Transcript Properties for a proportional electoral system

The Mathematics of Games:
Strategies, Cooperation and Fair
Division Theory
An Equitable Electoral System for the
Congress of Deputies
Prof. Dr. Victoriano Ramírez-González
University of Granada (Spain)
[email protected]
Seville, march 26th, 2011
OUTLINE
1. Introduction to electoral systems
2. Properties of an electoral system
3. Discordant allocations: Some illustrative examples
4. Proposal of a proportional electoral system. Empirical
applications to the cases of:
1. Spain,
2. Italy, Greece, Sweden, Germany.
Properties for a proportional electoral system
Introduction to electoral systems
• Size of the Parliament
– No problem in designing a E.S. It can have 300, 500,…seats.
• Constituencies
– Tradition.
– Geographic limitations.
– Gerrymandering is important when there are uninominal districts, but it
is not relevant if the total number of seats of the political parties
depends on their total number of votes.
Properties for a proportional electoral system
Introduction to electoral systems (cont.)
• Representation of political parties
– Sometimes it is calculated by applying a proportional method in each
constituency and, when doing so, discordant allotments frequently
emerge.
– In other cases the representation of political parties depends on the total
number of votes of each party. We can cite several examples, such as
Germany, Mexico, Greece and Italy (but with different criteria for each
country).
Properties for a proportional electoral system
Introduction to electoral systems (cont.)
• Thresholds
– Continuous thresholds are not oftenly used. I consider it is better not setting
thresholds or change.
o Classical thresholds imply obtaining a minimal number of votes or a minimum
percentage of votes. Hence:
• If the minimal is small, then the threshold provide non-practical consequences.
• If the minimal is large, unfair results can be obtained. For example, a change of
one vote can lead to a change in a big number of seats.
– E.g. In Italy, a difference of one vote between two parties leads to a
change of more than 60 seats from one party to another party.
• Therefore, classical thresholds are not logical.
o Moreover, a threshold is continuous if a change of one vote leads to a new
allotment which does not differ more than one seat from the previous allotment, for
any of the political parties.
Properties for a proportional electoral system
Hamilton Electoral Method: I
• Alabama paradox
(Firstly, to each political party the integer part of their exact proportion (quota) is
assigned. Next, the distribution is completed by assigning an additional seat to the political parties with greater
remainder)
Hamilton-12
Votes
Quota
Seats
A
433000
4.33
4
B
340000
3.4
3
C
240000
2.4
2
D
142000
1.42
2
E
45000
0.45
1
Hamilton-14
Votes
Quota
Seats
A
433000
5.05
5
B
340000
3.97
4
C
240000
2.8
3
D
142000
1.66
2
E
45000
0.53
0
Properties for a proportional electoral system
Hamilton Electoral Method: I
• Inconsistency
A
425000
4.25
4
B
135000
1.35
1
C
40000
0.40
1
Hamilton-2
Votes
Quotas
Seats
B
135000
1.54
2
C
40000
0.46
0
Hamilton-6
Votes
Quota
Seats
Properties for a proportional electoral system
Electoral methods obtained via optimization
•
We can find a method that minimizes the difference between the vectors of
quotas and allocations. We must use a norm for measuring the difference
between two vectors.
– With the norm:
n
q  a 1   qi  ai
i 1
– With the norm:
– With the norm:
qa 2 
qa

n
2
(
q

a
)
 i i
i 1
 Max qi  ai 
i 1,...n
– With other norm from an inner product
•
We can use other objetive functions. Such as:

vi 
 Min 
Max
ai  0,  ai  h  i 1,..., n ai 
Properties for a proportional electoral system
Huntington Methods
• The exact proportionality is:
vi v j
 ,
ei e j
ei e j
 ,
vi v j
vi * e j
ei
 v j , etc.
• Exactness is not possible. We can choose one of the equalities
and find a method that minimizes the difference between any
two political parties
D
ei e j

vi v j
D' 
ei  1 e j  1

vi
vj
Properties for a proportional electoral system
Divisor Methods
• If we Multiply the votes by a factor k appear fractions. How are
the fractions rounded to integers?
