### 5.1 The calculation

The calculation of new Felo ratings is a three-steps process:

- The result value for the fencer is calculated. This is the fraction of
single hits that he won. For example, if the fencer won 15:10, his
result value is 15/(15+10)=0.6.
- The difference of the Felo ratings of both fencers yields the
*expected* result value. The formula for this can be found
e.g. in the
Wikipedia article about the Elo rating system.
- Let us call the difference between the result value and the expected
result value the
*surprise*. The surprise is multiplied by the
total number of points in the bout, and the so-called k factor,
see List of all parameters. The result is added to the old Felo
rating. This yields the new one.

This k factor is some sort of damping parameter. If it is too
high, the Felo ratings will oscillate too heavily. If it is too low,
the Felo ratings will converge too slowly.

#### Why does the Felo rating take the score into account?

Some find it unusual or even irritating that the Felo ratings uses the
complete score rather than just using win or loss.

The rationale for this is very simple: It makes Felo ratings converge
very quickly, i.e., they find their true value much earlier. This
is advantageous when a fencing group starts with Felo ratings, so that
everybody has a real Felo rating quite soon. Similarly, it means that
the Felo ratings reflect changes in your abilities or condition rather
accurately. If Felo counted only wins and losses, such subtle
developments would not be noticeable.

There is another mathematical issue with pure win/loss ratings.
Normally, you have bouts with 5, 10, and 15 win points. Sometimes, even
other values are possible. Unfortunately, they are incompatible because
the win/loss probability in a 15 point bout is more extreme than
that in a 5 point bout. Consequently, the ratings calculated from
5 point bouts only are much closer together than for longer bouts.
Putting them all into one ratings would mean you measure tallnesses both
to the head and to the shoulders, and calculate an average tallness from
the whole set of values. This is ridiculous obviously.

Thus, you'd have to make the results compatible first. Mathematically,
this works, however, it means that you estimate the single-hit
probability for every bout. Thus, you end up with the single hit again,
just with less accuracy. So you gain little (if anything at all) and
lose much.

The Felo ratings – as they are – are a good combination of rapid
adaption and accuracy.