Steven Landsburg argued, in an oft-quoted article, that the rational way to donate to charity is to give everything to the charity you consider most effective, rather than diversify; and that this is always true when your contribution is much smaller than the charities' endowments. Besides an informal argument, he provided a mathematical addendum for people who aren't intimidated by partial derivatives. This post will bank on your familiarity with both.

I submit that the math is sloppy and the words don't match the math. This isn't to say that the entire thing must be rejected; on the contrary, an improved set of assumptions will fix the math and make the argument whole. Yet it is useful to understand the assumptions better, whether you want to adopt or reject them. 

And so, consider the math. We assume that our desire is to maximize some utility function U(X, Y, Z), where X, Y and Z are total endowments of three different charities. It's reasonable to assume U is smooth enough so we can take derivatives and apply basic calculus with impunity. We consider our own contributions Δx, Δy and Δz, and form a linear approximation to the updated value U(X+Δx, Y+Δy, Z+Δz). If this approximation is close enough to the true value, the rest of the argument goes through: given that the sum Δx+Δy+Δz is fixed, it's best to put everything into the charity with the largest partial derivative at (X,Y,Z).

The approximation, Landsburg says, is good "assuming that your contributions are small relative to the initial endowments". Here's the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small - what then? Why does it follow that the linear approximation works? There's no explanation, and if you think this is because it's immediately obvious - well, it isn't. It may sound plausible, but the math isn't there. We need to go deeper.

We don't need to go all that deep, actually. The tool which allows us to estimate our error is Taylor's theorem for several variables. If you stare into that for a bit, taking n=1, you'll see that the leftovers from U(X+Δx, Y+Δy, Z+Δz), after we take out U(X,Y,Z) and the linear terms with the partial derivatives, are a bunch of terms that are basically second derivatives and mixed derivatives of U times quadratic contributions. In other, scarier words, things like ∂U²/∂x²*(Δx)² and ∂U²/∂x∂y*(ΔxΔy). The values of the second/mixed derivatives to be used are their maximal values over all the region from (X,Y,Z) to (X+Δx, Y+Δy, Z+Δz). And I've also ignored some small constant factors there, to keep things simple.

These leftovers are to be compared with the linear terms like ∂U(X, Y, Z)/∂x*(Δx). If the leftovers are much smaller than the linear terms, then the approximation is pretty good. When will that happen?

Let's look at the mixed derivatives like ∂U²/∂x∂y first. A partial derivative like ∂U/∂x measures the effectiveness of charity X - how much utility it brings per dollar, currently. A mixed derivative measures how much the effectiveness of X changes as we donate money to Y. If charities X and Y operate in different domains, this is likely to be very close to zero, and the mixed derivative terms can be ignored. If X and Y work on something related, this isn't likely to be zero at all, and the overall argument fails. So, we need a new assumption: X and Y operate in different domains, or more generally do not affect each other's effectiveness.

(Here's an artificial example: say you have X working on buying food for hungry Eurafsicans, and Y working on delivering food aid to Eurafsica. If things are perfectly balanced between them, then your $100 contribution to either is not of much use, but a $50 contribution to both is helpful. This is because X's effectiveness rises if Y gets a little more money, and vice versa. Note that it may also happen that the interference between X and Y hinders rather than helps their common goal, and in that case it may still be better to give money to only one of them, but not because of Landsburg's argument).

Now that the mixed derivate terms are gone, let's look at the second derivative terms like ∂U²/∂x²*(Δx)²  and compare them to the linear terms ∂U(X, Y, Z)/∂x*(Δx). Cancelling out the common factor Δx, we see that for the linear approximation to be good, the effectiveness of the charity (first derivative) must be much greater than maximum rate of effectiveness change (over the given interval) times the contribution. This gives us the practical criterion to follow. If our contribution is very large, or if it is liable to influence the charity's effectiveness considerably - and "large", "considerably" here are words that can in principle be measured and made precise - then the linear approximation isn't so good, and we can't infer automatically that banking everything on one charity is the right thing to do. It may still be, but that requires a deeper analysis, one that may or may not be feasible with our amount of uncertainty.

