What is the "consensus" by the "experts" about the best single-winner voting system?

(Executive summary)

The unfortunate reality as of 2006 is that there is no visible consensus among either political scientists, economists, or mathematicians about the best single-winner voting system. But there should be, and we think there ultimately will be, a consensus for range voting.

The usual impression one gathered from political science books up to 2000 was there were 4 main branches of single-winner systems, namely, Borda, single transferable vote, Condorcet, and approval voting – with few people (and fewer, if any, books) willing to stand up and say one of them was the best – and almost nobody in that literature said anything about range voting whatever up to about 2004 (well, see this history for some examples...)

Now that indecision was justifiable since all four of those systems do seem best in computer simulation studies – depending on the "settings" of the simulations. (E.g. see Merrill's book.) In other words, computer simulation studies were incapable of reaching an unambiguous conclusion that one of these four was the best; you can construct plausible election situations where each seems better than the others. Furthermore, philosophical comparisons also were not capable of reaching such a conclusion, in the sense that people kept disagreeing.

That changed with my computer-sim study in 2000 (#56 here) which was the first to include range voting as a contender. It used Bayesian Regret as its yardstick, and found RV was always superior to all four of the above main voting system branches at every setting (of 720 tried) in my simulations. So that in some sense gets rid of the indecision and provides a clear best: Range Voting. My study also pointed out that "Arrow's impossibility theorem" which was often interpreted to mean "no 'best' voting system can exist" actually does not apply to range voting, since according to Arrow's (foolishly too-restrictive) definition RV is not a "voting system." But that information has not yet fully percolated into published political science books and the general zeitgeist.

Now besides me just telling you my impression about my view of the consensus thinking (above), there is this objective evidence. I examined the bylaws of professional associations in the field and found:

APSA (American Political Science Assoc):
its 1903 constitution mandates IRV (transferable vote) for presidential election (but so far, apparently no election in which IRV was actually invoked has happened, so it's been kind of irrelevant... voting method choice is not an issue unless at least 3 election outcomes are possible). Also we should note that this constitution was written before any political scientists first invented (in 1976) Approval Voting. Hence it cannot be interpreted as a judgement by the Political Science community that IRV is superior to Approval, although it might be interpreted as a judgement IRV is superior to Borda, Condorcet, and Plurality. APSA considered changing its nonpresidential elections (all conducted via plurality) to use IRV and/or STV voting, and in a 2005 report (pdf) decided against it. Thus one could equally well interpret the APSA stance as saying (oppositely!) that plurality is superior to IRV.
ACS (American Chemical Society):
As of 2013, uses IRV to elect president and district directors by mail ballot.
AAAS American Assoc. for Advancement of Science:
Bylaws specify "votes" but do not specify which voting method is to be used in situations with more than 2 choices available. (Such lack of specification is quite common in many organization bylaws.)
EPCS (European Public Choice Society):
Uses plurality voting. (Founded 1972.)
IEEE (Institute of Electrical and Electronics Engineers, world's largest professional society):
Used Approval Voting 1987-2002 but then reverted to plurality voting. (Story recounted here.)
Public Choice Society:
It has no formal membership. Although it always has a president, as far as I know until 2006 there never had been a competed election for president. But then the 2004-2006 president, Prof. Steven Brams (noted Approval Voting inventor and advocate) got them to adopt Approval Voting. They then held their first-ever genuinely-competed election (to elect their 2006-2008 president) using AV.
AMS (American Mathematical Society, ≈30,000 members):
Formerly used Transferable Vote, but then switched to Approval Voting to elect its officers.
INFORMS (Institute of Management Science and Operations Research, ≈12000 members):
MAA (Mathematical Association of America, ≈32000 members):
ASA (American Statistical Association, ≈15000 members):
SCWS (Social Choice and Welfare Society):
All of the above four organizations use Approval Voting to elect their officers.
United Nations:
Uses a form of Approval Voting to elect the Secretary General.
Econometric Society:
NAS (USA National Academy of Science):
Both of the above organizations use approval voting to elect "fellows." See also this list.
Range voting (essentially) for judging events.
Time Magazine:
Used 1-10 range voting to help select its "person of the year" in 2008. (In 2007, Vladimir Putin had been selected by a non-democratic process.) 25 candidates, winner was Barack Obama, also just elected US president, with average rating [rounded to integer] 10 based on 348416 ratings. Runner-up, also with 10 but based on only 15035 ratings, was Douglas Melton, a biologist who partially-cured diabetes in mice in what might be a revolutionary development. Every other candidate got average rating≤7.

As far as I know as of 2008, no professional society whatever uses Range, Borda, or Condorcet voting – but Debian (Linux software developers; and previously the wikimedia foundation) use Schulze's beatpath method (which is a complicated Condorcet method) conducted electronically via special software, and many internet sites use Range Voting, e.g. AllRecipes.com, Yahoo movies, Amazon.com, Hot or Not, Newegg.com, internet movie database, Cornell vegetable varieties agricultural rating service, etc; and academia worldwide long ago reached a "consensus" on 0-100 and A-F range voting for use in grading students and selecting valedictorians. RV is also used by the Seattle commercial co-operative Madison Market and various online polling services such as bigpulse.com and zohopolls.com; and Approval Voting is employed to elect Princeton University's USG president.

At the Voting Procedures workshop held in Normandy in July 2010, there was a vote (using approval voting) for which, among 18 possible voting systems, its participants (essentially all of whom were professional voting experts) preferred "for your town to use to elect the mayor." Result was: approval came top with 15 approvals. (Only system approved by majority. The runner-up system, IRV, enjoyed 10 approvals.) Plain plurality came (non-uniquely) bottom with zero.

Verdict: based on the above, if there is any professional consensus (which there isn't) it would be for Approval Voting, which is the maximally simple degenerate case of Range Voting (which is also fairly highly supported academically).

Further, it would appear from the above that if there is a professional consensus against something, Condorcet and Borda are them, with an especially bad knock against Borda. However Mathematics Professor Donald G. Saari is a noted Borda advocate and was infuriated by the SCWS's adoption of Approval Voting. (I do not know of any other noted Borda advocates besides Saari.)

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