It is a well-known argument that diversification of charitable giving is hard to justify philosophically [1, 2, 3]. Givewell, probably the gold standard in terms of impact maximisation, mostly suggests 4 top charities [4]. Arguably, this is either a fairly big number (bigger than 1) or a fairly small number (not at all diversified, if diversification matters). Obviously, this number will have practical reasons, related to available opportunities, real-world constraints and operational considerations. However, it prompted me to think about the theoretical foundations of what I actually think I should do, which I would like to write up here.
In this post, I am firstly aiming to make the non-diversification argument somewhat rigorous without obscuring it too much with math. Secondly I would like to explain, in the spirit of Bayesian model averaging, my current thinking of why I still like diversification somewhat more than these results suggest.
The latter is somewhat ad-hoc. One aim is to get pushback and further directions on this. I suspect my thinking will be related to some commonly held views and to philosophical arguments around worldview diversification (?). Perhaps there is a simple refutation I am not aware of.
Setup and the "fully concentrated" argument
Assume I have one unit of resources/wealth that I want to give/donate effectively. Let there be giving opportunities 1 to N, and wi>0 will be my allocation to the ith, where ∑iwi=1.
The marginal expected utility/goodness/... of me allocating wi to i will be uiwi. The total (still "marginal" in eg a counterfactual contribution sense) expected utility will be E[u⋅w]=E[∑iuiwi]=E[u]⋅w.
(Note that it is very plausible that this is linear, because my donations are relatively small and donation opportunities have a lot of capacity – so even if you don't believe in linear utility you can just Taylor expand.)
To maximise the impact my donation has, I need to solve w∗=argmaxwE[u]⋅w. Clearly, the solution is to allocate everything to the largest entry of the vector E[u], which has index k. So w∗k=1 and all others 0.
So, I should find the most worthwhile cause and allocate solely to it. This somewhat counterintuitive in some cases – what if there are two almost indistinguishable causes and I'm very uncertain which one is better? Should I not just diversify across both, if practical?
That is, this method seems very non-robust in the sense that as soon as my belief about the ranking of opportunities updates slightly, I will fully switch from k to another opportunity going forward. At the core of this is that "ranking" is an extremely non-linear tranformation of very "noisy" and continuous variables like lives saved per dollar.
Why giving is different from portfolio optimisation
In portfolio optimisation, eg in finance, I would typically solve something like argmaxw(E[u]⋅w−λ2w′E[uu′]w+h.o.),
where the second and higher order terms capture interactions and non-linearities related to risk aversion. This is not an irrational bias in investing, cf "volatility drag".
However, risk aversion is not justifiable in charitable giving – it would only serve to make me feel better and reduce the impact [3, 5].
Interactions are also mostly relatively implausible because giving opportunities are mostly independent, eg geographically separated.
If I deviate from allocating all to the best opportunity, I have to have some other reason.
Pulling some parameters out of the optimisation problem
The trick
I don't know the utility of different opportunities and their probability distributions for sure, so let's say there is a (hyper ?) parameter θ∈Ω that captures different models of the world, including my model around the uncertainty of my allocations' impact.
The aim is for this to try and reflect epistemic humility in some way and get rid of the hard cut-off that ranking creates.
For example, this could capture some of the model uncertainty around plausible distribution of lives saved per money per charity, but this could also capture moral uncertainty around what opportunity has which utility (or equivalently "choiceworthiness" when comparing theories).
This means that I've now split the expectation into E[u]=∫ΩEθ[u]dP(θ).
(This is just the law of iterated expectations.)
If I just solve the maximisation problem above, the result will be the same.
However, if I instead want to get a "model probability weighted best allocation", I need to first solve the problem for each value of θ and then average.
So, I compute w∗=∫Ωargmaxw(Eθ[u]⋅w)dP(θ).
This has the solution w∗i=pi where pi is the probability that i is the best allocation opportunity over the whole space of possible probability weighted models. In other words, if I was randomly placed in a world according to P(θ), optimised, and reported back the results, this would be the average allocation.
Conveniently, allocations are linear, so I can just take the probability weighted optimal allocation across models of the world instead of the optimum across all models of the world. In this sense, allocation problems are a very special type of decision problem when trying to include intertheoretic comparisons: We can always compare allocations and do not need to "normalise" theories.
What does this mean?
In this setting, I should allocate proportionally to the probability that the model where the cause is the most worthwhile in expectation is true.
This has some nice properties. Allocating proportionally to this kind of probability means that I still very strongly favour the best options in the typical sense. Opportunities that are not the best in any possible model of the world will get zero allocation. At the same time, charities that could plausibly be the best will get some allocation.
