back to what it was before. Black c, only threatening a ko, is trickier.

Gérard TAILLE wrote:On the contrary the feeling is certainly that the temperature drops to zero. In other words it looks like an artificial complication if we say that the temperature becomes very high beetween the black sente move and the white reply. For me the black sente move and the white reply have to be analysed globally. Of course I agree to say that the temperature will increase greatly in the case white does not reply immediatly to black move!

If you look at the evaluation of the sente position I posted above, you see that indeed the question of the sente and the reply to it is analyzed globally. That occurs, assuming an ideal environment, only when the temperature of the environment is greater than the gain from the reverse sente and less than the gain from replying locally to the sente. We don't know if the conditions for the sente exchange will exist, but they probably will, unless the difference is very small.

Bill,

t = (23-8)/2 = 7.5

Seeing your equation above. I understand that the estimated value of the initial position is -7,5 (for black point of view) and I understand also that (without taking into account the sente notion) a black or white play in this local area earns 7,5 points.

In addition to that, if we take into account the fact that a black play here is (almost?) sente then the value of a black move here becomes only 7 points.

I fully understand why the sente value (7 points) is less than the gote estimation (7,5 points) of a black play but I do not understand why you said a little farther in your post:

So Black's initial play at a also gains 8 pts

Ok for a value of 7 points or 7,5 points (if you take into account or not the sente notion), but how can you find the value 8 points for a play at a?

Gérard TAILLE wrote:Bill,

t = (23-8)/2 = 7.5

Seeing your equation above. I understand that the estimated value of the initial position is -7,5 (for black point of view) and I understand also that (without taking into account the sente notion) a black or white play in this local area earns 7,5 points.

In addition to that, if we take into account the fact that a black play here is (almost?) sente then the value of a black move here becomes only 7 points.

I fully understand why the sente value (7 points) is less than the gote estimation (7,5 points) of a black play but I do not understand why you said a little farther in your post:

So Black's initial play at a also gains 8 pts

Ok for a value of 7 points or 7,5 points (if you take into account or not the sente notion), but how can you find the value 8 points for a play at a?

Sorry for not being clear. We get the value of 8 pts. from our previous analysis of the position after Black's play. I left that out, sorry.

The position after the Black play and White reply with sente is worth 16 pts. for White. That is equal to the average value of the position before a play. So the average value of Black's play and White's reply is the same.

After Black's play we have a gote position with an average value of 8 pts. for White (the average of 16 and 0). So Black's play gained on average 16 - 8 = 8 pts.

Ok that is clear now:
In a sente analysis a black plays at "a" earns 7 points in sente.
In a gote analysis a black plays at "a" earns 7,5 points and then it remains a move which can earn 8 points for who is playing (black or white).

Coming back to your previous post you said:

The term, temperature, started out in Combinatorial Game Theory (CGT) as part of thermography. Hence the thermo- in thermograph. Temperature was a parameter associated with a tax on making a play. In 1998, to extend thermography to multiple kos (and superkos), I redefined thermography, not as considering a tax on making a play, but as considering the gain from making a different play in an ideal environment.

At some point on rec.games.go, go players adopted the term, temperature, to refer to the value of the largest play on the whole board. As adopted, temperature was not a technical term, but an informal term, with the usual ambiguity of everyday speech. See https://senseis.xmp.net/?Temperature . At some point, I and others started using this informal notion of temperature to refer not just to the whole board, but to local regions of the board. But it is still an informal term, not a technical one.

If I understand correctly instead of taking as definition a "tax" or the "largest play" you prefer considering the "gain from making a different play".
That sounds a good idea. In particularly I would see great difficulties with the "largest play" because it could be uneasy to eliminate ko threats as in the following example:

Click Here To Show Diagram Code

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

The evaluation of this position (for black point of view) is -4 points.
Let's now consider a black move at "a". The evaluation of the resulting position is now the average value between -4 (a white reply) and +34 (if black plays and capture the 16 white stones). After a black move at "a" the evaluation of the position is (-4 + 34)/2 = 15. And you can see that the value of the black "a" move is 15 - (-4) = 19 points.
Here you see the drawback of considering the "largest move" because all ko threats look like a quite large move and, as a consequence all ko threat areas, in terme=s of temperature, will hide all really interesting areas where you want really move.

