Post on maximizing EV by diversifying w/ declining marginal returns and uncertainty

Imagine you have two causes where you believe their cost-effectiveness trajectories cross at some point. Cause A does more good per unit resources than cause B at the start but hits diminishing marginal returns faster than B. Suppose you have enough resources to get to the crossover point. What do you do? Well, you fund A up to that point, then switch to B. Hey presto, you're doing the most good by diversifying.

It might be helpful to draw a dashed horizontal line at the maximum value for B, since you would fund A at least until the intersection of that and the curve for A, and start funding B from there (but possibly switching thereafter, and maybe back and forth). Basically, you want to start funding B once the marginal returns from A are lower than the marginal returns for B. It doesn't actually matter for whether you fund B at all that B hits diminishing marginal returns more slowly, only that A's marginal returns are eventually lower than B's initial marginal returns before you exhaust your budget.

If you're including more than just A and B, and A's marginal expected returns are eventually lower than B's initial marginal expected returns before you would exhaust your budget on A, then we can still at least say it wouldn't be optimal to exhaust your budget on A (possibly you would exhaust it on B, if B also started with better marginal returns, or some completely different option(s)).

Adding empirical uncertainty doesn't change the picture: empirical uncertainty basically means you should draw fuzzy lines instead of precise ones, and it'll be less clear when you hit the crossover.

If the uncertainty is precisely quantified (no imprecise probabilities), and the expected returns of each option depends only on how much you fund that option (and not how much you fund others), then you can just use the expected value functions.

For precisely quantified distributions of returns, and expected return functions which depend only on how much is allocated to their corresponding project (and not to other projects), you can just use the expected return functions and forget about the uncertainty, and I think the following is very likely to be true (using appropriate distance metrics):

(Terminology: "weakly concave" means at most constant marginal returns, but possibly sometimes decreasing marginal returns; "strictly concave" means strictly decreasing marginal returns.)

If the expected value functions for N different projects are all identical, positive, non-decreasing and (weakly) concave (and each only depends on the resources used for the corresponding project), then the even allocation is optimal, and if they're also strictly increasing and strictly concave, then the even allocation is the only optimal allocation. And if you fudge the functions a bit so that they're very similar but non-identical, then these conclusions are approximately true, too.

More precisely, for any given positive, increasing and (weakly) concave function, as the expected value functions approach the given function,

1. the maximum expected value approaches the expected value of the even allocation portfolio,
2. the minimum distance between an optimal portfolio and the even allocation portfolio approaches 0, and
3. under a strictly increasing and strictly concave function, the maximum distance between an optimal portfolio and the even allocation portfolio approaches 0.

Maybe you can claim a stronger result by making this more uniform over the function being approached (uniform convergence), or without specifying any function to be approached at all and just having the functions approach each other, assuming the functions come from (a subset of) a compact set of possible functions.

If you could precisely specify the expected value functions (and so had precise credences) as $f_i$ , then you could tractably optimize with the budget $B$.

Maximize the following over the allocation weights $w_i,0\le w_i\le 1,{\sum }_{i=1}^N{w}_i=1$:

I think you can just use a greedy allocation strategy, starting from 0 initial allocation, and allocating each marginal dollar (or X dollars) where it maximizes the expected marginal return until you exhaust your budget.

I think this would also be a convex optimization problem, so fairly well-behaved, and there are other specific optimization methods that would be practical to use (e.g. gradient ascent (+ noise and from random initial points to escape flat regions), maybe a greedy random walk if there aren't too many different projects).

If your credences aren't precise, you could check how much the optimal weights and expected values change under different specifications of the $f_i$.

Mathematical theorems you had no idea existed, cause they’re false...

This looks interesting, but I'd want to see a formal statement.

Is it the expected value that's logarithmic, or expected value conditional on nonzero (or sufficiently high) value?

tl;dr: I think under one reasonable interpretation, with logarithmic expected value and precise distributions, the theorem is false. It might be true if made precise in a different way.

If

1. you only care about expected value,
2. you had the expected value of each project as a function of resources spent (assuming logarithmic expected returns already assumes a lot, but does leave a lot of room), and
3. how much you fund one doesn't affect the distribution of any others (holding their funding constant),

then the uncertainty doesn't matter (with precise probabilities), only the expected values do. So allocating in proportion to your credence that each project will be best depends on something that doesn't actually matter that much, i.e. your credence that the project will be best, because you can hold the expected values for a project constant while adjusting the probability that it's best. 

To be more concrete, we could make all of the projects statistically independent and either return 0 or some high value with some tiny probability, and the value or probability of positive return scales with the amount of resources spent on the project, so that the expected values scale logarithmically. Let's also assume only two projects (or a number that scales sufficiently slowly with the inverse probability of any one of them succeeding). Then, conditional on nonzero impact, your impact will with probability very close to 1 come from whichever project you fund succeeds, since it'll be very unlikely that multiple will.

So, I think we've satisfied the stated conditions of the theorem, and it recommends allocating in proportion to our credences in each project being best, which, with very low independent probabilities of success across projects, is roughly the credence that the project succeeds at all. But we could have projects with the same expected value (at each funding level, increasing logarithmically with resources spent) with one 10x more likely to succeed than the rest combined, so the theorem claims the optimal allocation is to put ~91% into the most likely to succeed project, but the expected value-maximizing allocation is to give the same amount to each project.

I think the even allocation would still be optimal if the common expected value function of resources spent on each project was non-decreasing and (weakly) concave, and if it's strictly increasing and strictly concave (like the logarithm), then the even allocation is also the only maximizer.