A common belief within the Effective Altruism movement that you should not diversify charity donations when your donation is small compared to the size of the charity. This is counter-intuitive, and most people disagree with this. A Mathematical Explanation of Why Charity Donations Shouldn't Be Diversified has already been written, but it uses a simplistic model. Perhaps you're uncertain about which charity is best, charities are not continuous, let alone differentiable, and any donation is worthless unless it gives the charity enough money to finally afford another project, your utility function is nonlinear, and to top it all off, rather than accepting the standard idea of expected utility, you are risk-averse.

Standard Explanation:

If you are too lazy to follow the link, or you just want to see me rehash the same argument, here's a summary.

The utility of a donation is differentiable. That is to say, if donating one dollar gives you one utilon, donating another dollar will give you close to one utilon. Not exactly the same, but close. This means that, for small donations, it can be approximated as a linear function. In this case, the best way to donate is to find the charity that has the highest slope, and donate everything you can to it. Since the amount you donate is small compared to the size of the charity, a first-order approximation will be fairly accurate. The amount of good you do with that strategy is close to what you predicted it would do, which is more than you'd predict of any other strategy, which is close to what you'd predict for them, so even if this strategy is sub-optimal, it's at least very close.

Corrections to Account for Reality:

Uncertainty:

Uncertainty is simple enough. Just replace utility with expected utility. Everything will still be continuous, and the reasoning works pretty much the same.

Nonlinear Utility Function:

If your utility function is nonlinear, this is fine as long as it's differentiable. Perhaps saving a million lives isn't a million times better than saving one, but saving the millionth life is about as good as the one after that, right? Maybe each additional person counts for a little less, but it's not like the first million all matter the same, but you don't care about additional people after that.

In this case, the effect of the charity is differentiable with respect to the donation, and the utility is differentiable with respect to the effect of the charity, so the utility is differentiable with respect to the donation.

Risk-Aversion:

If you're risk-averse, it gets a little more complicated.

In this case, you don't use expected utility. You use something else, which I will call meta-utility. Perhaps it's expected utility minus the standard deviation of utility. Perhaps it's expected utility, but largely ignoring extreme tails. What it is is a function from a random variable representing all the possibilities of what could happen to the reals. Strictly speaking, you only need an ordering, but that's not good enough here, since it needs to be differentiable.

Differentiable is more confusing in this case. It depends on the metric you're using. The way we'll be using it here is that having a sufficiently small probability of a given change, or a given probability of a sufficiently small change, counts as a small change. For example, if you only care about the median utility, this isn't differentiable. If I flip a coin, and you win a million dollars if it lands on heads, then you will count that as worth a million dollars if the coin is slightly weighted towards heads, and nothing if it's slightly weighted towards tails, no matter how close it is to being fair. But that's not realistic. You can't track probabilities that precisely. You might care less about the tails, so that only things in the 40% - 60% range matter much, but you're going to pick something continuous. In fact, I think we can safely say that you're going to pick something differentiable. If I add a 0.1% chance of saving a life given some condition, it will make about the same difference as adding another 0.1% chance given the same condition. If you're risk-averse, you'd care more about a 0.1% chance of saving a life it's takes effect during the worst-case scenario than the best-case, but you'd still care about the same for a 0.1% chance of saving a life during the worst case as for upgrading it to saving two lives in that case.

Once you accept that it's continuous, the same reasoning follows as with expected utility. A continuous function of a continuous function is continuous, so the meta-utility of a donation with respect to the amount donated is continuous.

To make the reasoning more clear, here's an example:

Charity A saves one life per grand. Charity B saves 0.9 lives per grand. Charity A has ten million dollars, and Charity B has five million. One or more of these charities may be fraudulent, and not actually doing any good. You have $100, and you can decide where to donate it.

The naive view is to split the $100, since you don't want to risk spending it on something fraudulent. That makes sense if you care about how many lives you save, but not if you care about how many people die. They sound like they're the same thing, but they're not.

If you donate everything to Charity A, it has $10,000,100 and Charity B has $5,000,000. If you donate half and half, Charity A has $10,000,050 and Charity B has $5,000,050. It's a little more diversified. Not much more, but you're only donating $100. Maybe the diversification outweighs the good, maybe not. But if you decide that it is diversifying enough to matter more, why not donate everything to Charity B? That way, Charity A has $10,000,000, and Charity B has $5,000,100. If you were controlling all the money, you'd probably move a million or so from Charity A to Charity B, until it's well and truly diversified. Or maybe it's already pretty close to the ideal and you'd just move a few grand. You'd definitely move more than $100. There's no way it's that close to the optimum. But you only control the $100, so you just do as much as you can with that to make it more diversified, and send it all to Charity B. Maybe it turns out that Charity B is a fraud, but all is not lost, because other people donated ten million dollars to Charity A, and lots of lives were saved, just not by you.

Discontinuity:

The final problem to look at is that the effects of donations aren't continuous. The time I've seen this come up the most is when discussing vegetarianism. If you don't it meat, it's not going to make enough difference to keep the stores from ordering another crate of meat, which means exactly the same number of animals are slaughtered.

Unless, of course, you were the straw that broke the camel's back, and you did keep a store from ordering a crate of meat, and you made a huge difference.

There are times where you might be able to figure that out before-hand. If you're deciding whether or not to vote, and you're not in a battleground state, you know you're not going to cast the deciding vote, because you have a fair idea of who will win and by how much. But you have no idea at what point a store will order another crate of meat, or when a charity will be able send another crate of mosquito nets to Africa, or something like that. If you make a graph of the number of crates a charity sends by percentile, you'll get a step function, where there's a certain chance of sending 500 crates, a certain chance of sending 501, etc. You're just shifting the whole thing to the left by epsilon, so it's a little more likely each shipment will be made. What actually happens isn't continuous with respect to your donation, but you're uncertain, and taking what happens as a random variable, it is continuous.

A few other notes:

Small Charities:

In the case of a sufficiently small charity or large donation, the argument is invalid. It's not that it takes more finesse like those other things I listed. The conclusion is false. If you're paying a good portion of the budget, and the marginal effects change significantly due to your donations, you should probably donate to more than one charity even if you're not risk-averse and your utility function is linear.

I would expect that the next best charity you manage to find would be worse by more than a few percent, so I really doubt it would be worth diversifying unless you personally are responsible for more than a third of the donations.

An example of this is keeping money for yourself. The hundredth dollar you spend on yourself has about a tenth of the effect the thousandth does, and the entire budget is donated by you. The only time you shouldn't diversify is if the marginal benefit of the last dollar is still higher than what you could get donating to charity.

Another example is avoiding animal products. Avoiding steak is much more cost-effective than avoiding milk, but once you've stopped eating meat, you're stuck with things like avoiding milk.

Timeless Decision Theory:

If other people are going to make similar decisions to you, your effective donation is larger, so the caveats about small charities applies. That being said, I don't think this is really much of an issue.

If everyone is choosing independently, even if most of them correlate, the end result will be that the charities get just enough funding that some people donate to some and others donate to others. If this happens, chances are that it would be worth while for a few people to actually split their investments, but it won't make a big difference. They might as well just donate it all to one.

I think this will only become a problem if you're just donating to the top charity on GiveWell, regardless of how closely they rated second place, or you're just donating based purely on theory, and you have no idea if that charity is capable of using more money.