Base WDL model on material count and normalize evals dynamically

This PR proposes to change the parameter dependence of Stockfish's
internal WDL model from full move counter to material count. In addition
it ensures that an evaluation of 100 centipawns always corresponds to a
50% win probability at fishtest LTC, whereas for master this holds only
at move number 32. See also
https://github.com/official-stockfish/Stockfish/pull/4920 and the
discussion therein.

The new model was fitted based on about 340M positions extracted from
5.6M fishtest LTC games from the last three weeks, involving SF versions
from e67cc979fd (SF 16.1) to current
master.

The involved commands are for
[WDL_model](https://github.com/official-stockfish/WDL_model) are:
```
./updateWDL.sh --firstrev e67cc979fd
python scoreWDL.py updateWDL.json --plot save --pgnName update_material.png --momType "material" --momTarget 58 --materialMin 10 --modelFitting optimizeProbability
```

The anchor `58` for the material count value was chosen to be as close
as possible to the observed average material count of fishtest LTC games
at move 32 (`43`), while not changing the value of
`NormalizeToPawnValue` compared to the move-based WDL model by more than
1.

The patch only affects the displayed cp and wdl values.

closes https://github.com/official-stockfish/Stockfish/pull/5121

No functional change

src/evaluate.cpp

@@ -92,11 +92,11 @@ std::string Eval::trace(Position& pos, const Eval::NNUE::Networks& networks) {
 
     Value v = networks.big.evaluate(pos, false);
     v       = pos.side_to_move() == WHITE ? v : -v;
-    ss << "NNUE evaluation        " << 0.01 * UCI::to_cp(v) << " (white side)\n";
+    ss << "NNUE evaluation        " << 0.01 * UCI::to_cp(v, pos) << " (white side)\n";
 
     v = evaluate(networks, pos, VALUE_ZERO);
     v = pos.side_to_move() == WHITE ? v : -v;
-    ss << "Final evaluation       " << 0.01 * UCI::to_cp(v) << " (white side)";
+    ss << "Final evaluation       " << 0.01 * UCI::to_cp(v, pos) << " (white side)";
     ss << " [with scaled NNUE, ...]";
     ss << "\n";
 

src/nnue/nnue_misc.cpp

@@ -54,11 +54,11 @@ void hint_common_parent_position(const Position& pos, const Networks& networks)
 namespace {
 // Converts a Value into (centi)pawns and writes it in a buffer.
 // The buffer must have capacity for at least 5 chars.
-void format_cp_compact(Value v, char* buffer) {
+void format_cp_compact(Value v, char* buffer, const Position& pos) {
 
     buffer[0] = (v < 0 ? '-' : v > 0 ? '+' : ' ');
 
-    int cp = std::abs(UCI::to_cp(v));
+    int cp = std::abs(UCI::to_cp(v, pos));
     if (cp >= 10000)
     {
         buffer[1] = '0' + cp / 10000;
@@ -90,9 +90,9 @@ void format_cp_compact(Value v, char* buffer) {
 
 
 // Converts a Value into pawns, always keeping two decimals
-void format_cp_aligned_dot(Value v, std::stringstream& stream) {
+void format_cp_aligned_dot(Value v, std::stringstream& stream, const Position& pos) {
 
-    const double pawns = std::abs(0.01 * UCI::to_cp(v));
+    const double pawns = std::abs(0.01 * UCI::to_cp(v, pos));
 
     stream << (v < 0   ? '-'
                : v > 0 ? '+'
@@ -114,7 +114,7 @@ std::string trace(Position& pos, const Eval::NNUE::Networks& networks) {
         board[row][8 * 8 + 1] = '\0';
 
     // A lambda to output one box of the board
-    auto writeSquare = [&board](File file, Rank rank, Piece pc, Value value) {
+    auto writeSquare = [&board, &pos](File file, Rank rank, Piece pc, Value value) {
         const int x = int(file) * 8;
         const int y = (7 - int(rank)) * 3;
         for (int i = 1; i < 8; ++i)
@@ -125,7 +125,7 @@ std::string trace(Position& pos, const Eval::NNUE::Networks& networks) {
         if (pc != NO_PIECE)
             board[y + 1][x + 4] = PieceToChar[pc];
         if (value != VALUE_NONE)
-            format_cp_compact(value, &board[y + 2][x + 2]);
+            format_cp_compact(value, &board[y + 2][x + 2], pos);
     };
 
