Elo Rating Algorithm

Elo Rating Algorithm is widely used rating algorithm that is used to rank players in many competitive games.
Players with higher ELO rating have a higher probability of winning a game than a player with lower ELO rating. After each game, ELO rating of players is updated. If a player with higher ELO rating wins, only a few points are transferred from the lower rated player. However if lower rated player wins, then transferred points from a higher rated player are far greater.

Approach:
P1: Probability of winning of player with rating2
P2: Probability of winning of player with rating1.

P1 = (1.0 / (1.0 + pow(10, ((rating1 – rating2) / 400))));
P2 = (1.0 / (1.0 + pow(10, ((rating2 – rating1) / 400))));
Obviously, P1 + P2 = 1.



The rating of player is updated using the formula given below :-

rating1 = rating1 + K*(Actual Score – Expected score);

In most of the games, “Actual Score” is either 0 or 1 means player either wins or loose. K is a constant. If K is of a lower value, then the rating is changed by a small fraction but if K is of a higher value, then the changes in the rating are significant. Different organizations set a different value of K.

Example:

Suppose there is a live match on chess.com between two players
rating1 = 1200, rating2 = 1000;
P1 = (1.0 / (1.0 + pow(10, ((1000-1200) / 400)))) = 0.76
P2 = (1.0 / (1.0 + pow(10, ((1200-1000) / 400)))) = 0.24

And Assume constant K=30;

CASE-1 : Suppose Player 1 wins:
rating1 = rating1 + k*(actual – expected) = 1200+30(1 – 0.76) = 1207.2;
rating2 = rating2 + k*(actual – expected) = 1000+30(0 – 0.24) = 992.8;

Case-2 : Suppose Player 2 wins:
rating1 = rating1 + k*(actual – expected) = 1200+30(0 – 0.76) = 1177.2;
rating2 = rating2 + k*(actual – expected) = 1000+30(1 – 0.24) = 1022.8;

CPP

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// CPP program for Elo Rating
#include <bits/stdc++.h>
using namespace std;
  
// Function to calculate the Probability
float Probability(int rating1, int rating2)
{
    return 1.0 * 1.0 / (1 + 1.0 * 
           pow(10, 1.0 * (rating1 - rating2) / 400));
}
  
// Function to calculate Elo rating
// K is a constant.
// d determines whether Player A wins or Player B. 
void EloRating(float Ra, float Rb, int K, bool d)
{  
  
    // To calculate the Winning
    // Probability of Player B
    float Pb = Probability(Ra, Rb);
  
    // To calculate the Winning
    // Probability of Player A
    float Pa = Probability(Rb, Ra);
  
    // Case -1 When Player A wins
    // Updating the Elo Ratings
    if (d == 1) {
        Ra = Ra + K * (1 - Pa);
        Rb = Rb + K * (0 - Pb);
    }
  
    // Case -2 When Player B wins
    // Updating the Elo Ratings
    else {
        Ra = Ra + K * (0 - Pa);
        Rb = Rb + K * (1 - Pb);
    }
  
    cout << "Updated Ratings:-\n";
    cout << "Ra = " << Ra << " Rb = " << Rb;
}
  
int main()
{
    // Ra and Rb are current ELO ratings
    float Ra = 1200, Rb = 1000;
      
    int K = 30;
    bool d = 1;
    EloRating(Ra, Rb, K, d);
  
    return 0;
}

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Java














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// Java program to count rotationally  


// equivalent rectangles with n unit  


// squares 


class GFG { 


  


    // Function to calculate the Probability 


    static float Probability(float rating1,  


                               float rating2) 


    { 


        return 1.0f * 1.0f / (1 + 1.0f *  


                (float)(Math.pow(10, 1.0f *  


               (rating1 - rating2) / 400))); 


    } 


      


    // Function to calculate Elo rating 


    // K is a constant. 


