[EM] Poll, preliminary ballots

[Kristofer Munsterhjelm]

On 2024-04-19 01:44, Richard, the VoteFair guy wrote:
> On 4/18/2024 1:17 PM, Kristofer Munsterhjelm wrote:
>  > There doesn't need to be a fixed proportion of failures, does there?
> 
> You are correct.
> 
> What I'm taking into account are two factors:
> 
> * How deeply down into the pairwise counts does the method look?
> 
> * How convoluted is the counting process?
> 
> The Schulze method looks very deeply into the pairwise counts.  However, 
> its counting process is so convoluted that it's very difficult to 
> comprehend.
> 
> I suspect that that convolution causes lots of IIA failures.

I'm not sure, since the elegance of an object doesn't need to relate to 
the simplicity of the algorithm that finds them. E.g. it's very easy to 
say what a prime number is, but a polynomial time deterministic 
algorithm to determine if a number is prime can be quite complex.

> 
> Research is needed to measure failure rates.
> 
> I really don't know what those measurements will reveal.
> 
> Interestingly, Yee diagrams serve as a simple way to measure some IIA 
> failure rates.  They clearly reveal the IIA failures of IRV.
> 
> We need something even better to identify and measure the failure rates 
> of better counting methods.
> 
> The graph I generated and referred to is just a beginning.  We need lots 
> more research that measures failure RATES.  Just looking at 
> pass-versus-fail checkboxes is not looking deep enough.

Here's a simple result for IIA.

Define an election under a method M as failing IIA if:
	- The original winner according to M is X, and
	- It is possible to remove one or more candidates who aren't X, so that 
the winner instead becomes some other candidate Y.

Then, for a Condorcet method, an election fails IIA iff it has no 
Condorcet winner. Remove every candidate in the cycle, except for the 
one who beats X pairwise, and then that candidate is the Y who wins.

For a non-Condorcet method, if it works like majority rule when there 
are only two candidates, the election fails IIA either if there's no CW, 
or if there is a Condorcet winner that the method doesn't elect.

In either case, you just remove everybody except the current winner and 
a candidate who beats him pairwise.

So here for Condorcet methods, the IIA rate is simply the rate of non-CW 
elections under the election distribution in question (impartial 
culture, spatial, etc). It is not affected by other properties like 
monotonicity, reversal symmetry, clone dependence, or even LIIA.

And the rate for non-Condorcet methods depends directly on how often 
they fail Condorcet, and not on other properties either.

-km