Monroe's method: Difference between revisions
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==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 <math>v_a</math> and candidate <math>x_i</math>, the amount of misrepresentation <math>v_a</math> incurs were <math>x_i</math> to represent him, is known and defined as <math>\mu_{ia}</math>. If these <math>\mu</math> 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:
<blockquote>Our objective in pure FPR is to find the set of <math>m</math> representatives, each associated with an equally sized constituency (of <math>\frac{n}{m}</math> voters), for which the total misrepresentation (summed across all voters), <math>\mu</math>, 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.</blockquote>
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===Fully Proportional Representation with Ordinal Balloting===
In his 1995 paper "''Fully Proportional Representation''" <ref name="Monroe 1995 pp. 925–940"/>, Monroe defined an ordinal voting method where the value <math>\mu_{ia}</math> is defined as the rank that <math>v_a</math> ranks <math>x_i</math> (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 <math>\mu_{ia} = a_1 - a_{rank_{ia}}</math>, where <math>a</math> is the weight vector for that weighted positional method, and <math>rank_{ia}</math> is the rank that <math>v_a</math> gave <math>x_i</math>.
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===Cardinal method===
[[Warren D. Smith]] later defined a cardinal method based on Monroe's concept.<ref name="RangeVoting.org">{{cite web | title=Multiwinner election method based on optimum constrained-degree-subgraph problem | website=RangeVoting.org | url=https://rangevoting.org/MonroeMW.html | access-date=2020-02-09 |date=February 2010 |last=Smith |first=Warren D.}}</ref>
This method maximizes total representation instead of minimizing misrepresentation, with each elected candidate the representative of an equal number of voters, and the degree to which a voter is represented by
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 <math>\frac{n}{m}</math> 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 monotonicity|house monotone]].<ref>{{Cite journal|last=Sánchez-Fernández|first=Luis|last2=Fisteus|first2=Jesús A.|date=2019-02-22|page=7|title=Monotonicity axioms in approval-based multi-winner voting rules|url=http://arxiv.org/abs/1710.04246|journal=arXiv:1710.04246 [cs]}}</ref>
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