[EM] Consistency Criterion

[Blake Cretney]

On Fri, 17 Aug 2001 12:55:04 -0400
Douglas Greene <douggreene at earthlink.net> wrote:

> Multiple-districts paradox:
> ---------------------------
> also in the literature called the "consistency" problem:
> if 2 subsets of the voters agree on a winner, to be consistent, the
> combined election should produce that same winner - but
> with the IRV, that is not necessarily so.

Here's an interesting paradox along the same lines.  I'm getting this
from a book called "The Paradoxicon" by Nicholas Falletta.

Let's say you have two jars, a tall jar and a fat jar.  You also have
two kinds of candies, orange and mint.  The jars have the following
contents.

Tall:  50 orange 60 mint
Fat:   30 orange 40 mint

If you want an orange flavoured candy, and the candies all look the
same, then you'd want to pick from the tall jar, since it has a higher
proportion of orange:mint candies.

Now, imagine a different pair of jars with these proportions.

Tall:  60 orange 30 mint
Fat:   90 orange 50 mint

Once again, you would pick the tall jar for an orange candy.

OK, now imagine that you dumped the contents of both tall jars into
one giant tall jar, and both the fat jars into one giant fat jar. 
Now, the question is, which giant jar do you pick for the best chance
of an orange candy.

It would seem sensible that we could rely on a statistical consistency
criterion to give us an answer, similar to the criterion suggested for
elections.  That is, if we picked the tall jar both times, it is only
common sense that we should pick the giant tall jar with their
combined contents.  Unfortunately, this is wrong.  The giant jars
contain.

Tall:  110 orange 90 mint
Fat:   120 orange 90 mint

It turns out, the fat jar is the correct choice, in violation of my
proposed statistical consistency criterion.

So, I guess it makes sense to be cautious about what criteria one
adopts.

---
Blake Cretney