Puzzle #38: Nonadditive utilities
Puzzle:
Criticize the axioms that underlay the preceding two puzzles' proofs
that utility must be additive.
Then consider the following nonadditive utilitycombining method, suggested
by Clay Shentrup:
The "Shentrup Social Utility," as a function
of the personal utilities of each of V people, is:
SSU = Max_{x} 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 permutationinvariance, monotonicity, smoothness,
the existence of "zero," and closure (or "nonsilliness").
But there is room to be skeptical of these axioms:

Inverses: Suppose you die. Is there really any utility X
so large and positive that me getting it (making me happier) is enough to compensate
the two of us for you
dying? Most people would agree that there exists X such that some sufficiently large
constant number N of
people getting utility X, is enough to compensate for a death (otherwise, we
would demand a speed limit of ≤20 km/hour on all our roads, which nobody
is willing to accept). But there is room to be skeptical that N=1 is "sufficiently large."

Selfconsistency:
If you combine utilities a,b,c,d, then you ought to get the same result as
combining in pairs such as (a,b) and (c,d) and then combining the resulting two values.
Really?

Utilityorderinvariance under adding a constant:
If we add a constant to every utility of some person (a "golden meteorite" landed
in their back yard), then the societal relative rankings
of all the alternatives should be unaffected. Really? Perhaps that person
would now have different views and that would be enough of a change to tip
society's relative rankings of something.
(Incidentally:
The "Gini social welfare" of X is the expected minimum utility of X among 2 random people,
and it does not obey this axiom.)
Shentrup's SSU obeys permutationinvariance 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 selfconsistency (combining 1,2,4,5 yields 8 and selfconsistency 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 utilityorderinvariance 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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