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 ____________________________________________________________________________________ We won't tell. Get more on shows you hate to love (and love to hate): Yahoo! TV's Guilty Pleasures list. http://tv.yahoo.com/collections/265 ---- election-methods mailing list - see http://electorama.com/em for list info