For example, if I pull some of the uncertainty around which of GiveWell's top charities are slightly more or less effective out of the optimisation problem, I will allocate close to equally across them. If GiveWell introduced another recommended top charity that I think is only 5% likely to be the best among plausible world views, it seems reasonable to allocate 5% to it.
Similarly, if there are two very contradictory and extreme world views that I give some mildly different credence to, with this approach my allocation might be close to zero instead of "all in" on one of them. Likewise, low credence "fanatic" theories can only do limited damage.
Is this justified?
I don't know, but I think it is somewhat distinct from risk aversion-type arguments, and also different from a minimax-type argument. I also think this is similar to what many people do in practice. It is more closely related to ambiguity aversion.
Admittedly, it is fairly arbitrary which parameters I "pull outside" of the optimisation problem and hard to do this rigorously. However, it seems intuitively reasonable to average what I do over some possible models of the world and not maximise my impact in the average model of the world.
Another concern is that it could be some kind of "risk aversion introduced through the backdoor".
In the thought experiment in [3] (originally from [5]), for x<10 this method would allocate 50:50 between option A and B with a total expected "goodness" of 15+x/2, which is worse than 15+x, but better than 15. For large x>10 it flips to always preferring B, which is also reasonable.
References
[1] https://forum.effectivealtruism.org/posts/wHaj9zpoeraQqBG9H/statistical-foundations-for-worldview-diversification
[2] https://slate.com/culture/1997/01/giving-your-all.html
[3] https://www.lesswrong.com/posts/h7GSWD2eLXygdtuhZ/against-diversification
[4] https://www.givewell.org/charities/top-charities
[5] https://globalprioritiesinstitute.org/on-the-desire-to-make-a-difference-hilary-greaves-william-macaskill-andreas-mogensen-and-teruji-thomas-global-priorities-institute-university-of-oxford/
Maximizing a linear objective always leads to a corner solution. So to get an optimal interior allocation, you need to introduce nonlinearity somehow. Different approaches to this problem differ mainly in how they introduce and justify nonlinear utility functions. To me I can't see where the nonlinearity is introduced in your framework. That makes me suspect the credence-weighted allocation you derive is not actually the optimal allocation even under model uncertainty. Am I missing something?
Yes, agree with all your points.
The reason I get a different allocation is indeed because I ultimately don't maximise - the outermost step is just averaging.
This is hard to justify philosophically, but the intuition is sort of "if my maximiser is extremely sensitive to ~noise, I throw out the maximiser and just average over plausible optimal solutions", which I think is in fact what people often do in different domains. (Where "noise" does a lot of work - of course I am very vague about what part of the probability distribution I'm happy to integrate out before the optimisation and which part I keep.)
Just to add: This is similar to taking the average over what many rational utility maximising agents with slightly different models/world views would do, so in some sense if many people followed this rule the aggregate outcome might be very similar to everyone optimising.
I certainly agree that you're right about describing why people diversify but I think the interesting challenge is to try and understand under what conditions this behavior is optimal.
You're hinting at a bargaining microfoundation, where diversification can be justified as the solution arrived at by a group of agents bargaining over how to spend a shared pot of money. I think that's fascinating and I would explore that more.
Yes, I think understanding the microfoundations would be desirable. This need not necessarily be in the form of a proof of optimality, but could come in a different flavour, as you said.
Some concepts that would be interesting to explore further, having thought about this a little bit more (mostly notes to future self):
* Unwillingness to let "noise" be the tie-breaker between exactly equally good options (where expected utility maximisation is indifferent) --> how does this translate to merely "almost equally good" options? This is related to giving some value to "ambiguity aversion": I can have the preference to diversify as much as possible between equally good options without violating maximising utility, but as soon as there are slight differences I would need to trade off optimality vs ambiguity aversion?
* More general considerations around non-commutativity between splitting funds between reasonable agents first and letting them optimise or letting reasonable agents vote first and then optimising based on the outcome of the voting process. I seem to prefer the first, which seems to be non-utilitarian but more robust.
* Cleaner separation between moral uncertainty and epistemic & empirical uncertainty.
* Understand if and how this ties in with bargaining theory [1], as you said, in particular is there a case for extending bargain theoretical or (more likely) "parliamentary" [2] style approaches beyond moral to epistemic uncertainty?
* How does this interacts with "robustness to adverse selection" as opposed to mere "noise" – eg is there some kind of optimality condition assuming my E[u|data] is in the worst case biased in an adversarial way by whoever gives me data? How does this tie in with robust optimisation? Does this lead to a maximin solution?
[1] https://philpapers.org/rec/GREABA-8
[2] https://www.fhi.ox.ac.uk/wp-content/uploads/2021/06/Parliamentary-Approach-to-Moral-Uncertainty.pdf