Kenneth Arrow has worried that range-voting-type "score" votes might have no or unclear-to-Arrow "meaning." In contrast, he considers rank-ordering-style votes to have a clear meaning. Nic Tideman has also expressed similar worries in email, but now about the "lack of meaning" of an approval-style vote. In contrast, I think Tideman regards a plurality-style "name one candidate then shut up" vote as having a clear meaning.
E.g. "what does a score of 6.5 mean, as opposed to a score of 6.1, on some ballot?" But the Bayesian view is: whether or not Arrow or Tideman or somebody has a more-or-less muddled mental notion of the "meaning" of a ballot, is irrelevant. The only genuinely meaningful thing is "who won the election?" All meaning of any ballot therefore derives purely from the rules for mathematically obtaining the election-winner from the ballots. For a simple example of how ballots have no inherent meaning without voting system rules, consider plurality and AntiPlurality voting in which the meanings of a "name one candidate" ballot are pretty much opposite (plurality: most-named candidate wins; AntiPlurality: least-named candidate wins). Let us now enquire more deeply about ballot "meaning." In a non-monotone voting system like Instant Runoff, your vote A>B>C can cause A to lose, whereas your vote B>C>A would have caused A to win. Would Arrow be right if he said IRV is wonderful because "A>B>C" has a "clear meaning"? Or would a Bayesian be right in saying this example indicates the "meaning" Arrow had in mind, was not valid? Indeed the Gibbard-Satterthwaite theorem http://rangevoting.org/GibbSat.html shows that in essentially ANY rank-order ballot system and also in the plurality and AntiPlurality systems with "name one candidate" ballots -- i.e. exactly the systems Arrow & Tideman thinks "have meaning" -- there ALWAYS exist elections in which some voter's vote of A>B>C will cause a worse election winner (for the A>B>C notion of "better" and "worse") than some different dishonestly-ordered vote would have caused. (And with Plurality and AntiPlurality, "dishonestly" ranking your non-favorite candidate top or your really-not-worst candidate bottom, can be the only way for you to get an improved election result.) In such an election, what is the "clear meaning" of an A>B>C rank-order vote? Gibbard identified/invented exactly two rank-order ballot systems in which honest and strategic voting were the same thing (this required him to employ non-determinism), but stated that both of his systems were not good enough for practical use since they "leave too much to chance." In contrast, consider the "double range voting" system invented by F.Simmons and Warren D. Smith http://rangevoting.org/PuzzRevealU2.html This system (or others of the Simmons class) ARE good enough for practical use if any rank-order system is (since it leaves only an arbitrarily small amount of the deciding to chance, and deviates from your favorite system in an arbitrarily small way). In this voting system, each ballot contains a part on which the voter is urged to provide her honest scores (on, say, an 0-to-9 range) for each candidate. In this system, ONLY voting on this ballot portion in a unique honest manner is strategically best. Any deviation from perfect honesty (or omision of information) is a strictly worse voting strategy. That is, if your expected utility if A wins is 6.5 and your expected utility for B winning is 6.1 on an 0-to-9 scale (defining the utility scale so you've rated the best available candidate 9 and the worst 0) then you MUST score A=6.5 and B=6.1 EXACTLY, otherwise you are guaranteed to get in expectation a worse-utility election result. So contrary to assertions by the likes of Arrow that utility is "unmeasurable" or that range votes "lack meaning" it seems to me that we have a very clear, totally unique, not changeable by one iota, meaning for the scores 6.5 and 6.1 deriving wholy from the procedure the voting system uses to determine the winner from the votes. This is wholy unlike EVERY allegedly-practical rank-order voting system. So Arrow, and Tideman (and anybody else) are simply wrong if they assert scoring-style votes are inherently less-meaningful than rank-ordering-style or name-one-candiate-style votes. So now Arrow might perhaps riposte that to HIM, deep in the recesses of his brain, rank-order votes have more meaning, even though every voting system he and his colleagues have ever considered for practical use, disagrees with his private meaning in at least some situations, and even though (therefore) the true meaning of your vote really also depends on how the OTHER voters are voting, not just on the candidates and your evaluation of them in your private brain. (I would have to then counter-riposte: who cares?) Another riposte might be that under the assumption there are a large number of other voters all of whom vote TOTALLY RANDOMLY and independently, your vote can have a clear meaning, and in some rank-order systems (e.g. Borda) this meaning coincides with "honest ordering." I would then riposte that (a) that assumption is false, and (b) under the same assumption approval voting also has a "clear meaning" (namely: you should "approve" candidates above mean utility for you). Does our argument tell us that score-style votes inherently have MORE meaning than rank-order style votes? (Exactly contrary to Arrow?!?) Well... not necessarily. Yes, score-style votes certainly inherently convey more information than rank-ordering-style votes (strengths of preference as well as preferences). And I would claim that if they were employed for the honest-part of double range voting ballots, they inherently have more meaning. But if employed for plain range voting, then it is posible to construct 4-candidate election situations in which it is strategically best for a range voter to misorder, so we run into the same Gibbard-Satterthwaite-style issues (albeit only with 4, not 3, candidates). All this analysis really tells us is the Bayesian view is correct. And certainly that any dismissal of range- or approval-style voting on the grounds of their claimed "inherent lack of meaning", is hogwash. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) ---- Election-Methods mailing list - see http://electorama.com/em for list info