• Example if V = ( 90, 130, 360 ) and k = 0.01 we have the
fractions:
k V = ( 0.90,
0
1
1.30,
2
3.60 )
3
4
5
6
Threshold for rounding: 0.8, 1.4, 2.4, 3.1, 4.8, 5.2, ….
0

1

2

3
4

5
6
Rounding: 1, 1, 4. To assign 6 seats this is the solution, but whether to
allocate only 5 seats then we have to decrease k.
Properties for a proportional electoral system
Some Divisor Methods
•
Jefferson (d’Hondt). Rounding down .
The thresholds are: 1, 2, 3, 4, 5, 6, …
•
Webster (Sainte-Laguë). Rounding to the nearest whole number
•
The thresholds are: 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, …
•
Adams. Rounding up
The thresholds are: 0, 1, 2, 3, 4, 5, 6, …
Properties for a proportional electoral system
Jefferson method (or d’Hondt method)
• Example: To allot 24 seats
Votes   990, 430, 400, 270, 180, 80, 50 
Quota   9.9, 4.3, 4.0, 2.7, 1.8, 0.8, 0.5 
Hondt  Votes *0.0113 
 11.18, 4.86, 4.52, 3.05, 2.03, 0.90, 0.57   0
 11, 4,
4,
3,
2,
0,
0
• Lower quota.
• Penalizes the fragmentation of the political parties.
• Benefit the large political parties.
Properties for a proportional electoral system
Webster method (Sainte-Laguë method)
• Example: To allot 24 seats
Votes   990, 430, 400, 270, 180, 80, 50 
Quota   9.9, 4.3, 4.0, 2.7, 1.8, 0.8, 0.5 
Webster  Votes *0.01 
  9.9, 4.3, 4.0, 2.7, 1.8, 0.8, 0.5  
 10, 4,
4,
3,
2,
1, 0 
• It is impartial
Properties for a proportional electoral system
Adams method
• Example: To allot 24 seats
Votes   990, 430, 400, 270, 180, 80, 50 
Quota   9.9, 4.3, 4.0, 2.7, 1.8, 0.8, 0.5 
Adams  Votes *0.009  
  8.9, 3.8, 3.6, 2.4, 1.6, 0.7, 0.45  
  9, 4,
4,
3,
2,
1,
1
• Benefits small parties. In fact, It is not used to allocate seats
to parties. It can be used to allocate seats in the constituencies
• Cambridge Compromise: 5+Adams
Properties for a proportional electoral system
Criteria for choosing an electoral method
•
Desirable properties: Exactness, lower quota, impartial, monotonous, consistency,
punish schisms.
Hamilton
Adams
Webster
Hondt
Exacness
Si
Si
Si
Si
Lower Quota
Si
No
No
Si
Impartial
Si
No
Si
No
Monotonous
No
Si
Si
Si
Consistency
No
Si
Si
Si
Punish Schisms No
No
No
Si
d’Hondt is one of the most recommended methods for allocating seats to
parties. Webster should be used when impartiality is very important.
Properties for a proportional electoral system
Properties for an electoral system: I
• Applying acceptable methods of apportionment (consistency, no
paradoxes, exactness, etc.)
– Divisor methods (in general).
– Jefferson for allocating seats to the different political parties.
– Webster when impartiality is required.
Properties for a proportional electoral system
Properties for an electoral system: II
• Representativity (global and local)
– Large proportionality. For example, more than 95% with the
usual indexes to measure it.
– Equity. Two political parties with a similar number of votes must
be allocated an equal or almost equal number of seats.
– Important regional or local parties must obtain representation.
Properties for a proportional electoral system
Properties for an electoral system: III
• Governability
– Bonus in the representation of the winner party.
• Continuity
– Application of continuous methods to transform votes into seats.
– Application of continuous thresholds.
Properties for a proportional electoral system
Why Governability?
