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,
the maximum expected value approaches the expected value of the even allocation portfolio,
the minimum distance between an optimal portfolio and the even allocation portfolio approaches 0, and
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 fi , then you could tractably optimize with the budget B.
Maximize the following over the allocation weights wi,0≤wi≤1,∑Ni=1wi=1:
N∑i=1fi(wiB)
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 fi.
I have two related posts, but they’re about deep/model uncertainty/ambiguity, not precisely quantified uncertainty:
Even Allocation Strategy under High Model Ambiguity
Hedging against deep and moral uncertainty
It sounds closer to the first one.
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,
the maximum expected value approaches the expected value of the even allocation portfolio,
the minimum distance between an optimal portfolio and the even allocation portfolio approaches 0, and
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 fi , then you could tractably optimize with the budget B.
Maximize the following over the allocation weights wi,0≤wi≤1,∑Ni=1wi=1:
N∑i=1fi(wiB)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 fi.