The one charity theorem

The normal scheme for donating to charities is to divide money up among several different charities. The following argument shows why this strategy is often wrong. Both the statement and the proof will be extremely informal:

One charity theorem: Assume we have a fixed amount of money to divide between $n$ charities. Assume that utility is a smooth function of the performances of the charities, which in turn depend smoothly on the amount of money each receives. In the limit of a small amount of money, it is optimal to give to only one charity. Conversely, with overwhelming probability, it is never optimal to give to more than one charity.

Proof: Let the utility function be $u(X) = u(x_1, \ldots, x_n)$, where $x_1 + \cdots + x_n = T$ are the amounts of money given to each charity. Since $u$ is smooth and $T$ is small, we can linearize to get $$u(X) \approx u(0) + \nabla u \cdot X$$ where $\nabla u$ is the gradient of $u$. This linearized version is maximized by giving all money to the charity which maximizes $du/dx_i$. Moreover, as long as this maximum value is distinct, which occurs with overwhelming probability if we imagine that $u$ is a bit noisy, the maximum is unique.

Practical matters

It’s important to discuss when this result applies in practice and when it doesn’t. First, uncertainty does not matter, including uncertainty about “the values of $du/dx_i$“. All kinds of uncertainly are simply folded into the utility function $u$, which results in more smoothness rather than less. Thus, the result applies even if you don’t know what your preferences really are; in this case, just make a good guess and give all the money to that charity.

Similarly, multiplexing in time also does not matter to some extent: even if we don’t know about the future, we can simply list “charities in the future” as one of the entries and apply the theorem. In particular, saving up money and donating it in larger chunks may be better than numerous small donations over time if one expects to have more knowledge (and therefore more accurate utility) in the future.

There are two valid reasons the theorem may not apply: failure of smoothness and failure of smallness. The smoothness is easy: two charities need a small amount of money to meet a certain goal, donating a small amount of money to both may be optimal. However, this applies only if the discontinuity can be accurately predicted. For example, Kickstarter projects have a threshold amount which must be reached to take effect, but an individual donating a small amount of money will still have a smooth utility function due to the unknown amount of other people’s donations.

The more important condition is smallness, which turns utility into a fully nonlinear function and in particular increases the likelihood of discontinuities. Smallness can arise either because the charity itself is small, or because utility depends on something intrinsic to the donation rather than the performance of the charity. If you derive personal satisfaction or reputation based on the number of charities you donate to, independent of the amount, the result does not apply (and also you are part of a problem). If the charities themselves are small, so that smaller donations to several have both a significant effect and significant diminishing returns, great. This is one of the reasons why microfinance is such a great idea.

On the other hand, even if there are too many charities for the theorem to apply for each individually, it’s possible that it does apply to entire classes of charities. It may not be rational to donate to both microfinance and any other charity, for example.

Also note that same argument applies when money is replaced with donations of time, though it’s much easier for time to have personal utility terms which break smallness.

Rootstrikers

This argument is one of the main reasons why Rootstrikers is a good idea in principle: given a small amount of influence on a large system such as government, it is highly likely that focusing on one problem to the exclusion of all else is the only rational course. This applies to those with fulltime politically related jobs (e.g., Lessig), or to donations of money and (again to a lesser extent) donations of small amounts of time, thought, etc.

While nothing in the argument says that Rootstrikers is the best way to go about this task, or that Rootstrikers is the best organization with this plan, remember that uncertainty does not invalidate the theorem. If you are small, pick one. Do not hedge charities.

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