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Graph Utilitarianism:

People care about others, so their utility function naturally takes into account utilities of those around them. They may weight others' utilities by familiarity, geographical distance, DNA distance, trust, etc. If every weight is nonnegative, there is a unique global utility function (Perron-Frobenius).

Some issues it solves:

  • Pascal's mugging.
  • The argument "utilitarianism doesn't work because you should care more about those around you".

Big issue:

  • In a war, people assign negative weights towards their enemies, leading to multiple possible utility functions (which say the best thing to do is exterminate the enemy).

This is a very imprecise use of “utility”. Caring about others does not generally take their utility into account.

It takes one’s model of the utility that one thinks the others should have into account.

And, as you note, even this isn’t consistent across people or time.