A Model of Hits-Based Giving

harsimony

This is a linkpost for https://harsimony.wordpress.com/2022/03/24/a-model-of-hits-based-giving/

In a situation with high upside and high uncertainty, how should an altruist distribute their time and resources to different causes? Is it better to focus on a single cause, or spread attention across causes?

Considering these issues, Open Philanthropy has endorsed a hits-based giving approach:

We suspect that high-risk, high-reward philanthropy could be described as a “hits business,” where a small number of enormous successes account for a large share of the total impact — and compensate for a large number of failed projects.

Despite the high uncertainty that necessitates this approach, might we be able to say something precise about how to distribute a charitable portfolio? How does hits-based giving change how an altruist should act?

Here, I propose a simple model of hits-based giving. Specifically, I model an altruist who can fund a certain number of grants across different cause areas. Each grant awarded in a cause area is treated as an independent draw from from an exponential distribution, and the funder is assumed to care solely about the most successful grant in each cause area. As we will see, this creates diminishing returns to further awards within a single cause, leading the funder to spread their grants across areas.

Notation

The funder is considering different cause areas indexed by $i$ , each with their own impact distribution $X_{i}$ :

$X_{1}, . . ., X_{i}, . . ., X_{A}$

The funder must choose how many grants $n_{i}$ to award in each cause area.

$X_{(n_{i})}$ denotes the maximum draw from a set of $n_{i}$ samples:

$max (x_{1}, . . ., x_{n_{i}}) = X_{(n_{i})} v$

The Model

The funder only cares about the most successful project in each cause area

Their utility function within a cause is:

$U_{i} = X_{(n_{i})}$

Across all causes, the funder adds up the utility of the most successful projects:

$U = A \sum i = 0 U_{i} = A \sum i = 0 X_{(n_{i})}$

This is the key difference from marginal impact thinking. We are ignoring average impact and instead focusing on finding black swans. This also allows us to simplify the math by focusing on the maximum draw from a distribution. This “hits-based” assumption works best for fat tailed distributions where extreme tail events dominate the expected value calculation. This is not an accurate characterization of what organizations like OpenPhil endorse, but attempts to capture the spirit of hits-based altruism.

Each grant is an independent draw from the impact distribution

We assume that within a cause area, the funder is supporting disconnected projects with the same after-the-fact impact distribution. Specifically, impacts are drawn i.i.d. from the distribution. This assumption works best when the results of different projects are uncorrelated, such as when funding independent researchers. Each distribution has an expected value $v_{i}$ .

The impact distribution is represented by an exponential distribution

Within a cause area, grants have an after-the-fact impact drawn from an exponential distribution. Specifically, each $X_{i}$ is distributed according to an exponential distribution with rate parameter $r_i = 1/v_i $:

$X_{i} d = r_{i} e^{- r_{i} x}$

The exponential distribution has nice properties for this model. It has a simple formula for order statistics, impact is always non-negative, and the distribution has a long tail. Additionally, it’s at least possible that real life impact distributions might be described by an exponential distribution.

The General Case

Now we can determine the optimal allocation of grants across cause areas.

The distribution of the maximum of $n_{i}$ draws in this case is:

$X_{(n_{i})} d = \frac{1}{r_{i}} n_{i} \sum j = 1 \frac{Z_{j}}{n_{i} - j + 1}$

Where $Z_{j}$ are i.i.d. standard exponential distributions. Taking the expectation:

$E (X_{(n_{i})}) = \frac{1}{r_{i}} n_{i} \sum j = 1 \frac{1}{n_{i} - j + 1}$

The sum goes from $1$ to $n_{i}$ , meaning that the denominators range from $n_{i}$ to $1$ , so we can simplify the sum:

$E (X_{(n_{i})}) = \frac{1}{r_{i}} n_{i} \sum j = 1 \frac{1}{j} = H (n_{i})$

Where $H (n)$ is the $n$ -th harmonic number. We can use the Taylor series approximation for $H (n)$ (notice that we implicitly assume $n$ is reasonably large for this approximation to be accurate):

$\frac{H (n_{i})}{r_{i}} \approx \frac{γ}{r_{i}} + \frac{ln (n_{i})}{r_{i}}$

Since the funder only cares about the maximum draw, this is equivalent to the utility they receive from awarding $n_{i}$ grants in area $i$ .

