Consider three charities A,B,C and three voters X,Y,Z, who can donate $1 each. The matching funds are $3.
Voter Z likes charity C and thinks A and B are useless, and gives everything to C.
Voter Y likes charity B and thinks A and C are useless, and gives everything to B.
Voter X likes charities A and B equally and thinks C is useless.
Then voter X can get more utility by giving everything to charity B,
rather than splitting equally between A and B:
If voter X gives everything to charity B, the proportions for charities A,B,C are
02:(1+1)2:12
If voter X splits between A and B, the proportions are
√1/22:(1+√1/2)2:12
The latter gives less utility according to voter X.
The quadratic-proportional lemma works in the setting where there’s an unbounded total pool; if one project’s finding necessarily pulls from another, then I agree it doesn’t work to the extent that that tradeoff is in play.
In this case, I’m modeling each cause as small relative to the total pool, in which case the error should be correspondingly small.
This seems false.
Consider three charities A,B,C and three voters X,Y,Z, who can donate $1 each. The matching funds are $3. Voter Z likes charity C and thinks A and B are useless, and gives everything to C. Voter Y likes charity B and thinks A and C are useless, and gives everything to B. Voter X likes charities A and B equally and thinks C is useless.
Then voter X can get more utility by giving everything to charity B, rather than splitting equally between A and B: If voter X gives everything to charity B, the proportions for charities A,B,C are 02:(1+1)2:12 If voter X splits between A and B, the proportions are √1/22:(1+√1/2)2:12 The latter gives less utility according to voter X.
The quadratic-proportional lemma works in the setting where there’s an unbounded total pool; if one project’s finding necessarily pulls from another, then I agree it doesn’t work to the extent that that tradeoff is in play.
In this case, I’m modeling each cause as small relative to the total pool, in which case the error should be correspondingly small.