[EM] Fixing IRV

[Richard Moore]

Anthony Simmons wrote:
> The answer falls out of its own accord if I just
> reformulate the problem in sufficiently rigorous
> doubletalk.
> 
> Let the number of voters be N, and the number of ways of
> marking a ballot (valid ways plus one general way called
> "invalid") be m.  Then there are N^m distinct ways that all
> of the ballots can be marked.  (If you like, you could
> include all registered voters in N, and include one more way
> of marking the ballot, "unvoted".)
> 
> Each way of marking the ballots maps to an outcome, which is
> a ranking of the candidates, allowing for whatever
> peculiarities, such as ties, might be allowed.
> 
> Call this function, which maps sets of marked ballots to
> outcomes, an "election method".  Each election method, being
> a function, is just a set of ordered pairs; each pair
> contains one set of ballots and the outcome they map to.
> 
> Each election method is a finite set (with N^m elements) of
> ordered pairs.  Since every election method is a finite set,
> there is an effective, if somewhat tedious, method of
> comparing two of them to see if they are equivalent --
> compare their members.  One of those "at least in principle"
> things.  Since the number of election methods is also finite,
> it is, again at least in principle, possible to compare a
> given election method with all other election methods (or
> with a subset of them) to see if they are equivalent.
> 
> 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 does *not* mean there's a
> feasible method.
> 
> Life is so easy with finite sets.

This certainly could be used "in principle" to prove things 
about elections for N <= N1 and m <= m1, where N1 and m1 are 
abritrary limits. But unless you can construct a 
mathematical induction proof, this result doesn't apply for 
N > N1 or m > m1, and you have to repeat the process with 
these larger numbers. It then follows that the problem is 
infinite in scope.

Example: The method that says "elect the Borda winner unless
there are more than 1 million voters, in which case elect 
the plurality winner" is equivalent to Borda as long as the 
number of voters is less than or equal to 1 million.

In this light, of course, the number of methods is also 
infinite.

Richard