In regard to the current discussion about the appropriate gaps for the tables, I thought I would make some mathematical observations and set up a Quality-of-Opponent Adjustment (QOOA) that does not have to be re-calibrated. The QOOA and the TAPL that it modifies are discussed on my profile page.

The distribution of the number of tables of each level follow approximately the Golden Ratio (reference http://en.wikipedia.org/wiki/Golden_ratio), known from the time of the ancient Greeks. This ratio is 1.618034, and the conjugate and inverse of it is 0.618034. The number of kdice tables varies from time to time, but the ratio of the number of zero-limit tables to all higher tables, and the ratio of 1500-limit tables to all higher tables, and the ratio of 1700 limit tables to all higher tables, etc, seem to approximate the golden ratio. There is something aesthetically pleasing about this ratio for the tables, because regardless of your level of play, the proportion of the time that you can reach the next higher table is constant.

Therefore I will set up a QOOA that does not have to be re-calibrated when new levels of tables are added. It is designed for six levels of tables, which is sufficient when the proportion of games played at each level is rounded off to a whole percent. The QOOA is

QOOA = p0/371 + p1/142 + p2/54 + p3/21 + p4/08 + p5/03

where p0 is the percentage of games you have played at the zero limit tables, expressed as an integer, p1 for the next higher level of table, p2 next higher level, and so on. The table limits are to be disregarded; what is important is the level of table. The level of table is to be determined at the time you sit down. For example, if you sit down at a 1900 table when there are zero-limit, 1500-limit, and 1700 limit tables available, that particular game is to be counted as part of p3. If you sit at a 2000 table and a 1900 table is open, that game is counted in p4, but if you sit at a 2000 table and there are no 1900 tables open, that game would be counted in p3.

Some sample calculations:

Suppose you have played 50% of your games at the zero limit tables and 50% at the 1500-limit tables. Your QOOA would be 50/371 + 50/142 or 0.49. Suppose you have played 20% of your games at the zero limit tables and 80% of your games at the 1500-limit. Your QOOA would be 20/371 + 80/142 or 0.62. Your QOOA increases when you play more at the higher level tables. The QOOA multiplied by the TAPL gives you your TAPL adjusted for quality of opponents.

Incidentally, the denominators in the QOOA formula closely approximate alternative numbers from the Fibonacci series 1, 1, 2, 3, 5, 8, etc. The Fibonacci numbers and golden ratio are closely linked.