For that reason, because playing a ko threat does not gain anything, I prefer your wording "the gain from making a different play" even if this definition is still not quite clear (but is an informal term!).

Gérard TAILLE wrote:Ok that is clear now:
In a sente analysis a black plays at "a" earns 7 points in sente.
In a gote analysis a black plays at "a" earns 7,5 points and then it remains a move which can earn 8 points for who is playing (black or white).

All of these values are averages, useful for heuristic purposes. Fortunately, they work the vast majority of the time in go.

Gérard TAILLE wrote:I would see great difficulties with the "largest play" because it could be uneasy to eliminate ko threats as in the following example:

$$B
$$ -----------------------
$$ | . . a . O X . . . . .
$$ | O O O O O X . . . . .
$$ | O O O O O X Q Q . . .
$$ | O O O O O X Q . . . .
$$ | X X X X X X Q . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .

Click Here To Show Diagram Code

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

The evaluation of this position (for black point of view) is -4 points.
Let's now consider a black move at "a". The evaluation of the resulting position is now the average value between -4 (a white reply) and +34 (if black plays and capture the 16 white stones). After a black move at "a" the evaluation of the position is (-4 + 34)/2 = 15. And you can see that the value of the black "a" move is 15 - (-4) = 19 points.
Here you see the drawback of considering the "largest move" because all ko threats look like a quite large move and, as a consequence all ko threat areas, in terme=s of temperature, will hide all really interesting areas where you want really move.

Right. Since this ko threat costs one point to remove, it is a "-1 pt. sente". By itself it is worth nothing. However, in some ko fights it is possible that Black will play this threat and White will ignore it. That's unlikely, since the threat is so huge, but there are ko fights where it will still affect the analysis because it will effectively cancel one of the opponent's threats. There are also ko situations where it is right for White to eliminate the threat before the potential fight. We can now analyze all of these situations with thermography.

Gérard TAILLE wrote:For that reason, because playing a ko threat does not gain anything, I prefer your wording "the gain from making a different play" even if this definition is still not quite clear (but is an informal term!).

Formally we consider the temperature of an ideal environment. Reality may be different. When we combine games, such as kos and ko threats, we still make use of an ideal environment unless we can read things out, but the analysis can be rather complex.

Let me give an example of thermography for a simple ko with your ko threat. It's a large threat, so let's look at a ko with large moves where that is the only threat. Suppose that White to play takes the ko. If Black does not play the threat and White wins the ko the local score in the ko is -41 pts. (for Black). Subtracting the 4 pts. in the unplayed threat the net score is -45 pts. Or if Black to play wins the ko for 4 pts., the net score is 0. Each play in the ko gains 15 pts. on average. That makes the ko a little hotter than the empty go board. Such kos do arise in actual games.

When I was learning go the textbooks said that the size of the ko threat was bigger than the ko, i.e., that 19 > 15 (although they doubled those numbers), so White should answer the ko threat. Thermography tells a somewhat different story. Suppose that White takes the ko, Black plays the threat, White wins the ko, and then Black kills the corner. The net score will be -7, which indeed is worse for White than the average value of the ko plus the White corner, which is -15. White loses 8 pts, on average; how can that be right?

Well, suppose that White does answer the ko threat and Black takes the ko back. At this point White plays in the environment and gains t, and then Black wins the ko. The net result at temperature t will be 0 - t = -t.

White's strategy should be to get min(-7,-t), for t up to 15 (above which the ko is not fought). Solving for t we get

-t = -7
t = 7

So when the temperature of the environment is less than 7 White should ignore the ko threat and win the ko.

OC, that is a heuristic. But if we know more about alternative plays, we can add them to the game and analyze that. Besides, the textbook advice to simply compare the size of the ko play with the size of the threat is a heuristic, as well. Just not as good a heuristic.

Bill Spight wrote: When I was learning go the textbooks said that the size of the ko threat was bigger than the ko, i.e., that 19 > 15 (although they doubled those numbers), so White should answer the ko threat.

Ok for me. Seeing 19 > 15 it appears to me that white must answer to the ko threat.

