From cc27a395394c3b802bd8ff9dac5bd9fdde3cd455 Mon Sep 17 00:00:00 2001 From: H.G. Muller Date: Thu, 28 Feb 2013 15:42:50 +0100 Subject: [PATCH] Check in Pair-o-Max web page --- mating.html | 136 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 136 insertions(+), 0 deletions(-) create mode 100644 mating.html diff --git a/mating.html b/mating.html new file mode 100644 index 0000000..e769a6c --- /dev/null +++ b/mating.html @@ -0,0 +1,136 @@ + +

+Mating potential +

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Heuristics for accounting for mating potential

+ +

Introduction

+It is quite annoying to see a Chess program that is clearly winning trade its last Pawn, +to be left with a material combination that, although much stronger than the opponent, cannot deliver checkmate. +But computers are stupid, and when they are told that a Knight is worth 3 Pawns, +they think they have a very good deal when the can trade their remaining two Pawns for the Knight in K+N+2P vs K+N. +While every Chess player of course knows that K+N vs K is a reglementary draw without even so much as a help mate. +Even K+2N vs K would not be able to force a win, +but the engine counts it happily as +6, and would prefer it over K+R vs K when given the choice. +

+This kind of stupidity can affect the value of pieces that are susceptible to such mishaps. +Not only does the engine blunder easy wins away, +but it can also miss clever opportunities to secure a draw in a losing position, +by not sacrificing material to destroy mating potential before it is too late, +e.g. N for P in K+N vs K+N+P, or R for B in K+B+2N vs K+R. +

Pair-o-Max

+Pair-o-Max is a version of Fairy-Max that was enhanced to have some more awareness of such problems. +As this required keeping track of how many pieces of each type are still around, +it could also easily award an extra score bonus for having a pair of pieces of a certain type, such as the Bishop pair. +(Hence the name Pair-o-Max.) +The standard version valued Bishops higher than Knights to prevent it from squandering away its B-pair all too easily. +This can have the undesired effect that it clings to its Bishop too much when this is no longer needed +(making it too reluctant to trade a lone Bishop for a Knight in positions where the latter threatens to do damage). +But what is worse, is that it still gives up its advantage if it has destroyed the B-pair of the opponent, +but still has its own. +No matter how much it values the Bishop, a B vs B trade is always an equal trade, +so in B+B vs B+N it sees nothing wrong with trading B for B, +unwittingly giving up half a Pawn (lowering its score expectation by a not insignificant 8%). +

A heuristic to account for mating potential

+To prevent most of this misery, Pair-o-Max uses a heuristic to discount the score of 'drawish endings'. +One quite universally valid rule is that without Pawns you need about twice as much advantage to win. +So when the leading side has no Pawns, Pair-o-Max will divide the score by 2. +This had some undesirable effects for the K+R vs K ending, +and indeed there is no reason to discount endings that are simple theoretical wins even without Pawns. +Therefore an exception is made for cases when the opponent has a bare King, +or a King + Pawns, as it is usually also trivial to gobble up the Pawns in such a case. +

+This does not solve the problem of K + N or K + 2N vs bare King yet, however. +Especially the latter, with a nominal advantage of 6-6.5 Pawns, needs a huge discount +to express the fact that it offers less winning prospects than K + P. +Therefore (pawnless) piece combinations without mating potential on the leading side +will cause the score to be divided by 8. +The general heuristic used for this is that an advantage of less than 3.5 Pawn in terms of piece material +(covering the 'minors' B and N of orthodox Chess) +will not be enough to provide a forced win. +This rule is applied in 1:0, 1:1, 2:0 and 2:1 situations (attacking:defending pieces, ignoring King and Pawns). +So it covers K+N and K+B vs K, but also K+R vs K+N, K+R+N vs K+R, K+Q+B vs K+Q, K+B+N vs K+B etc. +This might miss some more complex cases, like K+R+B vs K+N+N, +but in 2:2 situations you will in gneral not be that much ahead, and (like the 1:1 situations) +they are still subject to the division-by-2 discount for being pawnless. +

