Glicko-2 C#

Glicko2.cs

// An implementation of the Glicko-2 system
// http://www.glicko.net/glicko/glicko2.pdf

// Copyright (c) 2023 Evan "cosmonaut" Hemsley

// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:

// The above copyright notice and this permission notice shall be included in all
// copies or substantial portions of the Software.

// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
// SOFTWARE.

using System;

// Note that this implementation assumes a rating period of 1
// This simplifies rating updates in arbitrary online matchmaking
public static class Glicko2
{
	// Algorithmic constants, don't touch
	const double ScaleConstant = 173.7178;
	const double ConvergenceTolerance = 0.000001;

	// Can modify defaults to suit your rating system's needs
	const double DefaultRating = 1500;
	const double DefaultDeviation = 350;
	const double DefaultVolatility = 0.06;

	// Constrains changes in volatility over time
	// Reasonable choices are values between 0.3 and 1.2
	const double Tau = 0.5;

	public readonly record struct Player(
		double Rating,
		double Deviation,
		double Volatility
	) {
		public static Player Unranked => new Player(DefaultRating, DefaultDeviation, DefaultVolatility);

		// The lower bound of the rating interval
		public double LowestRating => Rating - (Deviation * 2);

		// The upper bound of the rating interval
		public double HighestRating => Rating + (Deviation * 2);

		// The percentage confidence that the interval contains the true rating
		public double Confidence => 1 - Volatility;
	}

	// Score is 1 for player win, 0 for player loss
	public static Player UpdateRating(
		Player playerRating,
		Player opponentRating,
		double score
	) {
		var (scaledRating, scaledDeviation) = Scale(playerRating.Rating, playerRating.Deviation);
		var (opponentScaledRating, opponentScaledDeviation) = Scale(opponentRating.Rating, opponentRating.Deviation);

		var estimatedVariance = EstimatedVariance(scaledRating, opponentScaledRating, opponentScaledDeviation);
		var estimatedImprovement = EstimatedImprovement(estimatedVariance, score, scaledRating, opponentScaledRating, opponentScaledDeviation);

		var updatedVolatility = Volatility(playerRating.Volatility, scaledDeviation, estimatedVariance, estimatedImprovement);
		var updatedDeviation = RatingDeviation(updatedVolatility, scaledDeviation);
		var (updatedScaledRating, updatedScaledDeviation) =
			NewRating(scaledRating, updatedDeviation, estimatedVariance, score, opponentScaledRating, opponentScaledDeviation);

		var (rating, deviation) = Unscale(updatedScaledRating, updatedScaledDeviation);
		return new Player(rating, deviation, updatedVolatility);
	}

	private static (double, double) Scale(
		double rating,
		double deviation
	) {
		var mu = (rating - DefaultRating) / ScaleConstant;
		var phi = deviation / ScaleConstant;
		return (mu, phi);
	}

	private static (double, double) Unscale(
		double mu,
		double phi
	) {
		var rating = ScaleConstant * mu + DefaultRating;
		var deviation = ScaleConstant * phi;
		return (rating, deviation);
	}

	private static double G(double phi) => 1 / Math.Sqrt(1 + 3 * Math.Pow(phi, 2) / Math.Pow(Math.PI, 2));
	private static double E(double mu, double opponentMu, double opponentPhi) => 1 / (1 + Math.Exp(-G(opponentPhi) * (mu - opponentMu)));

	private static double EstimatedVariance(double scaledRating, double opponentScaledRating, double opponentScaledDeviation)
	{
		var gPhi = G(opponentScaledDeviation);
		var eValue = E(scaledRating, opponentScaledRating, opponentScaledDeviation);
		return 1 / (gPhi * gPhi * eValue * (1 - eValue));
	}

	private static double EstimatedImprovement(double estimatedVariance, double score, double scaledRating, double opponentScaledRating, double opponentScaledDeviation)
	{
		return estimatedVariance * G(opponentScaledDeviation) * (score - E(scaledRating, opponentScaledRating, opponentScaledDeviation));
	}

	private static double VolatilityFunction(double x, double scaledDeviation, double estimatedVariance, double estimatedImprovement, double a)
	{
		var phi2 = scaledDeviation * scaledDeviation;
		var d2 = estimatedImprovement * estimatedImprovement;

		var ex = Math.Exp(x);
		var a2 = phi2 + estimatedVariance + ex;
		var p2 = (x - a) / (Tau * Tau);
		var p1 = (ex * (d2 - phi2 - estimatedVariance - ex)) / (2 * a2 * a2);
		return p1 - p2;
	}

	private static double Volatility(double currentVolatility, double scaledDeviation, double estimatedVariance, double estimatedImprovement)
	{
		var a = Math.Log(currentVolatility * currentVolatility);
		var originalA = a;

		double b;
		if (estimatedImprovement * estimatedImprovement > scaledDeviation * scaledDeviation + estimatedVariance)
		{
			b = Math.Log(estimatedImprovement * estimatedImprovement - scaledDeviation * scaledDeviation - estimatedVariance);
		}
		else
		{
			var k = 1;
			var x = a - k * Tau;
			while (VolatilityFunction(x, scaledDeviation, estimatedVariance, estimatedImprovement, originalA) < 0)
			{
				k += 1;
			}

			b = a - k * Tau;
		}

		var fa = VolatilityFunction(a, scaledDeviation, estimatedVariance, estimatedImprovement, originalA);
		var fb = VolatilityFunction(b, scaledDeviation, estimatedVariance, estimatedImprovement, originalA);

		while (Math.Abs(b - a) > ConvergenceTolerance)
		{
			var c = a + (a - b) * fa / (fb - fa);
			var fc = VolatilityFunction(c, scaledDeviation, estimatedVariance, estimatedImprovement, originalA);

			if (fc * fb <= 0)
			{
				a = b;
				fa = fb;
			}
			else
			{
				fa /= 2;
			}

			b = c;
			fb = fc;
		}

		return Math.Exp(a / 2);
	}

	private static double RatingDeviation(double updatedVolatility, double scaledDeviation)
	{
		return Math.Sqrt(updatedVolatility * updatedVolatility + scaledDeviation * scaledDeviation);
	}

	private static (double, double) NewRating(
		double scaledRating,
		double updatedDeviation,
		double estimatedVariance,
		double score,
		double opponentScaledRating,
		double opponentScaledDeviation
	) {
		var phiPrime = 1 / Math.Sqrt(1 / (updatedDeviation * updatedDeviation) + (1 / estimatedVariance));
		var muPrime = scaledRating + phiPrime * phiPrime * (G(opponentScaledDeviation) * (score - E(scaledRating, opponentScaledRating, opponentScaledDeviation)));

		return (muPrime, phiPrime);
	}
}