I also looked at the math once, and at that time could not discover any reason why starting at a different rating would have impact on the overall system. However, I was not completely sure, and expected that drift may be a factor, therefore I thought to test it experimentally. I simulated 5 players in the system, assign each an initial strength (1 to 5), and let the chance of player A winning from player B equal 0.5 + arctan( strengthA - strengthB ) / π (the wider the gap in strength, the more likely the stronger player is to win).
I made sure that the random generator is deterministic, so that we can compare the effect of changing the variables without changing the games. There’s a constant tau, which affects how quickly volatility is changed, for which a value between 0.3 and 1.2 is recommended. I set it to 0.6.
I don’t think I understand how or why volatility works. It seems to steadily increase for the strongest and weakest players, and to steadily decrease for players who are closer to average in strength.
The effect of letting everyone start at a rank of their choice in the long run
It turns out that the initial rank distribution has little effect on the overall difference between the ranks, assuming the same players keep playing for a long time and no new players join.
Here’s 5000 games where each player started at 1500 rating, with an initial deviation of 350 and volatility of 0.06:
Here’s the same 5000 games where each player started with 500 rating instead:
There are no differences, except for the vertical scale of the first graph. This is mathematically expected, since the whole rating scale is relative: only (a linear transposition of) the difference in rating is used in the computation, thus the number 1500 itself is arbitrary, probably chosen to have some intuitive feel compared with Elo rating.
Next, let’s allow the players to choose their own starting rank: the strongest player chooses 2000, the weakest 1000 and the rest in between
Or, let’s have sandbaggers and airbaggers, where the weakest player chooses 2000 and the strongest 1000:
It seems clear to me that this has little to no effect after the first 30 or so matches.
We can even let one player start at a ridiculous 50000 rating:
Overall, this only affects the other ratings in the sense that the whole average is raised a bit, but it has barely any effect on the other players, and even this fixes itself over time without problems.
New accounts that misjudge their rank
Now let’s look at the effect of having a constant stream of new players, who on average choose the wrong rank. If the average new player has a tendency to sandbag, this will have the effect of pushing the rank down, and if the average new player has a tendency to airbag, this will have the effect of pushing the overall rank up.
To simulate this, let’s have new accounts consistently sandbag, for example, here I reset the rating of the strongest player each 50 games and choose a low rating of 1000:
Here it seems as though there is a clear drift in the rating: the initial idea that 1000 rating would be weak turns out to be false after 5000 games, since 1000 rating it has become about the strongest on the server. However, as we see later, this drift will eventually settle down and disappear.
What we could do (as an example of a bad decision), is make the initial rank selection based on the average rating on the server: if the average is 1500, then initial beginner rank could be 1000, and if the average is 500, the initial beginner rank could be 0:
Here the drift is naturally even stronger than before and what’s more problematic, it won’t settle down.
Apart from allowing people to choose rank, if everyone has to start at 1500, drift will still occur initially. In both cases it will be eventually solved. For example, let’s have the sandbagger scenario again, but the start at 1500 every 50 games:
It appears as though the rest of the users are drifting down because of this, which is true. But, if we increase the number of games (to 50000 for example), we see that in the end the drift is only an initial problem:
In conclusion
I don’t think that (in the long run) it matters if we allow users to choose an initial rank, as long as the ranks that they can choose stays fixed. The only snag, is that If on average new users choose too high a rank or too low a rank, then the scale of initially chosen ranks must necessarily be different from the actual ranks on the server, where the necessity is to prevent drift.
For example, in the sandbagger scenario above, the result of an abundance of sandbaggers is that the initially chosen ranks are “too strong” compared to where the user will eventually settle. Reversely, if there are on average more airbaggers, then the result will be that the initially chosen rank are “too weak” compared to where the user will eventually settle.