+The "minor ahead is not enough" rule needs some refinement, however. +Some unorthodox pieces are special by having mating potential against bare King. +The Woody Rook (WD) and Commoner (WF) are the best known examples of this. +If you have such a piece, being a minor ahead can be enough to secure a win; +you only have to trade the other material, to be left in a won 1:0 situation. +So the factor 8 is not applied if you have, for instance, K+WD+N vs N. +To this end light pieces can be marked as having mating potential in Pair-o-Max game-definition file, +by appending a 0,3 at the end of their move-descriptor list. +(Since 0 is an invalid step, not displacing the piece at all, this can be distinguished from a normal step,moveRights descriptor.) +

+Apart from exceptions, the general rule also needs extensions. +In particular to recognize cases like K+2N vs K, where even two minors cannot deliver mate. +The chances that this happens with a pair of pieces of the same type is far larger than that it would happen when they are of different type, +as in the latter case you have much more options for manoeuvring, +depending on which piece you use to trap the King, and which to check it. +Pair-o-Max allows at least the recognition of this "non-mating pair of equal pieces", +by marking the piece type with a 0,-4 suffix at the end of its move-descriptor list. +This makes them suffer the factor-8 discount in Pawnless end-games, +enough to suppress the advantage to below one Pawn, and thus prefer any other option where it still has an advantage. +Note that pairs of Ferzes or Wazirs, which also do not provide mating potential, +need not be marked as such, as even the value of such a pair does not exced the 350cP limit. +

+Another extension is required for 'heavy' color-bound pieces. +A single color-bound piece obviously has no mating potential, no matter how valuable it is. +And color-bound pieces can be pretty valuable. +The BD compound is worth about a Rook, and the BDD ('Adjutant' or leaky Queen) even ~7 Pawns. +Color-bound pieces also have a pair bonus, and Pair-o-Max automatically recognizes color-binding on simple pieces +(i.e. if they are not hoppers or alternators), +and awards them a pair bonus of 1/8 of their base value. +(This can be overruled by hand, by adding a 0,BONUS element at the end of their move-descriptor list, where BONUS is a bonus > 3 centi-Pawn.) +The presence of such a bonus is used by the mating-potential heuristic to recognize it as a color-bound piece, +and in 1:1 or 1:0 situations would also make it apply the factor 8. +Enough to reduce even an advantage of an Adjutant to below that of a Pawn. +

+Finally there is an exceptional case that I refer to as "tough defender". +The only example of this I am aware of is the Commoner (WF). +This is special, because it cannot be approached in any way by a King. +Due to this property it can be an excellent defender agains King + super-piece. +E.g. K+Q vs K+WF is a draw, where K+Q vs K+R is a general win. +All you have to do is keep the Commoner protected with your King, +and there is no way the Queen can harm it on her own. +In the mean time you can use it as a shield against checks, and to keep the enemy King at bay (or even chase it back with checks). +Defenders that draw in 1:1 situations even when behind far more than 350 cP in piece material +can be marked with 0,-1 at the end of their move-descriptor list. +This also is taken to imply mating potential against bare King. +

Mating potential in jeopardy

+When you still have Pawns it is never a done deal, +but you can nevertheless be in a pretty bad spot if the opponent can afford to sacrifice his least-valuable piece for your last one, +to leave you without mating potential. +Such a threat of sacrifice often can be used to hold off advance of the Pawn indefinitely. +This situation is recognized in 1:1, 1:2, 2:1 and 2:2 situations (discarding King and Pawns), +which after sacrifice become 1:0, 1:1, 2:0 and 2:1 situations that match the criteria described in the previous section. +These situations are therefore strongly discouraged for the leading side, +by dividing the score by 4. +Not as much as the factor 8 they would suffer when the last pawn is already sacrificed away +(because who knows, maybe they can shield it from that), +but stil pretty bad. +In any case, enough to make Pair-o-Max cling to its second-last Pawn like its life depended on it. +Even winning a minor for that Pawn would no longer seem an attractive deal, +e.g. in K+2N+2P vs K+B+N+P (+1) it would try to avoid conversion to K+2N+P vs K+N+P (0.75 after discount), +and prefer conversion to K+N+2P vs K+B+P (still +1) over it. +

+ + + -- 1.7.0.4