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) vramirez@ugr.es 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: qa 2 qa 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! vramirez@ugr.es