[EM] Induced Alabama paradox in Warren's new apportionment method?

Kristofer Munsterhjelm km_elmet at t-online.de
Thu Dec 23 01:37:45 PST 2021

Here's a thought. Consider the divisor-based method that Warren Smith
details in https://rangevoting.org/NewAppo.html. It consists of using a
divisor method with the divisor d = (1/K) ln( K/(1-exp(-K)) ), where K
is the number of states divided by the number of seats.

Warren then claims that this method is less biased than Webster while
also passing house monotonicity and population-pair monotonicity, as
every divisor method does.

But since d is a function of the number of seats, I was wondering: is it
possible to set up a house monotonicity violation when adding seats?
Suppose that say, state X and Y obtain an equal number of seats, and
then when adding another seat, d changes so that it favors X. Then X
will lose a seat when the number of seats increase, which is a violation
of house monotonicity?

Is this possible? It would have to be a very contrived situation, but if
so, one can't say that the new apportionment method is the best there is.

(In practice, plain old Webster with weighted votes, if required, would
probably suffice. Or the expected value of d over a sensible range of K,
e.g. suppose the law requires that K must be within a certain range,
then one could just fix d to be unbiased in expectation over some
reasonable distribution of the number of states.)


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