ALGORITAM ELO OCJENE - MATEMATIČKI

The Algoritam Elo ocjene je široko korišteni algoritam ocjenjivanja koji se koristi za rangiranje igrača u mnogim natjecateljskim igrama.

Igrači s višim ELO rejtingom imaju veću vjerojatnost da će pobijediti u igri od igrača s nižim ELO rejtingom.
Nakon svake utakmice ažurira se ELO rejting igrača.
Ako igrač s višim ELO rejtingom pobijedi, samo nekoliko bodova se prenosi od igrača s nižim rejtingom.
Međutim, ako igrač s nižim rejtingom pobijedi, tada su preneseni bodovi od igrača s višim rejtingom daleko veći.

Pristup: Da biste riješili problem, slijedite donju ideju:

P1: Vjerojatnost pobjede igrača s ratingom2 P2: Vjerojatnost pobjede igrača s ratingom1.
P1 = (1,0 / (1,0 + pow(10 ((ocjena1 - ocjena2) / 400))));
P2 = (1,0 / (1,0 + pow(10 ((ocjena2 - ocjena1) / 400))));

Očito P1 + P2 = 1. Ocjena igrača ažurira se pomoću formule dane u nastavku:-
ocjena1 = ocjena1 + K*(stvarna ocjena - očekivana ocjena);
U većini igara 'stvarni rezultat' je 0 ili 1 što znači da igrač pobjeđuje ili gubi. K je konstanta. Ako je K niže vrijednosti, tada se ocjena mijenja za mali dio, ali ako je K veće vrijednosti, promjene u ocjeni su značajne. Različite organizacije postavljaju različitu vrijednost K.

Primjer:

Pretpostavimo da postoji meč uživo na chess.com između dva igrača
ocjena1 = 1200 ocjena2 = 1000;
leksikografski
P1 = (1,0 / (1,0 + pow(10 ((1000-1200) / 400)))) = 0,76
P2 = (1,0 / (1,0 + pow(10 ((1200-1000) / 400)))) = 0,24
I Pretpostavimo konstantu K=30;
SLUČAJ-1:
Pretpostavimo da igrač 1 pobjeđuje: ocjena1 = ocjena1 + k*(stvarno - očekivano) = 1200+30(1 - 0,76) = 1207,2;
ocjena2 = ocjena2 + k*(stvarno - očekivano) = 1000+30(0 - 0,24) = 992,8;
Slučaj-2:
Pretpostavimo da igrač 2 pobjeđuje: ocjena1 = ocjena1 + k*(stvarno - očekivano) = 1200+30(0 - 0,76) = 1177,2;
ocjena2 = ocjena2 + k*(stvarno - očekivano) = 1000+30(1 - 0,24) = 1022,8;

Slijedite korake u nastavku da biste riješili problem:

Izračunajte vjerojatnost pobjede igrača A i B koristeći gornju formulu
Ako igrač A pobijedi ili igrač B pobijedi, ocjene se ažuriraju u skladu s tim pomoću formula:
- ocjena1 = ocjena1 + K*(stvarna ocjena - očekivana ocjena)
- ocjena2 = ocjena2 + K*(stvarna ocjena - očekivana ocjena)
- Gdje je stvarni rezultat 0 ili 1
Ispišite ažurirane ocjene

U nastavku je implementacija gornjeg pristupa:

CPP

#include    using namespace std; // Function to calculate the Probability float Probability(int rating1 int rating2) {  // Calculate and return the expected score  return 1.0 / (1 + pow(10 (rating1 - rating2) / 400.0)); } // Function to calculate Elo rating // K is a constant. // outcome determines the outcome: 1 for Player A win 0 for Player B win 0.5 for draw. void EloRating(float Ra float Rb int K float outcome) {  // Calculate the Winning Probability of Player B  float Pb = Probability(Ra Rb);  // Calculate the Winning Probability of Player A  float Pa = Probability(Rb Ra);  // Update the Elo Ratings  Ra = Ra + K * (outcome - Pa);  Rb = Rb + K * ((1 - outcome) - Pb);  // Print updated ratings  cout << 'Updated Ratings:-n';  cout << 'Ra = ' << Ra << ' Rb = ' << Rb << endl; } // Driver code int main() {  // Current ELO ratings  float Ra = 1200 Rb = 1000;  // K is a constant  int K = 30;  // Outcome: 1 for Player A win 0 for Player B win 0.5 for draw  float outcome = 1;  // Function call  EloRating(Ra Rb K outcome);  return 0; }