To optimally distribute their grants, the funder must ensure that the value of the marginal grant is equal across causes:

$\frac{d U_{i}}{d n_{i}} = \frac{1}{r_{i} n_{i}}$

$\frac{d U_{1}}{d n_{1}} = . . . = \frac{d U_{i}}{d n_{i}} = . . . = \frac{d U_{A}}{d n_{A}}$

Lets set the marginal utilities equal for two representative causes $i$ and $k$ :

$\frac{d U_{i}}{d n_{i}} = \frac{d U_{k}}{d n_{k}}$

$\frac{1}{r_{i} n_{i}} = \frac{1}{r_{k} n_{k}}$

Solving for $n_{i}$ in terms of $n_{k}$ :

$n_{i} = \frac{r_{k} n_{k}}{r_{i}} = r_{k} n_{k} v_{i}$

Now that we have the number of grants relative to each other, we can compute what fraction of the grants should go to cause $k$ :

$f_{k} = \frac{n_{k}}{\sum_{i} n_{i}} = \frac{n_{k}}{\sum_{i} r_{k} n_{k} v_{i}} = \frac{1}{r_{k} \sum_{i} v_{i}} = \frac{v_{k}}{\sum_{i} v_{i}}$

The funding a cause receives is proportional to its expected value.

A Numerical Example

A funder is considering awarding $30$ grants between three cause areas. Funding a grant in cause $w$ has an expected value of $8$ QALY’s (meaning that $r_{w} = \frac{1}{8}$ ), a grant in cause $y$ has an expected value of $5$ QALY’s ( $r_{y} = \frac{1}{5}$ ), and a grant in cause $z$ has an expected value of $2$ QALY’s ( $r_{z} = \frac{1}{2}$ ). Using the budgeting formula from before:

$f_{n_{w}} = \frac{8}{15}$

$f_{n_{y}} = \frac{5}{15}$

$f_{n_{y}} = \frac{2}{15}$

So the funder awards $16$ grants to cause $w$ , $10$ to cause $y$ , and $4$ to cause $z$ .

Contrast this with a funder focused on the expected value of each donation. In this case, all $30$ grants would go to cause $w$ . At the other extreme, a maximally uncertain funder might give $10$ grants to each cause.

Possible Extensions

There are several ways to extend this approach. It may be useful to consider other impact distributions, grants with different costs, or altruists who care about other order statistics. Alternatively, relaxing the i.i.d assumption could help model grants in areas where projects are complimentary to each other.

Notice that it is possible to include saving for future donation or personal consumption as “cause areas” within the budget, enabling one to consider a richer decision problem. When some causes are expected to become more valuable in the future, altruists will save more to spend on them later.

Ideally, this approach would be extended to allow for impact distributions to change over time. For example, learning more about a cause induces a change in the impact distribution. Investing in pilot projects within a cause area can help funders get a better sense of the impact distribution, providing valuable information. This becomes especially important when searching for new causes to fund.

It’s also important to take into account the choices of other donors. A cause that other donors are pouring money into is unlikely to benefit much from further funding. This gets especially interesting when considering donors time preferences. What does patient philanthropy look like under hits-based giving?

The analysis here assumes that each cause can be compared on a common scale like QALY’s. But funders may have multiple objectives they want to satisfy (e.g. QALY’s and x-risk reduction), and may need to consider trading off between two objectives.

Discussion

The hits-based approach is quite different from the case where we only consider the marginal impact of donations. In that case, altruists do best by donating to a single cause, in stark contrast to today’s giving.

This model seems particularly useful for microgrants programs, where uncertainty is high, typical returns for a project are low, and the goal is to find a highly valuable cause that traditional grantmaking organizations can scale up.

Though a grantmaking analogy was used to motivate the model, I think the same ideas could be productively applied to other areas such as innovation, charitable spending, and time investment.

The hits-based approach could be applied to s-process funding, allowing evaluators to specify the expected value of a grant in each area. Alternatively, charity prediction markets could be used to determine the expected value of different grants.

Charitable giving presents a rich decision problem with significant real-world implications. Studying the nuances of maximizing impact may improve funding decisions. I hope that further work in this area can improve how altruists spread their resources across causes.

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