--- /dev/null
+<html>
+<header>
+<title>Mating potential</title>
+</header>
+<body>
+<img src="./fairysmall.jpg">
+<h1>Heuristics for accounting for mating potential</h1>
+
+<h2>Introduction</h2>
+<p>
+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.
+</p>
+<p>
+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.
+</p>
+<h2>Pair-o-Max</h2>
+<p>
+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%).
+</p>
+<h2>A heuristic to account for mating potential</h2>
+<p>
+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.
+</p>
+<p>
+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.
+</p>
+<p>
+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 <b>0,3</b> 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 <b>step,moveRights</b> descriptor.)
+</p>
+<p>
+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 <b>0,-4</b> 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.
+</p>
+<p>
+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 <b>0,BONUS</b> 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.
+</p>
+<p>
+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.
+</p>
+<h2>Mating potential in jeopardy</h2>
+<p>
+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.
+</p>
+
+</body>
+</html>
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