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Post #221 Posted: Fri Sep 21, 2018 2:23 am 
Honinbo
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Hi Bill, :)
Thanks for the write-up.
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In two of the three cases switching gets the prize, in one it loses it, so switching is the superior strategy. Once you have found the superior strategy you can work out the probabilities. ;)
But this seems a brute forcing approach, no ?

As usual, it turns into work :blackeye:

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Post #222 Posted: Fri Sep 21, 2018 2:52 am 
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EdLee wrote:
Hi Bill, :)
Thanks for the write-up.
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In two of the three cases switching gets the prize, in one it loses it, so switching is the superior strategy. Once you have found the superior strategy you can work out the probabilities. ;)
But this seems a brute forcing approach, no ?

As usual, it turns into work :blackeye:


Sure, it's brute force. You are not relying upon any concepts, just strategies. What works? Once a person knows what works, they are open to learning the concept. :)

Edit: I realized a classroom demo for the two envelopes problem. Ask the students how much is a fair price for the privilege of switching. Then get two students who are willing to pay to switch, give each of them an envelope and tell them before they reveal the amount in their envelopes that if they are still willing to pay to switch you will give them the amount in the other student's envelope, minus their payment, even if the other student does not switch. ;) It should be obvious to the whole class that the overall payoff is more if they do not switch.

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 Post subject: Re: Sente, gote and endgame plays
Post #223 Posted: Fri Sep 21, 2018 6:50 am 
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Bill Spight wrote:
Many questions, which I will gloss over, if you don't mind too much.


Ok - but there is a reason why I pose them so carefully and detailed. In the current state, we cannot teach it as a general method. Somebody (i.e., you or I) would need to do the mathematical research to possibly create it as a general method. If I did it, I would need about 4 to 6 months (which I don't have) for formulating and proving the theorems. Therefore, I have hoped that you might already have completed the research. You haven't. No problem, except see above.

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Post #224 Posted: Fri Sep 21, 2018 9:19 am 
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RobertJasiek wrote:
Bill, you have shown the following two multiples:





You rely on the CGT definitions of local temperatures and mean. Can you explain the relation between the definitions and their application, please?


I'm not sure what you are asking.

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You show application for a local endgame with one player's simple follow-up and without kos. For such, application is straightforward. How about local endgames with more complicated follow-ups and without kos? Is the method exactly the same, except for needing more multiples?


You need a strategy at each decision point. So you end up with several comparisons.

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Can you write down your general method as a procedure applicable to all examples of a class?


Not sure what you are asking. It's a general method for comparing strategies and classifying positions, but more work needs to be done to verify means and temperatures. The method of multiples is inefficient.

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Do we know a priori how many multiples we need at least? How do we find out the minimal necessary number of multiples? (Why) is it always 4 for a local endgame with one player's simple follow-up and without ko?


The minimum is 2^d, where d is the depth of the tree.

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How would you define as a procedure the gote strategy and the sente strategy a) for a local endgame with one player's simple follow-up and without ko and b) for an arbitrary local endgame possibly with follow-ups and without ko?


Each decision point requires a different strategy, so the eventual strategy must say what to do at each node. Finding the best strategy for an arbitrary tree can take a lot of work.

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For a local endgame with both players' follow-ups, how to determine for whom to test a sente strategy? Must a procedure be more complicated by possibly having to test either player's gote strategy and either player's sente strategy?


Yes. In these cases we could eliminate a possible sente strategy for Black, but, OC, that is not always possible. Also, the final node is always a gote, so we don't have to test it for sente.

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Suppose we test one player's gote versus sente strategies with the resulting counts G versus S. Does the more favourable count for the starting player determine the correct strategy?


Each player must have made the same number of plays with each strategy, so that when you double the number of multiples the result will be the same ad infinitum. Also, you must have compared every possible strategy to find out which is correct. But you can compare strategies one by one.

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Is the count of one local endgame the resulting count of the more favourable strategy divided by the number of multiples?


If you have found the correct strategy, yes. But in the arbitrary case the correct strategy can be arbitrarily complex.

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Why is this so explained by the CGT definitions? If we determine a local sente, is the reverse sente gain S - G (with counts calculated in favour of the starting player)? Why is this so explained by the CGT definitions?


Thermography determines mean values and temperatures. These mean values must be the same (for non-ko games) as those found by the method of multiples, by the mean value theorem.

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Suppose we have a local endgame with follow-ups and without ko and have to test both players' gote versus sente strategies with the resulting counts Gb, Sb, Gw and Sw. How to determine the correct strategy, calculate the correct count, explain by the CGT definitions, possibly calculate a reverse sente gain and explain by CGT definitions?

