Eric Maskin (Economics Nobel co-Laureate 2007) gets into trouble on Arrow's theorem

QUOTE [this is from an online report, written by his hosts, describing a lecture "How Should We Elect Presidents?" Maskin gave at Georgetown University's Gaston Hall on 17 January 2008; the text of the lecture is (essentially) available here]:

Response

Let's check out how range voting performs on Maskin's 5 principles.

1. Consensus: Range obeys it.
2. Anonymity: Range obeys it.
3. Neutrality: Range obeys it. [ Note: Anonymity and Neutrality imply Arrow's "no dictator" property, which is the connection between the way Maskin is phrasing Arrow's theorem, and the more-usual formulation.]
4. Irrelevancy: not precisely defined in the lecture-summary or Maskin's text. However, if we define it to mean "removing a loser from the race and from all ballots, should not be able to alter the winner" – then range obeys this criterion, unlike, e.g, Borda voting. Maskin claims on page 3 of his text that all Condorcet systems obey this principle; that claim is true when a Condorcet winner exists, but is false for most (all?) Condorcet systems in some circumstances. (It is, in fact, false for every Condorcet system that, in some situation in which there is a strong but not perfectly-symmetric top-cycle, elects one of the cycle-members. To prove this, simply delete another cycle-member.) Schulze's "beatpath" Condorcet system is an example of a system that is "cloneproof"; Maskin was apparently unaware of Schulze's work. Cloneproofness is not the same thing as this irrelevancy-condition, but it goes in that direction.
5. Generic Decisiveness: If we use "continuum range voting" and every vote is a "generic real number" [which could be achieved by randomly perturbing every score by an amount of size 10^-30 – which would happen automatically if each range voter had an analog input device and was unable to control the slider to subatomic positional accuracy] then ties occur with probability 0. In other words, we get a clear winner. Maskin in his text claims his "majority rule" voting-method enjoys "generic decisiveness." That would be true if some additional rules were specified to produce a winner in circumstances in which there is a Condorcet top-cycle. As far as I can see Maskin does not specify those rules, which is why I say his "Majority Rule" method is "undefined." In the event that non-specification was intentional because he intended that there be no winner in such circumstances, then his claim of generic decisiveness was false because it is known that random 3-candidate ∞-voter elections are cyclic 8.78% of the time. (If decisiveness fails 8.78% of the time, then it is simply false to claim generic decisiveness.) So I presume Maskin did have additional winner-specifying rules in mind despite not clearly saying what they were, and from reading his Scientific American article I suspect he had in mind Black's system.

"No voting system satisfies all five principles" eh? Wrong. Or to quote Maskin on page 6 of his text: "We might enquire whether there is a voting procedure that actually satisfies these principles. The answer unfortunately is 'no'. That answer was provided by economist Kenneth Arrow..." This quote is simply flat-out false: range voting is such a procedure and Arrow's theorem does not apply to, and never was claimed to apply to, "all voting procedures" – including it does not apply to range voting. Here Maskin is simply repeating a common misunderstanding of what Arrow's theorem actually said. The resolution of this misunderstanding had already been published by Harsanyi in the 1950s almost immediately after Arrow's work, but Maskin missed that and did not cite or mention Harsanyi in either his lecture text or his 2007 paper. (Incidentally, Harsanyi also won an economics Nobel, in 1994.)

Moral

Just because you win a Nobel prize in economics, unfortunately does not mean you are either infallible or omniscient. Nor does it necessarily mean you know much about voting. You (the reader) have now verified that Maskin made a mistake.

Actually in my (and Nobel's) opinion, even having a "Nobel prize in Economics" is a mistake. (The "Nobel" prize in economics was created by the Swedish Bank, was first awarded in 1969, about 75 years after Nobel died. It was not approved in any way by Nobel, and its real title therefore is "Nobel Memorial Prize" not "Nobel prize" in economics.) I feel economics is not an advanced-enough field to deserve such a prize; the aura of infallibility and omniscience created by the awarding of these prizes has in several cases been a bad thing, and many Economics Nobels have been awarded for feats which I as a mathematician do not consider very impressive. Also, some Economics Nobelists have made some ludicrously false and damaging statements.

Indeed, the fact that Range Voting accomplishes what Arrow had deemed "impossible" had already been observed by, and published by (also Economics Nobel Laureate) John Harsanyi in the 1950s, but no mention of either range voting or Harsanyi was made by Maskin during his lecture half a century later or in his 2007 paper.

Further, as regards the final sentence in the Maskin-report-quote, about how his undefined "majority rule" system supposedly comes the closest to satisfying Arrow's impossible conditions – here Maskin was apparently unaware (never cited, never mentioned) that a theorem indicating that range voting comes the closest to satisfying a set of Arrow-like conditions, already had been published in 1999 (years before Maskin's voting work he was lecturing about), by Dhillon & Mertens.

And finally, as for Maskin's notion Condorcet winners ought to be elected – Maskin was unfortunately unaware that with strategic voters range voting can actually elect Condorcet winners more often than "official" Condorcet methods of the sort he is pushing! (This has been experimentally verified.)

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