[EM] Fixing IRV

[Richard Moore]

Anthony Simmons wrote:
> Fortunately for everyone who is eager to get started matching
> them up, election methods have to be compared only with
> others that have the same number of ballots.  N is an
> arbitrary variable, and the proof says that for any N, the
> result is true.  This amounts to having a separate proof for
> each N, and for each proof, the number of possible
> comparisons is finite.
> 
> "In principle" just means that the actual act of comparing
> sets only has a finite number of steps, and implies that it
> may not be practical.

Anthony,

Even if we limit ourselves to a finite number of cases, we 
still don't have a way to apply this to Markus' question, 
which was "How do you want to check whether a given 
elimination method can also be defined as a method that 
doesn't "eliminate alternatives prior to selecting a 
winner"?" For when we speak of "elimination" we are 
classifying methods according to their algorithms, not by 
their output. So for a given method, you can generate all 
possible input/output pairs (given a finite limit on the size).

You also wrote "Therefore, it is possible (yes, "at least in 
principle") to prove that a given election method is or is 
not equivalent to a non-elimination method." This assumes we 
are given both methods and want to compare them. But we were 
presented with a different problem, which is that if we are 
given one method that uses elimination, how do we either 
find an equivalent non-elimination method or else prove that 
one doesn't exist? To answer Markus' question using the 
large finite sets approach, you then have to turn to the 
question of "What algorithms could have generated these 
outputs for these inputs?", and then answer the question, 
"Are any of these algorithms non-elimination algorithms?"

I imagine you could generate an algorithm of arbitrary 
complexity for a finite set of inputs and outputs. So don't 
we still have an infinite number of equivalent algorithms to 
examine?

What we are looking for is a definition of "elimination" 
that will allow us to cleanly split the set of all possible 
methods into elimination methods and non-elimination 
methods. I agree with Markus that this isn't an easy thing 
to define, but should we conclude that no useful definition 
exists? (By "useful" I mean that you could build theorems 
around the definition.)

Richard