An Idea and it's conclusion

Recently after a lot of thought and with nothing much to do I watched a movie called "The Social Network". I was very speculative to watch this movie at first because it had not action and seemed like a drama, but I was wrong. The story, script and dialogues are so intense and fast paced that a hard core action movie can never provide. In the first few minutes our hero Mark discussed about a formula on the window of his room. This formula is used for his site facemash.com.

After watching the movie I wanted to check if this formula is for real or is it created by our director. To my surprise this formula is for real and it is known as Elo Rating system.

More details of it can be found here http://en.wikipedia.org/wiki/Elo_rating#Mathematical_details


If Player A has true strength R_A and Player B has true strength R_B, the exact formula ( the expected score of Player A is

Similarly the expected score for Player B is

The maximum possible adjustment per game (sometimes called the K-value) was set at K = 16 for masters and K = 32 for weaker players.

Supposing Player A was expected to score E_A points but actually scored S_A points. The formula for updating his rating is

My analysis for this is.

Suppose there are four players A, B, C, D, E.
We should assign the initial rating for all the players.

This initial rating should be proportional to the no of players playing the game.

Suppose there are 1000 players and you assign a initial player rating of 1000 then it should be fine. But if there are 100000 players and you assign initial player of 100o then there is high probaility for 2 or more players to get the same rating.

so in our example since there are only 5 players let us assign each player intial rating of 1000 points.

Let us assume Player A played played with the other 4 players and the following happened

A	1st	2nd
B	Lost	Win
C	Lost	Win
D	Win	Lost
E	Lost	Win

consider the 1st case
Ea=
1/1 + 10^(1000 - 1000 / 400) = 1/1 + 10^(0 / 400) = 1/1 + 10^0 = 1/(1 + 1) = 0.5

So the actual points scored Sa = 0 + 0 + 1 + 0 = 1
Expected Score Ea = 0.5 + 0.5 + 0.5 + 0.5 = 2 (Since all the 5 players have the same initial rating 1000)

Now the updated rating of our player A = 1000 + 32 (1 -2 ) = 1000 - 32 = 968 Points.

For case 2 :

Actual points Sa = 1 +1 +0 +1 = 3.

Updated Player A rating would be
1000 + 32 (3 -2 ) = 1000 + 32 = 1032 Points.

This case 2 is independent of case 1.

Consider another scenario where case 2 has occured after case1 and we have not updated the Points of B, C, D and E players.

Then
Ea = 1/1 + 10^(1000 - 968 / 400) = 1/1 + 10^(32 / 400) = 1/1 + 2.089 = 0.32

So the actual points scored Sa = 1 +1 +0 +1 = 3.
Expected Score Ea = 0.32 + 0.32 + 0.32 + 0.32 = 1.28 (Since all the 4 players have the same initial rating 1000)

Now the updated rating of our player A = 968 + 32 (3 -1.28 ) = 968 + 56.95 =~ 1025 Points

So if you have noticed the player A instead of getting 1000 points had got 25 points extra. This is because he has won against more stronger opponents and is been rewared more.