Well, here's an example that makes some sense to me. If I find another, less complicated situation which "should" provoke the same feelings as this situation, that's progress - and I can! Assume my utility function is approximately linear in dollars I lose at this scale, dollars my employer loses at this scale, and dollars puppies gain at this scale. Then maybe I can attribute to me my loss of $50 and half of my loss of the next $50, half of my employer's loss of $100, and puppies' gain of $50 and half of puppies' gain of $150, saying I should feel about the overall situation similar to how I'd feel if I solely caused a situation where:
That's an example of math that makes a lot of sense to me, regardless of my utility function except for the fairly reasonable linearity-at-small-scales assumption.
Assume my utility function is approximately linear in dollars I lose, dollars my employer loses, and dollars puppies gain at this scale.
If we made that assumption then you'd never stop giving to puppies - whatever you gained by giving $100 you'd gain twice over by giving $200. Assuming that both your and your favorite charity have a lot more money than you, then it's probably okay to assume that they experience changes in marginal utility which are locally linear in dollars, but at some point you're going to stop giving because your own utility function went noticeably nonlinear, e.g. that second $100 would have been a bigger loss to you than the first was.
Often we cooperate to extract surplus value from the government, hotels, the physics that makes operating cars cost money, or other sources - value that we could not extract individually. When I notice such a surplus I often wonder how the surplus should be split. What is fair? Purely cooperatively, without anyone trying to game the surplus-allocation-function, and assuming the stated coalitions are fixed rather than negotiable, how much of the surplus should be attributed to each contributing party?
Some concrete examples that have come up recently in real life*:
1. Matching donations. The company I work for will match donations to charity, dollar for dollar, up to a certain maximum. Viscerally, how should I feel about donating $100 to puppies**? More than $100, since puppies get $200, certainly. But less than $200, since my employer should feel puppy-love too, and presumably there's a conservation of visceral feeling law that should apply here. Further suppose that my employer's matching offer caused me to donate $100 instead of, say, $50. What math should be done and why?
2. Exemption splitting. An amicable divorce leaves two parents wondering who should claim their student daughter as a dependent. As a purely "what is fair?" financial question, how much of the tax savings from that exemption should be distributed to the father, mother, and daughter? Suppose the father's marginal tax rate is 25% and overall tax rate is 18%, and the mother's marginal rate is 15% and overall is 12%. What math should be done and why?
3. Refinancing. My friend has a debt at 12% and for silly reasons is obviously able to pay it off but cannot this year. I can pay it off, though, and so could several other people***. Assume there are 3 people including me who could pay it off, and our current expected returns on invested money are (say) 2%, 3.5%, and 6%, and for simplicity she will repay the loan plus any surplus due in one year. Who should pay off how much of the loan (say it's $5000)? I assume the 2% person should pay all of it. That's a 10% surplus - how much do each of the four of us get? What math should be done and why?
As in The Bedrock of Fairness, are there qualities of the solutions we have strong opinions on, even if we do not know the procedure which would generate solutions with those qualities?
*Details changed.
**I do not donate to puppies.
***Assume default risk is negligible.