Endgame Strategy (04)

[Robert Jasiek]

Preface: The now solved class of positions 3 has either player to
move, an arbitrary number of two-sided dame, arbitrary numbers of ko
threats, and a difference of kos open for Black and kos open for White
that is divisble by 3.

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Parity effect of D:

par(D) := 0 if D EVEN
1 if D ODD && Black to move
-1 if D ODD && White to move

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CLASS OF POSITIONS 3

Either player to move // either player
D > 0 // with two-sided dame
K > 0
Kb >= Kw >= 0
Tb >= 0
Tw >= 0
Kd MOD 3 = 0 // the Kd case

STRATEGY FOR EACH PLAYER

1. answer ko threat
2. answer dame play by dame play
3. answer ko play by connect ko
4. answer ko play by capture ko if legal
5. play dame
6. connect ko
7. capture ko if legal
8. pass

Remark: The strategy does not advise either player to play any ko
threat, however, either player's strategy has to react to the
opponent's ko threats if the opponent should not follow the strategy
but should try to confuse the player by making (possibly virtual) ko
threats.

Algorithm:
1 Repeat D times: play dame
2 Repeat 2 * Kw times: connect ko
3 Choose possible variation A or B and repeat Kd/3 times:
3A.1 Black connects ko 3B.1 White captures ko
3A.2 White captures ko 3B.2 Black connects ko
3A.3 Black connects ko 3B.3 White connects ko
3A.4 White connects ko 3B.4 Black connects ko
4 Passes

Proposition 3.1: The score due to the algorithm is: 2/3 * Kd +
par(D)

Proof: (1) produces the score par(D). For (2) to (4), see propositions
1.1.1 and 2.1.

Remarks: The score is independent of numbers of ko threats. A ko fight
does not occur.

Proposition 3.2: The score is optimal for either player.

Proof: By playing dame before ko plays, the starting player takes the
tedomari if D is ODD and later the moving player takes a tedomari
whenever D is ODD. By answering dame plays or ko plays, respectively,
all dame together and all ko intersections together are treated
separately from each other. Thereby the classes of positions 1.1 and
2.1, which have the same score regardless of the starting player, can
be applied to ko play here. Not answering a ko play might lead to a
disadvantage depending on Td, but it does not lead to an advantage;
details left as exercise. Likewise not answering a dame play
(especially interesting: if previously their remaining number has been
EVEN; for ODD see above), because a ko play instead will be answered
by a ko play and the player having not answered the dame play is then
the next with a turn to make that dame play. QED.

--
robert jasiek