Donor coordination under simplifying assumptions

Summary of findings

Owen Cotton-Barratt & Zachary Leather

The problem: There is a cooperation problem when multiple donors agree about the most-preferred charity but have different views about the second-best use for money, and there is more money available than the mutually preferred charity can productively absorb. There are challenges in trying to produce cooperation. However a complication is that it’s not obvious what the cooperative solution should look like.

Our work: This is preliminary work, mostly pursued by Zack over a week of an internship. We’re publishing this now because we don’t have a plan to come back and develop it in the near future.

We explore a scenario with two donors with well-defined preferences. We find the Nash Bargaining Solution for this scenario, which involves splitting the funding of the most-preferred charity dependent on funding capacity and strengths of preferences. Subject to standard conditions, this would be the stable (Pareto optimal) outcome of bargaining. We suggest it as a reasonable target for cooperative behaviour.

In the solution for two donors, each funds any gap of the preferred charity that the other donor is unable to fill. Then the remaining gap is split according to the relative strengths of preferences between the donors’ first- and second-choice charities. When the difference in preferences is marked enough, this can include one donor fully funding the gap.

More precisely, for donor X let t_X denote the ratio of (strength of preference for the preferred charity over the other donor’s second-choice charity) to (strength of preference for that donor’s second-choice charity over the other donor’s second-choice charity). In terms of utility functions, we are using the two free parameters to set the utility of a dollar to the least-preferred of the three charities at 0, and the utility of a dollar to the second-choice charity at 1, so t_X represents the utility of a dollar to the preferred charity.

Then the proportion of the gap that should be filled by donor A is (1+ t_A – t_B)/2, if this lies between 0 and 1 (and capped at 0 or 1 otherwise).[*]

Major limitations of our analysis:

We limit analysis to the two-donor scenario.
We assume a mutually agreed “funding gap”, an amount of money the preferred charity should receive before donations are better spent elsewhere.

Neither of these is a principled block, but the mathematics of finding solutions is more complicated (in both cases finding an optimum in a higher-dimensional space).

See Zachary Leather’s research notes for more detail and discussion. The research notes are somewhat technical, and haven’t been checked carefully—we apologise for any errors that remain.

Thanks to Daniel Dewey for originally suggesting the question and helping with an early whiteboard analysis.

[*] In fact the NBS would use this expression as it stands, without capping. However in this case it seems unreasonable to use “no donations to mutually preferred charity” as a disagreement point, as one of the donors does better by unilaterally fully funding the preferred charity than by submitting to the bargaining process.