• Are both representativity and governability mutually self-excluding?
– No, it is possible
governability.
to
obtain
large
representativity
• A country must:
– Be well represented.
– Enjoy governance.
Properties for a proportional electoral system
and
Governance in the current electoral systems
• The vast majority of electoral systems.
• Proportional electoral systems with plenty of small or median
constituencies (many countries).
• Electoral laws (e.g. Italy, Mexico, Greece).
• Large thresholds.
• Exceptions: Israel, Netherlands, Estonia (only one constituency and
small or null threshold).
Properties for a proportional electoral system
U.K. 2010-Election
U.K. 2010-Election
Political party
% votes
Conservative
36.1
Labour
29.0
Liberal
23.0
Democrat
UKIP
3.1
BNP
1.9
SNP
1.7
Green
1.0
Sinn Fein
0.6
Democratic
0.6
Unionist
Plaid Cymru
0.6
SDLP
0.4
Other parties
2.0
100.00
Seats
306
258
57
0
0
6
1
5
8
3
3
3
650
Properties for a proportional electoral system
Some current bonus for the winner
• Italy, 2008:
– Il PDL
37.64% votes
44.08% seats
• Germany, 2005:
– SPD
34.25% votes
40.67% seats
• Spain, 2008:
– PSOE
43.20% votes
48.28% seats
43.90% votes
53.33% seats
• Greece, 2009:
– PASOK
• Netherlands, 2010
– VVD
20.49% votes
20.67% seats
Fragmentation: 31 – 30 – 24 – 21 – 15 – 10 – 10 – 5 – 2 - 2
Properties for a proportional electoral system
Threshold: Proportionality
40
30
20
10
10 000
20 000
30 000
40 000
Properties for a proportional electoral system
Usual threshold (non-continuous)
50
40
30
20
10
10 000
20 000
30 000
40 000
Properties for a proportional electoral system
Continuous threshold
50
40
30
20
10
10 000
20 000
30 000
40 000
Properties for a proportional electoral system
Comparison Usual (non-continuous) vs
Continuous thresholds
50
40
30
20
10
10 000
20 000
30 000
40 000
Properties for a proportional electoral system
Representativity
• A good representativity involves that an electoral system must
meet the following properties:
– Local representativity (i.e. representation of the most voted parties).
– Global representativity (i.e. high proportionality).
– Equity. Two political parties with a similar number of votes must be
allocated an equal or almost equal number of seats.
• Usually several (sometimes even all) of these requirements are
not verified. WHY DOES THIS HAPPEN?
Properties for a proportional electoral system
Many constituencies and thresholds:
Discordant apportionments
When an electoral system is designed in a country, the State is usually
districted into a high number of constituencies.
The size of such constituencies is a function of the number of inhabitants
in the country:
• Sometimes proportional to its population.
• Sometimes, small constituencies are overrepresented (e.g.
Spain).
In the election, the seats of each constituency are normally allocated in
proportion to the votes that political parties (or coalitions) receive.
Properties for a proportional electoral system
Many constituencies and thresholds:
Discordant apportionments (cont.)
So, political parties receive seats in proportion to their votes in each
constituency. But the total number of seats received by the political
parties is not guaranteed to be proportional to the respective total
votes.
There are electoral systems, with higher degrees of complexity and
fairness, yielding proportionality between total votes and total seats, like
in Germany. In other cases discordant apportionments frequently arise.
Properties for a proportional electoral system
Many constituencies in several countries
Examples:
Country
Constituencies
Italy
27 and Estero
Chile
60
Argentina
24
Colombia
32
Brazil
27
Spain
52
Etc.
Properties for a proportional electoral system
Discordant apportionments
Italy, 2008-Election
Party
Votes
Seats
La Sinist.
1.093.415
0
La Destra
862.043
0
MPAS
410.487
8
Partito S.
347.923
0
Partito C.
