Endgame Strategy (03)
Robert Jasiek
Kd: #open kos for Black - #open kos for White, which by assumption is
>=0.
Td: #Black's ko threats - #White's ko threats.
STRATEGY FOR EACH PLAYER
1. answer ko threat
2. connect ko
3. capture ko if legal
4. ko threat if there is some ko
5. pass
STRATEGIC CASES
§ Move Case Score Ko Fight
(Winner)
1.1 Black Kd MOD 3 = 0 2/3 * Kd no
1.2 Black Kd MOD 3 = 1 2/3 * (Kd - 1) + 2 no
1.3 Black Kd MOD 3 = 2 && Td > 0 2/3 * (Kd - 2) + 4 Black
1.4 Black Kd MOD 3 = 2 && Td <= 0 2/3 * (Kd - 2) White
2.1 White Kd MOD 3 = 0 2/3 * Kd no
2.2 White Kd MOD 3 = 1 && Td > 0 2/3 * (Kd - 1) + 2 Black
2.3 White Kd MOD 3 = 1 && Td <= 0 2/3 * (Kd - 1) - 2 White
2.4 White Kd MOD 3 = 2 2/3 * (Kd - 2) no
**************************************************************************
Where traditional go theory speaks carelessly of "more ko threats",
one has to be more precise: Either it is "strictly more ko threats" or
"at least as many ko threats". E.g., in §1.3 Black has strictly more
ko threats than White; in §1.4. White has at least as many ko threats
as Black.
The scores seem to be great but recall that previously Black played Kd
stones more than White locally. If you subtract the excess Kd White
stones previously played elsewhere in alternation, then the scores
become negative, i.e. favouring White.
If there is a ko fight in a position without two-sided dame, then
winning or losing the ko equals a score difference 4.
The positional superko rule does not make strategy particularly
complex in positions with basic endgame kos (and, as studied so far,
without dame).
**************************************************************************
SCORES EXPRESSED DIFFERENTLY
§ Move Case Score Ko Fight
(Winner)
1.1 Black Kd MOD 3 = 0 2/3 * Kd no
1.2 Black Kd MOD 3 = 1 2/3 * Kd + 4/3 no
1.3 Black Kd MOD 3 = 2 && Td > 0 2/3 * Kd + 8/3 Black
1.4 Black Kd MOD 3 = 2 && Td <= 0 2/3 * Kd - 4/3 White
2.1 White Kd MOD 3 = 0 2/3 * Kd no
2.2 White Kd MOD 3 = 1 && Td > 0 2/3 * Kd + 4/3 Black
2.3 White Kd MOD 3 = 1 && Td <= 0 2/3 * Kd - 8/3 White
2.4 White Kd MOD 3 = 2 2/3 * Kd - 4/3 no
**************************************************************************
This is a nice table for extracting the miai value 4/3, isn't it?
--
robert jasiek
Bill Spight
Very nice work! :-) The tabular summary is quite clear.
I don't want to steal your thunder, but let me drop a hint and make a few
remarks.
Hint for readers: I expect that Robert has discovered what I have
dubbed "virtual ko threats", in this case a pair of dame that act in a
certain way like a ko threat for one player.
Other remarks: Robert is starting from scratch, but if you consider the
evaluation of plays you can make some simplifications.
. . # # O . . All stones outside the ko are alive.
. # # O O . . Open points outside the ko are dame.
. # O . O . .
-------------
This ko position, where each player has the same number of stones, is
worth 1/3 point (for Black) on average. Three of them are worth 1 point.
In that case there is no real ko fight. White will win two of the kos and
Black will win one.
. . # # O . . . . # # O . . . . # # O . .
. # # O O . . . # # O O . . . # # O O . .
. # O O O . . . # O O O . . . # # # O . .
---------------------------------------------
Similarly, six of them are worth 2 points, nine are worth 3, etc.
What about two of them? OC, they are worth 2/3 point, on average.
. . # # O . . . . # # O . .
. # # O O . . . # # O O . .
. # O . O . . . # O . O . .
-----------------------------
Suppose that Black plays first.
. . # # O . . . . # # O . .
. # # O O . . . # # O O . .
. # O 1 O . . . # O 2 O . .
-----------------------------
Since each player has made a play of the same size, the value remains the
same, 2/3, but there is only one ko left.
Suppose that White plays first.
. . # # O . . . . # # O . .
. # # O O . . . # # O O . .
. # O 2 O . . . # O 1 O . .
-----------------------------
These two positions look the same, but there is a difference. In the
second case White is kobanned. To achieve equality White needs a primary
ko threat (one that Black must answer).
That yields this simplification:
. . # # O . . . . # # O . .
. # # O O . . . # # O O . . + 1
. # O . O . . . # O . O . .
-----------------------------
equals
. . # # O . .
. # # O O . . + Black (primary) threat
. # . # O . .
-------------
If Black plays first, in the first case we get
. . # # O . . . . # # O . .
. # # O O . . . # # O O . . + 1
. # O 1 O . . . # O . O . .
-----------------------------
Each player will win a ko, with a result of +3.
In the second case we get
. . # # O . .
. # # O O . . + Black (primary) threat
. # 1 # O . .
-------------
The result is the same, +3.
If White plays first, in the first case we get
. . # # O . . . . # # O . .
. # # O O . . . # # O O . . + 1
. # O . O . . . # O 1 O . .
-----------------------------
which has a value of 1/3.
In the second case we get
. . # # O . . B2 = ko threat
. # # O O . . W3 = reply
. # 1 # O . .
-------------
which has a value of 1/3.
So the two are equivalent.
Because of these simplifications, we only need to consider the case of a
single ko. We can derive everything else from that.
(And as Robert has already indicated, we only need to consider the
difference in primary threats.)
Ciao,
Bill
Robert Jasiek
>I expect that Robert has discovered what I have
>dubbed "virtual ko threats", in this case a pair of dame that act in a
>certain way like a ko threat for one player.
Actually I have wanted to discuss dame threats a little later:)
>Other remarks: Robert is starting from scratch
Yes. This is my exercise so that I can understand application of other
rulesets better, it allows algorithmic construction of strategies
rather naturally (for my taste), and it shows that there are more
approaches to theory than CGT :)
> [White plays 1]
>
> . . # # O . .
> . # # O O . . + Black (primary) threat
> . # 1 # O . .
> -------------
>
>The result is the same, +3.
How do you get the +3? I cannot follow you here.
>if you consider the evaluation of plays you can make some simplifications.
[...]
>Because of these simplifications, we only need to consider the case of a
>single ko. We can derive everything else from that.
I appreciate your apparently efficient introduction of another method
for doing proofs about kobans combined with values but I fear that I
am not sufficiently used to this approach yet. I am more comfortable
with algorithmic thinking...
Best,
--
robert jasiek
Bill Spight
>> [White plays 1]
>>
>> . . # # O . .
>> . # # O O . . + Black (primary) threat
>> . # 1 # O . .
>> -------------
>>
>>The result is the same, +3.
>
> How do you get the +3? I cannot follow you here.
>
That's Black plays 1, not White. With the resolution of the ko, the ko
threat is worth 0, so we just have 7 Black stones vs. 4 White stones, for
a local score of 3.
Best,
Bill
Robert Jasiek
wrote:
>That's Black plays 1, not White. With the resolution of the ko, the ko
>threat is worth 0
Now it's clear to me, thanks! :)
--
robert jasiek