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How evaluate double sente moves ? http://lifein19x19.com/viewtopic.php?f=12&t=17810 
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Author:  Gérard TAILLE [ Mon Oct 19, 2020 6:26 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: Gérard TAILLE wrote: I see Bill. I drawed the mast with black colour but, due the ambiguity, you were allowed to use the blue one and my point appears hidden. Well I have to find a way to avoid ambiguity. Here it is Here again you can see that a white move at "b" here dominates a move in a simple 2 points gote area. But now you cannot paint the mast in blue because a black sente move at "b" looks a loss in an ideal environment. If I am not wrong the thermograph is thus now identical to a simple 2 points gote though we would prefer to see the domination of this area against a simple 2 points gote area. I do not know if this way I manage to clarify my point. Let's see. Another beautiful example, Gérard. Black a and Black b are incomparable, but White b dominates, and Black b loses points on average. So the thermograph is not affected by Black b. Let me see if I understand your point. Here is a possible example based upon my early study of the yose. In his book on yose aimed at a kyu level audience, the great Takagawa showed a similar position. He made the point that, although the hanetsugi at a was usual for White, the sagari at b had the same value and was thus playable, and also sometimes preferable. The same holds true for the Black hanetsugi at b and sagari at a. Three points Bill: First of all this example is quite different because it shows an ambiguity I tried to avoid in my example. Secondly black "a" is not sente. The point in my example is to have a sente move appearing a real loss in an ideal environment. Thirdy it appears in this example that black "b" dominates black "a" and that means that black "a" can only be preferable to black "b" in case of ko fight. In a nonko environment I do not see how black "a" can be preferable, do you? 
Author:  Bill Spight [ Mon Oct 19, 2020 10:05 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: Three points Bill: First of all this example is quite different because it shows an ambiguity I tried to avoid in my example. Secondly black "a" is not sente. The point in my example is to have a sente move appearing a real loss in an ideal environment. Thirdly it appears in this example that black "b" dominates black "a" and that means that black "a" can only be preferable to black "b" in case of ko fight. In a nonko environment I do not see how black "a" can be preferable, do you? Thanks for the clarification, Gérard. BTW, I started replying to your note, but somehow I clobbered the browser screen and lost my draft. {shrug} But that's OK. You have since posted another interesting position that is relevant. More below. I went back and took a look at your original post on this subject. Gérard TAILLE wrote: Let's take a very simple example: if you claim a go player that the position above has a miai value = 1 she will have some difficulty with the credibility of the theory. Why? Simply because instead of "a" she will clearly see the possibility black "b" which could very interesting if the environment looks like a tedomari situation with only one remaining 2 gote points (all 1 gote points being miai). How can you ignore the adding value of such possibility when, for other situations, you estimate a value with a precision of 1/16 if not still better? She is not wrong is she? OC, on such questions there is some degree of eyeofthebeholder, isn't there? Let me talk a little more about the purpose of thermography. Thermography arose out of the desire to find the mean value of a combinatorial game, e.g., a nonko go position. Methods of finding the mean value existed, but they were cumbersome. Conway proved that you can find it using thermography. The idea of temperature was not about an ideal environment. The idea of a rich environment came later, from Berlekamp. Still later I adopted that idea to redefine thermography. For over 30 years the temperature was conceived of as a tax upon plays. Here is a simple example. Take the game, {4  0}. Now tax each move by one point. That gives us the related game, {3  1}. Now let's tax each move by 2 points. That gives us the related game, {2  2}. Obviously, if we tax moves any more, neither player will wish to play. That means, as Conway showed, that we have found the mean value of the original game, {4  0}. Now, go players had figured out a couple of centuries earlier that 2 points was a good estimate of the territory of a corresponding go position, but Conway didn't know about that. Thermography confirmed the intuition of go players. Let's take a look at a simple sente. {9  1  2} Let's apply a tax of 1 point per move. {7  1  1} The right side is as expected, but what happened to the left side? The 9 is taxed by 2 points because it took 2 Black plays to get there. The 1 remains the same because Black made a play and then White made a play, so the taxes cancel out. Let's apply a tax of 3 points per move. {3  1  1} If we apply a higher tax White will not make a play. Conway recognized this as the number 1 plus the following game, which is an infinitesimal. {2  0  0} Further taxation will not get us any closer to the mean value, so the mean value of the original sente is 1. Again, this verified the traditional practice of go players, who estimated the territorial value of the corresponding go position as 1 point. Now let's take a look at a double sente. {Big  5  3  Big} Big refers to some positive value that's too big to count. Even if the Big on the left is not the same as Big on the right, that does not matter. They are just Big. Now let's apply a tax of 1 point. {Big  2  5  3  2  Big} OC, since Big is too big to count, that reduces to {Big  5  3  Big} Without the ability to assign values instead of Big, we cannot find the mean value of a double sente. But when we do assign values, the double sente is no longer a double sente. Time for a break. More later. 
