If both n and r get large, under what circumstances is it still true that the resource allocation is approximately uniform? I suppose that depends on how you define “approximately uniform” but let’s try looking at the ratio of r1 to r/n. If my scribbling is correct, this equals frac{r_i}{r/n}=1alphafrac{sumlogi}{r}. When n is large this is (very crudely) of order n/r log n. So for any reasonably definition of “approximately uniform” this requires that r be growing at least proportionally to n. Eg., for the ratio to be much below log(n) we require r >= alpha n. And the expected utility is, if some more of my scribbling is correct, 1 - exp(-r/n)/n.G/A where G,A are the geometric and arithmetic means of (i^-alpha). So, e.g., the expected utility is at least 1-exp(-r/n)/n unconditionally (so at least 1-exp(-alpha)/n if r >= alpha n), and if alpha is much bigger than 0 (so as to make it in any way unreasonable for the ratio to be far from uniform) then the expected utility is bigger because GM/AM is smaller.
Let’s take a very concrete example. Take the Zipfian alpha=1, and when n=100. So our probabilities for the classes are roughly 19%, 10%, 6%, 5%, …, 0.2%. If we take r = alpha n = 100 then our resource allocation is 4.64, 3.94, 3.54, …, 0.03. Perhaps that’s “approximately uniform” but it certainly doesn’t look shockingly so. The expected utilities conditional on the various classes are 0.99, 0.98, 0.97, …, 0.032. The overall expected utility is 0.81.
That doesn’t seem to me like an unreasonable or counterintuitive outcome, all things considered.
If both n and r get large, under what circumstances is it still true that the resource allocation is approximately uniform? I suppose that depends on how you define “approximately uniform” but let’s try looking at the ratio of r1 to r/n. If my scribbling is correct, this equals frac{r_i}{r/n}=1 alphafrac{sumlogi}{r}. When n is large this is (very crudely) of order n/r log n. So for any reasonably definition of “approximately uniform” this requires that r be growing at least proportionally to n. Eg., for the ratio to be much below log(n) we require r >= alpha n. And the expected utility is, if some more of my scribbling is correct, 1 - exp(-r/n)/n.G/A where G,A are the geometric and arithmetic means of (i^-alpha). So, e.g., the expected utility is at least 1-exp(-r/n)/n unconditionally (so at least 1-exp(-alpha)/n if r >= alpha n), and if alpha is much bigger than 0 (so as to make it in any way unreasonable for the ratio to be far from uniform) then the expected utility is bigger because GM/AM is smaller.
Let’s take a very concrete example. Take the Zipfian alpha=1, and when n=100. So our probabilities for the classes are roughly 19%, 10%, 6%, 5%, …, 0.2%. If we take r = alpha n = 100 then our resource allocation is 4.64, 3.94, 3.54, …, 0.03. Perhaps that’s “approximately uniform” but it certainly doesn’t look shockingly so. The expected utilities conditional on the various classes are 0.99, 0.98, 0.97, …, 0.032. The overall expected utility is 0.81.
That doesn’t seem to me like an unreasonable or counterintuitive outcome, all things considered.