glicko2.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
    glicko2
    ~~~~~~~

    The Glicko2 rating system.

    :copyright: (c) 2012 by Heungsub Lee
    :license: BSD, see LICENSE for more details.
"""
import math

__version__ = "0.0.dev"

from typing import List, Tuple

# pylint: disable=invalid-name,missing-function-docstring,missing-class-docstring
# pylint: disable=too-many-arguments,too-few-public-methods,consider-using-f-string

#: The actual score for win
WIN = 1.0
#: The actual score for draw
DRAW = 0.5
#: The actual score for loss
LOSS = 0.0


MU = 1500.0
PHI = 350.0
SIGMA = 0.06
TAU = 1.0
EPSILON = 0.000001


class Rating:
    def __init__(self, mu: float = MU, phi: float = PHI, sigma: float = SIGMA) -> None:
        self.mu = mu
        self.phi = phi
        self.sigma = sigma

    def __repr__(self) -> str:
        c = type(self)
        args = (c.__module__, c.__name__, self.mu, self.phi, self.sigma)
        return "%s.%s(mu=%.3f, phi=%.3f, sigma=%.3f)" % args

    def __lt__(self, other) -> bool:  # type: ignore
        return bool(self.mu < other.mu)

    def __le__(self, other) -> bool:  # type: ignore
        return bool(self.mu <= other.mu)

    def __gt__(self, other) -> bool:  # type: ignore
        return bool(self.mu > other.mu)

    def __ge__(self, other) -> bool:  # type: ignore
        return bool(self.mu >= other.mu)

    def __eq__(self, other) -> bool:  # type: ignore
        return bool(self.mu == other.mu)

    def __ne__(self, other) -> bool:  # type: ignore
        return bool(self.mu != other.mu)


class Glicko2:
    def __init__(
        self,
        mu: float = MU,
        phi: float = PHI,
        sigma: float = SIGMA,
        tau: float = TAU,
        epsilon: float = EPSILON,
    ) -> None:
        self.mu = mu
        self.phi = phi
        self.sigma = sigma
        self.tau = tau
        self.epsilon = epsilon

    def create_rating(
        self, mu: float = -1.0, phi: float = -1.0, sigma: float = -1.0
    ) -> Rating:
        if mu == -1:
            mu = self.mu
        if phi == -1:
            phi = self.phi
        if sigma == -1.0:
            sigma = self.sigma
        return Rating(mu, phi, sigma)

    def scale_down(self, rating: Rating, ratio: float = 173.7178) -> Rating:
        mu = (rating.mu - self.mu) / ratio
        phi = rating.phi / ratio
        return self.create_rating(mu, phi, rating.sigma)

    def scale_up(self, rating: Rating, ratio: float = 173.7178) -> Rating:
        mu = rating.mu * ratio + self.mu
        phi = rating.phi * ratio
        return self.create_rating(mu, phi, rating.sigma)

    @staticmethod
    def reduce_impact(rating: Rating) -> float:
        """The original form is `g(RD)`. This function reduces the impact of
        games as a function of an opponent's RD.
        """
        return 1.0 / math.sqrt(1 + (3 * rating.phi**2) / (math.pi**2))

    @staticmethod
    def expect_score(rating: Rating, other_rating: Rating, impact: float) -> float:
        return 1.0 / (1 + math.exp(-impact * (rating.mu - other_rating.mu)))

    def determine_sigma(
        self, rating: Rating, difference: float, variance: float
    ) -> float:
        """Determines new sigma."""
        phi = rating.phi
        difference_squared = difference**2
        # 1. Let a = ln(s^2), and define f(x)
        alpha = math.log(rating.sigma**2)

        def f(x: float) -> float:
            """This function is twice the conditional log-posterior density of
            phi, and is the optimality criterion.
            """
            tmp = phi**2 + variance + math.exp(x)
            _a = math.exp(x) * (difference_squared - tmp) / (2 * tmp**2)
            _b = (x - alpha) / (self.tau**2)
            return float(_a - _b)

