**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 un*able* 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.