# Monroe's method

Monroe's method is one of two proportional multi-winner voting methods based on a concept first proposed by Burt Monroe.[1]

## 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 ${\displaystyle v_a}$ and candidate ${\displaystyle x_i}$, the amount of misrepresentation ${\displaystyle v_a}$ incurs were ${\displaystyle x_i}$ to represent him, is known and defined as ${\displaystyle \mu_{ia}}$. If these ${\displaystyle \mu}$ 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 ${\displaystyle m}$ representatives, each associated with an equally sized constituency (of ${\displaystyle \frac{n}{m}}$ voters), for which the total misrepresentation (summed across all voters), ${\displaystyle \mu}$, 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.

## Methods

### Fully Proportional Representation with Ordinal Balloting

In his 1995 paper "Fully Proportional Representation" [1], Monroe defined an ordinal voting method where the value ${\displaystyle \mu_{ia}}$ is defined as the rank that ${\displaystyle v_a}$ ranks ${\displaystyle x_i}$ (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 ${\displaystyle \mu_{ia} = a_1 - a_{rank_{ia}}}$, where ${\displaystyle a}$ is the weight vector for that weighted positional method, and ${\displaystyle rank_{ia}}$ is the rank that ${\displaystyle v_a}$ gave ${\displaystyle x_i}$.

### Cardinal method

Warren D. Smith later defined a cardinal method based on Monroe's concept.[2]

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.

## Variants

### Chamberlin-Courant

The Chamberlin-Courant method works like Monroe, except that the constituencies can be of any size: they are not limited to ${\displaystyle \frac{n}{m}}$ 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.[3]

## Complexity

Determining the optimal Monroe outcome is NP-hard but fixed-parameter tractable.[4] There exist constant-factor approximation algorithms for the problem of maximizing representation, but not for minimizing misrepresentation.[5]

## References

1. a b Monroe, Burt L. (1995). "Fully Proportional Representation". American Political Science Review. Cambridge University Press (CUP). 89 (4): 925–940. doi:10.2307/2082518. ISSN 0003-0554. Retrieved 2020-02-09.
2. Smith, Warren D. (February 2010). "Multiwinner election method based on optimum constrained-degree-subgraph problem". RangeVoting.org. Retrieved 2020-02-09.
3. Sánchez-Fernández, Luis; Fisteus, Jesús A. (2019-02-22). "Monotonicity axioms in approval-based multi-winner voting rules". arXiv:1710.04246 [cs]: 7.
4. Procaccia, Ariel D.; Rosenschein, Jeffrey S.; Zohar, Aviv (2007-04-19). "On the complexity of achieving proportional representation" (PDF). Social Choice and Welfare. Springer Science and Business Media LLC. 30 (3): 353–362. doi:10.1007/s00355-007-0235-2. ISSN 0176-1714.
5. Skowron, Piotr Krzysztof; Faliszewski, Piotr; Slinko, Arkadii (2013-06-29). "Fully Proportional Representation as Resource Allocation: Approximability Results". Twenty-Third International Joint Conference on Artificial Intelligence: 357.