Social ChoiceLecture 21

John Hey

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.

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.

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

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.

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.

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.

Single Transferable Vote: Finding the Winners 2

• Define the DROOP QUOTA: m/(n+1)+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.

STV: A Wikipedia example• 57 voters, 4 candidates, 2 places. • Droop Quota = 57/(2+1)+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.

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.

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).

Proportional Representation 2.1: D’Hondt 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/(s+1), 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.

Proportional Representation 2.1: D’Hondt method Wikipedia example

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 D’Hondt method.

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).

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.

Proportional Representation 3: Mixed Member Problems

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

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/(n+1) 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.

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.

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.

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.

Some oddities

• Monotonicity• Tactical Voting• The Alabama Paradox

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.

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.

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.

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!!

‘Lecture’ tomorrow

• This is another example of an experiment.• Please be at the CESARE lab at 11.00 tomorrow

morning.• We will run Charlie Holt’s Voting Game. • You can check this out at

http://veconlab.econ.virginia.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!