[EM] Linear methods are consistent and monotonic

[Richard Moore]

Forest Simmons wrote:

> Very well done!

Thanks. I was wondering about my definition for the 
monotonicity criterion: It seems right but it's limited to 
methods that base the result on numerical scores (just as 
the other definitions I've seen seem to be limited to 
positional voting methods). Is there a definition general 
enough to take in all classes of methods or does 
monotonicity have to be defined separately for each class?

Also, I noticed a shortcoming in the MC definitions on 
various web sites. They all seem to be worded along the 
lines of, "If X wins and you increase the support for X then 
X still wins," but this seems to require that even if you 
increase the support for X and Y at the same time, Y will 
not win. Of course this definition, taken literally, is too 
strict for any method to meet. I know this is nit-picking, 
but this definition should include words such as "without 
increasing the support for any other candidate" after 
"increase the support for X". I included this condition in 
my definition.

Finally, regarding the Cal IRV site I posted the link to: 
They claim IRV is monotonic when adding preferences to the 
end of an incomplete list of preferences. But they are 
confusing monotonicity over a limited region with compliance 
to the monotonicity criterion. The monotonicity criterion is 
not met unless the method is *universally* monotonic. Every 
definition of the criterion has some words to this effect: 
For any possible set of ballots, there is no way to modify 
any subset of those ballots in a way that produces a 
non-monotonic response.

Richard