# [EM] In effect and in voting theory, Plurality is a rankings method.

Chris Benham chrisbenham at bigpond.com
Sat Feb 7 05:30:02 PST 2004

```Mike,
This is the full text of  Steve Eppley's  discussion/definition of   his
"Minimal  Defense" criterion from which  Marcus Schulze copied:

minimal defense
<http://www.alumni.caltech.edu/%7Eseppley/Proof%20MAM%20satisfies%20Minimal%20Defense%20and%20Truncation%20Resistance.htm>:
If more than half of the voters prefer alternative y over
alternative x, then that majority must have some way of voting that
ensures x will not be elected and does not require any of them to
rank y equal to or over any alternatives preferred over y. (Another
wording is nearly equivalent: Any ordering of the alternatives
must be
an admissible vote, and if more than half of the voters rank y
over x
and x no higher than tied for bottom, then x must not be elected.
This criterion, in particular the first wording, is promoted by
Mike
Ossipoff under the name Strong Defensive Strategy Criterion.
Satisfaction means a majority can defeat "greater evil"
alternatives
without having to pretend to prefer some compromise alternative
as much as or more than favored alternatives.  Since voters in
public
elections cannot be relied upon to misrepresent their preferences
in this way, non-satisfaction means that elites will sharply
limit the
set of nominees that voters are asked to vote on, by offering a
system
in which there are only two viable parties, each of which nominates
only one alternative.)

CB: I must say I greatly prefer the "nearly equivalent" wording as being
much more concise, simple and reasonable-sounding.

Earlier  (Fri.Feb.6) I wrote:

If the rules stated "voters rank the candidates, no equal first preferences allowed, and the winner is
the candidate with the most first preferences", wouldn't that be a "rank method"? A method that is
exactly equivalent to a rank method, in my book *is* a rank method.

To which you replied (Fri.Feb.6):

""Equivalent to [in some particular specified regard]"  most definitely does
not mean "is"."

"First Preference Plurality" (as Woodall calls it, a name I prefer to "ranked Plurality")  is not just
equivalent to Plurality "in some particular specified regard". It is equivalent in EVERY regard that
is of interest to voting theorists. It transmits voted preferences into a result identically, and is
therefore of course strategically equivalent.
This (Tue.Jul.15,03) quote from Alex Small in the "Arrow's Theorem" thread is of some relavence/interest:

"In the formal derivations of Arrow's Theorem that I've seen, an election
method is defined as a mapping from the set of voter preferences to the
set of candidates.  Show me the preference order of each individual voter,
and (barring the case of ties) I'll show you who the winner is.  No
ambiguity."

You wrote:
"Your unproposable method is "equivalent" to Plurality in the sense that both
only count indicated favorites. But because different voting is admissible
in the 2 methods, they are not the same method, in keeping with your own

For the purpose of rationally analysing voting methods, the fact that "different voting is admissable"
is only relavent if that can possibly give a different result. (And then it is only interesting if it
makes a different viable strategy available,like equal ranking in RP versus equal ranking not allowed
in RP.)
In FPP, the only restriction on vote admissabilty that is relavent is that the voter can mark as
favourite one candidate only. Obviously it makes no difference whether voters are not allowed to enter
lower rankings or are compelled to enter lower rankings or anything in between.

By way of a contrasting example,there is a horrible method called "the Supplementary Vote" (is or was
used to elect the Lord Mayor of London in the UK), which is a version of IRV that restricts voters to
voting a single first and a single second preference. Obviously, with four or more candidates, this is
not equivalent to IRV because it can give a different result.

Chris Benham

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