1. Seville, march 26th, 2011 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

2. 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

3. 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

4. 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.)

5. Thresholds –Continuous thresholds are not oftenly used. I consider it is better not setting thresholds or change. oClassical 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. oMoreover, 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

6. 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) Properties for a proportional electoral system

7. Hamilton Electoral Method: I Inconsistency Properties for a proportional electoral system

8. 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: –With other norm from an inner product We can use other objetive functions. Such as: Properties for a proportional electoral system

9. Huntington Methods The exact proportionality is: Exactness is not possible. We can choose one of the equalities and find a method that minimizes the difference between any two political parties Properties for a proportional electoral system

10. 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, 1.30, 3.60 ) 0 1 2 3 4 5 6 Threshold for rounding: 0.8, 1.4, 2.4, 3.1, 4.8, 5.2, …. Properties for a proportional electoral system 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.

11. 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

12. Jefferson method (or d’Hondt method) Example: To allot 24 seats Lower quota. Penalizes the fragmentation of the political parties. Benefit the large political parties. Properties for a proportional electoral system

13. Webster method (Sainte-Laguë method) Example: To allot 24 seats It is impartial Properties for a proportional electoral system

14. Adams method Example: To allot 24 seats 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

15. 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

16. 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

17. 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

18. 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

19. Why Governability? Are both representativity and governability mutually self-excluding? –No, it is possible to obtain large representativity and governability. A country must: –Be well represented. –Enjoy governance. Properties for a proportional electoral system

20. 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

21. U.K. 2010-Election Properties for a proportional electoral system

22. 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 Greece, 2009: –PASOK43.90% votes53.33% seats Netherlands, 2010 –VVD20.49% votes20.67% seats Fragmentation: 31 – 30 – 24 – 21 – 15 – 10 – 10 – 5 – 2 - 2 Properties for a proportional electoral system

23. Threshold: Proportionality Properties for a proportional electoral system

24. Usual threshold (non-continuous) Properties for a proportional electoral system

25. Continuous threshold Properties for a proportional electoral system

26. Comparison Usual (non-continuous) vs Continuous thresholds Properties for a proportional electoral system

27. 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

28. 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

29. 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

30. Many constituencies in several countries Examples: Country Constituencies Italy 27 and Estero Chile 60 Argentina24 Colombia32 Brazil27 Spain52 Etc. Properties for a proportional electoral system

31. 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

32. Discordant apportionment Chile, 1997-Election Party Votes Seats P. Comunista de Chile 393,523 0 P. Radical Social-D.179,701 4 Properties for a proportional electoral system

33. Discordant apportionment Argentina, 2005-Election Party VotesSeats Afirm. para una Rep. Igualitaria 1,215,1118 Alianza Propuesta Republicana 1,095,4949 Partido Unidad Federalista 394,3982 Alianza Frente Nuevo 349,1123 Alianza Frente Justicialista 146,2204 Others 2,916,8510 Properties for a proportional electoral system

34. Discordant apportionment Colombia, 2002-Election Party Votes Seats Radical Change 316,5160 7 Coalition Coal235,339011 http://pdba.georgetown.edu/Elecdata/Col/dip02.html Properties for a proportional electoral system

35. Discordant apportionment Brazil, 1994-Election Party Votes Seats Brazilian Social-Democracy Party (PSDB) 6,350,94162 Liberal Front Party (PFL) 5,873,37089 Workers' Party (PT)5,859,34749 Republican Progressive Party (PRP)4,307,87852 http://pdba.georgetown.edu/Elecdata/Brazil/legis1994.html Properties for a proportional electoral system

36. Discordant apportionment Spain, 2008-Election PartyVotes Seats IU969.946 2 CiU779.425 10 UPyD306.079 1 PNV306.128 6 ERC298.139 3 CC212.543 2 Properties for a proportional electoral system

37. The usual apportionment problem Party 1Party 2 Party 3 Size Constit. 1 v11 v12 v13 n1 O.K. Constit. 2 v21 v22 v23 n2 O.K. Constit. 3 v31 v32 v33 n3 O.K. Constit. 4 v41 v42 v43 n4 O.K. Total number of seats for all the political parties = Lottery? Properties for a proportional electoral system

38. 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

39. 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

40. 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

41. 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

42. Similar to the current allotment Similar to the current allotment: Application of Jefferson’s method in each of the 52 constituencies, to allot 350 seats. First stage: R1 Allotment to the political parties Properties for a proportional electoral system 2008-ELECTION PartyVotEs R1 PSOE11.289.335 (45,6%)168 (48,0%) PP10.278.010 (41,5%)152 (43,4%) IU969.946 (3,92%) 4 (1,14%) CiU779.425 (3,15%) 12 (3,43%) EAJ-PNV306.128 (1,24%) 4 (1,14%) UPyD306.079 (1,24%) 1 (0,29%) ERC298.139 (1,20%) 4 (1,14%) BNG212.543 (0,86%) 2 (0,57%) CC-PNC174.629 (0,70%) 2 (0,57%) CA68.679 (0,28%)- NA-BAI62.398 (0,25%) 1 (0,29%) Total 24.745.311 350

