House monotonicity criterion

From electowiki

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.[1]

The house monotonicity criterion for an apportionment method is:

If the number of seats increases with fixed populations, no state delegation decreases.

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:[2]

No candidate should be harmed by an increase in the number of seats to be filled, with no change to the profile.

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[3], 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.

Criterion incompatibility

Methods that are house monotone and pass the Droop proportionality criterion all fail the Condorcet criterion. A method may pass any two of these criteria, but not all three at once.



  1. Balinski, M. L.; Young, H. P. (1974-11-01). "A New Method for Congressional Apportionment". Proceedings of the National Academy of Sciences. Proceedings of the National Academy of Sciences. 71 (11): 4602–4606. doi:10.1073/pnas.71.11.4602. ISSN 0027-8424.
  2. Woodall, D. (1994). "Properties of preferential election rules". Voting matters (3): 8–15.
  3. Markus Schulze (2011-03-14). "Free Riding and Vote Management under Proportional Representation by the Single Transferable Vote" (PDF). p. 42.