By the difference in generality i meant the difficulty-based problem selection. (Or the possibility of some other hidden variable that affects the order in which we solve problems.)
I was assuming something roughly (locally) log-uniform. You assume a Pareto distribution.
On a closer examination of your 2014 post, i don’t think this is true. If we look at the example distribution
Assume that an area has 100 problems, the first of difficulty 1, and each of difficulty 1.05 times the previous one. Assume for simplicity that they all have equal benefits.
and try to convert it to the language i’ve used in this post, there’s a trick with the scale density concept: Because the benefits of each problem are identical, their cost-effectiveness is the inverse of difficulty, yes. But the spacing of the problems along the cost-effectiveness axis decreases as the cost increases. So the scale density, which would be the cost divided by that spacing, ends up being proportional to the inverse square of cost-effectiveness. This is easier to understand in a spreadsheet. And the inverse square distribution is exactly where i would expect to see logarithmic returns to scale.
As for what distributions actually make sense in real life, i really don’t know. That’s more for people working in concrete cause areas to figure out than me sitting at home doing math. I’m just happy to provide a straightforward equation for those people to punch their more empirically-informed distributions into.
Of course you’re right; my “log uniform” assumption is in a different space than your “Pareto” assumption. I think I need to play around with the scale density notion a bit more until it’s properly intuitive.
Howdy. I appreciate your reply.
By the difference in generality i meant the difficulty-based problem selection. (Or the possibility of some other hidden variable that affects the order in which we solve problems.)
On a closer examination of your 2014 post, i don’t think this is true. If we look at the example distribution
and try to convert it to the language i’ve used in this post, there’s a trick with the scale density concept: Because the benefits of each problem are identical, their cost-effectiveness is the inverse of difficulty, yes. But the spacing of the problems along the cost-effectiveness axis decreases as the cost increases. So the scale density, which would be the cost divided by that spacing, ends up being proportional to the inverse square of cost-effectiveness. This is easier to understand in a spreadsheet. And the inverse square distribution is exactly where i would expect to see logarithmic returns to scale.
As for what distributions actually make sense in real life, i really don’t know. That’s more for people working in concrete cause areas to figure out than me sitting at home doing math. I’m just happy to provide a straightforward equation for those people to punch their more empirically-informed distributions into.
Of course you’re right; my “log uniform” assumption is in a different space than your “Pareto” assumption. I think I need to play around with the scale density notion a bit more until it’s properly intuitive.