202.382
0
Svp
147.666
2
Properties for a proportional electoral system
Discordant apportionment
Chile, 1997-Election
Party
P. Comunista de Chile
P. Radical Social-D.
Votes
393,523
179,701
Seats
0
4
Properties for a proportional electoral system
Discordant apportionment
Argentina, 2005-Election
Party
Afirm. para una Rep. Igualitaria
Alianza Propuesta Republicana
Partido Unidad Federalista
Alianza Frente Nuevo
Alianza Frente Justicialista
Others
Votes
1,215,111
1,095,494
Seats
8
9
394,398
349,112
146,220
2
3
4
2,916,851
0
Properties for a proportional electoral system
Discordant apportionment
Colombia, 2002-Election
Party
Radical Change
Coalition Coal
Votes
316,5160
235,3390
Seats
7
11
http://pdba.georgetown.edu/Elecdata/Col/dip02.html
Properties for a proportional electoral system
Discordant apportionment
Brazil, 1994-Election
Party
Brazilian Social-Democracy Party (PSDB)
Liberal Front Party (PFL)
Votes
6,350,941
5,873,370
Seats
62
89
Workers' Party (PT)
Republican Progressive Party (PRP)
5,859,347
4,307,878
49
52
http://pdba.georgetown.edu/Elecdata/Brazil/legis1994.html
Properties for a proportional electoral system
Discordant apportionment
Spain, 2008-Election
Party
IU
CiU
Votes
969.946
779.425
UPyD
PNV
ERC
CC
306.079
306.128
298.139
212.543
Seats
2
10
1
6
3
2
Properties for a proportional electoral system
The usual apportionment problem
Constit. 1
Constit. 2
Constit. 3
Constit. 4
Party 1
Party 2
Party 3
Size
v11
v21
v31
v41
v12
v22
v32
v42
v13
v23
v33
v43
n1
n2
n3
n4
Total number of seats for all the political parties =
Lottery?
Properties for a proportional electoral system
O.K.
O.K.
O.K.
O.K.
Is it possible to meet all the properties
mentioned before?
Yes, it is possible to design electoral systems verifying:
» High proportionality and representativity.
» Bonus for the winner (governability).
» Continuity.
» Etc.
Properties for a proportional electoral system
How?
By allocating the seats to the political parties in several stages and
several levels.
First, we will show how it can be done for the case of Spain. The
procedure can be applied to any country whose constituencies are not
very small-sized.
If the constituencies are uninominal-district type (e.g. U.K.) or very
small (e.g. Chile) we can use a complementary regional list.
Properties for a proportional electoral system
Properties of the current electoral system
in Spain
Acceptable methods. Hamilton’s method is used in order to allocate
the 350 seats of the Parliament to the constituencies. Consequently,
we must replace this method by Webster’s method.
Governability. Yes
Continuity. Yes
Representativity
Local. Yes
Global. No
Equity. No
(NOTE: This is a common situation in many countries)
Properties for a proportional electoral system
Keeping governability and getting
representativity in Spain
• Representativity
– Allocate part of the seats to the political parties according to their
local results (in the constituencies). (Allotment R1)
– Allocate another part of the seats to the political parties in
proportion to their total votes. (Allotment R2)
• Governability
– Allocate the remaining seats rewarding to the winner party
(Allotment R3)
• Continuity
– It is obtained by using a continuous function to transform votes
into seats.
Properties for a proportional electoral system
First stage: R1 Allotment to the political parties
Similar to the current allotment:
Application of Jefferson’s method in each of the
52 constituencies, to allot 350 seats.
2008-ELECTION
Party
PSOE
PP
IU
CiU
EAJ-PNV
UPyD
ERC
BNG
CC-PNC
CA
NA-BAI
Total
VotEs
11.289.335 (45,6%)
10.278.010 (41,5%)
969.946 (3,92%)
R1
168 (48,0%)
152 (43,4%)
4 (1,14%)
779.425 (3,15%)
12 (3,43%)
306.128
306.079
298.139
212.543
174.629
68.679
62.398
24.745.311
(1,24%)
(1,24%)
(1,20%)
(0,86%)
(0,70%)
(0,28%)
(0,25%)
4
1
4
2
2
(1,14%)
(0,29%)
(1,14%)
(0,57%)
(0,57%)
1 (0,29%)
350
Properties for a proportional electoral system
Second stage: R2 Allotment to the political parties
We apply Jefferson’s method to allot 370 seats in proportion
to the total votes.