Author:  Bill Spight [ Mon Oct 19, 2020 5:14 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Well, I got into explaining thermography and ended up showing how you cannot evaluate a double sente position when the difference between the two thermographic walls is positive. What makes it easy to see is that those walls, after sente, must be vertical. So they never meet and therefore the double sente has no average territorial value. Of course, on a real, finite go board every nonko position has an average territorial value. That means that sooner or later the thermographs come together, and that means that at least one of the walls must turn towards the other one. If they meet where one of the walls is vertical, the position is like sente; if they meet where both are inclined, the position is like gote. The vertical line extending upwards from where they meet is the mast. It is possible for one or both of the walls to extend upwards from where the walls meet. If so the mast is colored, otherwise it is black. A colored mast is sentelike, and black mast is gotelike. If how the lines meet and the color of the mast are both like sente, the position is sente, if both are like gote, the position is gote, and if they are different, the position is ambiguous. Of course, during a go game a position may arise which, because of its relation with the rest of the board, may be played with sente by either player. Informally we may call that a double sente, but it is still a sente, gote, or ambiguous position. 
Author:  Bill Spight [ Mon Oct 19, 2020 7:19 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
OK. Well, finally, back to the question of confidence in thermography. Thermography is a proven method for finding the average value of any finite combinatorial game. In go terms, that means any nonko position. Berlekamp extended thermography to most ko and superko positions, and I extended it to multiple ko and superko positions. However, the values of ko and superko thermography are not guaranteed to be average values. Berlekamp called them mast values, not average values. But that's not the question before us, which is about the nonko theory. The theory concerning the average values is proven. So any real doubts must be about the walls of the thermographs. The walls of the thermograph are defined for each temperature as the result of minimax play at that temperature, depending on who plays first. Here is the game which first raised doubts. Gérard TAILLE wrote: Let's take a very simple example: if you claim a go player that the position above has a miai value = 1 she will have some difficulty with the credibility of the theory. We'll get to the difficulty below. First, let's write the game using slash notation. White to play plays the hanetsugi, for a net score on the top of 4; i.e., 4 points for White from Black's point of view. The right side of the game looks like this: 4} Black to play can play this hanetsugi, for a net score of 2. So far the game looks like this. {24} The vertical slash separates the Black follower or followers from the White follower or followers. Black can also play the sagari with sente, for a net score of 3. Edit: It turns out that the sagari is dominated by the oki at a. When I wrote this, I was under the misapprehension that they were equivalent. kills the White group, for a net score of +21. Edit: My mistake, which Schachus caught. The oki threatens to kill, but not the sagari. Edit: The correct analysis of the game follows. threatens to kill. captures the stone for a net local score of 3. White can also live with this . If Black saves the stone the net local score is 0, but Black takes gote. If Black omits , White can play the hanetsugi for a net local score of 4. Finally, if Black kills the net local score is 21. Here is how we write the game. {{213,{04}},24} This looks a bit complicated. I have added some slashes to help me keep track of the number of moves. Let's construct the thermograph. The easy way is to do that graphically, but for the purpose of illustration, let's figure out minimax play at different temperatures. We start at temperature 0, i.e., no tax. OC, White to play moves to a net score of 4. Black to play does better with the gote option instead of the sente option and moves to a net score of 2. Now let's apply a tax of ½ point per move. The resultant game is {{203,{½3½}},2½3½} White to play gets a net score of 3½, Black to play gets a net score of 2½. We can see where this is going. Let's apply a tax of 1 point per move. The resultant game is {{193,{13}},33} Bingo. We have found the average value of the game, 3, at temperature 1. Below temperature 1 Black prefers the gote option, but above temperature 1, rising 11 points more, Black can play the sente. We indicate that fact by coloring the mast blue up to temperature 12. This game is ambiguous. You can see the thermograph at #49. https://www.lifein19x19.com/viewtopic.p ... 