There are two aspects to the answer:
1. defining "the theoretical value of the first move of the game"
2. computing or estimating it

1. there's a lot between the quotation marks; possible rewordings
a) "a fair compensation for moving second"
b) "the difference between black moving first and white moving first"
c) "the contribution of the first move to the final score"

2. we know a few things:
a) this is the definition of komi and it has historically shifted from 5,5 to 7,5 while also maybe being dependent on the rule set used and with (or without) the goal to avoid a draw. It's the expected score of a game without such compensation, or rather the mean of the normal distribution of the results of a large number of games played. Let's agree there's a 90% probability that a fair compensation is between 6 and 8
b) this difference is twice the number of points under a)
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.

Thanks for reminding me. I had evaluation in mind. In that context he doesn't even mention sente, much less double sente.

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

Visualize whirled peas.

Everything with love. Stay safe.

For evaluation purposes the rest of the board is represented by a single parameter, t, called the temperature of the environment, or the ambient temperature. If you have more information you can incorporate that as part of the game. (Games can be added together.) At sufficiently high temperature neither player will play in this game, with correct play. As you reduce the temperature, at some point both players will be indifferent between playing locally or playing elsewhere. That temperature is the average gain of this game for a gote play or reverse sente play.

At lower temperatures the payoff for each player moving first is determined by minimax at that temperature, with the proviso that the second player moves last. Let me illustrate with a simple gote and a simple sente.

Suppose that there is a simple gote such that whoever plays first gets a local score of 10 pts. Let's say that the ambient temperature is 14. Now if Black takes the gote White replies on the rest of the board (in the environment). Black has a score of 10, from which we subtract 14 for a net score of -4. Whether that is good or bad we don't know yet. If Black plays in the environment and White takes the gote the result is a net score of 14 - 10 = 4, which is plainly better for Black, so Black plays elsewhere. The same goes for White. Now suppose that the ambient temperature is 10. If either player takes the gote and the other player replies in the environment, the net score is 0, and the same is true if the first player plays in the environment first. Each player is indifferent between playing in the game and playing in the environment. So a play in the gote gains 10 pts. We can find t by solving the equation, 10 - t = t - 10. We have also determined the average value of the gote position, which is 0.

Now suppose that the ambient temperature is 5. An unusual situation, but there we are. If Black plays first the result at temperature 5 is 10 - 5 = 5, as Black will prefer to take the gote. Similarly, the result if White plays first will be -5 from Black's point of view. In a more complicated position Black might have a choice of plays, one of which might have a better result for Black at temperature 5, in which case she will play it. (OC, we have simplified the situation by using a single parameter for the rest of the board. This theory is for heuristics.)

Now let's look at a Black sente. Suppose that Black to play can move to the simple gote above, and that White to play can move to a position with a local score of -15, from Black's point of view. Obviously, above a temperature of 10 neither play will play. What if the temperature is 8? If Black plays first to the gote, we already know that White will reply in the gote instead of the environment, so the result will be -10 for Black. If White plays first locally, the result will be -15 + 8 = -7, which is better for Black than -10, and worse for White. White will play in the environment. At temperatures below 10 Black can guarantee a result of -10 by playing locally. She can play locally, or not, as she pleases, since White will not play the reverse sente. Now let the temperature be 5. Black to play still gets -10, while if White plays locally the result will be -15 + 5 = -10. At this temperature both players are indifferent between playing locally or in the environment. The value of the position is -10 and the gain from the reverse sente is 5. Edit: We could have found this by solving the equation, -10 = -15 + t.

This is not the traditional way of finding these values. The method is mine, but it is based upon thermography, which is presented in Conway's On Numbers and Games (ONAG), as well as Winning Ways by Berlekamp, Conway, and Guy. Berlekamp extended thermography to cover kos, and I redefined it in 1998 to cover multiple kos. This method is a simplification of my method back then. The ONAG version uses the idea of tax instead of the value of plays elsewhere.

On the question of the choice of plays, Robert Jasiek and I have discussed that often here on L19. Those discussions will probably show up if you search for thermography.



Plainly there is a value to breaking down the problem of play into smaller problems. It is not so easy to do rigorously, but Martin Mueller has tackled the problem, although I don't know of anything recent.

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

It is not easy for to understand how the environment can me summarized by a sigle number t (for temperture). Can we take an example in order for me to understand your point.

Let's take as the local region the upper left corner in the diagram above.
First of all I try to count the position:
If it is white to play, white plays "a" and the score (for black point of view) is -23.
If is is black to play, black plays "a" and we reach a position where white or black can then play "b".
if now black plays "b" the score is +0
and if now white plays "b" the score is -6
As a consequence after black "a" the position is estimated -3 (the average value between +0 and -6)
Finally the initial position is evaluate -13 (the average value between -23 and -3).
On average the "a" move earns 10 points and then the "b" move earns 3 points.

Now let me take two scenarios for the environment:
S1 : four gote areas which earn : 11, 8, 5 and 2 points
S2 : five gote area which earns : 10, 8, 6, 4, 2 points

In the S1 scenario black has to play in the environment but in the S2 scenario black has to play locally.
I do not know what is the temperature of S1 or S2 but my feeling is that, depending of the temperature definition, I will probably be able to build two different environments with the same temperature and different play for black.

May be I could agree temperature may be a great "help" to guess the correct move but in any case you have to read a great part of the yose, haven't you?
In my example the "standard" method to guess the correct move is "simply" to estimate the local situation (here 10 points) and to verify this guess by reading the yose. For the time beeing I do nont really see the advantage of the temperature notion. I suspect my example is not that significant for the temperature notion.

