Phragmen's voting rules: Difference between revisions
Definition tightened up and the "odd step" is explained
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# The ballots are [[Approval voting]] i.e. each ballot lists the set of candidates that voter "approves."
# Later on, we shall associate a "cost" with each ballot. (Phragmén used the Swedish word "belastning." Other people often prefer to translate this into the English word "load" rather than "cost.") All ballots initially have cost=0.
# Seats are elected sequentially. Perform steps 4-
# As soon as any candidate is elected, the N ballots
# The candidate who wins the next seat is the one whose N supporters' ballots will each have the
# Once a candidate has been elected and the cost distributed among the voters, this cost is fixed and cannot be redistributed once later candidates are elected. If this redistribution were allowed, then the result would be further optimized and closer to the non-sequential max-Phragmén method.
▲# The candidate who wins the next seat is the one whose N supporters' ballots will have the least average cost. (So, for example, the first winner is simply the most-approved candidate, because if he is approved by N voters the average cost per approving-ballot is 1/N, which is minimal because N is maximal.)
In the case of a single candidate to be elected, Phragmén's method reduces to [[Approval voting]], because the candidate resulting in the minimal load on each voter is the one with most voters to share the load. When all votes are in party list order, it reduces to the [[D'Hondt method]].<ref name="Janson 2016">{{cite arXiv | last=Janson | first=Svante | title=Phragmén's and Thiele's election methods | date=2016-11-27 | eprint=1611.08826|class=math.HO}}</ref>
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* '''seq-Phragmén''': Phragmén's original method, described above. It attempts to minimize the maximal load by sequentially electing candidates.
* '''max-Phragmén''': The non-sequential variant of seq-Phragmen: the objective is the same (minimize the maximal load), but it's treated as a global optimization problem. The load from each candidate does not have to be evenly spead across their
* '''var-Phragmén''': This variant minimizes the variance of the load distribution. As with max-Phragmén, the load from each candidate can be spread unevenly across their voters.
* '''[[Ebert's method]]''': As var-Phragmén but where a candidate's load is evenly spread across their voters.
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