For all cases, how to calculate the move value and explain it by CGT definitions?

In summary, your examples look convincing but how and why can they be generalised as methods?


I think your questions here can be answered by the fact that the method of multiples finds mean values, and temperatures can be derived from them. As for why generalize, why indeed? Thermography is more efficient.

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 Post subject: Re: Sente, gote and endgame plays
Post #225 Posted: Fri Sep 21, 2018 10:10 am 
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Bill Spight wrote:
RobertJasiek wrote:
You rely on the CGT definitions of local temperatures and mean. Can you explain the relation between the definitions and their application, please?

I'm not sure what you are asking.


Your answer "the method of multiples finds mean values, and temperatures can be derived from them" goes in the intended direction of my question.

Quote:
You need a strategy at each decision point. So you end up with several comparisons.
[...] The method of multiples is inefficient. [...]
Each decision point requires a different strategy, so the eventual strategy must say what to do at each node. Finding the best strategy for an arbitrary tree can take a lot of work.
[...] you must have compared every possible strategy to find out which is correct [...]
in the arbitrary case the correct strategy can be arbitrarily complex [...]
why generalize, why indeed? Thermography is more efficient.


OIC. Looking at your examples, I thought that it was a 1 ply decision-making at the top level. You explain that it is iterative strategic decision-making bottom-up. If not thermography, we may use methods for value comparisons to distinguish gote from sente for short or long sequences worth playing successively. I also suspect that this would be more efficient than methods involving multiples, at least when we would need 8+ multiples.

Quote:
Quote:
Can you write down your general method as a procedure applicable to all examples of a class?

Not sure what you are asking. It's a general method for comparing strategies and classifying positions, but more work needs to be done to verify means and temperatures.


If it is a general method, state its procedure explicitly - otherwise, it is not general:)

For the class of all examples with one player's simple follow-up, stating the procedure is straightforward. Can you also state it so that it applies to iterative follow-ups? (If you have the time for doing so and think it is worthwhile.)

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Post #226 Posted: Fri Sep 21, 2018 11:07 am 
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RobertJasiek wrote:
Bill Spight wrote:
RobertJasiek wrote:
You rely on the CGT definitions of local temperatures and mean. Can you explain the relation between the definitions and their application, please?

I'm not sure what you are asking.


Your answer "the method of multiples finds mean values, and temperatures can be derived from them" goes in the intended direction of my question.

Quote:
You need a strategy at each decision point. So you end up with several comparisons.
[...] The method of multiples is inefficient. [...]
Each decision point requires a different strategy, so the eventual strategy must say what to do at each node. Finding the best strategy for an arbitrary tree can take a lot of work.
[...] you must have compared every possible strategy to find out which is correct [...]
in the arbitrary case the correct strategy can be arbitrarily complex [...]
why generalize, why indeed? Thermography is more efficient.


OIC. Looking at your examples, I thought that it was a 1 ply decision-making at the top level. You explain that it is iterative strategic decision-making bottom-up. If not thermography, we may use methods for value comparisons to distinguish gote from sente for short or long sequences worth playing successively. I also suspect that this would be more efficient than methods involving multiples, at least when we would need 8+ multiples.

Quote:
Quote:
Can you write down your general method as a procedure applicable to all examples of a class?

Not sure what you are asking. It's a general method for comparing strategies and classifying positions, but more work needs to be done to verify means and temperatures.


If it is a general method, state its procedure explicitly - otherwise, it is not general:)

For the class of all examples with one player's simple follow-up, stating the procedure is straightforward. Can you also state it so that it applies to iterative follow-ups? (If you have the time for doing so and think it is worthwhile.)


The overall strategy must say what each player plays at each choice point. Each option that is part of the strategy must be equally represented in the combined instances. These simple examples have only two choice points. The sente strategy gives only one option at each choice point, and each player makes the same number of plays, so all it requires is a single instance. The gote strategy gives an option for each player at each choice point, and requires four instances. So to compare the results of the two strategies requires four instances.

Suppose that we had another follower, with only one obvious choice for each player. Then we would have these possible strategies with White playing first: White sente, first play by White gote and the second play by White sente, each play by White gote. The comparisons would require 8 instances.

