312 lines
9.4 KiB
Elixir
312 lines
9.4 KiB
Elixir
defmodule Glicko do
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@moduledoc """
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Provides the implementation of the Glicko rating system.
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See the [specification](http://www.glicko.net/glicko/glicko2.pdf) for implementation details.
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## Usage
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Get a player's new rating after a series of matches in a rating period.
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iex> results = [Result.new(Player.new_v1([rating: 1400, rating_deviation: 30]), :win),
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...> Result.new(Player.new_v1([rating: 1550, rating_deviation: 100]), :loss),
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...> Result.new(Player.new_v1([rating: 1700, rating_deviation: 300]), :loss)]
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iex> player = Player.new_v1([rating: 1500, rating_deviation: 200])
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iex> Glicko.new_rating(player, results, [system_constant: 0.5])
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{1464.0506705393013, 151.51652412385727}
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Get a player's new rating when they haven't played within a rating period.
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iex> player = Player.new_v1([rating: 1500, rating_deviation: 200])
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iex> Glicko.new_rating(player, [], [system_constant: 0.5])
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{1.5e3, 200.27141669877065}
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Calculate the probability of a player winning against an opponent.
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iex> player = Player.new_v1
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iex> opponent = Player.new_v1
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iex> Glicko.win_probability(player, opponent)
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0.5
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Calculate the probability of a player drawing against an opponent.
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iex> player = Player.new_v1
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iex> opponent = Player.new_v1
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iex> Glicko.draw_probability(player, opponent)
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1.0
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"""
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alias __MODULE__.{
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Player,
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Result
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}
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@default_system_constant 0.8
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@default_convergence_tolerance 1.0e-7
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@type new_rating_opts :: [system_constant: float, convergence_tolerance: float]
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@doc """
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Calculates the probability of a player winning against an opponent.
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Returns a value between `0.0` and `1.0`.
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"""
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@spec win_probability(player :: Player.t(), opponent :: Player.t()) :: float
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def win_probability(player, opponent) do
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win_probability(
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player |> Player.rating(:v2),
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opponent |> Player.rating(:v2),
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opponent |> Player.rating_deviation(:v2)
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)
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end
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@doc """
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Calculates the probability of a player winning against an opponent from a player rating, opponent rating and opponent rating deviation.
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Values provided for the player rating, opponent rating and opponent rating deviation must be *v2* based.
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Returns a value between `0.0` and `1.0`.
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"""
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@spec win_probability(
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player_rating :: Player.rating(),
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opponent_rating :: Player.rating(),
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opponent_rating_deviation :: Player.rating_deviation()
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) :: float
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def win_probability(player_rating, opponent_rating, opponent_rating_deviation) do
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calc_e(player_rating, opponent_rating, calc_g(opponent_rating_deviation))
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end
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@doc """
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Calculates the probability of a player drawing against an opponent.
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Returns a value between `0.0` and `1.0`.
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"""
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@spec draw_probability(player :: Player.t(), opponent :: Player.t()) :: float
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def draw_probability(player, opponent) do
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draw_probability(
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player |> Player.rating(:v2),
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opponent |> Player.rating(:v2),
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opponent |> Player.rating_deviation(:v2)
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)
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end
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@doc """
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Calculates the probability of a player drawing against an opponent from a player rating, opponent rating and opponent rating deviation.
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Values provided for the player rating, opponent rating and opponent rating deviation must be *v2* based.
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Returns a value between `0.0` and `1.0`.
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"""
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@spec draw_probability(
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player_rating :: Player.rating(),
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opponent_rating :: Player.rating(),
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opponent_rating_deviation :: Player.rating_deviation()
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) :: float
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def draw_probability(player_rating, opponent_rating, opponent_rating_deviation) do
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1 -
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abs(win_probability(player_rating, opponent_rating, opponent_rating_deviation) - 0.5) / 0.5
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end
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@doc """
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Generate a new rating from an existing rating and a series (or lack) of results.
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Returns the updated player with the same version given to the function.
