4.2. AT-LEAST-DRAW ANDAT-MOST-DRAW 47
0
= 0
0
=
max
min
Figure 4.7: With simple prop- agation the start position of this graph is not solved.
≥
= 0
=
=
max
min
Figure 4.8: With
at-least-draw /
at-most-draw values the start position of this graph is solved.
≥
0
=
≥
+9
=
max max
Figure 4.9: The start position on the left is probably adraw, the start position on the right is probably a win, but their values are the same.
≥
|
0= 0
≥
|
+9= +9
max max
Figure 4.10: The same graphs as in Figure 4.9, now with attributed at-least-draw values.
Figure 4.11 shows the Hasse diagram of a value representation with at- tributedat-least-draw andat-most-draw values, and withhmin =−2 and hmax = +2. ≥|h means that the player to move has a forced draw, but has the option to move into a position with value h instead. ≤|h means that the opponent has a forced draw, but has the option to move into a position with value h. The values [≥|hmin, . . . ,≥|hmax] and [≤|hmin, . . . ,≤|hmax] are continuous ranges of values between draw and win and loss and draw re- spectively, just in the same way as [hmin, . . . , hmax] is a continuous range of values between loss and win. Therefore these new values give us a more accurate estimate of the exact value than what we had before.
+
0 -1 +2
-2 +1
-
=
≥|-1≥|0
≥|+2
≥|-2
≥|+1
≤≤|0|-1
≤|+2
≤|-2
≤|+1
Figure 4.11: The Hasse diagram for a value representation with attributed at-least-draw and at-most-draw values, and with hmin =−2 and hmax = +2. ≥|hmeans that the player to move has a forced draw, but has the option to move into a position with value h instead. ≤|h means that the opponent has a forced draw, but has the option to move into a position with valueh.
The term ‘at least draw’ is sometimes used with a different semantics.
For example Chess positions with a king and a pawn against a king (KPK) are said to be ‘at least draw’ for the player with the pawn: whatever he does, no mistake is bad enough to make him lose.
Our definition of at-least-draw is weaker. The value ≥|hmust be read as ‘draw exclusive-or h’, and if a player chooses h then he might still lose.
However, the name at-least-draw is still justified, because whatever the exact value of the position with the currently heuristic value h will be, the exact value of≥|h will be either win ordraw.
4.2. AT-LEAST-DRAW ANDAT-MOST-DRAW 49 Although the problem of incomparability between drawand heuristic val- ues is now solved, Figure 4.11 shows that now two new types of incompara- bility call for a propagation heuristic:
• ≥|a and h are incomparable if a < h. This case is simple to solve.
Remember that ≥|a means that the player to move can force a draw, but has the option to move to a position with a heuristic value a (and a is the best heuristic value that is reachable). Therefore it is natural to propagate≥|max(a, h) in this case.
• ≤|a and h are incomparable if a > h. We know that the propagated value of≤|a andh must be a heuristic value, because we cannot derive any bounds on the exact value from ≤|a and h. Now ≤|a means that the opponent can force adraw, but has the option to move to a position with a heuristic valuea (andais the best heuristic value that is reach- able). Therefore when we compare ≤|a and h, a is an option for the opponent andh is an option for the player to move, the two cannot be related to each other. To solve this problem we devised the following propagation heuristic: we assume that the opponent will always play the forced draw if a≥0, but will always avoid the draw if a < 0. The propagated value will be max(h, min(a,0)).
Table 4.3 summarizes the propagation rules for a value representation with attributed at-least-draw and at-most-draw values.
propagate(+, x) → +
propagate(−, x) → x
propagate(h1, h2) → max(h1, h2) propagate(=,≥|a) → ≥|a
propagate(=,≤|a) → =
propagate(≥|a1,≥|a2) → ≥|max(a1, a2) propagate(≥|a1,≤|a2) → ≥|a1 propagate(≤|a1,≤|a2) → ≤|min(a1, a2)
propagate(=, h) → ≥|h
propagate(h,≥|a) → ≥|max(h, a) propagate(h,≤|a) → max(h, min(a,0))
Table 4.3: Propagation rules withat-least-draw andat-most-draw values.
The last three rules define the propagation for incomparable values.
4.2.1 At-least-draw, At-most-draw and Opening Book Construction
There are two goals in opening book construction: to improve the playing strength at tournament games, and to solve a game. In a situation like in Figure 4.12 we have a conflict of interests.
≤ | a h
0
max
Figure 4.12: Conflicting expansion priorities (for h < 0). To improve the playing strength use drop-out expansion. To solve the start node use con- spiracy numbers.
Improve playing strength The at-most-draw position is the most likely move and should get the higher priority for expansion. This is exactly what drop-out expansion does: ≤|a is treated as 0, and the expansion priority of the two moves depends onhand the depths of the subtrees.
However, the value of the start node does not depend on a, therefore expanding the left node is unlikely to influence the value of the start node.
Solve the game There are several ways how the start node of Figure 4.12 can be partially or completely solved. For a partial solution we can prove the left successor to be a draw, or prove that the right successor is either at-most-draw or a loss. For a complete solution we can either prove the right move to be awinor adraw. What we would like to do is to minimize the number of nodes that have to be expanded.
This is best done with the conspiracy numbers heuristic [31, 43].