43. 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. Second stage: R2 Allotment to the political parties Properties for a proportional electoral system 2008-ELECTION PartyVotes – quota 370 R1 +R2 PSOE11.289.335 - 168.81682170 PP10.278.010 - 153.71523155 IU969.946 - 14.5 410 14 CiU779.425 - 11.6 12 EAJ-PNV306.128 - 4.6 4 4 UPyD306.079 - 4.6 13 4 ERC298.139 - 4.5 4 4 BNG212.543 - 3.2 21 3 CC-PNC174.629 - 2.6 2 2 CA68.679 - 1.0-1 1 NA-BAI62.398 - 0.9 1 1 Total24.745.311 - 370.035020370

44. 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 Third stage: R3 Allotment to the political parties Properties for a proportional electoral system 2008-ELECTION Party Votes – Quota 400 R2+R3 PSOE11.289.335 - 182.517024194 PP10.278.010 - 166.2155 6161 IU 969.946 - 15.714 CiU 779.425 - 12.612 EAJ-PNV 306.128 - 4.94 4 UPyD 306.079 - 4.94 4 ERC 298.139 - 4.8 4 4 BNG 212.543 - 3.53 3 CC-PNC 174.629 - 2.82 2 CA 68.679 - 1.11 1 NA-BAI 62.398 - 1.01 1 Total24.745.311 - 400.0.37030400

45. PSOE PP IU CiU PNV UPyD ERC BNG CC CA N-Bai 194 161 14 12 4 4 4 3 2 1 1. Madrid 48 1.401 1.737 164 0 0 132 0 0 0 0 0 Barcelona 42 1.309 470 155 547 0 5 184 0 0 0 0 Valencia20 599 770 46 0 0 10 3 0 0 0 0 Sevilla 15 626 339 58 0 0 13 0 0 0 0 0 Alicante 14 Málaga 12 Murcia 11 Cádiz 10 Vizcaya 10 Coruña 10 Asturias 9 326 289 50 0 0 9 0 0 0 0 0 Las Palmas 9 Islas Baleares 9 S. C. Tenerife 9 Pontevedra 8 Zaragoza 8 Granada 8 246 182 27 0 0 5 0 0 0 0 0 Bi-proportional allotment Properties for a proportional electoral system

46. 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 UPyD- representative 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

47. 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

48. 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

49. 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

50. Italy, 2008-Election with RG: 537+20+60 Threshold: -50.000 votes Party Votes QuotaR1R2R3 Current Il Popolo 13.628.865 232.26 234 234277272 Partito D. 12.092.998 206.08201201218211 Lega N. 3.024.522 51.54 46 46 46 60 Unione C. 2.050.319 34.94 25 25 25 36 Di Pietro 1.593.67527.16 14 18 18 28 La Sinist. 1.093.41518.63 8 12 12 0 La Destra 862.04314.69 3 9 9 0 MPAS 410.487 7.00 3 4 4 8 Partito S. 347.923 5.93 0 3 3 0 Partito C. 202.382 3.45 0 1 1 0 Sinistra C. 162.974 2.78 0 1 1 0 SVP 147.666 2.52 3 3 3 2 Total537557617617 Properties for a proportional electoral system

51. Greece, 2009-Election with RG: (R1+R2=270)+30 Threshold: -50.000 votes Party Votes QuotaRepres. Govern. Current Pasok 3.012.373 134.87 126 +30 = 152160 N.D. 2.295.967 102.79 95 95 91 KKE 517.154 23.15 19 19 21 LAOS 386.152 17.29 14 14 15 Syriza 315.627 14.13 11 11 13 Ecologist Green 157.449 7.77 5 5 0 Total6.700.722300.00 270300300. An Equitable Electoral System for the Congress of Deputies

52. Germany, 2005-Election: 299+251+50 RG-1: threshold = - 100.000 votes. RG-2: threshold = -200.000 votes. PartyVotes%VotesQuotaDistrictCurrentRG-1RG-2 SPD16.194.665 34.58207.47145215241244 CDU13.136.740 28.05168.29106174158160 FDP 4.648.144 9.92 59.55 0 61 54 54 Die Linke 4.118.194 8.79 52.76 3 54 48 48 GRÜNE 3.838.326 8.20 49.17 1 50 45 44 CSU 3.494.309 7.46 44.76 44 46 44 44 NPD 748.568 1.60 9.59 0 0 7 6 REP 266.101 0.57 3.41 0 0 1 0 GRAUE 198.601 0.42 2.54 0 0 1 0 FAMILLIE 191.842 0.41 2.46 0 0 1 0 Total47.054.698 100.0600.00 299 600600600 An Equitable Electoral System for the Congress of Deputies

53. Thank you very much for your attention! vramirez@ugr.es An Equitable Electoral System for the Congress of Deputies