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"""
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@spec new_rating(player :: Player.t(), results :: list(Result.t()), opts :: new_rating_opts) ::
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Player.t()
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def new_rating(player, results, opts \\ [])
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def new_rating(player, results, opts) when tuple_size(player) == 3 do
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do_new_rating(player, results, opts)
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end
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def new_rating(player, results, opts) when tuple_size(player) == 2 do
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player
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|> Player.to_v2()
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|> do_new_rating(results, opts)
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|> Player.to_v1()
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end
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defp do_new_rating({player_r, player_pre_rd, player_v}, [], _) do
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player_post_rd = calc_player_post_base_rd(:math.pow(player_pre_rd, 2), player_v)
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{player_r, player_post_rd, player_v}
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end
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defp do_new_rating({player_pre_r, player_pre_rd, player_pre_v}, results, opts) do
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sys_const = Keyword.get(opts, :system_constant, @default_system_constant)
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conv_tol = Keyword.get(opts, :convergence_tolerance, @default_convergence_tolerance)
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# Initialization (skips steps 1, 2 and 3)
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player_pre_rd_sq = :math.pow(player_pre_rd, 2)
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{variance_est, results_effect} = result_calculations(results, player_pre_r)
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# Step 4
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delta = calc_delta(results_effect, variance_est)
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# Step 5.1
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alpha = calc_alpha(player_pre_v)
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# Step 5.2
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k = calc_k(alpha, delta, player_pre_rd_sq, variance_est, sys_const, 1)
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{initial_a, initial_b} =
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iterative_algorithm_initial(
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alpha,
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delta,
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player_pre_rd_sq,
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variance_est,
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sys_const,
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k
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)
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# Step 5.3
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initial_fa = calc_f(alpha, delta, player_pre_rd_sq, variance_est, sys_const, initial_a)
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initial_fb = calc_f(alpha, delta, player_pre_rd_sq, variance_est, sys_const, initial_b)
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# Step 5.4
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a =
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iterative_algorithm_body(
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alpha,
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delta,
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player_pre_rd_sq,
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variance_est,
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sys_const,
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conv_tol,
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initial_a,
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initial_b,
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initial_fa,
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initial_fb
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)
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# Step 5.5
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player_post_v = calc_new_player_volatility(a)
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# Step 6
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player_post_base_rd = calc_player_post_base_rd(player_pre_rd_sq, player_post_v)
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# Step 7
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player_post_rd = calc_new_player_rating_deviation(player_post_base_rd, variance_est)
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player_post_r = calc_new_player_rating(results_effect, player_pre_r, player_post_rd)
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{player_post_r, player_post_rd, player_post_v}
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end
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defp result_calculations(results, player_pre_r) do
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{variance_estimate_acc, result_effect_acc} =
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Enum.reduce(results, {0.0, 0.0}, fn result, {variance_estimate_acc, result_effect_acc} ->
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opponent_rd_g =
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result
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|> Result.opponent_rating_deviation()
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|> calc_g
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win_probability = calc_e(player_pre_r, Result.opponent_rating(result), opponent_rd_g)
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{
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variance_estimate_acc +
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:math.pow(opponent_rd_g, 2) * win_probability * (1 - win_probability),
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result_effect_acc + opponent_rd_g * (Result.score(result) - win_probability)
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}
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end)
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{:math.pow(variance_estimate_acc, -1), result_effect_acc}
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end
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defp calc_delta(results_effect, variance_est) do
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results_effect * variance_est
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end
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defp calc_f(alpha, delta, player_pre_rd_sq, variance_est, sys_const, x) do
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:math.exp(x) *
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(:math.pow(delta, 2) - :math.exp(x) - player_pre_rd_sq - variance_est) /
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(2 * :math.pow(player_pre_rd_sq + variance_est + :math.exp(x), 2)) -
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(x - alpha) / :math.pow(sys_const, 2)
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end
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defp calc_alpha(player_pre_v) do
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:math.log(:math.pow(player_pre_v, 2))
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end
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defp calc_new_player_volatility(a) do
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:math.exp(a / 2)
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end
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defp calc_new_player_rating(results_effect, player_pre_r, player_post_rd) do
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player_pre_r + :math.pow(player_post_rd, 2) * results_effect
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end
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defp calc_new_player_rating_deviation(player_post_base_rd, variance_est) do
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1 / :math.sqrt(1 / :math.pow(player_post_base_rd, 2) + 1 / variance_est)
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end
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defp calc_player_post_base_rd(player_pre_rd_sq, player_pre_v) do
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:math.sqrt(:math.pow(player_pre_v, 2) + player_pre_rd_sq)
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end
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defp iterative_algorithm_initial(alpha, delta, player_pre_rd_sq, variance_est, sys_const, k) do
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initial_a = alpha
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initial_b =
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if :math.pow(delta, 2) > player_pre_rd_sq + variance_est do
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:math.log(:math.pow(delta, 2) - player_pre_rd_sq - variance_est)
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else
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alpha - k * sys_const
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end
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{initial_a, initial_b}
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end
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defp iterative_algorithm_body(
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alpha,
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delta,
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player_pre_rd_sq,
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variance_est,
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sys_const,
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conv_tol,
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a,
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b,
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fa,
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fb
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) do
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if abs(b - a) > conv_tol do
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c = a + (a - b) * fa / (fb - fa)
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fc = calc_f(alpha, delta, player_pre_rd_sq, variance_est, sys_const, c)
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{a, fa} =
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if fc * fb < 0 do
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{b, fb}
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else
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{a, fa / 2}
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end
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iterative_algorithm_body(
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alpha,
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delta,
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player_pre_rd_sq,
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variance_est,
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sys_const,
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conv_tol,
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a,
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c,
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fa,
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fc
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)
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else
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a
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end
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end
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defp calc_k(alpha, delta, player_pre_rd_sq, variance_est, sys_const, k) do
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if calc_f(alpha, delta, player_pre_rd_sq, variance_est, sys_const, alpha - k * sys_const) < 0 do
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calc_k(alpha, delta, player_pre_rd_sq, variance_est, sys_const, k + 1)
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else
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k
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end
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end
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# g function
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defp calc_g(rd) do
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1 / :math.sqrt(1 + 3 * :math.pow(rd, 2) / :math.pow(:math.pi(), 2))
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end
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# E function
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defp calc_e(player_pre_r, opponent_r, opponent_rd_g) do
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1 / (1 + :math.exp(-1 * opponent_rd_g * (player_pre_r - opponent_r)))
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end
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end
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