Biproportional Apportionment Methods: Lessons Learned from … · 2011. 5. 31. · Votes are given as a n ×m matrix (parties = rows, ... vali" ver" ur tskriva"!! Sainte-Lagu deilarah

Biproportional Apportionment Methods:Lessons Learned from

Two Real-Life Benchmark Studies

Martin Zachariasen

Department of Computer ScienceUniversity of Copenhagen

May 31, 2011

Purpose of This Talk?

Discuss the issue of quality in biproportional apportionment

Summarize some of the main findings of two fairly recent simulationstudies on biproportional apportionment

Provide some recommendations for a new electoral formula for theSwedish Riksdag

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Notation and Basic Assumptions

Electoral system with n parties and m districts (or constituencies)

Votes are given as a n ×m matrix (parties = rows, districts = columns)

Assumed goals of biproportional apportionment for the Swedish Riksdag:

1 District sizes should be proportional to district population sizes(fairly important?)

2 Total party seat numbers should be proportional to total vote counts(very important)

3 Seat numbers within districts should be proportional to vote counts(important)

4 Seat numbers within parties should be proportional to vote counts(fairly important?)

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Biproportional Apportionment: Three Sub-Problems

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Uppskot um n!ggjan útrokningarhátt

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!! Tinggátt: Flokkar vi" minni enn 3% av atkvø"unum fáa ongan tingsess

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!! Tvílutfalsligi Sainte-Laguë deilarahátturin n!ttur

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Allocate seats to districts

Allocate seats to parties

Allocate seats within districts/parties

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WHAT IS QUALITY ININ BIPROPORTIONALAPPORTIONMENT?

...or how do we distinguish a good apportionment from a less good one?

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WHAT IS QUALITY ININ BIPROPORTIONALAPPORTIONMENT?

...or how do we distinguish a good apportionment from a less good one?

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Hard and Soft Measures of Quality

Should distinguish between properties that

must be fulfilled (axioms or hard constraints)

should be fulfilled (soft constraints)

A number of hard constraints (axioms) were proposed by Balinski andDemange in 1989 — constraints that only divisor-based methods are ableto fulfill

In this talk I focus on two types of soft constraints:

1 Properties related to the quota (or fair share)

2 Properties related to the ranking of parties within districts

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Four Definitions of Quota

Overall quota: Vote count related to the overall vote total

+ intuitively appealing, correct overall sum

– does not necessarily result in the required district or party sums

District quota: Vote count related to the district vote total

+ backward ”compatible”, correct district sum

– does not necessarily result in the required party sums

Party quota: Vote count related to the party vote total

+ correct party sum

– does not necessarily result in the required district sums

Fair share quota: Output of the continuous matrix scaling problem

+ all sums correct

– less intuitive

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+ correct party sum



+ all sums correct

– less intuitive

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+ correct party sum



+ all sums correct

– less intuitive

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+ correct party sum



+ all sums correct

– less intuitive7

Classical Quality Measures Related to Quota

Gallagher index: Basically the L2-distance between seat numbers andquota (where both have been scaled to a sum of 100)

Quota breaches: Number of seat assignments where the seat number ismore than one unit away from quota

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Classical Quality Measures Related to Quota

Gallagher index: Basically the L2-distance between seat numbers andquota (where both have been scaled to a sum of 100)

Quota breaches: Number of seat assignments where the seat number ismore than one unit away from quota

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Discordant Seat Allocations (or Seat Reversals)

No biproportional apportionment method can guarantee elementaryproportionality within districts or within parties — except in trivial cases

Consider some district:

Discordant seat allocation: Party i has more votes than party j , butparty i has received less seats than party j

Strongest party discordant: Party i has received most votes, butparty i has received less seats than some other party j

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Discordant Seat Allocations (or Seat Reversals)

No biproportional apportionment method can guarantee elementaryproportionality within districts or within parties — except in trivial cases

Consider some district:

Discordant seat allocation: Party i has more votes than party j , butparty i has received less seats than party j

Strongest party discordant: Party i has received most votes, butparty i has received less seats than some other party j

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FIRST SIMULATION STUDY

P. Zachariassen and M. Zachariasen. A Comparison of Electoral Formulae for the

Faroese Parliament (The Løgting). In Mathematics and Democracy. Recent

Advances in Voting Systems and Collective Choice, pp. 235–251, Springer, New

York, 2006.

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The Faroese Parliament (Løgting) - 2004 Election

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Løgtingsvali! 2004

Atkvø!ur Kvota Atkvø!utal b"tt vi! me!altali av

atkvø!um fyri tingmannin. Dømi: 11771 / ( 31711 / 32) ! 11,9

Tingsessir Sambært núverandi útrokningarskipan

NO EY NS SS VA SA SU Tils.!