        # 2. Set the initial values of the iterative algorithm.
        a = alpha
        if difference_squared > phi**2 + variance:
            b = math.log(difference_squared - phi**2 - variance)
        else:
            k = 1
            while f(alpha - k * math.sqrt(self.tau**2)) < 0:
                k += 1
            b = alpha - k * math.sqrt(self.tau**2)
        # 3. Let fA = f(A) and f(B) = f(B)
        f_a, f_b = f(a), f(b)
        # 4. While |B-A| > e, carry out the following steps.
        # (a) Let C = A + (A - B)fA / (fB-fA), and let fC = f(C).
        # (b) If fCfB < 0, then set A <- B and fA <- fB; otherwise, just set
        #     fA <- fA/2.
        # (c) Set B <- C and fB <- fC.
        # (d) Stop if |B-A| <= e. Repeat the above three steps otherwise.
        while abs(b - a) > self.epsilon:
            c = a + (a - b) * f_a / (f_b - f_a)
            f_c = f(c)
            if f_c * f_b < 0:
                a, f_a = b, f_b
            else:
                f_a /= 2
            b, f_b = c, f_c
        # 5. Once |B-A| <= e, set s' <- e^(A/2)
        return float(math.exp(1) ** (a / 2))

    def rate(self, rating: Rating, series: List[tuple]) -> Rating:
        # Step 2. For each player, convert the rating and RDs onto the
        #         Glicko-2 scale.
        rating = self.scale_down(rating)
        # Step 3. Compute the quantity v. This is the estimated variance of the
        #         team's/player's rating based only on game outcomes.
        # Step 4. Compute the quantity difference, the estimated improvement in
        #         rating by comparing the pre-period rating to the performance
        #         rating based only on game outcomes.
        variance_inv = 0.0
        difference = 0.0
        if not series:
            # If the team didn't play in the series, do only Step 6
            phi_star = math.sqrt(rating.phi**2 + rating.sigma**2)
            return self.scale_up(self.create_rating(rating.mu, phi_star, rating.sigma))
        for actual_score, other_rating in series:
            other_rating = self.scale_down(other_rating)
            impact = self.reduce_impact(other_rating)
            expected_score = self.expect_score(rating, other_rating, impact)
            variance_inv += impact**2 * expected_score * (1 - expected_score)
            difference += impact * (actual_score - expected_score)
        difference /= variance_inv
        variance = 1.0 / variance_inv
        # Step 5. Determine the new value, Sigma(*), ot the sigma. This
        #         computation requires iteration.
        sigma = self.determine_sigma(rating, difference, variance)
        # Step 6. Update the rating deviation to the new pre-rating period
        #         value, Phi*.
        phi_star = math.sqrt(rating.phi**2 + sigma**2)
        # Step 7. Update the rating and RD to the new values, Mu' and Phi'.
        phi = 1.0 / math.sqrt(1 / phi_star**2 + 1 / variance)
        mu = rating.mu + phi**2 * (difference / variance)
        # Step 8. Convert ratings and RDs back to original scale.
        return self.scale_up(self.create_rating(mu, phi, sigma))

    def rate_1vs1(
        self, rating1: Rating, rating2: Rating, drawn: bool = False
    ) -> Tuple[Rating, Rating]:
        return (
            self.rate(rating1, [(DRAW if drawn else WIN, rating2)]),
            self.rate(rating2, [(DRAW if drawn else LOSS, rating1)]),
        )

    def quality_1vs1(self, rating1: Rating, rating2: Rating) -> float:
        expected_score1 = self.expect_score(
            rating1, rating2, self.reduce_impact(rating1)
        )
        expected_score2 = self.expect_score(
            rating2, rating1, self.reduce_impact(rating2)
        )
        expected_score = (expected_score1 + expected_score2) / 2
        return 2 * (0.5 - abs(0.5 - expected_score))