A Mathematical Explanation of Why Charity Donations Shouldn't Be Diversified

Diversification makes sense in a preservation context, where any asset you have could devalue by a near-arbitrary factor - and then it only matters because the utility of going near 0 assets is really really negative, so you try to steer around that by not letting one catastrophe sink you.

When handing out goodies, there is no such extreme penalty for it ending up worthless. It would be bad, but since it's only linearly bad, that's taken into account fully by taking the expected return.

What would you say to someone who told you that the second assumption is reasonable because most of humanity is irrational?

(Not exactly a hypothetical -- someone did attempt this rebuttal recently.)

I don't think the argument is parallel. Instead, consider:

If you're giving to charity anyway, give to the charity that has the highest expected impact. If you're voting anyway, vote for the candidate with the highest expected impact.

Here, you have optimal philanthropy plus voting against lizards.

I agree that this is a simple statement that shouldn't need this kind of elaboration. Unfortunately, some people don't agree with the statement. I'm hoping that when written out in such excruciating detail, the argument gets a better chance of finally getting communicated, in those cases where a much more lightweight explanation, such as the one you've cited, doesn't do the trick. (Added more disclaimers to the first paragraph, describing when there is probably no point in reading the post.)

Thinking further about my own question, it would depend on whether one values not just QALYs, but confidence that one had indeed bought some number of QALYs -- perhaps parameterised by the mean and standard deviation respectively of the effectiveness estimates. But that leads to an argument for diversification, for the same reasons as in investing: uncorrelated uncertainties tend to cancel out.

The logic requires that your donations are purely altruistically motivated and you only care for good outcomes.

E. g. take donating to one of the organizations A, or B for cancer research. If your donations are purely altruistic and the consequences are the same you should have no preference on which of the organizations finds a new treatment. You have no reason to distinguish the case of you personally donating $ 1000 to both organizations and someone else doing the same from you donating $2000 to A and someone else donating $2000 to B. And once the donations are made you should have no preference between A or B finding the new treatment.

So the equivalent to your personal portfolio when making investments aren't your personal donations, but the aggregate donations of everyone. And since you aren't the only one making donations the donations are already diversified, so you are free to pick something underrepresented with high yield (which will almost certainly still be underrepresented afterwards). If you manage 0.1% of a $ 10,000,000 portfolio with 90% in government bonds it makes no sense to invest any of that 0.1% in government bonds in the name of diversification.

Why would charities behave any differently than profit-making assets? Do you think that charities have less uncertainties?

The confusion concerns whose risk is relevant. When you invest in stocks, you want to minimize the risk to your assets. So, you will diversify your holdings.

When you contribute to charities, if rational you should (with the caveats others have mentioned) minimize the risk that a failing charity will prove crucial, not the risk that your individual contribution will be wasted. If you take a broad, utilitarian overview, you incorporate the need for diversified charities in your utility judgment. If charity a and b are equally likely to pay off but charity a is a lot smaller and should receive more contributions to avoid risk to whatever cause, then you take that into account at the time of deciding on a and b, leading you to contribute everything to a for the sake of diversification. (It's this dialectical twist that confuses people.)

If your contribution is large enough relative to the distinctions between charities, then diversification makes sense but only because your contribution is sufficient to tip the objective balance concerning the desirable total contributions to the charities.

I don't think philanthropists are risk-adverse in lives saved or quality-adjusted life years.

There's additional issue concerning imperfect evaluation.

Suppose we made a charity evaluator based on Statistical Prediction Rules, which perform pretty well. There is an issue though. The charities will try to fake the signals that SPR evaluates. SPR is too crude to resist deliberate cheating. Diversification then decreases payoff for such cheating; sufficient diversification can make it economically non viable for selfish parties to fake the signals. Same goes for any imperfect evaluation scheme, especially for elaborate processing of the information (statements, explanations, suggestions how to perform evaluation, et cetera) originating from the donation recipient.

You just can not abstract the imperfect evaluation as 'uncertainty' any more than you can abstract a backdoor in a server application as noise in the wire.

Why would charities behave any differently than profit-making assets?

The difference (for some) isn't in uncertainty, it's in utility, which isn't really made clear in the OP.

Risk aversion for personal investment stems from diminishing marginal utility: Going from $0 to $1,000,000 of personal assets is a significantly greater gain in personal welfare than going from $1,000,000 to $2,000,000. You use the first million to buy things like food, shelter, and such, while the second million goes to less urgent needs. So it makes sense to diversify into multiple investments, reducing the chance of severe falls in wealth even if this reduces expected value. E.g. for personal consumption one should take a sure million dollars rather than a 50% chance of $2,200,000.

If one assesses charitable donations in terms of something like "people helped by anyone" rather than something like "log of people helped by me" then there isn't diminishing utility (by that metric): saving twice as many people is twice as good. And if your donations are small relative to the cause you are donating to, then there should be significantly diminishing returns to money in terms of lives saved: if you donate $1,000 and increase the annual budget for malaria prevention from $500,000,000 to $500,001,000 you shouldn't expect that you are moving to a new regime with much lower marginal productivity.

But you might care about "log of lives saved by me" or "not looking stupid after the fact" or "affiliating with several good causes" or other things besides the number of people helped in your charitable donations. Or you might be donating many millions of dollars, so that diminishing impact of money matters.

It is common knowledge that diversification is a strategy used by risk-adverse agents to counter the negative effects of uncertainty.

When one is risk averse, one trades some expected gain to minimize potential loss. The relevant question is whether it makes any sense to be risk averse with respect to your charity donations.

I'd say no. Risk aversion for my own pile of money comes largely from decreasing marginal utility of each dollar in my pile when spent on me and mine. My first and last dollar to most charities are two drops in a bucket, with the same marginal "problem solving" power.

This doesn't take into account the other benefits of charitable giving, such as signaling and good feelings. In both cases, I'd say that others and you respond more favorably to you the more charities you donate to. In that respect, at least, there is decreasing marginal utility for each dollar more spent on a particular charity. But I think that feel good aspect was not part of the assumed utility calculation. If your goal is to best solve problems, take aim at what you consider the best target, and shoot your wad.

This is not intended as a complete argument, rather it's an elaboration of a point whose understanding might be helpful in understanding a more general argument. If this point, which is a simplified special case, is not understood, then understanding the more general argument would be even less likely. (Clarified this intent in the first paragraph.)

One difference: Utility in money is non-linear; utility in lives saved is linear.

Not sure if the argument is worth it, seems like all it does is to prove that in a simple linear programming model the solution corresponds to a vertex (a single charity). It does not deal with uncertainties and risk at all.

By assuming the linear approximation is good enough, you beg the question. Consider that if that is to be assumed, you could just take U, A and B to be linear functions to begin with...

It's not so much an assumption, as an intermediate conclusion that follows from the dependence of U on A and B being smooth enough (across the kind of change that dM is capable of making) and nothing else. My only claim is that under this assumption, the non-diversification conclusion follows. U(A,B) being linear is a hugely more restrictive assumption.