[EM] Monotonicity and the Muller-Satterthwaite Theorem

[matt at tidalwave.net]

On 25 Oct 2002 at 12:38, Alex Small wrote:

> 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?"

D R Woodall defines various versions of monotocity in his article Properties of 
Preferential Election Rules.  A still wins if A's ranking is raised is called Mono-
Raise.  It is usefull to divide monotocity into narrower versions not only for greater 
clarity, but also because not all variations of monotonicity appear to have equal 
merit and because different variations of monotonicity are members of sets of 
fairness criteria that conflict with different alternative sets of fairness criteria.  

In particular, Woodall claims that any method that satisifies Condorcet will fail 4 
different versions of monotonocity (raise-random, sub-top, raise-delete, and sub 
plump), Participation, Later-no-help and Later-no-harm (some STV advocates 
regard these latter two as being important, however violating both does not 
necessary create any practical strategy opportunity due to unpredictability of which 
violation will occur and equal probability of both occuring).  He also claims that if a 
Condorcet method passes Plurality then it fails Mono-add-top.

I recommend that election method researchers 1) argue the relative merit of these 
different versions of monotonicity 2) identify which of these versions of monotonicity 
the particular methods they are interested in pass and 3) discuss the 
frequency/impact of the failures for those methods.  I have not yet seen a 
discussion at this level of detail for the leading Condorcet methods.

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