# How the IECG Rating Scheme works: an Example

An Example of how to calculate Ratings under the new IECG Rating Scheme.

This example refers to the
IECG Rating Rules.

Let us assume the following situation: Player A plays against Player B. Before the game, we have the following entry in the old rating lists:

```
Player A:     1800    2    1   2  (2000)
Player B:     2200   15   18   7
```

What do these entries mean?

Player A has a Starting Rating of 1800, but has already played five games. His opponents' average currently is 2000. Player B has an established rating of 2200 with 40 games played under the new scheme. Both player finished the first game in this rating period.

In the following 1-0 means a victory for A, = means a draw and 0-1 means victory for B.

We now calculate the provisional rating for Player A (see
section IV.1 of the Rating Rules):

Rp = Opponents Average + Expected Rating Change*Correction.

The Average of the opponents' ratings is (2000*5+2200)/6 = 12200/6 = 2033

A's percentage result is given by p=(2*W+D)/2*N

```
1-0: p = (2*3+1)/(2*6) = (6+1)/12 = 7/12 = 0.58
=:   p = (2*2+2)/(2*6) = 6/12 = 0.50
0-1: p = (2*2+1)/(2*6) = 5/12 = 0.42
```

Then the correction is given by F = -2*p*p + 2*p + 0.5, i. e.

```
1-0:    F = -2*0.58 * 0.58 + 2*0.58 + 0.5 = 0.9872
=:      F = -2*0.50 * 0.50 + 2*0.50 + 0.5 = 1.0
0-1:    F = -2*0.42 * 0.42 + 2*0.42 + 0.5 = 0.9872
```

The expected Rating Change is then given by:

```
1-0:    D(0.58) = -400*log10((1-0.58)/0.58) = -400*(-0.14) = 56
=:      D(0.50) = -400*log10((1-0.5)/0.5) = -400* (0) = 0
0-1:    D(0.42) = -400*log10((1-0.42)/0.42) = -400*0.14 = -56
```

Therefore:

```
1-0:    Rp = 2033 + 56 * 0.9872 = 2088
=:      Rp = 2033 + 0 * 1       = 2033
0-1:    Rp = 2033 - 56 * 0.9872 = 1978
```

This is the first provisional rating of Player A.

Player B has an established rating. So we have to use the following formula to calculate the rating change for this game:

```
dR = k*(W-We)
```

k is given by the product of two coefficients, determined by the old rating of B and his experience (i.e. the number of games played). Following
section V.3 of the Rating Scheme Document, we have

```
k = r*P with

r = 70-2200/40 = 70-55 = 15
```

and because he already played 40 games

```
P = 1.4 - 40/200 = 1.4 - 0.2 = 1.2
```

therefore k = 1.2*15 = 18.

"We" in the above formula is the probability that B wins, given by the rating difference of B and A. Since the difference D is given by

```
D = 2200-1800 = 400
```

we get (see
section I.3)

```
P(D) = P(400) = 1/(1+10^(-400/400)) = 1/(1+10^(-1)) = 1/(1+0.1)
= 1/1.1 = 0.91
```

Now we can easily calculate the rating change dR:

```
1-0: dR = 18*(0-0.91)   = 18*(-0.91) = -16.38
=:   dR = 18*(0.5-0.91) = 18*(-0.41) = -7.38
0-1: dR = 18*(1.0-0.91) = 18*0.09    = 1.62
```

Then the Performance of B is (since the rating period has not ended yet):

```
1-0: 2200-16 = 2184
=:   2200- 7 = 2193
0-1: 2200+ 2 = 2202
```

During a rating period all dR for each game are added up and the new rating is then calculated.

Created by Dr. Ortwin Pätzold, last modified by Vania Mascioni on June 9, 1998.

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