     // We estimate the value of each piece by doing a differential evaluation from
@@ -180,13 +180,13 @@ std::string trace(Position& pos, const Eval::NNUE::Networks& networks) {
     {
         ss << "|  " << bucket << "        ";
         ss << " |  ";
-        format_cp_aligned_dot(t.psqt[bucket], ss);
+        format_cp_aligned_dot(t.psqt[bucket], ss, pos);
         ss << "  "
            << " |  ";
-        format_cp_aligned_dot(t.positional[bucket], ss);
+        format_cp_aligned_dot(t.positional[bucket], ss, pos);
         ss << "  "
            << " |  ";
-        format_cp_aligned_dot(t.psqt[bucket] + t.positional[bucket], ss);
+        format_cp_aligned_dot(t.psqt[bucket] + t.positional[bucket], ss, pos);
         ss << "  "
            << " |";
         if (bucket == t.correctBucket)

src/search.cpp

@@ -155,7 +155,8 @@ void Search::Worker::start_searching() {
     {
         rootMoves.emplace_back(Move::none());
         sync_cout << "info depth 0 score "
-                  << UCI::value(rootPos.checkers() ? -VALUE_MATE : VALUE_DRAW) << sync_endl;
+                  << UCI::to_score(rootPos.checkers() ? -VALUE_MATE : VALUE_DRAW, rootPos)
+                  << sync_endl;
     }
     else
     {
@@ -1898,10 +1899,10 @@ std::string SearchManager::pv(const Search::Worker&     worker,
 
         ss << "info"
            << " depth " << d << " seldepth " << rootMoves[i].selDepth << " multipv " << i + 1
-           << " score " << UCI::value(v);
+           << " score " << UCI::to_score(v, pos);
 
         if (worker.options["UCI_ShowWDL"])
-            ss << UCI::wdl(v, pos.game_ply());
+            ss << UCI::wdl(v, pos);
 
         if (i == pvIdx && !tb && updated)  // tablebase- and previous-scores are exact
             ss << (rootMoves[i].scoreLowerbound

src/uci.cpp

@@ -28,6 +28,7 @@
 #include <memory>
 #include <optional>
 #include <sstream>
+#include <utility>
 #include <vector>
 
 #include "benchmark.h"
@@ -44,9 +45,8 @@
 
 namespace Stockfish {
 
-constexpr auto StartFEN             = "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1";
-constexpr int  NormalizeToPawnValue = 356;
-constexpr int  MaxHashMB            = Is64Bit ? 33554432 : 2048;
+constexpr auto StartFEN  = "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1";
+constexpr int  MaxHashMB = Is64Bit ? 33554432 : 2048;
 
 
 namespace NN = Eval::NNUE;
@@ -338,15 +338,43 @@ void UCI::position(Position& pos, std::istringstream& is, StateListPtr& states)
     }
 }
 
-int UCI::to_cp(Value v) { return 100 * v / NormalizeToPawnValue; }
+namespace {
+std::pair<double, double> win_rate_params(const Position& pos) {
 
-std::string UCI::value(Value v) {
+    int material = pos.count<PAWN>() + 3 * pos.count<KNIGHT>() + 3 * pos.count<BISHOP>()
+                 + 5 * pos.count<ROOK>() + 9 * pos.count<QUEEN>();
+
+    // The fitted model only uses data for material counts in [10, 78], and is anchored at count 58.
+    double m = std::clamp(material, 10, 78) / 58.0;
+
+    // Return a = p_a(material) and b = p_b(material), see github.com/official-stockfish/WDL_model
+    constexpr double as[] = {-185.71965483, 504.85014385, -438.58295743, 474.04604627};
+    constexpr double bs[] = {89.23542728, -137.02141296, 73.28669021, 47.53376190};
+
+    double a = (((as[0] * m + as[1]) * m + as[2]) * m) + as[3];
+    double b = (((bs[0] * m + bs[1]) * m + bs[2]) * m) + bs[3];
+
+    return {a, b};
+}
+
+// The win rate model is 1 / (1 + exp((a - eval) / b)), where a = p_a(material) and b = p_b(material).
+// It fits the LTC fishtest statistics rather accurately.
+int win_rate_model(Value v, const Position& pos) {
+
+    auto [a, b] = win_rate_params(pos);
+
+    // Return the win rate in per mille units, rounded to the nearest integer.
+    return int(0.5 + 1000 / (1 + std::exp((a - double(v)) / b)));
+}
+}
+
+std::string UCI::to_score(Value v, const Position& pos) {
     assert(-VALUE_INFINITE < v && v < VALUE_INFINITE);
 
