Quick Explanation of Ranking system

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Quick Explanation of Ranking system

Post by dave14810 on February 20th 2013, 5:41 am

I won't go into too long an explanation, if you want more information you can read it on http://en.wikipedia.org/wiki/Elo_rating_system

The first calculations are Ea and Eb which are the expectations for team A and team B win a head to head ( basically how many times out of a 100 each team wins the match). The Ra and Rb in the formulas are the ranking of the team before they play. Every team will start with a ranking of 1500.

Ea = 1/1+(10 to the power(Rb-Ra)/400)
Eb= 1/1+(10 to the power(Ra-Rb)/400)

Then the second calculations are what the ranking of each team are after the game.

R'a = Ra + K(Sa - Ea)

where Ra is the starting ranking, K is the maximum possible points adjustment per game and we will be using K=32, Sa is what the result of the match was either 1 for a win or 0 for a loss. The other team is calculated the exact same way with the winner gaining x amount of points and the loser losing negative x amount of points. For those who are already confused here is a few examples of how wins and losses can potentially affect your ranking

Example 1: Team A, the stronger team, and Team B, the weaker team. Under this scenario the weaker team wins 4 games in a row and then the initial stronger team wins(which isnt plausible but its just an eg)

Team A Ranking Team B Ranking
1555.801 1444.199
1534.831 1465.169
1515.665 1484.335
1498.226 1501.774
1482.389 1517.611
1500.005 1499.995


You can see the 1st time Team B wins their ranking goes up approx 21 and Team A goes down about 21, then approx 19, then approx 17, then approx 16. Then when Team A wins the last game they are both basically the same score. The thing to take away from the example is that it takes a weaker team a good few games to catch up with a strong team. The stronger teams eventually risk losing more ranking points if they lose to a weaker team as opposed to playing a strong team.


Example 2: 2 teams starting off and they play a number of games

Team A Ranking Team B Ranking Difference in Points
15001500 0
1516 1484 32
1530.53 1469.47 61.06
1543.747 1456.253 87.494
1523.801 1476.199 47.602
1537.622 1462.378 75.244
1518.21 1481.79 36.42


In this example Team A wins the 1st, 2nd and 3rd games against team B, then loses the 4th, wins the 5th and loses the 6th. This example shows how you can greatly catch up with a team near you in the rankings, as you can see by beating a team 3 times in a row you can take an 87 point lead over team if you were initially equal in points.

TLDR; I will update the leaderboard rankings every weaker initially. I will probably update more regularly in the last week or two and possibly every day of the final week. Feel free to ask any concerns you may have.


Last edited by dave14810 on February 20th 2013, 10:23 pm; edited 2 times in total

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Re: Quick Explanation of Ranking system

Post by JibJabJessie on February 20th 2013, 9:13 pm

Very good post Cool
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Re: Quick Explanation of Ranking system

Post by Rain on February 20th 2013, 9:53 pm

You should correct the Eb formula, it's (Ra - Rb) and also format the equation correctly. I got confused when working out the formula.

Some parenthesis would help or refer to the images below.

i.e. Ea = 1 / (1 + (10^( (Rb-Ra) / 400) ))


Expected Score (Ea and Eb) equations (usually between 0.0 - 1.0) aka your probability of winning the match:





New rank equation:




If people are still confused by the tables above provided by dave look below for an example.

Example

Team A's starting rank (Ra): 1500
Team B's starting rank (Rb): 1500

Probability of Team A winning (Ea): Ea = 1 / (1 + (10^( (Rb-Ra) / 400) )) = 0.5 or 50% chance
Probabiility of Team B winning (Eb): Eb = 1 / (1 + (10^( (Ra-Rb) / 400) )) = 0.5 or 50% chance

Both teams have a 50% change of winning the first match due to their rank both being 1500.

If Team A wins, we use the the formula: R'a = Ra+ K(Sa-Ea)

Team A's new rank (R'a) is:

R'a = 1500 + 32(1-0.5) = 1516
Sa = 1 for the win.

Team A (Ra) gains 16 points for the 1st match.


Now since Team B lost, we use the formula: R'b = Rb + K(Sb-Eb)

Team B's new rank (R'b) is:

R'b = 1500 + 32(0-0.5) = 1484
Sb = 0 for the loss.

Team B lost 16 point for losing the 1st match.


Conclusion

After 1st match:

Team A's Rank: 1516
Team B's Rank: 1484

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Re: Quick Explanation of Ranking system

Post by dave14810 on February 20th 2013, 10:35 pm

Rain that was a good post and good to see that people want to understand the math behind the rankings and not just accept what we will be posting, I made a few corrections to my original post. My post was just an introduction to the system and the examples I gave are basically just to show teams how your ranking will change depending on who you play and how winning/losing to certain teams will affect it. It was also to show it might take a week or two for the rankings to show some separation between teams as well as if teams are wondering how when they went 6-2, for eg, how they aren't further up the leaderboard which might be that they are playing too many lower ranked teams and might encourage them to play the stronger teams more.

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