Monroe's method

Monroe's method is one of two proportional multi-winner voting methods based on a concept first proposed by Burt Monroe.

Concept
In the 1995 paper, Monroe first defined what he calls "pure fully proportional representation" or "pure FPR". Pure FPR assumes that for each voter $$v_a$$ and candidate $$x_i$$, the amount of misrepresentation $$v_a$$ incurs were $$x_i$$ to represent him, is known and defined as $$\mu_{ia}$$. If these $$\mu$$ values are known, then pure FPR consists of assigning a representative to each voter to minimize the total misrepresentation, and then electing these representatives. To quote:

"Our objective in pure FPR is to find the set of $m$ representatives, each associated with an equally sized constituency (of $\frac{n}{m}$ voters), for which the total misrepresentation (summed across all voters), $\mu$, is minimized. In effect, we want voters to define for themselves the group membership or identities that they wish to have represented and to have maximum flexibility in doing so."

Because the actual misrepresentation values usually aren't known, they must be inferred somehow, which is where the two voting methods differ.

Fully Proportional Representation with Ordinal Balloting
In his 1995 paper "Fully Proportional Representation", Monroe defined an ordinal voting method where the value $$\mu_{ia}$$ is defined as the rank that $$v_a$$ ranks $$x_i$$ (e.g. 1 for a top-ranked candidate, 2 for a candidate ranked second, and so on). Equal-rank is symmetrically completed: each candidate gets a misrepresentation value equal to the average he would've got if the candidates were ranked strictly in a random order. In the single-winner case, FPR with ordinal balloting reduces to the Borda count.

In principle, any weighted positional method can be generalized to an ordinal FPR variant by letting $$\mu_{ia} = a_1 - a_{rank_{ia}}$$, where $$a$$ is the weight vector for that weighted positional method, and $$rank_{ia}$$ is the rank that $$v_a$$ gave $$x_i$$.

Cardinal method
Warren D. Smith later defined a cardinal method based on Monroe's concept.

This method maximizes total representation instead of minimizing misrepresentation, and the degree to which a voter is represented by a candidate is simply that voter's rating of the candidate. In the single-winner case, this Monroe method reduces to either Range voting or Approval voting depending on the ballot format.

Following this work a sequential Cardinal method, Sequential Monroe voting, was later invented to simplify this methodology but keep the key requirements.

Chamberlin-Courant
The Chamberlin-Courant method works like Monroe, except that the constituencies can be of any size: they are not limited to $$\frac{n}{m}$$ voters each. This variant is useful for election to a council where each representative has a weighted vote, or for party list PR without thresholds.

Egalitarian Monroe and Chamberlin-Courant
Instead of optimizing representation (maximizing representation or minimizing misrepresentation), the egalitarian variants optimize worst misrepresentation: they produce an outcome so that the least represented voter is most represented. Egalitarian Chamberlin-Courant tends to produce consensus outcomes, similar to Minimax approval, while for egalitarian Monroe, the tendency to do so is balanced by its constituency restriction.

Criterion compliances
Monroe's method is not house monotone.

Complexity
Determining the optimal Monroe outcome is NP-hard but fixed-parameter tractable. There exist constant-factor approximation algorithms for the problem of maximizing representation, but not for minimizing misrepresentation.