Endgame Strategy (05)
Robert Jasiek
Black to move // Black
D > 0 // with two-sided dame
K > 0
Kb >= Kw >= 0
Tb >= 0
Tw >= 0
Kd MOD 3 = 1 // the Kd case
Case 1: D ODD && Td <= (D - 1) / 2
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Strategy for each player:
If it is Black's first move, then connect ko.
Afterwards see the class of positions 3.
Algorithm:
1. Black connects ko.
Afterwards see the class of positions 3.
Proposition 4.1.1: The score due to the algorithm is: 2/3 * (Kd - 1)
+ 1
Proof: During (1), Black gets 2 points more than White. Then for the
remaining dame and kos, proposition 3.1 is applied. Because already 1
ko is connected, (Kd - 1) instead of Kd has to be inserted. Because D
is ODD and it is White to move after Black's first move (when
proposition 3.1 is being applied), par(D) = -1. The small numbers 2
and -1 have the sum 1. QED.
Proposition 4.1.2: The score is optimal for Black.
Proof: Within the kos, Black cannot do better than to start with
connect ko, see the class of positions 1.2. So to improve on the
score, Black has to keep the extra ko open for him by starting with
playing a dame. White cannot use ko plays as ko threats for that ko
because after a few plays that resolve three other kos it is still
White with a need for finding a ko threat that is not a ko play. Since
it is Black's intention to get an extra dame, White can use a dame
play as a ko threat and Black, trying to fulfil his intention, has to
answer that kind of ko threat also by a dame play. Each ko threat and
answer that are dame plays fill together 2 of the dame. Since
initially Black has filled already 1 dame, the initially remaining (D
- 1) dame provide (D - 1) / 2 ko threats. White can use dame as ko
threats. Black cannot because then White connects the extra ko and
Black, although he gets 2 extra dame, does not improve on the score,
as it has been his intention. So all the dame ko threats belong
exclusively to White. Because Black starts by filling a dame, White
captures in the extra ko first, and Black has to make the first ko
threat, after (D - 1) / 2 non-dame ko threats of Black (if Tb >= (D -
1) / 2 at all) and (D - 1) / 2 dame ko threats of White (WLOG, White
can choose to use dame as ko threats before non-dame as ko threats)
all the dame are filled, Black has still not won the ko fight, but
Black has already used up (D - 1) / 2 of his Td (hopefully positive at
all) excess of ko threats. By assumption of this case 1, however, Td
is at most (D - 1) / 2. So Black fails to improve on the score because
he misses at least 1 further ko threat that he would need to win the
ko fight. QED.
Proposition 4.1.3: The score is optimal for White.
Proof: After Black's first move, see proposition 3.2. QED.
Case 2: D ODD && Td > (D - 1) / 2
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Strategy for Black:
1. play dame if it is his first move
2. answer non-dame ko treat
3. answer dame ko threat
4. capture ko if the previous White move is an answer to a ko threat
5. answer non-ko-threat dame play by dame play
6. answer ko play by connect ko
7. answer ko play by capture ko if legal
8. play non-dame ko threat if currently at least one ko capture is
illegal for Black
9. connect ko if this option is called for its first time
10. play dame
12. connect ko
12. capture ko if legal
13. pass
Remarks: Can this be compressed a little? It is not optimized for
greater positional contexts in that retaining ko threats might be an
advantageous objective. In the currently discussed class of positions,
retaining ko threats is not advantageous.
Strategy for White:
1. answer non-dame ko threat
2. capture ko if the previous Black move is an answer to a ko threat
or Black's first move
3. play non-dame ko threat if currently at least one ko capture is
illegal for White
4. play dame ko threat if currently at least one ko capture is illegal
for White
5. connect ko if this option is called for its first time
6. answer dame play by dame play
7. answer ko play by connect ko
8. answer ko play by capture ko if legal
9. play dame
10. connect ko
11. capture ko if legal
12. pass
Remarks: There are other possible perfect strategies for Black or
White. E.g., White might admit defeat in the ko fight earlier or White
might first force connections in some triples of the kos before
starting the ko fight. In the strategy for White, White plays non-dame
threats before dame threats to threaten a possibly greater exchange if
Black should be short of ko threats and make a strategic mistake.
Since Black plays perfectly and answers all threats, this does not
have consequences though. In a greater positional context, one would
rather play the smallest sufficient ko threats first to retain the
bigger ones for later ko fights. - Both strategies are to be verified.
Possibly more options have to be added if the opponent passes
prematurely.
Algorithm:
1. Black plays dame
2. Repeat (D - 1) / 2 + 1 times:
2.1 capture ko
2.2 ko threat
(White prefers non-dame ko threats to dame ko threats, Black plays
only non-dame ko threats)
2.3 answer ko threat
(Black answers also dame ko threats, White does not)
3 Repeat 2 * Kw + 1 times: connect ko
4. Repeat (Kd - 1) / 3 times:
4.1 White captures ko
4.2 Black connects ko
4.3 White connects ko
4.4 Black connects ko
5 Passes
Remarks: At (3), all dame are already filled. The "+1" during (3)
connects the extra ko of the ko fight.
Proposition 4.2.1: The score due to the algorithm is: 2/3 * (Kd - 1)
+ 3
Proof: The difference to 4.1.1 is another dame filled by Black instead
of White. QED.
Proposition 4.2.2: The score is optimal for either player.
Proof: See the proof for 4.1.2, except that now Black has at least 1
further ko threat. QED.
Case 3: D EVEN
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Strategy for each player:
If it is Black's first move, then connect ko.
Afterwards see the class of positions 3.
Algorithm:
1. Black connects ko.
Afterwards see the class of positions 3.
Proposition 4.3.1: The score due to the algorithm is: 2/3 * (Kd - 1)
+ 2
Proof: During (1), Black gets 2 points more than White. Then for the
remaining dame and kos, proposition 3.1 is applied. Because already 1
ko is connected, (Kd - 1) instead of Kd has to be inserted. Because D
is EVEN after Black's first move (when proposition 3.1 is being
applied), par(D) = 0. QED.
Proposition 4.3.2: The score is optimal for Black.
Proof: If Black tries to improve by playing a dame before connecting
his first ko, White answers by playing a dame and Black has not
achieved a score improvement because the parity of the number of
remaining dame is still EVEN. QED.
Proposition 4.3.3: The score is optimal for White.
Proof: After Black's first move, see proposition 3.2. QED.
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robert jasiek