The Logic of Different Electoral Formulae With Examples

Introduction to political science – Seminar: The logic of various electoral formulae (with examples)Roxana Marin, PhDc., T.A.

Plurality voting (first-past-the-post)

Scenario: 1 seat to fill; 250,000 votes cast; 5 candidates

Candidate No. of votes A 31,000B 49,000C 110,000D 17,000E 43,000

The candidate who gathers a „relative majority” of the votes cast wins the seat

=>

Candidate No. of votes received Percentage of votes receivedA 31,000 12.4%B 49,000 19.6%C 110,000 44%D 17,000 6.8%E 43,000 17.2%

=> C gathers a „relative majority” of 44% of the total no. of votes cast => C wins the seat

Single majority voting with Borda count


Candidate No. of votes Preference Points received (1st method, when n = no. of candidates)

Points received (2nd

method, when formula is calculated as n-1)

Points received (3rd

method, when formula is calculated as 1/n)

A 31,000 4 2 (n-3) 1 (n-4) 0.25 (1/4)B 49,000 2 4 (n-1) 3 (n-2) 0.50 (1/2)C 110,000 1 5 (n) 4 (n-1) 1 (1/1)D 17,000 5 1 (n-4) 0 (n-5) 0.20 (1/5)E 43,000 3 3 (n-2) 2 (n-3) 0.33 (1/3)

Majority voting with Condorcet method


the Condorcet winner = the candidate preferred by a majority of voters, in every imaginary one-on-one contests

no. of possible pairings, for N candidates = 12N (N−1)

1


When N = 5 => = 125 (5-1) = 10 possible pairings

Pairing WinnerA (31,000) vs. B (49,000) BA (31,000) vs. C (110,000) CA (31,000) vs. D (17,000) AA (31,000) vs. E (43,000) EB (49,000) vs. C (110,000) CB (49,000) vs. D (17,000) BB (49,000) vs. E (43,000) BC (110,000) vs. D (17,000) CC (110,000) vs. E (43,000) CD (17,000) vs. E (43,000) E => C wins every other possible candidate => C wins the seat

Proportional representation (PR) with d’Hondt formula (highest averages)

Scenario: 3 seats to fill; 250,000 votes cast

Party Initial no. of votes (s= 0) s = 1 s= 2 s= 3A 31,000 15,500 10,333 7,750B 49,000 24,5000 16,333 12,250C 110,000 55,000 36,666 27,500D 17,000 8,500 5,666 4,250E 43,000 21,500 14,333 10,750

d’Hondt quotient = Vs+1 , where V – no. of votes one party receives, and s = number of seats allocated

to that party

Who wins? Select the first three highest values in the table. Turn them into three seats.

Alternative voting (majority/ plurality voting)

Scenario: 1 seat to fill, 4 candidates

the distribution of votes and preferences is the following:

Percentage of votes

31% 12% 5% 6% 12% 22% 12%

Distribution of preferences

CBAD

BDAC

ADCB

ADBC

ACDB

DBCA

BADC

Ist round: NO majority A = 23% BUT D – eliminated => D’s votes go to B (because B is the B = 24% 2nd preference of those voting D)

C = 31%

2


D = 22%

IInd round => A = 23% NO majority, BUT A – eliminated => A’s votes go to C B = 24 + 22 = 46% (for D – already eliminated)

C = 31%

IIIrd round => B = 46%

C = 31 + 23 = 54% - MAJORITY => C wins the seat.

Single transferable vote (STV), with Droop formula

q=( valid votes castseats ¿fill+1¿)+1 - Droop formula

Scenario: 250,000 votes cast, 3 seats to fill

The votes are distributed as in the example with PR, but the preferences are the following:

Candidate Initial no. of votes Second preferenceA 31,000 D, EB 49,000 C, DC 110,000 BD 17,000 A, BE 43,000 A

=> q=(250,0003+1 )+1=62,501 – minimum no. of votes one has to gather in order to win the seat

Ist round => C meets the quota and wins 1 seat => his surplus = 47,499 – transferred to others, according to preference (let’s say equitably): D gets 23,750 and E gets 23,749

IInd round => A = 31,000

B = 49,000

C – win

D = 17,000 + 23,750 = 40,750

E = 43,000 + 23,749 = 66,749

=> E meets the quota and wins 1 seat => his surplus = 4,248 – transferred to the 2nd preference: A gets 4,248

=> NO elimination, since one candidate met the quota

IIIrd round => A = 31,000 + 4,248 = 35,248

B = 49,000

3


C – win

D = 40,750

E – win

=> no candidate met the quota => D – eliminated => his votes – transferred to others, according to preference (let’s say randomly): A gets 24,875 and B gets 15,875

IVth round => A = 35,248 + 24,875 = 60,123

B = 49,000 + 15,875 = 64,875

=> B meets the quota and wins 1 seat => C, E, and B win the 3 seats

Single Non-transferrable vote (SNtV)

Scenario: 3 seats to fill, 25,000 votes cast; 5 candidates, representing 3 parties

Candidate Party Initial no. of votesA Z 31,000B X 49,000C Y 110,000D Y 17,000E X 43,000

According to the largest number of votes received, candidates C, B, E win the 3 seats. This means that: party X wins 2 seats, while party Y wins 1 seat.

=>

Party No. of votes received Percentage from the total no. of votes

Seats won

X 92,000 36.8% 2Y 127,000 50.8% 1Z 31,000 12.4% 0

PR with the largest remainder voting (Hare quota)

q= total no .of votes casttotalno .of seats allocated

Scenario: 3 seats to fill; 250,000 votes cast, distributed as follows:

Candidate Initial no. of votesA 31,000B 49,000C 110,000D 17,000E 43,000

4


=> q=250,0003

=83,333, representing 33.33% of the total number of votes cast – minimum no. of

votes one has to gather in order to win the seat

Ist round: the seats are allocated to each party by dividing the no. of votes each party received to the Hare quota =>

Party Initial no. of votes Seats allocatedA 31,000 0.372B 49,000 0.588C 110,000 1.32D 17,000 0.204E 43,000 0.516

Ist round => C gathered 1.32 seats => C wins 1 seat => his remainder = 0.32

IInd round: the remainders are calculated =>

Party Initial no. of votes Seats allocated after Ist round

Remainders

A 31,000 0 0.372B 49,000 0 0.588C 110,000 1 0.32D 17,000 0 0.204E 43,000 0 0.516

=> the 2 seats are allocated to the parties gathering the largest remainders => B (0.588) and E (0.516) win the seats

=> C, B, E win the seats

5

Documents

The Logic of Different Electoral Formulae With Examples