Social Choice Lecture 21

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Social ChoiceLecture 21

• John Hey

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Voting Systems

• This and the previous two lectures concern voting
  systems.
• Voting is used to choose between alternatives.
• We note that we have exactly the same problems
  (though rather more obviously manifested here) as
  we had when we discussed Arrow.
• Basically if people have different preferences
  then it is difficult/impossible to aggregate
  them.
• If we are trying to choose the most preferred,
  for example, President, if people have different
  preferences, what do we mean by most preferred?
• I note, without shame, that most of this material
  I have taken from Wikipedia
• http//en.wikipedia.org/wiki/Voting_system
  and linked pages.

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Lectures on Voting Systems

• Lecture 19, Thursday the 3th of December
  Single-Winner Voting Systems.
• Lecture 20, Friday the 4th of December
  Historical and Factual description of voting
  systems.
• Lecture 21, Thursday the 10th of December
  Multiple-Winner Voting Systems.

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Lecture 21 Multiple-Winner Voting Systems

• Proportional representation
• Single transferable vote (2 sub-headings)
• Party-list  (open or closed) (4 sub-headings)
• Mixed-member
• Semi-proportional representation
• Cumulative voting
• Limited voting (Single Non-transferable vote)
• Parallel voting (Not discussed)
• Non-proportional representation
• Plurality-at-large
• Preferential block voting

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Proportional Representation

• Proportional representation (PR), sometimes
  referred to as full representation, is aimed at
  securing a close match between the percentage of
  votes that groups of candidates (grouped by a
  certain measure) obtain in elections and the
  percentage of seats they receive. PR is a
  democratic principle rather than an electoral
  system in itself. It is often contrasted to
  plurality voting systems, where disproportional
  seat distribution results from the division of
  voters into multiple electoral districts,
  especially "winner takes all" plurality
  districts.
• Various forms of proportional representation
  exist, such as party-list proportional
  representation, where the above-mentioned groups
  correspond directly with candidate lists as
  usually given by political parties. Within this
  form a further distinction can be made depending
  on whether or not a voter can influence the
  election of candidates within a party list (open
  list and closed list respectively).
• Another kind of electoral system covered with the
  term proportional representation is the single
  transferable vote (STV), which, in turn, does not
  depend on the existence of political parties (and
  where the above-mentioned "measure of grouping"
  is entirely left up to the voters themselves.

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Proportional Representation 1 Single
Transferable Vote

• The Single transferable vote (STV) is a system of
  preferential voting designed to minimize "wasted"
  votes and provide proportional representation
  while ensuring that votes are explicitly
  expressed for individual candidates rather than
  for party lists.
• In STV, each voter ranks the list of
• candidates in order of preference. In other
• words they place a '1' beside their most
• preferred candidate, a '2' beside their
• second most preferred, and so on. The
• ballot paper submitted by the voter
• therefore contains an ordinal list of candidates.

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Single Transferable Vote Finding the Winners 1

• There are clearly many methods. One can select
  winners sequentially, or one can select losers
  sequentially, and in either case re-distribute
  the votes. There is no unambiguously correct way
  to re-distribute the votes nor choose a method.
• Many of these ways are very complicated and
  difficult to describe and understand.
• They are prone to various problems including
  non-monotonicity, tactical voting and other
  paradoxes.
• A common way is described on the following slide.

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Single Transferable Vote Finding the Winners 2

• Define the DROOP QUOTA m/(n1)1 where m is the
  number of valid votes and n is the number of
  winners.
• The Droop quota is the smallest number that
  guarantees that no more candidates can reach the
  quota than the number of seats available to be
  filled.
• Any candidate who has reached or exceeded the
  required quota is declared elected.
• If not enough candidates have been elected, the
  count continues.
• If an elected candidate has more votes than the
  quota, then that candidate's surplus is
  transferred to other candidates according to the
  next preference on each voter's ballot.
• If no one meets the quota, the candidate with the
  fewest votes is eliminated and that candidate's
  votes are transferred.
• This process repeats from step 1 until the
  required number of candidates have been elected.
• Whether votes are transferred to elected
  candidates depends on the particular counting
  system chosen systems that allow this
  subsequently redistribute the surplus.