Java

import java.lang.Math; public class EloRating {  // Function to calculate the Probability  public static double Probability(int rating1 int rating2) {  // Calculate and return the expected score  return 1.0 / (1 + Math.pow(10 (rating1 - rating2) / 400.0));  }  // Function to calculate Elo rating  // K is a constant.  // outcome determines the outcome: 1 for Player A win 0 for Player B win 0.5 for draw.  public static void EloRating(double Ra double Rb int K double outcome) {  // Calculate the Winning Probability of Player B  double Pb = Probability(Ra Rb);  // Calculate the Winning Probability of Player A  double Pa = Probability(Rb Ra);  // Update the Elo Ratings  Ra = Ra + K * (outcome - Pa);  Rb = Rb + K * ((1 - outcome) - Pb);  // Print updated ratings  System.out.println('Updated Ratings:-');  System.out.println('Ra = ' + Ra + ' Rb = ' + Rb);  }  public static void main(String[] args) {  // Current ELO ratings  double Ra = 1200 Rb = 1000;  // K is a constant  int K = 30;  // Outcome: 1 for Player A win 0 for Player B win 0.5 for draw  double outcome = 1;  // Function call  EloRating(Ra Rb K outcome);  } }

Python

import math # Function to calculate the Probability def probability(rating1 rating2): # Calculate and return the expected score return 1.0 / (1 + math.pow(10 (rating1 - rating2) / 400.0)) # Function to calculate Elo rating # K is a constant. # outcome determines the outcome: 1 for Player A win 0 for Player B win 0.5 for draw. def elo_rating(Ra Rb K outcome): # Calculate the Winning Probability of Player B Pb = probability(Ra Rb) # Calculate the Winning Probability of Player A Pa = probability(Rb Ra) # Update the Elo Ratings Ra = Ra + K * (outcome - Pa) Rb = Rb + K * ((1 - outcome) - Pb) # Print updated ratings print('Updated Ratings:-') print(f'Ra = {Ra} Rb = {Rb}') # Current ELO ratings Ra = 1200 Rb = 1000 # K is a constant K = 30 # Outcome: 1 for Player A win 0 for Player B win 0.5 for draw outcome = 1 # Function call elo_rating(Ra Rb K outcome)

using System; class EloRating {  // Function to calculate the Probability  public static double Probability(int rating1 int rating2)  {  // Calculate and return the expected score  return 1.0 / (1 + Math.Pow(10 (rating1 - rating2) / 400.0));  }  // Function to calculate Elo rating  // K is a constant.  // outcome determines the outcome: 1 for Player A win 0 for Player B win 0.5 for draw.  public static void CalculateEloRating(ref double Ra ref double Rb int K double outcome)  {  // Calculate the Winning Probability of Player B  double Pb = Probability((int)Ra (int)Rb);  // Calculate the Winning Probability of Player A  double Pa = Probability((int)Rb (int)Ra);  // Update the Elo Ratings  Ra = Ra + K * (outcome - Pa);  Rb = Rb + K * ((1 - outcome) - Pb);  }  static void Main()  {  // Current ELO ratings  double Ra = 1200 Rb = 1000;  // K is a constant  int K = 30;  // Outcome: 1 for Player A win 0 for Player B win 0.5 for draw  double outcome = 1;  // Function call  CalculateEloRating(ref Ra ref Rb K outcome);  // Print updated ratings  Console.WriteLine('Updated Ratings:-');  Console.WriteLine($'Ra = {Ra} Rb = {Rb}');  } }

JavaScript

// Function to calculate the Probability function probability(rating1 rating2) {  // Calculate and return the expected score  return 1 / (1 + Math.pow(10 (rating1 - rating2) / 400)); } // Function to calculate Elo rating // K is a constant. // outcome determines the outcome: 1 for Player A win 0 for Player B win 0.5 for draw. function eloRating(Ra Rb K outcome) {  // Calculate the Winning Probability of Player B  let Pb = probability(Ra Rb);  // Calculate the Winning Probability of Player A  let Pa = probability(Rb Ra);  // Update the Elo Ratings  Ra = Ra + K * (outcome - Pa);  Rb = Rb + K * ((1 - outcome) - Pb);  // Print updated ratings  console.log('Updated Ratings:-');  console.log(`Ra = ${Ra} Rb = ${Rb}`); } // Current ELO ratings let Ra = 1200 Rb = 1000; // K is a constant let K = 30; // Outcome: 1 for Player A win 0 for Player B win 0.5 for draw let outcome = 1; // Function call eloRating(Ra Rb K outcome);

Izlaz

Updated Ratings:- Ra = 1207.21 Rb = 992.792

Vremenska složenost: Vremenska složenost algoritma najviše ovisi o složenosti pow funkcije čija složenost ovisi o arhitekturi računala. Na x86 ovo je operacija u konstantnom vremenu: -O(1)
Pomoćni prostor: O(1)