So phi is increased depending on your current level?
Sorry, I was not clear: with "level" I meant the level of phi not the "level" that is calculated from (mu,phi) for the leaderboard. And it increases by more if it's currently low (not playing is more unusual when previously playing 10 games/day than when playing 1 game before.)
For example not playing increases phi from
0.2 --> 0.2088
0.5 --> 0.5036
And phi is decreased depending on current phi value? Or did you mean something else?
Exactly. If if's high, one game will decrease it more than when it's low. You can also see that from above numbers: going from 1 to 3 games/day does more than going from 3 to 10 games/day.
Interesting. You would then think that with an unequal opponent (poor match), if the result is the opposite of the expected, phi would fall more, because that is even more informative...?
The result doesn't directly matter for the updating of phi. This is a second order effect that would be captured by sigma (i.e. an usual result will increase sigma and that will actually increase
phi over time. The intuition is the following: daily increase of phi captures that we're more uncertain about somebody's current skill if there's no recent evidence of their playing. If, however, unlikely results come in, we should become more uncertain about the current skill. But for practical purposes the algorithm can't capture that well and sigma is almost the same for everyone.)
I once calculated roughly
So playing 1 game instead of 10 per day will cost you 2.5 levels but not more.
So these values are what phi will converge to eventually?
Yes, with the caution that it depends on your match quality. I think I calculated those numbers assuming that the win probability is 80%-20% back when that was more common. I'm setting narrower bounds for my opponents now.
Therefore, with on average 7.4 games/day over the last month (
), my phi is only 0.20. So treat the numbers as rather upper bounds.