(Here's an artificial example: say X is a non-profit that has a program to match your contribution until they reach a set goal of donations, and they're $50 short of that goal. Say Y is another non-profit or charity that you consider to be 1.5 more effective than X, normally. Then, if your budget is $100, it's optimal to give $50 to X and $50 to Y, rather than everything to one of them. This is because X will undergo a radical change in effectiveness after the first $50, and the second derivative will be large).

Where is the original criterion "keep Δx/X small", then? It failed to materialize; instead, the math tells us that we need to keep Δx small and ∂U²/∂x² small. The endowment size, as expected, is irrelevant; but let's try to make a connection with it anyhow. We can do it with a heuristic argument that goes something like this. The second derivative measures the way our donations influence effectiveness of the charity, rather than the utility. If the charity is large and well-established, probably the way your money splits into administrative costs and actual good has stabilized and is unlikely to change; whereas if the charity is small and especially just starting out, your money may help it set up or change its infrastructure which will change its effectiveness. So the size of the endowment probably correlates well with the second derivative being small. And then the recipe becomes "keep Δx small and X large", which can be rephrased as the original "keep Δx/X small". 

Does this re-establish the correctness of the original argument, then? To some extent, yet to my mind not a large one. For one thing, the correlation is not ideal, it's easy to think of exceptions, and in fact if you're dealing with real charities that tell you something about how they operate, it may be easier for you to estimate that the rate of effectiveness change is or isn't zero, than it is to look at endowment size. But more importantly, the heuristic jump through the correlative hoop strips the argument of any numeric force that the correct version does have. Since we don't know how exactly X and ∂U²/∂x² are related, e.g. what is the correlation factor, we can't give any specific meaning to "keep Δx/X small". We can't say how small, even approximately: 1/10? 1/1000? 1/10⁶? This makes me suspect that the heuristic argument is not a good way to approach the truth, but may be a good way to convince someone to put everything into one charity, because normally Δx/X will appear to be rather small.

Once we've worked out the math, some deficiencies of the original informal article become clear. For instance, the talk about "making a dent" in the problem is a little off the mark:

So why is charity different? Here's the reason: An investment in Microsoft can make a serious dent in the problem of adding some high-tech stocks to your portfolio; now it's time to move on to other investment goals. Two hours on the golf course makes a serious dent in the problem of getting some exercise; maybe it's time to see what else in life is worthy of attention. But no matter how much you give to CARE, you will never make a serious dent in the problem of starving children. The problem is just too big; behind every starving child is another equally deserving child.

But it isn't making the dent in the problem that's the issue; it's making the dent in effectiveness. It's conceivable that my donation doesn't make a noticeable dent in the problem, but changes the rate of dent-making enough so the argument falls through. The words don't match the math. Similarly, consider the analogy with investment. Why doesn't it in fact work - why doesn't the math argument work on investment portfolios? If you want to maximize profit, your small stock purchase is unlikely on its own to influence the stock (change its effectiveness) much. Landsburg's answer - putting it in terms of "making a dent in the problem of adding high-tech stocks" - is flawed: it presupposes that diversification is good by cordoning off different investment areas from each other - "high-tech stocks". The real reason is, of course, risk aversion - our utility function for investment is likely to be risk-averse. You may want to apply risk aversion to your charity donation as well, but in that case, Eliezer's advice to purchase fuzzies and utilons separately is persuasive. 

To sum up, these assumptions will let you conclude that the rational thing to do is donate all your charity budget to the one charity you consider most effective currently:

1. Assume that a unified single-currency utility function describes your idea of these charities' utilities. The crucial thing you're assuming here is that X, Y and Z can be compared in terms of the good they're doing; that their amounts of good can be computed in the same "utilons". This is a strong assumption that may work for some and not others. Certainly most people reject it when all goals are at stake, and not just charity-giving; Complexity of Value discusses this. Why are things different when we restrict to helping others, and are they necessarily different? Yvain has presented an excellent argument for the latter position in Efficient Charity, while Phil Goetz provided an excellent argument against it in the comments. 
2. Assume that the charities you're choosing between operate in different domains, or more generally speaking do not affect each other's effectiveness. If they do, putting all into one may still be the right thing to do, but finer analysis is needed.
3. Assume that the effectiveness of the charities is influenced very little or not at all by your donation, and the donation itself is not too large. In case of doubt, apply more precise math: rate of effectiveness change times donation must be much smaller - say, a few percent of - effectiveness itself.