[EM] When and how can we speak of "individual utility" and "social utility"?

[David Cary]

I commend Jobst for his essay [
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2007-February/019584.html
] on utilities.  It helps clarify some issues about utilities and the
often abused notion of social utilities.

Utility functions are just a way of representing preferences and
choice functions, either for an individual or socially.  The social
utility function of a group, if it exists, reflects not only by the
preferences of the individuals in the group, but also by the
processes those preferences are assimilated into a group choice.  The
availability of utility functions does not automatically supply a
notion of what are the "best", "ideal", or "optimal" preferences or
choice functions.  Applying those concepts to social utility
functions involves similar complexity and ambiguity as deciding which
election methods are better than others.

When trying to determine the relative merits of various election
methods, it is circular logic to merely stipulate one particular
(equivalence class of) social utility function, and declare it the
standard by which all election methods should be judged.  The folly
is only compounded when the standard is ill-defined.

With regards to comparing or summing utility function values across
individuals, I'd go a step further than Jobst and say that such
things are either nonsense, because they are not well defined, or
that they involve such a high degree of arbitrariness (for each
individual, picking a specific utility function from among the
uncountably infinite equivalent utility functions), as to be
essentially vacuous of any meaning.

It is the burden for anyone who proposes comparing or summing utility
function values to justify why such combinations have any implicit
importance.

Utility functions can only represent a limited subclass of the
possible preferences and choice functions.  Any preferences
represented by a utility function will be transitive and any choice
function represented by a utility function will satisfy IIA
(Independence of Irrelevant Alternatives) (the preference-based
version, not the voting-based version – see
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2006-November/018802.html
).

Jobst's attempt at defining a social utility function that
disambiguates achieving “the greatest good for the greatest number”
is commendable for its analysis, but noteworthy for its failure. 
While some of the options could produce well-defined social choice
functions, none of those can be represented by a utility function. 
Option (i) can result in social preferences that are not transitive,
and hence can not be represented by a utility function.  Option (ii)
is  vague about what the threshold is, but if for each individual it
is the mean or median of the individual's utilities of the options
under consideration, then the social choice function does not satisfy
IIA and can not be represented by a utility function.  Option (iii)
is nonsense or essentially meaningless, because it apparently
involves summing utility values across individuals.  Option (iv) can
not be represented by a utility function because, as Jobst notes, it
can result in non-transitive social preferences.

Finally, I'll mention a couple of small but important details that
are missing from Jobst's essay:
  -- any utility equivalence transform of the form v = r + s * u
requires the restriction that s > 0.
  -- the (Decomp) property for individual utilities [ If 
(p?a:c)R(p?b:c)  then  aRb ] should be restricted to cases where p >
0.

-- David Cary


 
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