# [EM] Monotonicity and the Muller-Satterthwaite Theorem

Alex Small asmall at physics.ucsb.edu
Fri Oct 25 12:38:23 PDT 2002

```I figured out the Muller-Satterthwaite Theorem, which says that monotonic
and pareto efficient election methods are dictatorial.  The definition of
monotonicity in that theorem is not the one I've normally heard.  For the
details, see Reny, _Econ Lett._, v. 70, p.99 (2001).  I'll sketch out the
idea:

Consider a set of voters with preference orders {L_i}, where L_i is the
preference order of the i'th voter and {L_i} is the set of all preference
orders.  Say that the winner of the election is A.

The method is monotonic if the following holds:  Consider a new set of
voters with preference orders {M_i}.  If L_i ranks A above a particular
candidate, then M_i also ranks A above that candidate.  If the election
method once again elects A then it is monotonic.

Plurality flunks this definition.  Consider two electorates:

34 voters:  A>B>C
33 voters:  B>C>A
33 voters:  C>B>A

A is the plurality winner.

Now, shift 2 voters from the B>C>A category to the C>B>A category.  Every
voter who ranked A above B still does so, and every voter who ranked A
above C still does so.  However, C now wins.  By the above definition
plurality is not monotonic.

This definition of monotonicity, although certainly valid and perhaps
useful for some analyses, is more restrictive than the one I've always
heard.  Roughly speaking, this definition states that as long as A does
not _lose_ any support he should still win.  This is more restrictive than
saying that A should still win if voters upgrade their ranking of A
without changing the relative rankings of other candidates.

Hmm, I'm not sure if any ranked methods pass that criterion.  I'd have to
think about it.  Anyway, it's something to be aware of when discussing
election methods with other people:  If we say "Condorcet is monotonic"
they might come back with a definition of monotonicity that Condorcet does
not pass.  The argument then becomes "What does this word mean?" rather
than "Is this feature of a given method undesirable?"

Alex

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