Bill Spight wrote: Thermography tells a somewhat different story. Suppose that White takes the ko, Black plays the threat, White wins the ko, and then Black kills the corner. The net score will be -7, which indeed is worse for White than the average value of the ko plus the White corner, which is -15. White loses 8 pts, on average; how can that be right?

Nothing wrong with that. White gains 2 * 15 but black gains 2 * 19. As a consequence, White loses 8 points in this sequence.

Bill Spight wrote: Well, suppose that White does answer the ko threat and Black takes the ko back. At this point White plays in the environment and gains t, and then Black wins the ko. The net result at temperature t will be 0 - t = -t.

White's strategy should be to get min(-7,-t), for t up to 15 (above which the ko is not fought). Solving for t we get

-t = -7
t = 7

So when the temperature of the environment is less than 7 White should ignore the ko threat and win the ko.

Here I do not understand your calculation.
Notation :
x = value of move in the ko (here x = 15)
y = value of a move in the ko threat (here y = 19)
t = temperature of the environment (t =14 on an empty board)

To simplying the reasonning let's take two gobans:
a first goban G1 on which it remains only two areas : the ko area itself (white can take the ko or black can connect the ko) and the ko threat area
a second goban G2 which is empty.

First of all, in an ideal environment, a player will play in the ko only if x >= t, and ideally when x = t.
If white takes the ko and ignores the ko threat then, on G1, black score is -2x + 2y and then white plays first on the G2 and black loses the game by t/2 points (7 points which is the supposed ideal komi). Finally black score in this first scenario is b1 = -2x + 2y - t/2
Suppose now white answers to the ko threat and black retakes the ko and then connects. On G1 black score is x points and on G2 white plays two moves before black plays its first move. Because with a pass move black loses t points, that means that, on G2, black loses the game by 3t/2 points. Finally black score in this second scenario is b2 = x - 3t/2.

Comparing b1 and b2, white has to ignore the black ko threat if
b1 < b2 => -2x + 2y - t/2 < x - 3t/2 => 2y < 3x - t
and taking into account x = t :
b1 < b2 => 2y < 2x => y < x
and I confirm the common sense : you do not answer the ko threat if the value of the ko threat (y = 19) is less if then the value of the ko (x = 15).

In an ideal environment (obviously we would avoid an evironment with only one big gote point or an environment with complete miai points) I do not see why you have to take into account the temperature in order to know if you have to answer a ko threat. Comparing x and y seems enough isn't it?

Where is the difference between our two approches : you expect to win t points when playing in the environment where I consider you will win only t/2 points. The point is to be well aware of the difference between the value of a move (temperature) and the score expected for the game (komi = t/2).

Maybe I am wrong but, for the time beeing, I like the common sense result of my calculation.

Gérard TAILLE wrote:

Bill Spight wrote: When I was learning go the textbooks said that the size of the ko threat was bigger than the ko, i.e., that 19 > 15 (although they doubled those numbers), so White should answer the ko threat.

Ok for me. Seeing 19 > 15 it appears to me that white must answer to the ko threat.

Bill Spight wrote: Thermography tells a somewhat different story. Suppose that White takes the ko, Black plays the threat, White wins the ko, and then Black kills the corner. The net score will be -7, which indeed is worse for White than the average value of the ko plus the White corner, which is -15. White loses 8 pts, on average; how can that be right?

Nothing wrong with that. White gains 2 * 15 but black gains 2 * 19. As a consequence, White loses 8 points in this sequence.

Bill Spight wrote: Well, suppose that White does answer the ko threat and Black takes the ko back. At this point White plays in the environment and gains t, and then Black wins the ko. The net result at temperature t will be 0 - t = -t.

White's strategy should be to get min(-7,-t), for t up to 15 (above which the ko is not fought). Solving for t we get

-t = -7
t = 7

So when the temperature of the environment is less than 7 White should ignore the ko threat and win the ko.

Here I do not understand your calculation.

IIUC, you are estimating the final minimax result at the end of the game. That works, too. It's how I started out years ago.