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STV A Wikipedia example

• 57 voters, 4 candidates, 2 places.
• Droop Quota 57/(21)1 20
• In the first round, Andrea receives 40 votes and
  Delilah 17. Andrea is elected with 20 excess
  votes. Her 20 excess votes are reallocated to
  their second preferences. For example, 12 of the
  reallocated votes go to Carter, 8 to Brad.
• As none of the remaining candidates have reached
  the quota, Brad, the candidate with the fewest
  votes, is excluded from the count. All of his
  votes have Carter as the next-place choice, and
  are reallocated to Carter. This gives Carter 20
  votes and he is elected, filling the second seat.

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Problems with STV

• Defining the Quota (other quotas proposed).
• Redistributing the excess votes of the elected
  candidates.
• Redistributing the votes of the excluded
  candidates.
• Complexity.
• When STV is used for single-winner elections, it
  is equivalent to the non-proportional
  instant-runoff voting method. This is
  non-monotonic and hence so is STV.
• It also does not satisfy the independence of
  irrelevant alternatives criterion.
• It also seems vulnerable to tactical voting.

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Proportional Representation 2 Party List

• Party-list proportional representation systems
  are a family of voting systems emphasizing
  proportional representation (PR) in elections
  returning multiple candidates.
• In these systems, parties make lists of
  candidates to be elected, and seats get allocated
  to each party in proportion to the number of
  votes the party receives. Voters may vote
  directly for the party, as in Israel, for
  candidates and that vote will pool to the party,
  as in Turkey and Finland, or for a list of
  candidates, as in Hong Kong.
• The order in which a party's list candidates get
  elected may be pre-determined by some method
  internal to the party or the candidates (a closed
  list system) or it may be determined by the
  voters at large (an open list system).

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Proportional Representation 2.1 DHondt method

• In a closed list system, each voter casts a
  single vote for the party of their choice. In an
  open list system, the voter votes for a candidate
  personally, but the vote is principally counted
  as a vote for the candidate's party.
• After all the votes have been tallied, successive
  quotients or 'averages' are calculated for each
  list. The formula for the quotient is V/(s1),
  where
• V is the total number of votes that list
  received and
• s is the number of seats that party has been
  allocated so far (initially 0 for all parties)
• Whichever list has the highest quotient or
  average gets the next seat allocated, and their
  quotient is recalculated given their new seat
  total. The process is repeated until all seats
  have been allocated.
• The order in which seats allocated to a list are
  then allocated to individuals on the list is
  irrelevant to the allocation procedure. It may be
  internal to the party (a closed list system) or
  the voters may have influence over it through
  various methods (an open list system).
• The rationale behind this procedure (and the
  Sainte-Laguë procedure) is to allocate seats in
  proportion to the number of votes a list
  received, by maintaining the ratio of votes
  received to seats allocated as close as possible.
  This makes it possible for parties having
  relatively few votes to be represented.

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Proportional Representation 2.1 DHondt method
Wikipedia example
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Proportional Representation 2.2 Highest average
method

• The highest averages method is one way of
  allocating seats proportionally for
  representative assemblies with party list voting
  systems.
• The highest averages method requires the number
  of votes for each party to be divided
  successively by a series of divisors, and seats
  are allocated to parties that secure the highest
  resulting quotient or average, up to the total
  number of seats available. The most widely used
  is the d'Hondt formula, using the divisors
  1,2,3,4... The Sainte-Laguë method divides the
  votes with odd numbers (1,3,5,7 etc). The
  Sainte-Laguë method can also be modified, for
  instance by the replacement of the first divisor
  by 1.4, which in small constituencies has the
  effect of prioritizing proportionality for larger
  parties over smaller ones at the allocation of
  the first few seats.
• Notice that this is a generalisation of the
  DHondt method.

