The total wealth of the kdice system (the amount drawn from the bank by all zero-score players when they lose) will determine when the higher-level tables will come into play. Here is a rough estimator of the total wealth of the system that can be calculated from data on hand.
Suppose that players are ranked from lowest to highest and their scores (for the nonzero ones) follow a quadratic distribution a x squared + b x. The slope of the function would be 2 a x + b. I will fit the left end to a slope and the right end to a score.
To determine b, find a player whose score is 1 and subtract his rank from a player whose score is zero. Call this R0 – R1. The value of b when x is zero is approximately 2/(R1-R0). Next, go to the list of top 25 kdicers. Take the middle player of this group (the one ranked 13th)as an approximator for the upper end of the distribution. Set this equal to a x squared + b x, where x is the total number of players who have a positive score (which happens to be R0-1) less 12, and b is what we just figured out. Then we would have P13 = a (r0-12) squared + bx. This gives us the value of a.
The total wealth of the system would be the area under the quadratic, or ((R0-1) squared / 6 ) ( 2 a (R0-1) + 3 b).
So let’s get some data.
As of the time of this post,
The rank of score 0 is 4093, and the rank of score 1 is 3988. So the value of b is 2/(4093-3988) = 0.019. The value of x is 4080.
The thirteenth ranked player (who, by strange chance, happened to be rnd when I looked) has a score of 592. We thus get a value for a of .000031.
Our value of the wealth would then be
(4092 squared /6)( (2 x .000031 x 4092 )+ 3 x .019) = 867,095 points.
So over 10 days of play, roughly 85,000 points have been generated per day.
Now having generated this number, we don’t have to use the quadratic formula again to re-estimate it later. We can use a simpler linear formula. Suppose the distribution of points was linear. The total wealth would be the area of a triangle whose height is P1 and base is R0-1. The total wealth would be half of the product of 4092 and 1103, or 2,311,888 points. The quadratic formula estimate is 37.5% of the linear estimate. So for a quick math fix, calculate P1 x (R0-1)/2 and take 37.5% of it.
Perhaps Ryan will weigh in and let me know how far off I am in my estimate.