bernd-wechner
commented
4 years ago This page caught my eye:
http://www.beigebag.com/case_sigmoid_half.htm
or
https://web.archive.org/web/20200203173206/http://www.beigebag.com/case_sigmoid_half.htm
I think we can model this as a Sigmoid. Which of course is in fact the same thing as the Laplace CDF. I guess it's nice to have another name for it in the transition space, as we're considering a transition from one Sigma to another Sigma over time (this is what Tau modifies). If we imagine Sigma_1 is the Sigma last known after a recorded game session, and Sigma_2 is is the steady state uncertainty we have knowing someone once played at a certain level a long time ago, but we have no idea if they've played since.
Sigma_2 is arguably Sigma_default the standard Sigma we apply to any new player. And so we have a transition form Sigma_current to Sigma_default.
Without much evidence just gut feel we would expect exponential decay (nature graces us with that almost everywhere) but the Sigmoid transition allows us to tune the exponent and ideally we'd tune it based on some real world data or study, lacking that, we still want to model Sigma growth which translates to ranking decline over time because this is what encourages replay of games ... without replay our confidence in your skill goes down and you fall off the leaderboard ...
zass30
There's a fundamental design issue with TrueSkill that emerged from my study of it.
Every time a new game session is logged it adds Tau to the standard deviation (uncertainty) of skill.
This is intended to model the rising uncertainty in your skill over the time since you played last. This uncertainty contains two major known components:
A decline in skill over time without practice. In fact I have a paper on this and it's studied and understood and could in a sense be modeled but is immature research and not specific to games and the return on modelling it is likely to be negligible if present at all.
A possible rise in skill over due to unrecorded practice (times you played this game but did not log results) or cross-learning (similar games you played and did log experience with which positively impacts your skill at the game concerned). None of this can be modeled as it relates to unknown activity between recorded sessions.
As a consequence there is some rise in uncertainty (or loss in confidence) in our assesment of your skill, over time. for lack of incoming strengthening evidence of it.
And TrueSkill models this by adding a fixed value Tau to the uncertainty on each run.
But we know that this is time varying. That our uncertainty grows with time. And so a better model would add some uncertainty that grows over time.
One way to model this is to redefined Tau as a per diem value (fancy Latin term for per day). If we assumed their Xbox models are based on say an average gape between plays of a certain duration, say a day or a week, we could redefine the default Tau to get the ball rolling as an equivalent per diem value.
This impacts leaderboard presentation also. Currently Tau is added whenever results are calculated (a session is logged), and that is easy to model as we know the last time each player logged a session for this game so can get the delta diem and multiply that by the per diem Tau to get a Tau to add to the variance.
The gotcha is that for "current" leaderboards, the delta diem is now minus the last session date. So the rating which is currently simply a function of mu and sigma must be a function of mu, sigma, tau and the delta diem. In short as time passes a player will slip down the leaderboard.
This is actually a good thing and improves TrueSkill dramatically I suspect (it remains a suspiscion until tested). It solves a longstanding problem that TrueSkill has. Namely someone can waltz in, play a 6 player game against the top 5 players in a leaderboard, win and now dominate the laderboard, and years later they are still there, with zero evidence, because as the other five players continue to lay they are now and again losing (as happens in 6 player games) and their rating more or less stabilises over time, but the reward they receive from recording more sessions is that they lost rating strength in this way.
By moving Tau to per diem, old players who simply aren't maintaining their skills slide slowly down the leaderboard.
This does also introduce a new problem namely of unchecked decline, or growth in uncertainty. There is a case for example not to let uncertainty ever grow beyond it's initial default value (that which we grant newcomers) and in fact with one game recorded or n games recorded to have some minimum value higher than that default.
There may also be an argument to approach that not linearly with time, but probably more like nature dictates in an exponential drop (worth reviewing that paper I have to see if this proves to be the case in studied examples). An exponential drop sees the value drop fast initially and flatten out asymptotic to that minimum or limit.
There's reason to suspect that reality is the decay is probably more like CDF of a Laplace distribution:
https://en.wikipedia.org/wiki/Laplace_distribution
than exponential, namely skill is fairly stable for a while, then declines with a known rate towards an asymptotic minimum.
We could model Tau not as a per diem, but a time function that follows this CDF pattern with the minimum being the default plus some credit for the n games logged (evidence at hand) and teh initial value being uncertainty at last recorded session and we look delta diem up and get a value between them.
This would produce a nice decline in confidence that far better model expectations, and is within the bounds of utility in a sense for TrueSkill.