Endgame positions can be studied during the early endgame with temperature of the environment or during the late endgame when the correct solution can be found.

During the early endgame and at a moment of quiescence, one can make the assumption of linearly decreasing values of simple local gote endgames in the environment. The largest such gote is said to have the temperature T as its value. We have proven that T/2 is a good approximation for the value of starting play in the global environment. Therefore, the temperature is useful.

Robert, I understand that T/2 is quite good approximation.
In my scenarios S1 and S2 I imagine you can calculate the exact value of this temperature. If my understanding is correct this temperature is here equal to 6 but I am not completly sure.

The main point is that the theory is heuristic. As I am sure you know, the best play is not always the biggest play.

The other point is that games add and subtract. This is the basis for combinatorial game theory (CGT). Go players did not come up with that idea, but they do realize that the interaction between independent positions matters, even without ko. The thing is, then, that if you have information about other games that is relevant, it is not part of the environment. It is foreground knowledge, not background. (BTW, I came up with the idea of the environment independently, long before I heard about CGT, just from studying go endgames. )



Right.

Furthermore, if the ambient temperature is less than 3 and Black plays first, White replies locally. Even though this is a gote position, Black can play with sente under those conditions.



The term, environment, has more than one sense. For evaluation purposes we only care about its temperature. But if we want to include more information, then we add together all the other areas of interest to the original game to get a new game.


The temperature of S1 is 11, the temperature of S2 is 10. Your second statement is true, because the theory is heuristic. It's not always right to play the averages.

For instance, suppose that your original position plus a simple gote with an average value of 0 in which a play gains 11 pts. are the only unplayed positions left on the board with Black to play. Even though Black's play gains only 10 pts. in your original position Black should play there. Then after White replies in the other position, Black can take the last play for an additional 3 pts. Black's total gain is then 10 - 11 + 3 = 2 pts. instead of 11 - 10 = 1 pt.

Note that this calculation is at temperature 0. If we include an environment for the combined game, Black's total gain at temperature t when she takes the 10 pt. play first is 2 - t pts. That is better, on average, than taking the 11 pt. play first when 2 - t > 1, i.e., when 1 > t. So above temperature 1 Black's average gain is better by taking the 11 pt. gote.



The theory gives you a good guess to the best play in most circumstances. That's why go players came up with it in the first place. Tristan Cazenave has even done some research on the effectiveness of always playing the biggest play in actual games. It works well. Big duh.

But also, when you are reading extensively, the theory gives you a good first sequence of plays to read.



In a real game the ambient temperature is usually close to that of the gain from taking the biggest play. However, there are occasions when it is rather less. In such cases, as my example of the combined game illustrates, you may be able to figure out when the biggest play is probably not best.

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

Visualize whirled peas.

Everything with love. Stay safe.

Trying to understand what you said I have now a difficulty:

In your previous post you wrote :

Suppose that there is a simple gote such that whoever plays first gets a local score of 10 pts. Let's say that the ambient temperature is 14. Now if Black takes the gote White replies on the rest of the board (in the environment). Black has a score of 10, from which we subtract 14 for a net score of -4. Whether that is good or bad we don't know yet. If Black plays in the environment and White takes the gote the result is a net score of 14 - 10 = 4, which is plainly better for Black, so Black plays elsewhere. The same goes for White.

Let's take this local score of 10 points and the environment made of 14 gote areas with values 14, 13, 12 .., 3, 2, 1. If I understand your last post the ambiant temperature for this environment is equal to the biggest gote area t = 14.
If now black plays first the local 10 points it appears to me that black will get a net score of 10 - 14/2 = +3 and not 10 - 14 = -4 ?

Second example. We saw that, by playing the first move of a game, black will earn 14 points. Does that mean the temperature is 14 ? Finally we know that the result of the game is expected to be 7 points for black. My conclusion : starting from a (quiet?) position in which the evaluation is 0 (an empty board is only a trivial example) if it is black to play and if the temperature is equal to t then the expected result is about equal to t/2.
Is it the idea?

What is missing from the quote is something I said earlier, which is that we find the minimax result at the ambient temperature, with the second player playing last, not the result at the end of play. In my 1998 paper I did use the result at the end of play, but doing so required a bit of cleverness. The current method is simpler.

If we start from a position that has a certain value and one player makes a play that gains x and then the second player makes a play that gains x, the new position has the same value as before. So if Black plays locally and gains 10 pts. and then White plays in the environment and gains 14 pts., the result is a net loss of 4 pts. for Black. True, if we played everything out to the end, Black might gain 3 pts. instead of losing 4 pts., but Black would have done better to play in the environment and get a final gain of 7 pts. Anyway, sticking to the current temperature simplifies things.



I suppose that you are talking about the first play on an empty 19x19 board. Well, we guess that the first move gains around 14 pts., based on komi. Anyway, whatever it gains on average is the temperature of the empty board. Now, if the drops in temperature of the whole board, except for sente sequences, are approximately uniformly small, then the expected gain from playing first at temperature t is t/2. That seems to usually be the case.

An exception is temperature 1 at territory scoring. Fairly often the effective temperature drop is from 1 to 0, which is a large drop by comparison with the typical drop at other temperatures. Sometimes you can take advantage of that drop by getting the last play at temperature 1. Also, ko fights can keep the temperature elevated, so that winning the ko happens before a significant drop in temperature.

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

Visualize whirled peas.

Everything with love. Stay safe.

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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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!

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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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:



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?


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



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.



A matter of judgement.



We don't have enough information.



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.



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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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!).