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Proportional Representation 2.3 Largest
remainder method

• The largest remainder method requires the number
  of votes for each party to be divided by a quota
  representing the number of votes required for a
  seat, and this gives a notional number of seats
  to each, usually including an integer and either
  a vulgar fraction or alternatively a remainder.
  Each party receives seats equal to the integer.
  This will generally leave some seats unallocated
  the parties are then ranked on the basis of the
  fraction or equivalently on the basis of the
  remainder, and parties with the larger fractions
  or remainders are each allocated one additional
  seat until all the seats have been allocated.
  This gives the method its name.
• Notice that it is relatively easy to understand.
• However it is prone to the Alabama Paradox in
  which increasing the number of seats may cause a
  party to lose a seat (see end).

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Proportional Representation 3 Mixed Member

• Mixed member proportional representation, also
  termed mixed-member proportional voting and
  commonly abbreviated to MMP, is an 'additional
  member' voting system used to elect
  representatives to numerous legislatures around
  the world. MMP is similar to other forms of
  proportional representation (PR) in that the
  overall total of party members in the elected
  body is intended to mirror the overall proportion
  of votes received it differs by including a set
  of members elected by geographic constituency who
  are deducted from the party totals so as to
  maintain overall proportionality. Therefore, the
  additional party seats are compensatory they top
  up the local results.
• In most models the voter casts two votes one for
  a constituency representative and one for a
  party. If a candidate is on the party list, but
  wins a constituency seat, they do not receive two
  seats they are instead crossed off the party
  list and replaced with the next candidate down.
  In the original variant used at first in Germany,
  still used by two States of Germany, both votes
  were combined into one, so that voting for a
  representative automatically means also voting
  for the representative's party. Most of Germany
  changed to the two-vote variant to make local MPs
  more personally accountable. Voters can vote for
  the local person they prefer for local MP without
  regard for party affiliation, since the partisan
  make-up of the legislature is determined only by
  the party vote. In the 2005 New Zealand election,
  20 of local MPs were elected from electorates
  (constituencies) which gave a different party a
  plurality of votes.

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Proportional Representation 3 Mixed Member
Problems

• Used in many countries.
• Prone to tactical voting.
• Difficult to understand.

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Semi Proportional Representation 1 Cumulative
Voting

• An alternative method called Cumulative voting
  (CV) is a semi-proportional voting system in
  which each voter has n votes, where n is the
  number of seats to be elected. Voters can
  distribute portions of their vote between a set
  of candidates, fully upon one candidate, or a
  mixture. It is considered a proportional system
  in allowing a united coalition representing a
  m/(n1) fraction of the voters to be guaranteed
  to elect m seats of an n-seat election. For
  example in a 3-seat election, 3/4 of the voters
  (if united on 3 candidates) can guarantee control
  over all three seats.

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Semi Proportional Representation 2 Limited Voting

• Limited voting is an electoral system used in
  multi-member constituency elections in which
  electors have fewer votes than there are
  positions available.
• The positions are awarded to the candidates who
  receive the most votes absolutely. In a n-seat
  constituency, the n candidates receiving the
  largest numbers of votes would win office.
• In the special case in which the voter may vote
  for only one candidate and there are two or more
  posts, this system is called the single
  non-transferable vote or sometimes the strictly
  limited vote.

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Non Proportional Representation 1
Plurality-at-large

• Plurality-at-large voting (commonly referred to
  as block voting) is a voting system for electing
  several representatives from a single multimember
  electoral district using a series of check boxes
  and tallying votes similar to a plurality
  election. Although multiple winners are elected
  simultaneously, block voting is not a system for
  obtaining proportional representation instead,
  the usual result is that the largest single group
  wins every seat by electing a slate of
  candidates, resulting in a landslide.
• In a block voting election, all candidates run
  against each other for n number of positions.
  Each voter selects up to n candidates on the
  ballot, and the n candidates with the most votes
  win the positions. Often, voters are said to have
  "n votes", however they are unable to vote for
  the same candidate more than once as in
  cumulative voting.

