Elo Rating Algorithm - GeeksforGeeks

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

// 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;
}

Java

// 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.

Python3

# 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

C#

// 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.

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/

My Personal Notes

gaurav _purohit

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

CPP

Java

Python3

C#

Recommended Posts: