N.R. Miller

10/17/01

rev. 10/07/04

VOTING TO ELECT SEVERAL CANDIDATES (MMDs)

Important Note. This discussion pertains not to elections for a single office but an office for which several candidates are to be simultaneously elected. A familiar U.S. example would be the simultaneous “at-large” election of (several or all) members of a local school board or county council. The most common example worldwide is the use of multi-member districts (MMDs) to elect members of a national parliament.

District magnitude refers to the number of representatives from the district — that is, the number of winning candidates — and is given by some whole number m.

Single-Member Districts (SMDs)      m = 1 (Voting To Elect a Single Candidate handout)

Multi-Member Districts (MMDs)       m > 1

            Typical MMDs

            “small” MMDs:                     m ≈ 3 - 6 (perhaps not uniform, local districts)

            “large” MMDs:                       m ≈ 15 - 50 (probably not uniform, regional/state districts)

            “national” MMD:                   m ≈ 100 - 250 (single national district)

            A candidate-oriented ballot lists a number of candidates and allows each voter to express some kind of preference with respect to these candidates. (Ballots in SMD systems are invariably candidate-oriented; however, some [as in the U.S.] show the party affiliation of the candidates while others [as in the U.K. until recently] do not.) Given a nominal (and candidate-oriented) ballot, a voter either votes “for” a candidate (e.g., by putting an “X” beside the candidate’s name) or not. (As discussed in the Voting To Elect a Single Candidate handout, Simple Plurality, Plurality Runoff, and Approval Voting use nominal ballots.) Given an ordinal ballot, a voter ranks the candidates in order of preference, typically by putting a “1” beside the name of the first-ranked candidate, etc.). (As discussed in the Voting To Elect a Single Candidate handout, Instant Runoff, Borda Point, and Condorcet Voting use ordinal ballots.)

            In contrast, a party-oriented ballot lists a number of parties and gives each voter an opportunity to express some kind of preference with respect to these parties. Such ballots are almost always nominal, and each voter simply votes for one party (e.g., by putting an “X” beside the party’s name). However, some primarily party-oriented ballots also allow voters to express certain candidate preferences as well.

Here is some further notation pertaining to nominal candidate-oriented ballots that we will use :

            n =    the number of candidates in a given district

            m =    the number of candidates to be elected (district magnitude)

            k =      the maximum number of votes each voter can cast (typically the maximum number of candidates each voter can vote for)

            v =     the number of votes each voter actually casts (v < k)

            N =     total number of voters casting ballots

Candidate-Oriented Quasi-Proportional Systems

            In the handout on Voting To Elect a Single Candidate, several voting systems used in SMDs were discussed. As noted above, Simple Plurality, Plurality Runoff, and Approval Voting use nominal ballots, while Instant Runoff, Borda Point, and Condorcet Voting use ordinal ballots. Approval, Borda Point, and Condorcet Voting can be directly applied to MMDs, with the top m candidates with respect to approval votes, Borda points, or the Condorcet ranking being elected.

            Plurality voting can be generalized to MMDs in several ways, as outlined below. Here are some (more or less) common voting systems that use nominal ballots and are typically applied in “small” MMDs. Ballots with v > k are “spoiled” and disqualified (e.g., the “overvotes” in Florida in 2000). Under each system, candidates are ranked by the number of votes received, and the top m candidates are elected. (These definitions are stated very concisely; you will need to stop and think out what expression each means.)

            Single Non-Transferable Vote (SNTV)                      v = k = 1

            Limited Vote (LV) [SNTV being a special case] 1 < v < k < m

            Generalized Plurality Voting (GPV)                          1 < v < k = m

            Block Vote or Plurality with “Full Slate” (BV)          v = k = m

            Approval Voting (AV)                                                1 < v < k = n -1

            Cumulative Voting (CV)                                            v < k = m but votes may be

                                                                                            “plumped” on fewer than m candidates

Note. A party-oriented ballot can turn a MMD election into one that is logically equivalent to an SMD single-winner election. This results when each party runs a slate of m candidates and voters choose among party slates (usually on the basis of Simple Plurality Voting) rather than individual candidates. A notable example is provided by the general ticket system (that might be called the “Block Vote” but is to be distinguished from the system with the same name defined above) used by almost all U.S. states to select Presidential electors.

            The principal system that uses ordinal ballots in (small) MMDs is the Single Transferable Vote (STV).

            (1)       The total number of ballots cast (N) is determined.

            (2)       The Droop Quota Q is calculated:                  Q = N      + 1

m + 1

Actually Q is the quotient rounded up to the next integer. Q is the smallest number of votes such that no more than m candidates get Q votes (can “meet the quota”).

             (3)       The ballots are then sorted and counted according their first preference votes. Any candidate who meets the quota at this stage, i.e., receives Q first-preference votes, is elected. If m candidates meet the quota, those m candidates are elected and vote counting stops.

            (4)       Otherwise votes are “transferred” and counting continues. First preference votes received by elected candidates in excess of the quota Q are “surplus” and are transferred to unelected candidates according to the second (or lower) preferences expressed on these ballots. (Since it is arbitrary which votes are surplus, votes are usually transferred in proportion to all the second preference votes on the elected candidate’s ballots. This means that either fractional votes may be transferred or transferred votes must be rounded to integer totals.) As a result of the transfer of surplus votes, additional candidates may now meet the quota and be elected. If m candidates now meet the quota and have been elected, vote counting stops.

            (5)       Otherwise candidates are eliminated, more votes are transferred, and counting continues. Specifically, the candidate with the fewest (first preference plus transferred) votes is eliminated and all of the eliminated candidate’s votes are transferred according to their highest preference for any remaining (non-eliminated and non-elected) candidates. As a result of this transfer, additional candidates may meet the quota and be elected. If m candidate now meet the quota, vote counting stops.

            (6)       The vote transfer process continues until m candidates have met the quota and are elected.

Note 1. Even though this description seems complicated, it does not address all the complexities that can arise in the STV vote counting process.

Note 2. STV is often justified as a means of implementing the principle of free association (or “self-defined constituencies”), according to which any group of voters of size Q should be able to elect a candidate of their choice, regardless of their geographical distribution (within the MMD). As a practical matter, STV can be implemented only in small magnitude MMDs, so application of this principle must be quite limited in practice.

Note 3. STV can be applied to SMDs (with m = 1), in which case Q is a simple majority of (N+1)/2. Since only one candidate can be elected, votes are transferred from eliminated candidates only. This system is usually called the Alternative Vote or the Majority Preferential Vote and is equivalent to the second variant of Instant Runoff Voting. (See footnote 1.)

            In MMD elections in which candidates are affiliated with parties and voters are likely to vote for candidates largely on the basis of their party affiliations, SNTV, CV, and STV tend to produce roughly proportional results. Under STV, a party’s success in electing its candidates pretty clearly depends on the (approximate) number of quotas that the party’s electoral support within the MMD adds up to, and this is more or less true under SNTV and CV as well. For example, in an MMD with m = 5, Q ≈ V/6 ≈17% of the vote. So something like the following would hold.

            A party with this amount of support should be able to elect this number of candidates

                        ∼0-15%                                                                                   none

                        ∼15-30%                                                                                 one

                        ∼30-45%                                                                                 two

                        ∼45-60%                                                                                 three

                        ∼60-75%                                                                                 four

                        ∼75-100%                                                                               all five

Accordingly, these are commonly referred to as quasi-proportional voting systems.

            In contrast, GP and BV do not produce proportional results. In fact, the party supported by the most voters will probably win all the seats (i.e., “winner take all,” as is necessarily true in all SMD elections and under a general ticket system ). LV (with k > 2) and AV (applied to MMDs) tend to produce subproportional results (i.e., the leading party probably will not win all the seats but does tend to win more than its proportionate share).

Party-Oriented List Proportional Representation Systems

            Under “true” (list) proportional representation (PR), legislators are elected from “large” — possibly a nationwide — MMDs, using a simple nominal party-oriented ballot. The share of the vote received by each party is calculated, and then some mathematical apportionment formula (there are many such formulas, and they apportion seats in somewhat different ways) is used to allocate the m seats among the parties according to their respective vote shares (in the same manner in which such formulas are used to fulfill the constitutional requirement that “Representatives [U.S. House seats] shall be apportioned among the states according to their respective numbers [populations].” Prior to the election, each party will have drawn up an ordered list of m candidates (nominees). If (according to its vote share and the apportionment formula) Party A is entitled to k seats, the top k candidates on its list are elected. However, proportional representation can never work perfectly, because perfect proportionality essentially always would require that parties be awarded fractional seats, which is impossible. For the most part, different apportionment formulas differ in the way they round off such fractions. List PR is by far the most common electoral system used by countries with free elections. (See Table 1.2 in Comparing Democracies 2 and the Figures in Chapter 2 of CD2.) However, no two of these national systems are identical (and individual PR countries frequently modify the details of their PR systems from election to election). Here are some of the principal variations.

(a)       Different systems use different apportionment formulas, of which there are two main types: divisor (or “highest average”) methods and quota (or “largest remainder”) methods (see CD2, pp. 48-51). Each main type includes a multiplicity of particular methods. All apportionment methods have quirks (see the following note), and different methods maximize proportionality with respect to different criteria for proportionality.

(b)       Different systems use different and often varied magnitudes and “tiers” of MMDs, the effect of which may be that, with respect to the overall allocation of seats in parliament, there is effectively a single nationwide MMD, although most members are elected from smaller (regional) MMDs.

(c)       Different systems impose different vote share thresholds (5% is common) that (small) parties must meet before they qualify for any seats under the apportionment formula.

(d)       Some systems allow parties to form alliances, by pooling their lists and vote shares in order to meet a threshold requirement.

(e)       Some systems allow voters to express preferences over candidates and possibly change the order of candidates on party lists.

Partly because list PR systems use explicit apportionment formulas, but mostly because it is practicable to use list PR in high magnitude MMDs (in which seats can be divided up “finely”), such systems can produce highly proportional results.

Note. Just as single-winner voting rules run into “problems” once there are three or more candidacies, apportionment formulas run into “problems” once there are three of more candidates. Here are some conditions we would probably like any apportionment formula to meet.

(a)       Intra-Election Monotonicity. If party A gets a bigger vote share in an election than party B, A should get no fewer seats than B does.

(b)       Inter-Election Monotonicity. If party A gets a bigger vote share in Election 2 than in Election 1 (the number of seats remaining constant), A should get no fewer seats in Election 2 than in Election 1.

(c)       Staying in Quota. The number of seats a party gets should be its vote share multiplied by the number of seats either rounded down or rounded up to the next whole number.

(d)       Seat Monotonicity. For given party vote shares, no party should lose seats if the total number of seats to be apportioned increased.

(e)       Vote Monotonicity. If party A’s vote increases relative to party B’s vote from Election 1 to Election 2, A’s seats relative to B’s seats should not decreases from Election 1 to Election 2.

Every (halfway reasonable) apportionment formula satisfies condition (a), but no formula can meet condition (b) without violating (a). Every quota method satisfies condition (c) but may violate (d) and (e). Every divisor method meets conditions (d) and (e) but may violate (c). No method satisfies all three conditions.

            It is well known that SMD-Plurality system can produce strange or unfortunate outcomes both at the district level (e.g., failure to elect Condorcet winners, spoiler effects, incentives for strategic voting, etc., as noted in Voting To Elect a Single Candidate) as well as at the national level (extreme disproportionality between party seats and vote shares, “reversal of winner” effects, etc., to be discussed in the handout on Votes and Seats in Districted Elections). PR elections likewise can produce strange or unfortunate outcomes. Here are a couple of examples.

(a)       Imagine a society divided among three or more relatively hostile ethnic, language, religious, etc., groups, none of which constitutes a majority of the population and each of which has formed its own political party which is the most preferred party of almost all group members. List PR is commonly recommended for such a “plural society.” However, suppose that an encompassing cross-group (i.e., multi-ethnic, multi-lingual, and/or secular) “alliance” party tries to form. Such a party would likely be the second preference of almost every voter but the first preference of almost none. If so, the “alliance” party is the Condorcet winner among political parties but likely will win few if any seats in parliament under PR.

(b)       A threshold requirement under PR produces an extreme discontinuity between seats and votes in the vicinity of the threshold. For example, given the 5% threshold and parliament size of about 650 in Germany, a party that gets 4.99% of the votes wins no seats, whereas a party that gets 5.01% wins about 32 seats. If a party is predicted to win less than 5% of the votes, many of its normal supporters may defect and “strategically” vote for their more preferred larger party. On the other hand, if a major and minor party form an alliance (such as the CDU+FDP or SPD+Greens in Germany), the major party may urge some of its voters to vote “strategically” for its minor party partner (so as to help it meet the 5% threshold). In the absence of such strategic maneuvers, a PR system with a threshold can produce a “reversal of winners” (in the manner of the U.S. Electoral College). For example, in the next German general election voters will (almost certainly) be choosing between two prospective coalition governments: the incumbent center-left SPD+Green coalition and the opposition CDU+FDP coalition. Suppose the vote shares are as follows: CDU 47%, SPD 43%, Greens 5.5%, and FDP 4.5%. The opposition coalition wins 51.5% of the popular votes, but the incumbent coalition wins a majority of seats because the opposition’s votes are “unstrategically” split between its two parties, with the result that the FDP falls below the 5% threshold and fails to win any seats.

Mixed Systems

            Many countries use mixed electoral systems. Most notably, in Germany half members of the Bundestag (parliament) are elected from SMDs on the basis of Simple Plurality, while the other half are elected by list PR that is (in effect) applied on a national basis. Thus each German voter cast two votes: a candidate-oriented vote for a local SMD candidate and a party-oriented vote for the national PR election. (Prior to 1953, a German voter cast only one vote for a local candidate listed on the ballot by party affiliation [as candidates are listed in the U.S.], and this voted also counted as the party-oriented vote in the PR election. But since 1953, German voters have been able to “split” their votes by supporting a candidate of one party in the SMD election and a different party in the PR election.)

            In Germany, the national PR election is corrective (or compensating) in its effect — that is, the seats in the national PR election are allocated so as to produce an overall distribution of seats in parliament that is proportional to the national PR party vote shares (above the 5% threshold). However, in some countries, the local SMD and national PR components of the electoral system operate with independent (or parallel) effects, with the result that the overall allocation of seats in parliament among parties may deviate considerably from their national vote shares.