I am attaching another SGF file with comments that might make things clearer. :)


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 Post subject: Re: Sente, gote and endgame plays
Post #227 Posted: Wed Oct 10, 2018 12:59 am 
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Code:
          A
         / \
        /   \
       /     \
      B       C
     / \     / \
    D   E   F   G


1) Does a local endgame position exist with (B-C)/2 = E-C = B-F > 0? That would be a doubly ambiguous local endgame with short sequences and without other options having long sequences.

2) Does a local endgame position exist with B-F = (B-C)/2 > E-C > 0? That would be Black's sente and ambiguous on White's sente sequence despite the only short sequences. If you have the opposing colour preference, it would be 0 < B-F < (B-C)/2 = E-C.

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Post #228 Posted: Wed Oct 10, 2018 2:18 am 
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RobertJasiek wrote:
Code:
          A
         / \
        /   \
       /     \
      B       C
     / \     / \
    D   E   F   G


1) Does a local endgame position exist with (B-C)/2 = E-C = B-F > 0? That would be a doubly ambiguous local endgame with short sequences and without other options having long sequences.


(B-C)/2 = E-C = B-F

(B+C)/2 = E = F = A

This is miai or double ko threat.

Quote:
2) Does a local endgame position exist with B-F = (B-C)/2 > E-C > 0? That would be Black's sente and ambiguous on White's sente sequence despite the only short sequences. If you have the opposing colour preference, it would be 0 < B-F < (B-C)/2 = E-C.


F = (B+C)/2 > E

If F > 0 > E then A = 0; this is seki.

If F > E ≥ 0 then A = E.

If 0 ≥ F > E then A = F.

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 Post subject: Re: Sente, gote and endgame plays
Post #229 Posted: Wed Oct 10, 2018 4:29 am 
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Bill Spight wrote:
(B-C)/2 = E-C = B-F

(B+C)/2 = E = F = A

This is miai or double ko threat.


Do you say that the first is miai and the second a mutual ko threat?

How does a connected local endgame that is a miai look like? Note the required short sequences: we shall have numbers after the first two plays. Do such positions exist on the board?

A local endgame that is a mutual ko threat has the move value 0 and therefore does not fulfil my > 0 assumption.

Quote:
F = (B+C)/2 > E

If F > 0 > E then A = 0; this is seki.


A seki has the move value 0 and therefore does not fulfil my > 0 assumption.

Quote:
If F > E ≥ 0 then A = E.

If 0 ≥ F > E then A = F.


My problem is: how do such actual connection local endgame positions on the board look like with only short sequences before reaching numbers? Do they exist as positions on the board?

If we allow long sequences, there are such positions. The trouble is finding some for short sequences or proving their inexistence.

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Post #230 Posted: Wed Oct 10, 2018 7:43 am 
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RobertJasiek wrote:
Bill Spight wrote:
(B-C)/2 = E-C = B-F

(B+C)/2 = E = F = A

This is miai or double ko threat.


Do you say that the first is miai and the second a mutual ko threat?


Except for the equation indicating equality with A, the two are equivalent.

E-C = (B-C)/2
E = (B-C+2C)/2 = (B+C)/2

(B-C)/2 = B-F
F = (2B-B+C)/2 = (B+C)/2

Quote:
How does a connected local endgame that is a miai look like? Note the required short sequences: we shall have numbers after the first two plays. Do such positions exist on the board?


One example. Many similar positions go positions exist.
Code:
          A
         / \
        /   \
       /     \
      B       C
     / \     / \
    9   5   5   1


Quote:
A local endgame that is a mutual ko threat has the move value 0 and therefore does not fulfil my > 0 assumption.


This one satisfies your requirement. (B-C)/2 = B-F = E-C = 4 > 0.

Quote:
Quote:
F = (B+C)/2 > E

If F > 0 > E then A = 0; this is seki.


A seki has the move value 0 and therefore does not fulfil my > 0 assumption.


Seki example.
Code:
          A
         / \
        /   \
       /     \
     16      -6
     / \     / \
   37  -5   5  -17


Your required relation: B-F = (B-C)/2 > E-C > 0

16-5 = 11 = (16-(-6))/2 > -5-(-6) = 1 > 0

Quote:
Quote:
If F > E ≥ 0 then A = E.

If 0 ≥ F > E then A = F.


My problem is: how do such actual connection local endgame positions on the board look like with only short sequences before reaching numbers? Do they exist as positions on the board?

If we allow long sequences, there are such positions. The trouble is finding some for short sequences or proving their inexistence.


I am unaware of any example on the go board with F > E which satisfies all of your requirements. However, if such examples exist, they will in theory have the values given. (As we know from the latest Japanese rules, the scores may be different from the theoretical values. ;))

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