House monotonicity criterion

The House monotonicity criterion is a criterion for apportionment/party list methods, and by extension, for multi-member methods in general. The term was first used by Balinski and Young in 1974.

The house monotonicity criterion for an apportionment method is:

That is, a state must never lose a seat from the number of total seats increasing. When used as a party list system, no party can lose a seat in this way, either. The Alabama paradox is an example of a house monotonicity failure.

By extension, the house monotonicity criterion for a multi-member system is:

That is, increasing the number of winners should never evict anyone from the winner set who is already in it.

House monotone multi-member methods are sometimes called proportional orderings or proportional rankings, and James Green-Armytage's Proportional Ordering is such a method. Sequential methods that are based on highest-average methods and don't have deletion steps, such as sequential Ebert and sequential Phragmén, are also house monotone.

Related

 * Balinski–Young theorem
 * Population monotonicity