A. Fólkafl. 1434 1217 531 2041 532 259 516 6530

B. Sambandsfl. 602 2012 663 2638 663 105 818 7501

C. Javna"arfl. 741 1254 489 2267 373 326 1471 6921

D. Sjálvst#risfl. 264 529 60 527 39 22 20 1461

E. Tjó"veldisfl. 849 1254 582 3076 263 285 581 6890

H. Mi"fl. 148 466 78 790 79 26 74 1661

K. Hin Stuttligi Fl. 29 144 89 432 24 29 747

Tils. 4067 6876 2492 11771 1973 1023 3509 31711

1,45 1,23 0,54 2,06 0,54 0,26 0,52 6,59

0,61 2,03 0,67 2,66 0,67 0,11 0,83 7,57

0,75 1,27 0,49 2,29 0,38 0,33 1,48 6,98

0,27 0,53 0,06 0,53 0,04 0,02 0,02 1,47

0,86 1,27 0,59 3,10 0,27 0,29 0,59 6,95

0,15 0,47 0,08 0,80 0,08 0,03 0,07 1,68

0,03 0,15 0,09 0,44 0,02 0,00 0,03 0,75

4,10 6,94 2,51 11,9 1,99 1,03 3,54 32

2 1 1 2 1 7

2 1 2 1 1 7

1 1 2 1 2 7

1 1

1 1 1 3 1 1 8

1 1 2

0

4 7 3 10 2 2 4 32

For each (party,district): Votes, overall quota, seats

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Focus of Study: Quality of Marginals for DifferentApportionment Methods

The electoral formula from 2004 and many alternatives work ”inside out”— compute the inside of the matrix first, which leads to the resultingdistrict and party sums

The focus of the simulation study was to look at the effect on the”outside” of the matrix (marginal sums) for eight different apportionmentmethods

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Experimental Approach

In this talk

Simulation results based on 2004 election, ±50% uniform perturbation oneach vote count, 100 random instances

Algorithmic parameters

Threshold: 1/27 threshold for obtaining adjustment seats

Constrained: district size lower bounds from the 2004 electoral systemare enforced

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Experimental Results: Gallagher Index

Mean Min Max1 Current electoral formula 3.31 1.90 5.602 Current electoral formula with std.round. 2.83 1.46 5.353 Hylland method with threshold 2.86 1.46 5.504 Constrained Balinski with threshold 2.68 1.73 5.505 Hylland method without threshold 2.30 0.73 5.236 Constrained Balinski without threshold 1.82 0.73 2.647 Balinski (biproportional divisor method) 1.82 0.73 2.648 Controlled Rounding 1.92 0.73 2.83

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Experimental Results: Quota Breaches

Mean Min Max1 Current electoral formula 0.60 0 32 Current electoral formula with std.round. 0.30 0 23 Hylland method with threshold 0.34 0 44 Constrained Balinski with threshold 0.22 0 45 Hylland method without threshold 0.16 0 26 Constrained Balinski without threshold 0 0 07 Balinski (biproportional divisor method) 0 0 08 Controlled Rounding 0 0 0

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SECOND SIMULATION STUDY

S. Maier, P. Zachariassen, and M. Zachariasen. Divisor-Based Biproportional

Apportionment in Electoral Systems: A Real-Life Benchmark Study. Management

Science, 56:373–387, 2010.

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Focus of Study: Quality of Matrix for Divisor-BasedApportionment

The focus of the simulation study was to look at the effect on the ”inside”of the matrix for the divisor-based biproportional method (Balinski)

Simulation results based on a range of European parliaments,±20% uniform perturbation on each vote count, 10000 random instances

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Benchmark Instances

Electoral Short House Num. of District sizes Parties Partiessystem name size districts Avg Min–Max participating rec. seats

Faroese Parliament Fo04 32 7 4.6 1− 12 7 7Bavarian Parliament Bav03 180 7 25.7 17− 57 15 3Zurich City Council ZhCC06 125 9 13.9 10− 19 14 8Danish Folketing Dk05 175 17 10.3 2− 22 9 8Zurich Canton Parliament ZhCP07 180 18 10.0 4− 16 11 9Swiss National Assembly Ch03 200 26 7.7 1− 34 43 15Italian National Assembly It06 617 26 23.7 3− 44 13 13

Strongest party constrained biproportional rule (SPC): The strongest partyin a district always receives a seat (instances Fo04SPC and Ch03SPC)

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Experimental Results: Gallagher Index (District)

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4SP

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C06

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P07

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Experimental Results: Quota Breaches (District)

Fo0

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4SP

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C06

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P07

Ch0

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Ch0

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White = no quota breaches, Black = more than four quota breaches24

Experimental Results: Discordant Seat Allocations

Fo0

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White = none, Black = more than four districts25

Experimental Results: Strongest Party Discordant

Fo0

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C06

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ZhC

P07

Ch0

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Ch0

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It06

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White = none, Black = more than four districts26

Conclusions

Marginal sumsBiproportional divisor-methods provide the best results — basically perfectproportionality (if that is the goal)

Inside of the matrixBiproportional divisor-methods perform well wrt. the Gallagher index andquota breaches

Discordant seat allocations cannot be avoided — however, other methodsthat strive for the best possible proportionality on the national level dohave the same problem

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Conclusions

Marginal sumsBiproportional divisor-methods provide the best results — basically perfectproportionality (if that is the goal)

Inside of the matrixBiproportional divisor-methods perform well wrt. the Gallagher index andquota breaches

Discordant seat allocations cannot be avoided — however, other methodsthat strive for the best possible proportionality on the national level dohave the same problem

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Recommendations concerning the Swedish Riksdag

If you decide to keep the current system with adjustment seats:

Increase the number of adjustment seats

Decrease the number of districts

Make the districts more homogenous in size

Reduce the threshold

Use divisor-based methods with (pure) standard rounding everywhere

If you wish (or dare) to scrap the current system:

Use a biproportional divisor-method

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Use a biproportional divisor-method28

Documents

Biproportional Apportionment Methods: Lessons Learned from … · 2011. 5. 31. · Votes are given as a n ×m matrix (parties = rows, ... vali" ver" ur tskriva"!! Sainte-Lagu deilarah