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<P>Forest et al.</P>
<P>I won't have time for several days to think harder about your ideas and interpretations, except that I must at least note several things. </P>
<P>First, I enthusiastically endorse your generlized idea, as it amounts to the essence of the concept I was going to post: for a given voter, his or her contribution to the score for a panel of K candidates (in an election for a panel of K winners) should be a convex isotone function f(x) of the sum x of the separate individual grades the voter gives to the panel members. Note that this could be done using ANY given 1-winner grading method, e.g. a high-resolution method such as centile grading (grades 0.00 through 1.00), not just with using pass-fail grading (i.e. grades 0 and 1, 'approval'). More below on a key consequence of this. Various functions f might be good candidates, for various reasons, as you have been exploring. </P>
<P>For technical convenience let's assume that everything is 'normalized'. Namely, assume that the given 1-winner method has normalized its grades so that grades range between 0 and 1, inclusive (pass-fail, i.e. 'approval', is already thus normalized); that a voter's raw grade for a panel of K candidates is taken as the average of the voter's grades for the individual candidates (so that this raw grade again must range between 0 and 1, inclusive); and that f(0)=0, f(1)=1. </P>
<P>As you point out, this method of choosing a panel offers an important advantage: the winning panel's diversity (which will be abetted by the convex-isotone nature of f) reflects directly the voters actual support of the actual individual candidates, without being tied to labels, pre- or post-organized lists, party machineries, etc. Here in the USA - as versus some other polities - voter and candidate freedom from party categorizations and organizations is widely deemed a virtue, indeed a matter of civil liberties and rights. </P>
<P>There is an important rationale (or anyhow rationalizing terminology) other than 'layers of protection' for being open to using a nontrivial transform f, i.e. where f(x) is not identically x. This rationale is in terms of utility (again taken as normalized). Namely, it is a widespread myth that aggregate or 'social' utility (of a given panel or single candidate) should be taken as the sum of individual voter utilities (for that panel or candidate). This myth would have us believe that - for a given single candidate or panel - if each grade does (or could) express the voter's utility (in some sense) then the sum total of the grades should express social utility. Now, for many of us this myth is clearly erroneous, because we perceive social DISutility arising from unequal allocation of aggregate utility. In brief, a situation in which everyone is equally satisfied seems, on balance, somewhat more satisfactory overall than a situation in which, though average satisfaction is a bit higher, a large fraction of the people are far less satisfied than average. </P>
<P>That is, for a given average approval rating by voters, a panel's overall score should be diminished by a penalty which reflects variation among voters' ratings. This tactic - which promotes diversity of represented voter viewpoints - is precisely what is achieved by first transforming each voter's rating by a convex isotone function f before adding the ratings to score a panel or single candidate. </P>
<P>What happens when K=1 - a single-winner election - and a nontrivial f is used? Nothing different at all, if our 1-winner grading method is pass-fail ('approval'), but something quite interesting and useful if we use a higher-resolution method such as centile grading. A candidate will then be rewarded not only for attaining a high average grade but also for this average being the result of a broad base of consistent support (rather than strong support from some segments of the electorate and weak support from others). In short, depending on the particular convex function f used, we can craft a victory criterion which may be an attractive compromise between the extremes of high average (but possibly badly distributed) support and majority preference (which may be tepid and be more than averaged out by strong minority disapproval). </P>
<P>Joe Weinstein (Long Beach CA USA)</P>
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