    // d determines whether Player A wins 


    // or Player B.  


    static void EloRating(float Ra, float Rb, 


                            int K, boolean d) 


    {  


      


        // To calculate the Winning 


        // Probability of Player B 


        float Pb = Probability(Ra, Rb); 


      


        // To calculate the Winning 


        // Probability of Player A 


        float Pa = Probability(Rb, Ra); 


      


        // Case -1 When Player A wins 


        // Updating the Elo Ratings 


        if (d == true) { 


            Ra = Ra + K * (1 - Pa); 


            Rb = Rb + K * (0 - Pb); 


        } 


      


        // Case -2 When Player B wins 


        // Updating the Elo Ratings 


        else { 


            Ra = Ra + K * (0 - Pa); 


            Rb = Rb + K * (1 - Pb); 


        } 


      


        System.out.print("Updated Ratings:-\n"); 


          


        System.out.print("Ra = " + (Math.round( 


                   Ra * 1000000.0) / 1000000.0) 


                     + " Rb = " + Math.round(Rb  


                      * 1000000.0) / 1000000.0); 


    } 


      


    //driver code 


    public static void main (String[] args) 


    { 


          


        // Ra and Rb are current ELO ratings 


        float Ra = 1200, Rb = 1000; 


          


        int K = 30; 


        boolean d = true; 


          


        EloRating(Ra, Rb, K, d); 


    } 


} 


  


// This code is contributed by Anant Agarwal. 



















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Python3














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# Python 3 program for Elo Rating 


import math 


  


# Function to calculate the Probability 


def Probability(rating1, rating2): 


  


    return 1.0 * 1.0 / (1 + 1.0 * math.pow(10, 1.0 * (rating1 - rating2) / 400)) 


  


  


# Function to calculate Elo rating 


# K is a constant. 


# d determines whether 


# Player A wins or Player B.  


def EloRating(Ra, Rb, K, d): 


   


  


    # To calculate the Winning 


    # Probability of Player B 


    Pb = Probability(Ra, Rb) 


  


    # To calculate the Winning 


    # Probability of Player A 


    Pa = Probability(Rb, Ra) 


  


    # Case -1 When Player A wins 


    # Updating the Elo Ratings 


    if (d == 1) : 


        Ra = Ra + K * (1 - Pa) 


        Rb = Rb + K * (0 - Pb) 


      


  


    # Case -2 When Player B wins 


    # Updating the Elo Ratings 


    else : 


        Ra = Ra + K * (0 - Pa) 


        Rb = Rb + K * (1 - Pb) 


      


  


    print("Updated Ratings:-") 


    print("Ra =", round(Ra, 6)," Rb =", round(Rb, 6)) 


  


# Driver code 


  


# Ra and Rb are current ELO ratings 


Ra = 1200


Rb = 1000


K = 30


d = 1


EloRating(Ra, Rb, K, d) 


  


# This code is contributed by 


# Smitha Dinesh Semwal 



















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C#














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// C# program to count rotationally equivalent 


// rectangles with n unit squares 


using System; 


  


class GFG { 


      


    // Function to calculate the Probability 


    static float Probability(float rating1,  


                                 float rating2) 


    { 


        return 1.0f * 1.0f / (1 + 1.0f *  


               (float)(Math.Pow(10, 1.0f * 


                 (rating1 - rating2) / 400))); 


    } 


       


    // Function to calculate Elo rating 


    // K is a constant. 


    // d determines whether Player A wins or 


    // Player B.  


    static void EloRating(float Ra, float Rb, 


                                int K, bool d) 


    {   


       


        // To calculate the Winning 


        // Probability of Player B 


        float Pb = Probability(Ra, Rb); 


       


        // To calculate the Winning 


        // Probability of Player A 


        float Pa = Probability(Rb, Ra); 


       


        // Case -1 When Player A wins 


        // Updating the Elo Ratings 


        if (d == true) { 


            Ra = Ra + K * (1 - Pa); 


            Rb = Rb + K * (0 - Pb); 


        } 


       


        // Case -2 When Player B wins 


        // Updating the Elo Ratings 


        else { 


            Ra = Ra + K * (0 - Pa); 


            Rb = Rb + K * (1 - Pb); 


        } 


       


        Console.Write("Updated Ratings:-\n"); 


          


        Console.Write("Ra = " + (Math.Round(Ra  


                     * 1000000.0) / 1000000.0)  


                    + " Rb = " + Math.Round(Rb 


                     * 1000000.0) / 1000000.0); 


    } 


      


    //driver code 


    public static void Main() 


    { 


          


        // Ra and Rb are current ELO ratings 


        float Ra = 1200, Rb = 1000; 


           


        int K = 30; 


        bool d = true; 


        EloRating(Ra, Rb, K, d); 


    } 


} 


  


// This code is contributed by Anant Agarwal. 



















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Output:



Updated Ratings:-
Ra = 1207.207642 Rb = 992.792419



Time Complexity

Time complexity of algorithm depends mostly on the complexity of pow function whose

complexity is dependent on Computer Architecture.

On x86, this is constant time operation:-O(1)


References

http://www.gautamnarula.com/rating/





    

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