UCI_Elo implementation (#2225)

This exploits the recent fractional Skill Level, and is a result from some discussion in #2221 and the older #758.

Basically, if UCI_LimitStrength is set, it will internally convert UCI_Elo to a matching fractional Skill Level.
The Elo estimate is based on games at  TC 60+0.6, Hash 64Mb, 8moves_v3.pgn, rated with Ordo, anchored to goldfish1.13 (CCRL 40/4 ~2000).
Note that this is mostly about internal consistency, the anchoring to CCRL is a bit weak, e.g. within this tournament,
goldfish and sungorus only have a 200Elo difference, their rating difference on CCRL is 300Elo.

I propose that we continue to expose 'Skill Level' as an UCI option, for backwards compatibility.

The result of a tournament under those conditions are given by the following table, where the player name reflects the UCI_Elo.

   # PLAYER          :  RATING  ERROR  POINTS  PLAYED   (%)  CFS(%)
   1 Elo2837         :  2792.2   50.8   536.5     711    75     100
   2 Elo2745         :  2739.0   49.0   487.5     711    69     100
   3 Elo2654         :  2666.4   49.2   418.0     711    59     100
   4 Elo2562         :  2604.5   38.5   894.5    1383    65     100
   5 Elo2471         :  2515.2   38.1   651.5     924    71     100
   6 Elo2380         :  2365.9   35.4   478.5     924    52     100
   7 Elo2289         :  2290.0   28.0   864.0    1596    54     100
   8 sungorus1.4     :  2204.9   27.8   680.5    1596    43      60
   9 Elo2197         :  2201.1   30.1   523.5     924    57     100
  10 Elo2106         :  2103.8   24.5   730.5    1428    51     100
  11 Elo2014         :  2030.5   30.3   377.5     756    50      98
  12 goldfish1.13    :  2000.0   ----   511.0    1428    36     100
  13 Elo1923         :  1928.5   30.9   641.5    1260    51     100
  14 Elo1831         :  1829.0   42.1   370.5     756    49     100
  15 Elo1740         :  1738.3   42.9   277.5     756    37     100
  16 Elo1649         :  1625.0   42.1   525.5    1260    42     100
  17 Elo1558         :  1521.5   49.9   298.0     756    39     100
  18 Elo1467         :  1471.3   51.3   246.5     756    33     100
  19 Elo1375         :  1407.1   51.9   183.0     756    24     ---

It can be observed that all set Elos correspond within the error bars with the observed Ordo rating.

No functional change

Readme.md

@@ -55,7 +55,16 @@ Currently, Stockfish has the following UCI options:
     Leave at 1 for best performance.
 
   * #### Skill Level
-    Lower the Skill Level in order to make Stockfish play weaker.
+    Lower the Skill Level in order to make Stockfish play weaker (see also UCI_LimitStrength).
+    Internally, MultiPV is enabled, and with a certain probability depending on the Skill Level a
+    weaker move will be played.
+
+  * #### UCI_LimitStrength
+    Enable weaker play aiming for an Elo rating as set by UCI_Elo. This option overrides Skill Level.
+
+  * #### UCI_Elo
+    If enabled by UCI_LimitStrength, aim for an engine strength of the given Elo.
+    This Elo rating has been calibrated at a time control of 60s+0.6s and anchored to CCRL 40/4.
 
   * #### Move Overhead
     Assume a time delay of x ms due to network and GUI overheads. This is useful to

src/search.cpp

@@ -271,7 +271,7 @@ void MainThread::search() {
   // Check if there are threads with a better score than main thread
   if (    Options["MultiPV"] == 1
       && !Limits.depth
-      && !Skill(Options["Skill Level"]).enabled()
+      && !(Skill(Options["Skill Level"]).enabled() || Options["UCI_LimitStrength"])
       &&  rootMoves[0].pv[0] != MOVE_NONE)
   {
       std::map<Move, int64_t> votes;
@@ -335,11 +335,18 @@ void Thread::search() {
   beta = VALUE_INFINITE;
 
   multiPV = Options["MultiPV"];
+
   // Pick integer skill levels, but non-deterministically round up or down
   // such that the average integer skill corresponds to the input floating point one.
+  // UCI_Elo is converted to a suitable fractional skill level, using anchoring
+  // to CCRL Elo (goldfish 1.13 = 2000) and a fit through Ordo derived Elo
+  // for match (TC 60+0.6) results spanning a wide range of k values.
   PRNG rng(now());
-  int intLevel = int(Options["Skill Level"]) +
-        ((Options["Skill Level"] - int(Options["Skill Level"])) * 1024 > rng.rand<unsigned>() % 1024  ? 1 : 0);
+  double floatLevel = Options["UCI_LimitStrength"] ?
+                        clamp(std::pow((Options["UCI_Elo"] - 1346.6) / 143.4, 1 / 0.806), 0.0, 20.0) :
+                        double(Options["Skill Level"]);
+  int intLevel = int(floatLevel) +
+                 ((floatLevel - int(floatLevel)) * 1024 > rng.rand<unsigned>() % 1024  ? 1 : 0);
   Skill skill(intLevel);
 
   // When playing with strength handicap enable MultiPV search that we will

src/ucioption.cpp

@@ -74,6 +74,8 @@ void init(OptionsMap& o) {
   o["nodestime"]             << Option(0, 0, 10000);
   o["UCI_Chess960"]          << Option(false);
   o["UCI_AnalyseMode"]       << Option(false);
+  o["UCI_LimitStrength"]     << Option(false);
+  o["UCI_Elo"]               << Option(1350, 1350, 2850);
   o["SyzygyPath"]            << Option("<empty>", on_tb_path);
   o["SyzygyProbeDepth"]      << Option(1, 1, 100);
   o["Syzygy50MoveRule"]      << Option(true);