[John Paul]

This post is a result of Irieguy's posts/ideas about the abilities of players at different levels. He posited that at any one time there would be a number of players at a given level who do not play well enough to win at that level as some are folks trying to move up, and others are folks falling down from a higher limit. I was interested in this idea, so I have tried to put in mathematical terms as follows:

2 Limits
Assume that there is poker site with one game (NL SnGs or whatever) and two limits. Assume no one ever quits playing. We will call the two limits A (lower limit) and B (higher limit). All of the players would like to be winning at limit B, so if they do well at limit A they will move up. However, if they do poorly at B they will drop back down. So lets assume that the top U percent (expressed as a decimal) of A limit players will move up from A to B, and the bottom D percent (expressed as a decimal) of players will move down from B to A. So - what proportion of players are staying in A, going from A up to B, falling from B or staying at B?

We can figure out what these proportions will be assuming that the system runs long enough to reach an equilibrium.

The change in A will be:
-UA+DB
and the change in B will be
+UA-DB

we also know that A+B=1 (i.e. every player is either playing A or B)

So, at equilibrium, UA= DB, that is the same number of players will move up as are moving down. A little algebra will tell you that the proportion of players in A = D/(U+D) and the proportion in B is 1-[D/(U+D)]. (For brevity, I am only outlining the math here).

So, lets put in some numbers for U and D. Lets assume the top 25% of the A players at any time will try to move up, and the bottom 50% of the B players will drop back down.

Then the proportion playing at the two limits will be:
A=.5/(.25+.5)=.667
B = 1-.666=.333

Looking at the players by their percentile (0=worst, 100=best):
0-49.5th = always playing at A(=0.66*0.75)
49.5 to 83.5th = bouncing between A and B, beating A but failing at B (.66*.25*+.33*.50)
83.5-100th=staying at level B.

So who thinks they are a good (=better than average) player? Lets assume that anyone on level B thinks they are good, as well as half the folks on level A. The folks at B have already proven themselves at A, and half of the folks at A are better than their average opponent. If that is the case, 66.7% of all players will think they are better than average - which would be good for a site that wanted to keep its clients happy.

3 Limits
Now, let's say that the site now offers a third limit C, which is higher than the B. Winning B players move up to C and losing C players move down to B. For simplicity, we will assume that they do this at the same rates as the switch between A and B. Then, the change at each level will be
A: -UA+DB
B: +UA-DB-UB+DC
C: +UB-DC
Again A+B+C=1

Doing some algebra,
A=DB/U
C=UB/D
B=(UD)/(D^2+UD+U^2)

Assuming U=.25 and D=.5 again
A=.572
B=.286
C=.143

What does this mean? Because C has drawn good players from B, B is now easier to beat, so more A players are moving up, making A easier to beat as well.

By percentile:
0-42.9th = Staying at A
42.9-71.5th =Bouncing between A and B
71.5-78.2th =Staying at B
78.2-92.5st = Bouncing between B and C
92.5-100th = Staying at C.

So some players who used to be stuck at level A can now bounce back and forth to level B, and some who could not stay at B now can, or are even taking shots at level C.

No who thinks they are better than average? Lets assume everyone at B and C do - less than half the folks play at those levels. Again half of the A's think they are good as they are better than their average level A opponent. As a result, 71.4% of all players will think they are better than average.

So what does this mean?

Obviously I have made a lot of simplifying assumptions, and a much more detailed analysis could be done. However, I think a few things stand out. Adding a higher limit game attracts the best players, which in turn makes every game at a lower limit easier - which should make all the other players happy. Secondly, there may bucket-brigading of money going on - players who win money at one level and then lose it to the folks on the next highest level and drop back down. In the model the amount of this depends on the values you stick in for U and D, which I don't have real world estimates for. Also, in the real world, there are folks with a lot of money who just start at higher levels, but I suspect that a fair amount of money is working its way up from the bottom as well. Finally, I think Irieguy is on the right track with why competition is unusually hard at the 33's. I have no experience with either level myself, but from what folks are saying, it is harder to move up to the 55's due to the higher buy in and other factors he mentioned (U is lower compared to other levels) but it is just as easy to lose money and move down (D is the same).

Hope folks found this interesting,
John Paul