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What you wrote looks clean and correct and, indeed, i used the Pareto distribution  parameter incorrectly and will change that line of the post. Thank you!

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.

I was finding it hard to keep track of all the different organizations posting about their marginal funding plans recently, so i made a simple spreadsheet:

Feel free to add any other EA orgs or fix errors or re-arrange everything or whatever.

See Chapter 5 of this amazing textbook for more.

This link takes me to an error 404 page.

I'm assuming from the url and https://roberttrivers.com/Books.html that this is supposed to go here: Burt and Trivers' Genes in Conflict: The Biology of Selfish Genetic Elements.

Following up on this, i did work out a solution for the case where we have an annual income, there is outside funding, there's no discounting, and there's a consumption option where instead of funding a project we can just collect K utils per dollar. The work behind this is a mess, partly because the equations get long and partly because it was mostly just me doing the same thing repeatedly for slightly different situations until i stopped being confused. Since either declining to show my work or putting 14 kilobytes of garbage in a forum comment would both be bad, here it is in a pastebin link: https://pastebin.com/raw/PnDZ2rTZ

The result is if there are two projects, X and Y, and our income I is such that we can't affect which of two projects gets done first, that is, if C_X / F_X < C_Y / (F_Y + I), then project X will always be finished before project Y and there's nothing we can do about it, then we should fund whichever project has higher U/F. But if we are able to affect which project gets done first, we should fund whichever has higher (U - K*F) / C.

And after thinking about it more and writing more equations, i think U/F really does give us a direct comparison of project-like interventions (utility over time forever once it's fully funded) to consumption-like interventions (utility per dollar). And it gives us a direct comparison of project-like interventions if we can spend money to complete a project in zero time. And it gives us a direct comparison of project-like interventions in the case where making/spending money takes time, but that time doesn't matter because we can't change the order in which things get done. The case where it does not work is when we have an annual income as opposed to a one-shot budget and we're comparing two project-like interventions and the one that we fund first is the one that gets done first.

I think what makes the result for the case where we have an income and can determine which intervention gets done first so qualitatively different from the case where have a stack of cash and can choose between two projects to knock out is that we have to take into account how much choosing to fund the first project delays our ability to fund the second project. And that delay is proportional to C. (Everything here is assuming no diminishing returns to rate of funding, so it's always best to concentrate funding on one project to knock it out as soon as possible and never makes sense to split funding between projects and get neither one done.)

In your equation where future benefits are discounted, using the derivative makes sense if your donation is small relative to the total cost of the project. I was doing the opposite and assuming that we pay for the whole thing. Given that the estimated cost is in the billions of dollars and a lot of that funding is from people we can't coordinate with, your assumption seems closer to reality than mine.

Without discounting, things are less straightforward and i've got a messy page of math/notes right now that i'll try to turn into something post-able this weekend. But basically when i wrote that we should first fund whichever option has higher U/C, i had forgotten the assumptions that that result was based on. U/C is the right metric if there is no outside funding and you have a constant income per year. The way you're approaching the problem is as if we have a limited budget to use today and no income and each project has a constant, positive amount of outside funding per year. I wrote out a bunch of equations under those assumptions and am convinced that, given two projects, we should choose whichever has higher U/F. At least the way i derived them, these metrics come from calculations that work around the whole infinite utility thing by looking at the difference in the finite amounts of utility that we miss out on (relative to having both projects done at time zero) depending on which of two projects we fund first, so this stuff does not generally result in "utility per dollar" numbers that can be compared to other cost-effectiveness estimates where everything is nice and finite.

If there's no outside funding and your money is limited, some projects will never be completed, which causes infinity-related issues that break everything. With finite utilities, i think that's just a knapsack problem.

I have not yet figured out a solution for the case where we have an annual income, there is outside funding, and no discounting, but that's the plan.

Gotcha. I was thinking about a much simpler situation where we're comparing two interventions to accomplish equally valuable goals, rather than two interventions to accomplish the same goal, where finishing one makes the other obsolete. I was also assuming that we are able to coordinate on what to fund. But in the situation you described, it makes sense to fund the cheaper intervention only if we can put together enough money for it to overtake the one that's already being funded, like 555,555,555 euros in your example. But that number is assuming we can just linearly spend money to make stuff happen sooner.