84#p260684  Now let's look at the doubts. Gérard TAILLE wrote: Why? Simply because instead of "a" {the Black hanetsugi} she will clearly see the possibility black "b" {the Black sente} which could very interesting if the environment looks like a tedomari situation with only one remaining 2 gote points (all 1 gote points being miai). Edit: Now we know that the sente is the oki instead of the sagari, but that does not alter the question. IOW, if there is another gote on the board that gains 1 point for either player, such as another first line hanetsugi, then Black will prefer to play the sente and then take that gote at temperature 0. The problem being that the thermograph does not reflect that possibility. Quote: How can you ignore the adding value of such possibility when, for other situations, you estimate a value with a precision of 1/16 if not still better? She is not wrong is she? I replied that the thermograph does indicate the possibility of playing the sente by the fact that the mast is colored blue up to temperature 12. I think I should have done more, however. I should have simply added the hanetsugi to the current game and derived the thermograph for it. Maybe this thermograph by itself does not reflect the possibility that the skeptic brought up, but thermography is part of combinatorial game theory, and combinatorial game theory sure does. Combinatorial games add and subtract, as the name indicates. Let's add the two games together. Edit: Derivation corrected for the oki. {{213,{04}},24} + {13} I have chosen to make the hanetsugi {13}, because Gérard added that position himself in #51. https://www.lifein19x19.com/viewtopic.p ... 89#p260689 Now let's find the sum at temperature 1, by applying the tax. That give us {{193,{13}},33} + {22} The average value of this game is 3  2 = 5, and the mast is blue up to temperature 12. To draw the thermograph all we have to do now is to find the minimax values at temperature 0. Let's do that on the go board. Gérard pointed out that dominates , so this is the only result: a net score of 5. He has also pointed out that the sente is best at temperature 0, so this is the result: a net score of 4. The thermograph is in #53. https://www.lifein19x19.com/viewtopic.p ... 92#p260692 Note that the sente is reflected in the left wall of the thermograph from temperature 0 to temperature 12, not just in the mast. This is a better answer to the skeptic, because, while the sente was not reflected below temperature 1 in that thermograph, it was in this one. The point is that each thermograph is for only one game. Every nondominated option in a game will be represented in at least one thermograph, if not in the thermograph of that game, then in the thermograph of that game plus or minus some other game. 
Author:  Schachus [ Mon Oct 19, 2020 11:55 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Does 3 really kill? How about 4@e6? 
Author:  Bill Spight [ Tue Oct 20, 2020 2:00 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Schachus wrote: Does 3 really kill? How about 4@e6? Thanks. How silly of me! Of course. Back to writing the game in slash notation. If White lives, the net score is 0. To recap, if White replies to the net score is 3. If not, then is the sente here. And if White passes again, kills. for a score of 20. The game is thus {{2003},24} OK. Let's tax it by 1 point per move. At temperature 1 we get {{1713},33} The average territorial value is still 3, OC. But now the mast is blue only up to temperature 3. At that point the taxed sente game will look like this. {1133} Here is the corrected thermograph. Edit: This analysis is incomplete. It is corrected in the next few notes. The correct thermography for this position is in #49: https://www.lifein19x19.com/viewtopic.p ... 84#p260684 Then combination of this game and the hanetsugi looks like this. {{2003},24} + {13} The minimax result at temperature 0 is the same. When we tax each move 1 point we get this. {{1713},33} + {22} The average territorial value remains the same, 5. The correct thermograph is in #53: https://www.lifein19x19.com/viewtopic.p ... 92#p260692 Many thanks, Schachus. 
Author:  Bill Spight [ Tue Oct 20, 2020 2:48 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Schachus wrote: Does 3 really kill? How about 4@e6? Oh! Now I remember why I thought it killed. Here is what I saw before looking at Gérard's sagari. This sequence does threaten to kill. So I haven't completely lost my marbles. Even if I have to make sure I don't step on one. It is possible that White might answer at in order to take sente? I'll have to check that out. If so, this position has become quite interesting, hasn't it? Thanks again, Schachus. 
