dhu163 wrote:
a bonus of T/2 for sente.
"Sente" does not express it clearly. You mean "having the turn" or "moving first".
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We can think of T as the largest move in the environment of the ko.
If the environment is ko-less.
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NTE
So normally we might have
A = 3T/2, B = K/3+T/2, C=2K-T/2, D = K-3T/2
When K<3T, A>D, so it isn't worth playing in the ko.
Ko master
A = 3T/2, B = K/2+T/2 , C = K-T/2, D = K-3T/2
Ko monster
This means that RIGHT has so many ko threats that they can even keep playing in the environment (respond to every move in the environment that the opponent resorts to playing as a ko threat) until the temperature goes down to t<T.
Now we might have something more like
A = 3T/2, B = K/2+(3T-t)/4 , C = K-t/2, D = K-3t/2
where all the values have shifted in RIGHT's favour.
Please explain how to calculate these [RIGHT-]counts of A, B, C, D!
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[hide]Research problem
Can you construct a theory of how to draw the thermograph of the sum of two games? Or some principles beyond the obvious?
I have proved that it is impossible given only the information: thermograph of each subgame.
What is your proof? :)
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Direct ko + RIGHT enlargement option
What is a player's enlargement option? As a go player, I somewhat understand this concept. However, what is it mathematically?
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NTE
If RIGHT's best play is to end the ko with D, then since D leaves a threat of {0|L-T} behind (if L>T)
Is this what you mean as a player's (here: RIGHT's) enlargement option?
Previously, you have stated the count K of D. Now, K = ((L - T) + 0) / 2 = (L - T) / 2. Do I understand this correctly?
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then we can assume playing L is the largest move. (if L<T, we can forget the possibility of playing E).
Is E the position and count after RIGHT moving from the position D to the count L - T?
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If L>T, LEFT must defend
At which position must LEFT defend? Why?
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and at D, RIGHT expects to gain a move
In mathematical terms, what is "to gain a move"? Do you mean "move" or [a] "play"?
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even though they don't get L.
What do you mean?
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A = 3T/2, B = (K+T)/3 + T/2, C = 2(K+T)/3 - T/2 , D = K+T - 3T/2
How to calculate these counts?
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If RIGHT's best play is to play one move towards E from C,
I suppose "in 2 net plays, one from C to D, then the second from D to E".
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then we just write E on the right,
Do you suggest writing A - B - E? Or A - B - E - E? Or A - B - E - E - E? Otherwise, what do you mean by writing the count E on the right?
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and enter a different stage of ko where RIGHT has spent a move (like an approach ko) but has hooked on E.
What is a "stage of a ko" in your context? What is "spending a move"?
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A2 = T/2, B2 = (K+L)/3 - T/2, C2 = 2(K+L)/3 - 3T/2 , E = K+L - 5T/2
How and why to calculate like this?
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Since the transition occurred at state C, we want to know when equilibrium occurs, namely when C=C2 (with a transition)
Is equilibrium a CGT term or your informal description? If it is a term, what is its definition?
What transition?
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we have C2-C = 2(L-T)/3 - T = 0
C2 - C = 2(K+L)/3 - 3T/2 - (2(K+T)/3 - T/2) = 2K/3 + 2L/3 - 3T/2 - 2K/3 - 2T/3 + T/2 = 2L/3 - 2T/3 - 3T/2 + T/2 = 2(L-T)/3 - T, which you then compare to 0. I see.
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so L=5T/2
2(L-T)/3 - T = 0 <=>
2(L-T)/3 = T <=>
L - T = 3T/2 <=>
L = 5T/2.
Right.
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is the critical loss above which RIGHT wants to play towards E.
Please explain what you mean with critical loss and why then he wants to play to E!
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We can see (as expected) that this is pretty large.
Large in comparison to what?
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This is because playing normally to D would have gotten profit from the existence of E anyway.
Please explain this in greater detail!