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Non Proportional Representation 2 Preferential
Block Voting

• Preferential block voting is a voting system for
  electing several representatives from a single
  multimember constituency. Unlike the single
  transferable vote, preferential block voting is
  not a method for obtaining proportional
  representation, and instead produces similar
  results to plurality block voting. Under both
  systems, a single group of like-minded voters can
  win every seat, making both forms of block voting
  non-proportional.
• In preferential block voting, a preference voting
  ballot is used, ranking candidates from most to
  least preferred. Alternate ballot forms may have
  two groupings of marks, first giving n votes for
  an n seat election (as in traditional bloc
  voting), but also allowing the alternate
  candidates to be ranked in order of preference
  and used if one or more first choices are
  eliminated.
• Candidates with the smallest tally of first
  preference votes are eliminated (and their votes
  transferred as in instant runoff voting) until a
  candidate has more than half the vote. The count
  is repeated with the elected candidates removed
  and all votes returning to full value until the
  required number of candidates is elected.
• Both this and the plurality method are subject to
  tactical voting (see later). While many criticize
  block voting's tendency to create landslide
  victories, some cite it as a strength. Since the
  winners of a block voting election generally
  represent the same slate or group of voters,
  there is greater agreement amongst those elected,
  potentially leading to a reduction in political
  gridlock.

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Some oddities

• Monotonicity
• Tactical Voting
• The Alabama Paradox

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Monotonicity

• A candidate x should not be harmed (i.e., change
  from being a winner to a loser) if x is raised on
  some ballots without changing the orders of the
  other candidates.
• Or conversely a candidate should not be
  benefitted if x is lowered in some ballots
  without changing the order of other candidates.
• Example with Instant runoff. Between Left and
  Right below, Andrea has more support. On left,
  Cynthia is eliminated and then Andrea wins. On
  right, Belinda is eliminated and then Cynthia
  wins.

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Tactical Voting

• Suppose 100 voters vote as follows
• Far-Left candidate 10 Centre-Left
  candidate 41
• Centre-Right candidate 40 Far-Right
  candidate 9
• Provided we assume that the second preference of
  Far-Left voters is the Centre-Left candidate, and
  the second preference of Far-Right voters is the
  Centre-Right candidate, then the result of the
  second round of a runoff election will be
• Centre-Left candidate 51 Centre-Right
  candidate 49
• In this election tactical voting will be an
  unnecessary and ineffective tactic. This is
  because once the Far-left Candidate is eliminated
  his supporters have the opportunity to vote for
  the Centre-Left candidate in the second round, so
  it is unnecessary for Far-Left supporters to vote
  tactically for the Centre-Left candidate as a way
  of ensuring she survives to the second round. For
  the same reason the outcome will not be altered
  if Far-Right supporters vote tactically in the
  first round for Centre-Right.
• Were the election conducted using the plurality
  system compromising would be an effective
  strategy. For example if Far-Right supporters
  voted tactically for Centre-Right then he would
  be elected instead of Centre-Left. To counteract
  this tactic Far-Left supporters would also have
  to vote tactically. In this example, therefore,
  runoff voting removes the potential for tactical
  voting that would be there under the plurality
  system.

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The Alabama Paradox

• After the 1880 census, C. W. Seaton, chief clerk
  of the United States Census Bureau, computed
  apportionments for all House sizes between 275
  and 350, and discovered that Alabama would get 8
  seats with a House size of 299 but only 7 with a
  House size of 300. Here is a simplified example
  using the largest remainder rule.

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Conclusions

• After Lecture 19, on single-winner systems, we
  concluded that there does not appear to be a
  perfect voting system.
• We were not surprised.
• We asked then Might going to a Multiple-Winner
  System help?
• It does not appear so! In fact, perhaps the
  contrary things become more complicated.
• What a surprise!!

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Lecture tomorrow

• This is another example of an experiment.
• Please be at the CESARE lab at 11.00 tomorrow
  morning.
• We will run Charlie Holts Voting Game.
• You can check this out at http//veconlab.econ.vir
  ginia.edu/vt/vt.php.
• You will log in at
• http//veconlab.econ.virginia.edu/login.htm
• You will have to register (tomorrow in the lab)
  as a participant in the session with a name I
  will give you tomorrow. ENJOY!