Author:  Bill Spight [ Tue Oct 20, 2020 3:17 am ] 
Post subject:  Re: How evaluate double sente moves ? 
I have to say I am having fun. Thanks again to everyone. You have brightened my year. > The Kingston Trio wrote: If the skeeters don't get you then the gators will.

Author:  Bill Spight [ Tue Oct 20, 2020 6:03 am ] 
Post subject:  Re: How evaluate double sente moves ? 
OK. Well, I don't really have to check anything out. As Berlekamp said, the thermograph will help to find best play. So let's recap. Nice little corner you got there. White plays the hanetsugi for a net local score of 4. Black has three options. At least. Black plays the hanetsugi for a net local score of 2. plays the sagari. If replies, the net local score is 3. If does not reply, threatens to kill. (Thanks, Schachus. ) White saves his group for a net local score of 0. If Black kills, the local score is 20. plays the oki and then captures the Black stone for a net local score of 3. This is sente to kill. kills for a local score of 21. White also has the option of living in sente, for a net local score of 0. Is a mistake? It looks like it, doesn't it? It certainly loses points, on average. But you don't always play the averages in go. But we don't have to answer that question. It will all come out in the wash. You don't have to worry about including mistakes in thermography, as long as you include correct play. Let's write the game, adding the oki option and White's two replies. {{213,{0 Uhoh. What do we write here? White threatens the hanetsugi for a net local score of 4. That gives us this. {{213,{04}},{2003},24} Not too fearsome. I have added extra slashes to help me keep track of the number of moves played. What about the other Black options? The 21 threatens to kill. Nothing new here, in terms of net scores. But White has this atari for , which shows that is a mistake. White still has the original hanetsugi, but Black has lost her hanetsugi, and the sagari is no longer sente. We can ignore . Thank goodness! I don't want to try your patience. Or mine! We can ignore Black a, as well, since White can reply to it, too. Here is the game again. {{213,{04}},{2003},24} Adding the new option for Black hasn't changed the minimax results at temperature 0. Now let's apply a tax of 1 point per move. {{193,{13}},{1713},33} The minimax result is 3, no matter who plays first. If Black the oki, White will capture it for a score of 3, instead of playing to the game {13}, which Black will play to 1. So the average value of 3 remains the same. The next question is how to color the mast. Let's apply a tax of 2 points per move. {{173,{22}},{1423},42} Since 2 > 3, White will not play at temperature 2, and since 4 < 3, Black will not play the hanetsugi. Looking at the Black oki, since 3 < 2, White will capture it. That reduces the game at temperature 2 to this. {{173},{1423} } We can see where this is going. Let's apply a tax of 3 points per move. {{153},{1133} } Above temperature 3 the sagari will drop out, leaving the oki. So the blue mast goes up to temperature 12, after all. The thermograph is in #49: https://www.lifein19x19.com/viewtopic.p ... 84#p260684 BTW, all this would be easier graphically, wouldn't it? Edit: I have tried to simplify this game, and unfortunately, the sagari is dominated by the oki. So we can write the game this way. {{213,{04}},24} 
Author:  Gérard TAILLE [ Tue Oct 20, 2020 8:51 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: First let me say what they got right. From time to time, and quite often, positions arise on the go board such that, given the rest of the board, either player can play locally with sente. Everybody calls these positions double sente. In addition, we are told to hasten to play in those positions. (Some translations say to play them "early", which is problematic. It implies leaving them on the board for some time, albeit short.) If we leave them on the board, we could lose points with zero compensation, offering a free lunch to our opponent. What they get wrong — and let me say again that the NogamiShimamura book is a shining example of not doing so — is identifying certain positions as double sente without regard for the rest of the board. Two common examples are the double hanetsugi on the first line, each with a large followup, and the double kosumi on the second line, said to be worth 2 points and 6 points respectively. Each of these can arise fairly early in the game and remain on the board for some time. As you know, the beginning of wisdom for me about double sente came when I was 4 or 5 kyu and observed the formation of a double kosumi position in the game record of a pro game. After a while one player played the kosumi and his opponent did not answer it. Tilt! How can that be double sente when it is not even sente! I thought. The theory you defend (I mean CGT and a lot of concepts like thermograph, difference game, ideal environment, etc) is really a great progress I am very fond of it but we have not to forget that a theory is almost never a perfect approimation of the real life. One a of the main mathematical tool used in a lot of theories is the "average" tool. As you will see through my