Tapering the evaluation

Understanding the concept

In the previous chapters where we talked about the PSQT's it was already mentioned a few times that we would be 'tapering the evaluation.' This chapter discusses this concept using the PSQT's, but it can be applied to any other evaluation term in the future. Let's say, we have a bonus for having the bishop pair:

const BISHOP_PAIR: i16 = 10;

In this case, having both bishops will increase the evaluation for that side by +10 centipawns. However, the more the board opens up, the more advantageous the bishop pair becomes. In the endgame, the advantage may be (for example) as high as 40 centipawns. Thus the importance of this evaluation term increases the more we are going into the endgame. If we want to encode this, we could do it as follows, so we don't have to make an extra variable:

pub struct W(pub i16, pub i16);
const BISHOP_PAIR: W = W(10, 40);

Here we defined a type called W.

Sidenote: The type declaration comes from "weight", which is the chess engine/machine learning name of how much impact a certain feature has. We're using "W" instead of "Weight" because we need it later, in the PSQT, and the table would become extremely wide. You'll see in a moment.

The type can hold two i16 values, so we can now put TWO values in the BISHOP_PAIR constant. The first value can be read by using "BISHOP_PAIR.0", and the second one is "BISHOP_PAIR.1"; The first value is the one for the opening, the second one is the one for the endgame.

When the game is in the opening, we use 100% of the first value and 0% of the second value. This means that in the opening the BISHOP_PAIR value is calculated like this:

let value = BISHOP_PAIR.0 * 1 + BISHOP_PAIR.1 * 0

This boils down to 10 + 0 = 10.

However, when we are half-way through the game, so exactly in the middle game, we would use 50% of the first value and 50% of the second, which would give us this:

let value = BISHOP_PAIR.0 * 0.5 + BISHOP_PAIR.1 * 0.5

This will come down to (10 * 0.5) + (40 * 0.5), which is 5 + 20 = 25. As you can see, the value of the bishop pair is increasing as we approach the endgame. You can guess that the value will end up at 0% opening and 100% endgame, which will turn out to be 40, at some point in the game.

This process of calculating in-between values is called interpolation. The chess engine world also calls it tapering, because the opening value gradually tapers into the endgame value as the game progresses.

This works exactly the same within a PSQT. The only difference is that the PSQT encodes a piece-on-square relationship. Thus, the QUEEN PSQT has 64 elements (because of 64 squares), and each element has two values (opening and endgame), and there will be 6 PSQT's (because there are 6 piece types). The QUEEN PSQT may look like this:

pub struct W(pub i16, pub i16);
const QUEEN: Psqt = 
    W(865,918), W(902,937), W(922,943), W(911,945), W(964,934), W(948,926), W(933,924), W(928,942),
    W(886,907), W(865,945), W(903,946), W(921,951), W(888,982), W(951,933), W(923,928), W(940,912),
    W(902,896), W(901,921), W(907,926), W(919,967), W(936,963), W(978,937), W(965,924), W(966,915),
    W(881,926), W(885,944), W(897,939), W(894,962), W(898,983), W(929,957), W(906,981), W(915,950),
    W(907,893), W(884,949), W(899,942), W(896,970), W(904,952), W(906,956), W(912,953), W(911,936),
    W(895,911), W(916,892), W(900,933), W(902,928), W(904,934), W(912,942), W(924,934), W(917,924),
    W(874,907), W(899,898), W(918,883), W(908,903), W(915,903), W(924,893), W(911,886), W(906,888),
    W(906,886), W(899,887), W(906,890), W(918,872), W(898,916), W(890,890), W(878,906), W(858,879),

Sidenote: See what I meant above? If we had used "Weight" instead of "W", this table would have been massively wide.

Remember that the PSQT is shown from white's point of view, so the lower left corner is A1. When the queen is in the corner on A1 in the opening, it's worth is 906 cp (centipawns), but if its there in the endgame, its worth drops to 886 cp. This means that your queen will lose value if you leave it in the corner in the endgame. When you look at square D4 (4th from the left, 4th from the bottom), you'll see that the queen's worth is 896 in the opening, but 970 cp in the endgame. This makes sense: you DON'T want to have your queen in the middle of the board in the middle game because it's vulnerable there, but in the endgame, the queen would control the entire board from that square.

So, by applying the PSQT to the position, we now have an opening value and an endgame value for each piece. During the game we can calculate the value of the piece, on a certain square, by interpolating between these two values. We do so for every piece on every square. Now let's look at how to implement this.


I am assuming you are writing your evaluation in such a way that material values are folded into the PSQT (like with the queen above), and that each element in the PSQT holds two values. You could maintain separate material values and use a QUEEN_OT and QUEEN_ET table (for OpeningTable and EndgameTable), but all those extra variables just makes it more laborious to work with. Putting the material and both values into one array is much more convenient, and that is the reason why Rustic switched to this format between Alpha 3 and version 4.