U3A Web Groups
Counting Heads (2): Voting Systems
In a forthcoming referendum,
we will be asked to decide whether to change our current system for electing British MPs.
Different systems are used throughout the world in different circumstances, many of which are
described in Wikipedia. "Since voting
involves counting, it is algorithmic in nature", so software simulation can be used to
show that "different voting systems may give very different results" from the same electoral choices.
- Here is a program to demonstrate and compare methods.
It represents five different candidates and 100 electors, each having chosen three
candidates and ranked them in order of preference.
- One model assumes that the candidates (and most voters' choices) are spread across
a "left-right" political spectrum. Initially the choices are distributed fairly evenly
between the candidates, but there is a user option to "swing" them right or left to
change the distribution and see the result.
- The alternative model assumes no relationship between the candidates, and that all
voters' choices are made independently, based on approval of the candidates as individuals.
- Four different types of election can be run, producing one or several winners.
Fresh data distributions can be chosen at any time.
- The familiar first past the post is currently used in British elections.
It is very simple: the candidate with the highest vote wins, regardless of the size
of the margin.
- One extension involves a run-off: if the first round doesn't produce an
overall majority, a second election is run between the top two candidates only.
- That system is used in French presidential elections. Voters actually go to the
poll twice, possibly changing their minds in between! Our run-off simulation assumes
that the top two candidates benefit from half their second- or third- choice
votes from the first election.
- Another real-life method involves repeated polls until a majority is
achieved, with the lowest-placed candidate dropping out each time. This is practical
only in special circumstances, for instance when the cardinals are shut up together
in order to elect the next pope.
- When electors are asked to rank a set of candidates in order of
preference, repeated "rounds" can be played without the need for re-voting.
- A simple example is the alternative voting system (aka instant "run-off")
which may adopted for our parliamentary elections in future. Some ballot-papers may
need to be re-counted — perhaps several times.
- When no candidate achieves an overall majority, the one with fewest votes is
eliminated and his or her votes are redistributed to the relevant second / third
/ fourth etc. choice candidate. The process continues until a majority is produced.
- Under this system, second and later preferences don't always influence the result,
as the chosen candidate(s) may already be eliminated. Where one candidate has a very
large majority it can be argued that even some first-choice votes are essentially
Multiple winner systems
- Some elections involve more than one "winner", for instance in the large
multi-member constituencies represented by members of the European
Parliament. This is one way to achieve proportional representation.
- The single transferable vote is used in this situation. In order
to be elected, candidates must receive a quota of votes, which varies
according to the number of seats to be filled.
- Voters rank the candidates as usual and an initial count is made. Candidates
reaching the quota are declared elected. Their "surplus" votes (i.e. the number
above those needed to reach the quota) are redistributed to the relevant
second / third / fourth etc. choice candidates.
- In a manual system, the distribution is performed by taking a random
sample of physical ballot papers. With electronic voting systems, and in our
simulation, it is done by calculating what fractional part of the
vote can be added in each case.
- If no candidates reach the quota, the one with fewest votes is eliminated as
already described, the counts recalculated, and the tests repeated.
- To take all voters' preferences into account, it is necessary to
compare every candidate with every other.
- We can construct a matrix showing, for each pair of candidates X
and Y how many voters prefer X to Y and how many prefer
Y to X. Comparing the two values produces a "winner" for that
- The final score for each candidate is the number of contests won.
The scores can be used to select one or several overall winners.
- In principle this method should identify the candidate(s) whom voters
prefer to all others. In practice the ordering may turn out to be
circular (X beats Y beats Z beats X)
so there is no overall winner.
- With a large electorate, and using manual counting methods, pairwise
comparison might be fairer, but certainly more complex to operate!