following example the average tool is one the most strange object build by the mathematician. First example: Take a dice and throw it a number of times. You get each time a number in the set {1, 2, 3, 4, 5, 6} right. What about the average value? What is this number we call 3.5 ? I konw only six numbers {1, 2, 3, 4, 5, 6} and I never saw a dice giving me such strange 3.5 value as result. Is it a average dice? Is it a monster? Second example: Let's take a great number of squares with a side length between 1 and 2. The average side length is 1.5 OC. What about the area of the squares? They are between 1 and 4 for an average of 2.5 Well the average object looks like having a side length equal to 1.5 for an area equal to 2.5 Surely it is not a square isn't it. Maybe it is a monster. Third example: Let's study the behaviour of the ragdolls (it is a cat race). You may find a good average result for their behaviour but does an average ragdoll exist. Maybe not because male and female ragdolls may have different behaviour and the average ragdoll may here again be a monster Fourth example may be more surprising Human beeing may be caracterized by her weight, her height, her waist circumference ... Question : what about an average human? Is it a monster? Now let's come back to Go theory I mentioned above. In various part of the theory we use an ideal environment which is an average object over a large number of real environment. As I showed you such average object cannot be a real enviroment which cannot be ideal. this "ideal" environment looks like a monster doesn't it? Really it does not harm ... providing you do not claim that the result of the theory is the real life and cannot be constested. Inside the theory "double sente" move does not exist at least without regard for the rest of the board. Fine indeed but in the real life they really exist and they cause difficult timing problems for the player. The behaviour of a real "double sente" is in practice quite different from the behaviour of a gote point. Let's remind you my ragdoll example: you may draw a certain behaviour of the ragdolls but you cannot ignore that, if you accept to take into account the male and the female ragdoll then you will reach a better understanding of the behaviour of the ragdolls. Why not trying to identify what is commonly called "double sente"? Remenber that a go player do not say that you must in any case answer to a double sente move. On contrary all go player look always for a way to not answer it. The definition could be based on the value of the threat (something like b,w > 2n in my notation) and then you can enrich the theory by analysing such specific point. My feeling : a "double sente" exists as soon as you accept to define it! 
Author:  Bill Spight [ Tue Oct 20, 2020 9:08 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: Now let's come back to Go theory I mentioned above. In various part of the theory we use an ideal environment which is an average object over a large number of real environment. The concept of the ideal environment is only necessary for multiple ko and superko positions. I know, because I am the guy who came up with the theory for them. Since we are not talking about those, I have dispensed with the concept for this discussion. Quote: Inside the theory "double sente" move does not exist at least without regard for the rest of the board. Everybody agrees that double sente moves exist. The question is whether double sente nonko positions (finite combinatorial games) that gain points for the first player exist. Quote: Why not trying to identify what is commonly called "double sente"? You know that I have done so for double sente moves. As for what are generally considered double sente positions, I have to rely upon examples. Every example I have found is a sente, gote, or ambiguous position. (That's not an argument.) 
Author:  Gérard TAILLE [ Tue Oct 20, 2020 10:49 am ] 
Post subject:  Re: How evaluate double sente moves ? 
This time I will try my very first attempt to use slash notation After white is killed for a score of +22 After the score is 6 After the score is 3 After the score is 7 The game looks like {{+226},37} with tax = 1 the game becomes {{+206},46} with tax = 2 the game becomes {{+186},55} The average territorial value is 5 but what can I do with the sente {+186} ? can I change the color of the mast from black color to a part in blue color? If not that confirms that the possiblity of the sente black move is completly hidden by analysis. Is it correct Bill? 
Author:  Bill Spight [ Tue Oct 20, 2020 11:31 am ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: This time I will try my very first attempt to use slash notation After white is killed for a score of +22 After the score is 6 After the score is 3 After the score is 7 The game looks like {{+226},37} Very good, Gérard. I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first? Quote: with tax = 1 the game becomes {{+206},46} Same question. Quote: with tax = 2 the game becomes {{+186},55} Same question. Quote: The average territorial value is 5 but what can I do with the sente {+186} ? What is the average territorial value of {+186 } ? 