No party can receive less seats than those obtained in the
R1 allotment.
2008-ELECTION
Party
PSOE
PP
IU
CiU
EAJ-PNV
UPyD
ERC
BNG
CC-PNC
CA
NA-BAI
Total
Votes – quota 370
11.289.335 - 168.8
10.278.010 - 153.7
969.946 - 14.5
779.425 - 11.6
306.128 - 4.6
306.079 - 4.6
298.139 - 4.5
212.543 - 3.2
174.629 - 2.6
68.679 - 1.0
62.398 - 0.9
24.745.311 - 370.0
R1
168
152
4
12
4
1
4
2
2
1
350
+
R2
2
3
10
170
155
14
12
4
4
4
3
2
1
1
370
3
1
1
20
Properties for a proportional electoral system
Third stage: R3 Allotment to the political parties
We apply Jefferson’s method to allot 400 seats in proportion
to the square of the total votes.
No party can receive less seats than those obtained in the
R2 allotment. R3 is the final allotment to the political parties
2008-ELECTION
Party
PSOE
PP
IU
CiU
EAJ-PNV
UPyD
ERC
BNG
CC-PNC
CA
NA-BAI
Total
Votes – Quota 400
11.289.335
10.278.010
969.946
779.425
306.128
- 182.5
- 166.2
- 15.7
- 12.6
-
4.9
306.079 - 4.9
298.139 - 4.8
212.543 - 3.5
174.629 - 2.8
68.679 - 1.1
62.398 - 1.0
24.745.311 - 400.0 .
R2
+
R3
170
155
14
12
24
6
194
161
14
12
4
4
4
3
2
1
1
370
4
30
4
4
3
2
1
1
400
Properties for a proportional electoral system
Bi-proportional allotment
PSOE PP
194 161
Madrid
Barcelona
Valencia
Sevilla
Alicante
Málaga
Murcia
Cádiz
Vizcaya
Coruña
Asturias
Las Palmas
Islas Baleares
S. C. Tenerife
Pontevedra
Zaragoza
Granada
48
42
20
15
14
12
11
10
10
10
9
9
9
9
8
8
8
IU CiU PNV UPyD ERC BNG CC
14 12
4
4
4
3
2
1.401
1.737
164
0
0
132
1.309
470
155
547
0
5
599
770
46
0
0
626
339
58
0
326
246
289
50
0
182
27
0
CA N-Bai
1
1 .
0
0
0
0
0
184
0
0
0
0
10
3
0
0
0
0
0
13
0
0
0
0
0
0
9
0
0
0
0
5
0
0
0
0
Properties for a proportional electoral system
0
0
0
The representation in the regions
•
When the size of the constituencies is not uniform, as in the Spanish case,
the seats corresponding the small political parties are allocated according to
the biproportional method in the large constituencies.
•
For example, UPyD has obtained 131.242 votes in Madrid and 172.000 in
the other 51 constituencies. The 4 seats corresponding to UPyD are
allocated in Madrid. Then, the 40.261 votes obtained by UPyD at the 8
constituencies belonging to the region of Andalucia provide a UPyDrepresentative out of Andalucia.
•
Similarly, IU has obtained more than 50.000 in the Basque Country, but IU
has not got any seats in the Basque Country.
•
Nowadays, the regions in Spain has high importance. If the constituencies
would be the regions in Spain, UPyD would obtain one seat in Andalucia
and IU would obtained one seat in the Basque Country.
Properties for a proportional electoral system
How to obtain a correct representation in the
regions by using the current constituencies?
Answer: By using biproportional allotment twice.
In the first stage we apply biproportional allotment to know the number
of representatives belonging to each political party in each region.