     std::stringstream ss;
 
     if (std::abs(v) < VALUE_TB_WIN_IN_MAX_PLY)
-        ss << "cp " << to_cp(v);
+        ss << "cp " << to_cp(v, pos);
     else if (std::abs(v) <= VALUE_TB)
     {
         const int ply = VALUE_TB - std::abs(v);  // recompute ss->ply
@@ -358,6 +386,30 @@ std::string UCI::value(Value v) {
     return ss.str();
 }
 
+// Turns a Value to an integer centipawn number,
+// without treatment of mate and similar special scores.
+int UCI::to_cp(Value v, const Position& pos) {
+
+    // In general, the score can be defined via the the WDL as
+    // (log(1/L - 1) - log(1/W - 1)) / ((log(1/L - 1) + log(1/W - 1))
+    // Based on our win_rate_model, this simply yields v / a.
+
+    auto [a, b] = win_rate_params(pos);
+
+    return std::round(100 * int(v) / a);
+}
+
+std::string UCI::wdl(Value v, const Position& pos) {
+    std::stringstream ss;
+
+    int wdl_w = win_rate_model(v, pos);
+    int wdl_l = win_rate_model(-v, pos);
+    int wdl_d = 1000 - wdl_w - wdl_l;
+    ss << " wdl " << wdl_w << " " << wdl_d << " " << wdl_l;
+
+    return ss.str();
+}
+
 std::string UCI::square(Square s) {
     return std::string{char('a' + file_of(s)), char('1' + rank_of(s))};
 }
@@ -383,41 +435,6 @@ std::string UCI::move(Move m, bool chess960) {
     return move;
 }
 
-namespace {
-// The win rate model returns the probability of winning (in per mille units) given an
-// eval and a game ply. It fits the LTC fishtest statistics rather accurately.
-int win_rate_model(Value v, int ply) {
-
-    // The fitted model only uses data for moves in [8, 120], and is anchored at move 32.
-    double m = std::clamp(ply / 2 + 1, 8, 120) / 32.0;
-
-    // The coefficients of a third-order polynomial fit is based on the fishtest data
-    // for two parameters that need to transform eval to the argument of a logistic
-    // function.
-    constexpr double as[] = {-1.06249702, 7.42016937, 0.89425629, 348.60356174};
-    constexpr double bs[] = {-5.33122190, 39.57831533, -90.84473771, 123.40620748};
-
-    // Enforce that NormalizeToPawnValue corresponds to a 50% win rate at move 32.
-    static_assert(NormalizeToPawnValue == int(0.5 + as[0] + as[1] + as[2] + as[3]));
-
-    double a = (((as[0] * m + as[1]) * m + as[2]) * m) + as[3];
-    double b = (((bs[0] * m + bs[1]) * m + bs[2]) * m) + bs[3];
-
-    // Return the win rate in per mille units, rounded to the nearest integer.
-    return int(0.5 + 1000 / (1 + std::exp((a - double(v)) / b)));
-}
-}
-
-std::string UCI::wdl(Value v, int ply) {
-    std::stringstream ss;
-
-    int wdl_w = win_rate_model(v, ply);
-    int wdl_l = win_rate_model(-v, ply);
-    int wdl_d = 1000 - wdl_w - wdl_l;
-    ss << " wdl " << wdl_w << " " << wdl_d << " " << wdl_l;
-
-    return ss.str();
-}
 
 Move UCI::to_move(const Position& pos, std::string& str) {
     if (str.length() == 5)

src/uci.h

@@ -42,11 +42,11 @@ class UCI {
 
     void loop();
 
-    static int         to_cp(Value v);
-    static std::string value(Value v);
+    static int         to_cp(Value v, const Position& pos);
+    static std::string to_score(Value v, const Position& pos);
     static std::string square(Square s);
     static std::string move(Move m, bool chess960);
-    static std::string wdl(Value v, int ply);
+    static std::string wdl(Value v, const Position& pos);
     static Move        to_move(const Position& pos, std::string& str);
 
     static Search::LimitsType parse_limits(const Position& pos, std::istream& is);