Author:  Bill Spight [ Tue Oct 20, 2020 12:42 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: My feeling : a "double sente" exists as soon as you accept to define it! How about this? A double sente is a combinatorial game such that both sides of its thermograph are vertical. 
Author:  Gérard TAILLE [ Tue Oct 20, 2020 1:42 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: Gérard TAILLE wrote: My feeling : a "double sente" exists as soon as you accept to define it! How about this? A double sente is a combinatorial game such that both sides of its thermograph are vertical. The definition should be as near as possible as the common understanding of go players (which is not really defined in real life is it?). We can also take this other suggestion: Attachment: double sente.png [ 7.86 KiB  Viewed 1097 times ] Other possible definition : by definition we have a double sente area if x,y >= n Surely, with your fine knowledge of the problem, you will find more easily than me, what definition will fit the unclear but common understanding of double sente move in real life of go players. I am quite convinced that, by just adding such defintion, a lot of new players will adhere to the theory, especially if you find some interesting behaviour of such double sente move. Because adding such pure defintion cannot harm why not trying our best? 
Author:  Gérard TAILLE [ Tue Oct 20, 2020 1:56 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first? {{+226},37} tax = 0 black plays first => minimax = 3 white plays first => minimax = 7 {{+206},46} tax = 1 black plays first => minimax = 4 white plays first => minimax = 6 {{+186},55} tax = 2 black plays first => minimax = 5 white plays first => minimax = 5 
Author:  Bill Spight [ Tue Oct 20, 2020 2:00 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: Bill Spight wrote: Gérard TAILLE wrote: My feeling : a "double sente" exists as soon as you accept to define it! How about this? A double sente is a combinatorial game such that both sides of its thermograph are vertical. The definition should be as near as possible as the common understanding of go players (which is not really defined in real life is it?). We can also take this other suggestion: Attachment: double sente.png Other possible definition : by definition we have a double sente area if x,y >= n Surely, with your fine knowledge of the problem, you will find more easily than me, what definition will fit the unclear but common understanding of double sente move in real life of go players. Oh, double sente move is easy. I have already defined that. Quote: I am quite convinced that, by just adding such defintion, a lot of new players will adhere to the theory, especially if you find some interesting behaviour of such double sente move. Because adding such pure defintion cannot harm why not trying our best? No harm done? What do you think of the two double sente examples from Kano, 9 dan's, Yose Dictionary? Do you think they are good examples upon which to understand double sente? In thermography a necessary condition for sente is a vertical wall, because that's is what indicates that the second player made a local reply to the first playe's move. A double sente therefore requires two vertical walls, one for each player. True, most go players are not familiar with thermography, but explain why walls are vertical or inclined, and they will get it. So what is the problem with requiring two vertical walls for double sente? Sine qua non, n'estce pas? 
Author:  Bill Spight [ Tue Oct 20, 2020 2:03 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Gérard TAILLE wrote: Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first? {{+226},37} tax = 0 black plays first => minimax = 3 white plays first => minimax = 7 {{+206},46} tax = 1 black plays first => minimax = 4 white plays first => minimax = 6 {{+186},55} tax = 2 black plays first => minimax = 5 white plays first => minimax = 5 Bueno. What about this combination? {+226},37} + {100} 
Author:  Gérard TAILLE [ Tue Oct 20, 2020 2:15 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: Gérard TAILLE wrote: Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first? {{+226},37} tax = 0 black plays first => minimax = 3 white plays first => minimax = 7 {{+206},46} tax = 1 black plays first => minimax = 4 white plays first => minimax = 6 {{+186},55} tax = 2 black plays first => minimax = 5 white plays first => minimax = 5 Bueno. What about this combination? {+226},37} + {100} well I would say: black plays first => minimax = +3 = 10 + (7) white plays first => minimax = 3 = 0 + (3) 
Author:  Bill Spight [ Tue Oct 20, 2020 2:19 pm ] 
Post subject:  Re: How evaluate double sente moves ? 
Bill Spight wrote: Bueno. What about this combination? {+226},37} + {100} Gérard TAILLE wrote: well I would say: black plays first => minimax = +3 = 10 + (7) white plays first => minimax = 3 = 0 + (3) Black can do better at temperature 0. 
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