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Curiously, it is independent of K.
Interesting indeed.
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Similarly, if LEFT's best play is to play one move to F from C,
What is F? Do you mean if, instead of RIGHT's extension, we have LEFT's extension?
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we again enter a different stage of ko
A3 = 5T/2, B3 = K/3 - 3T/2, F = C3 = 2K/3 + T/2 , D3 = K - T/2
How to calulate these counts?
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Equating at C again,
Why?
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we have C3-C = 2T/3 - T = 0. So T=0.
C3 - C = 2K/3 + T/2 - (2(K+T)/3 - T/2) = 2K/3 + T/2 - 2K/3 - 2T/3 + T/2 = -2T/3 + T so what you calculate is rather C - C3 :) Anyway, you then compare to 0 and find T = 0.
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So only when T=0 does LEFT consider to block and wipe out the possibility of E completely.
Interesting.
However, I do not understand: When you consider F at all, do you assume the existence of E? I thought you would be studying F instead of E.
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(unless C2 is relevant)
Please explain.
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C3-C2 = 2T - 2L/3 = 0.
C3 - C2 = 2K/3 + T/2 - (2(K+L)/3 - 3T/2) = 2K/3 + T/2 - 2K/3 - 2L/3 + 3T/2 = 2T - 2L/3, which you then compare to 0. I see.
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So L=3T.
2T - 2L/3 = 0 <=>
2T = 2L/3 <=>
T = L/3 <=>
3T = L.
Right.
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When L>3T, LEFT prefers to block the possibility of E.
Please explain!
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Summary
- T=0: LEFT can block and move to F at no cost if RIGHT might win the ko
Why?
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-L<T: Both sides should treat the threat of E as a bonus of (L/2) added onto the size of the ko (with K->K+L/2). As usual, moves gain K/3. If K_old + L/2 = K_new>3T, the ko is worth fighting. So the ko is a bit bigger with higher temperature. RIGHT's profit from the existence of E is always rho x L/2 (which appears at T=L/2), where rho = 0 in A, 1/3 in B, 2/3 in C, 1 in D. C=2(K+L/2)/3-T/2
-T<L<5T/2: In this case, connecting at D is sente and K_new = K_old+T. RIGHT's profit from the existence of E is rho x T. C = 2(K+T)/3 - T/2
-5T/2<L<3T: RIGHT ignores D and fights the ko directly. C = 2(K+L)/3 - 3T/2
-L>3T: LEFT blocks the possibility of RIGHT enlarging the ko. C = 2K/3 +T/2
Please explain each bit of these statements in great detail, thanks!
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It is most interesting that there is a sweet spot at 5T/2<L<3T where fighting over E occurs. This is because it is large enough to be interesting for RIGHT but also LEFT has to pay a penalty of T to prevent it.
Please explain with more words why this is so in this range!
Why are equalities excluded?
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Only if the ko is really big should LEFT spend a move to prevent it.
Why is this so in terms of values?
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RIGHT Ko master
A = 3T/2, B = (K+T)/2 + T/2, C = K+T - T/2 , D = K+T - 3T/2
A2 = T/2, B2 = (K+L/2 - T/2, C2 = K+L - 3T/2 , E = K+L - 5T/2
A3 = 5T/2, B3 = K/2 - 3T/2, F = C3 = K + T/2 , D3 = K - T/2
C2=C when L=2T
C3=C always
C3=C2 when L=2T
Please explain these calculations! How to calculate the values? Why are these the calculations to do? What shall the comparisons express and how are they derived?
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Summary
We have C = K+T/2 in all cases
-L<2T: RIGHT shouldn't try to enlarge the ko (RIGHT will win the ko and threaten L anyway). Similarly, LEFT should not both stopping RIGHT from enlarging the ko.
-L>2T: RIGHT should try to enlarge the ko. However LEFT should never give RIGHT the chance and should immediately block the ko enlargement by spending a move.
Why?