For this allocation we use the total votes of the parties in the
regions.
In the second stage, we apply biproportional allotment into each
region to determine the number of seats assigned to each political
party in each constituency.
A double biproportional allotment must be applied in all Federal
States to obtain a good result.
Properties for a proportional electoral system
How many seats in R1, R2 and R3?
• R2 allotment must obtain high proportionality (near to 100%).
• Then, if we use near to 8% of the total seats for the governability and
the winner party has 40% of votes (more or less) we can expect a
proportionality of 95% (or more). Therefore a number of seats
equivalent to the 8% (of the total seats) to get governability can
represent a very realistic election in many cases (for example, in
Spain). Then R1+R2=92%.
• How many seats in R2? Different answer for different electoral
systems. Each country must be analyzed. When there are many
constituencies with large or median size, a percentage between 5%
and 10% can be enough.
• We can investigate other countries.
Properties for a proportional electoral system
Italy
The Italian Constitution establishes the constituencies and their sizes.
The Italian Electoral Law sets the total number of representatives for
the political parties.
Biproportional allotment is the only method able to yield an allotment
compatible with the Italian Constitution and Electoral Law.
The current Italian allotment for the Camera (as well as in the previous
2006 election) does not verify the Italian Law. In addition, the
electoral system for the Italian camera is neither continuous, nor
representative, etc.
The same technique applied to Spain before gives the next result for
Italy:
An Equitable Electoral System for the Congress of Deputies
Italy, 2008-Election with RG: 537+20+60
Threshold: -50.000 votes
Party
Il Popolo
Partito D.
Lega N.
Unione C.
Di Pietro
La Sinist.
La Destra
MPAS
Partito S.
Partito C.
Sinistra C.
SVP
Total
Votes
13.628.865
12.092.998
3.024.522
2.050.319
1.593.675
1.093.415
862.043
410.487
347.923
202.382
162.974
147.666
Quota R1
232.26 234
206.08 201
51.54
46
34.94
25
27.16
14
18.63
8
14.69
3
7.00
3
5.93
0
3.45
0
2.78
0
2.52
3
537
R2
234
201
46
25
18
12
9
4
3
1
1
3
557
R3 Current
277
272
218
211
46
60
25
36
18
28
12
0
9
0
4
8
3
0
1
0
1
0
3
2
617
617
Properties for a proportional electoral system
Greece, 2009-Election with RG: (R1+R2=270)+30
Threshold: -50.000 votes
Party
Votes
Quota Repres. Govern. Current
Pasok
3.012.373
N.D.
2.295.967
KKE
517.154
LAOS
386.152
Syriza
315.627
Ecologist Green 157.449
134.87
102.79
23.15
17.29
14.13
7.77
126
95
19
14
11
5
+30 = 152
95
19
14
11
5
160
91
21
15
13
0
Total
.
300.00
270
300
300
6.700.722
An Equitable Electoral System for the Congress of Deputies
Germany, 2005-Election: 299+251+50
RG-1: threshold = - 100.000 votes.
RG-2: threshold = -200.000 votes.
Party
Votes
%Votes Quota
SPD
CDU
FDP
Die Linke
GRÜNE
CSU
NPD
REP
GRAUE
FAMILLIE
16.194.665 34.58
13.136.740 28.05
4.648.144
9.92
4.118.194
8.79
3.838.326
8.20
3.494.309
7.46
748.568
1.60
266.101
0.57
198.601
0.42
191.842
0.41
207.47
168.29
59.55
52.76
49.17
44.76
9.59
3.41
2.54
2.46
Total
47.054.698 100.0
600.00
District
Current RG-1
RG-2
145
106
0
3
1
44
0
0
0
0
215
174
61
54
50
46
0
0
0
0
241
158
54
48
45
44
7
1
1
1
244
160
54
48
44
44
6
0
0
0
600
600
600
299
An Equitable Electoral System for the Congress of Deputies
An Equitable Electoral System for the
Congress of Deputies
Thank you very much for your attention!
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