Puzzle #38: Nonadditive utilities

Puzzle: Criticize the axioms that underlay the preceding two puzzles' proofs that utility must be additive. Then consider the following non-additive utility-combining method, suggested by Clay Shentrup:
The "Shentrup Social Utility," as a function of the personal utilities of each of V people, is:

SSU = Maxx   x · (#people whose utility is at least x).
Which of those axioms does SSU obey and which does it disobey?

Answer: I have no criticism of permutation-invariance, monotonicity, smoothness, the existence of "zero," and closure (or "non-silliness"). But there is room to be skeptical of these axioms:

Shentrup's SSU obeys permutation-invariance and monotonicity. It disobeys smoothness – increasing some individual's utility smoothly can cause the SSU to have a "corner" (but it is still continuous, so this is not very severe – i.e. smoothness is still obeyed by SSU if "smooth" means "continuous" as opposed to its more common meaning "infinitely differentiable"). What about "zero"? Is there a utility Z such that SSU(Z,X)=X? Yes: Z=-∞ works. (However, I must admit that the idea that -∞ is "zero" rather worries me. SSU seems to have no concept of the notion that making somebody's life get very horrible, ought to decrease social utility versus a situation where it is only mildly horrible. Similarly Gini social welfare is unaffected by making the happiest person in the world even happier.)

SSU disobeys inverses. (There is no inverse of +5.) SSU often obeys self-consistency (combining 1,2,4,5 yields 8 and self-consistency holds no matter how you split these four numbers into two pairs) but not always: SSU(0,4,5,7)=12, SSU(0,4)=4, SSU(5,7)=10, SSU(4,10)=10≠12. Finally, SSU disobeys utility-order-invariance under adding a constant: SSU(0,9)=9<SSU(5,7)=10; but if we add 2 to the second arguments in both cases, we get SSU(0,11)=11>SSU(5,9)=10. So in summary, SSU disobeys every axiom from the preceding two problems that I had flagged for possible skepticism, and obeys (perhaps in a weakened sense, but obeys) the others.

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