Notation :
x = value of move in the ko (here x = 15)
y = value of a move in the ko threat (here y = 19)
t = temperature of the environment (t =14 on an empty board)

To simplying the reasonning let's take two gobans:
a first goban G1 on which it remains only two areas : the ko area itself (white can take the ko or black can connect the ko) and the ko threat area
a second goban G2 which is empty.

First of all, in an ideal environment, a player will play in the ko only if x >= t, and ideally when x = t.

OK. First suppose that White makes a mistake and plays in the environment, and then Black makes a mistake and wins the ko, and then play continues in the environment. Both plays in the environment gain approximately 14 pts., so we estimate the final score as

1) -15 - 14 + 15 - 14/2 = -21

Now suppose that White takes the ko and ignores Black's threat. We estimate the final score this way:

2) -15 - 15 + 19 - 15 + 19 - 14/2 = -14

Now suppose that White answers Black's threat and Black takes and wins the ko. We estimate the final score this way:

3) -15 - 15 + 19 - 19 + 15 - 14 + 15 - 14/2 = -21

Plainly White should answer the ko threat, given that we know nothing else about the environment.

Now let play continue in the environment until play ends as expected with a score there of -7, and there are no additional ko threats. t = 0. At this point let White take the ko and Black play the threat.

Suppose that White answers the threat and Black takes and wins the ko. Then the final score will be this.

4) (-7 - 15) - 15 + 19 - 19 + 15 - 0 + 15 = -7

That's the best result for Black so far. Suppose instead that White does not answer the threat but wins the ko. The final score will be this.

5) (-7 - 15) -15 + 19 - 15 + 19 = -14

This is better for White by 7 pts. So when t = 0 White's best play is different from best play when t = 14, as a rule. White should as a rule answer the threat at a high ambient temperature but ignore it at a low temperature. Where is the crossover point?

Suppose now that play continues in the environment until t = 7, with no additional ko threats. We estimate that White has gained (14 - 7)/2 = 3½ pts.

First, suppose that White takes the ko and answers Black's ko threat and Black takes and wins the ko. This is our estimate of the final score.

6) (-3½ - 15) - 15 + 19 - 19 + 15 - 7 + 15 - 7/2 = -14

Next, suppose that White takes the ko and ignores Black's ko threat. This is our estimate of the final score.

7) (-3½ - 15) - 15 + 19 - 15 + 19 - 7/2 = -14

The result is the same, so we have found our crossover point. Below an ambient temperature of 7 White should ignore Black's ko threat, as a rule. QED.

Note also that Black gains when White does not play the ko fight, so White should play the ko fight before the ambient temperature drops. As a rule, OC.

Good news Bill, I think we are completly in line!

With your last post I now understand where is our previous misunderstanding

When introducing the ko threat problem you wrote:

Bill Spight wrote: Let me give an example of thermography for a simple ko with your ko threat. It's a large threat, so let's look at a ko with large moves where that is the only threat. Suppose that White to play takes the ko. If Black does not play the threat and White wins the ko the local score in the ko is -41 pts. (for Black). Subtracting the 4 pts. in the unplayed threat the net score is -45 pts. Or if Black to play wins the ko for 4 pts., the net score is 0. Each play in the ko gains 15 pts. on average. That makes the ko a little hotter than the empty go board. Such kos do arise in actual games.

When reading the last sentence I understood you started the ko as soon as the value of a move in the ko (here 15) was just a little hotter than the temperature of the environmment (here 14).In this hypothesis I explained (remember I supposed x = t) you always has to answer a ko threat if this ko threat is greater than the ko (19 > 15).
I understand now that your hypothesis was different. You do not analyse a ko with the value of the ko just a little hooter than the temperature but also in the case where the value of the ko move (15) is far highter than the temperature (7 in your analysis).

Taking my calculation with your hypothesis it remains
b1 < b2 => -2x + 2y - t/2 < x - 3t/2 => 2y < 3x - t

but now I cannot put x = t.
I that case:
b1 < b2 => t < 3x - 2y
if x = 15 and y = 19 then b1 < b2 => t < 7 which is exactly your result.

When you look at the formula
t < 3x - 2y
you see that you have to compare the temperature to the global value of the ko (3x) minus the global value of the ko threat (2y) and that does not hurt my common sense though it seems not a known rule.

Do you agree with that?