The Penrose’s Law and Decision-Making Processes in the Council of the European Union. Case Study: the Impact of the Square Root’ Rule on Formation of Romania’s Coalitions in the Council of Ministers
Constantin Chiriac The Bucharest Academy of Economic Studies [email protected]
Abstract. Enlargement of the European Union has led to much discussion on the need to reform institutions so that the various inequalities between the member states to be removed. Such Treaties of Nice and Lisbon do not grant to citizens of member the same influence in the decision-making in the Council of Ministers. One of the solutions proposed to remedy this problem is to establish a voting scheme that gives each country a share of votes proportional to the square root of the population and determining the optimum of a threshold, known as the Jagiellonian Compromise. Starting from the analysis of Penrose square root law in the decision-making Council of Ministers, the article pre- sents a case study on the influence of approximation by rounding the share of votes on the coalition’s formation in the case of Romania.
Keywords: Council of Ministers; decision-making processes; Jagiellonian Com- promise; coalition’s formation.
JEL Codes: C70, D70, D72, D74.
REL Codes: 20C, 7L, 9A. The Penrose’s Law and Decision-Making Processes in the Council of European Union
17 18 Theoretical and Applied Economics “is atthesametable”withGermany,which of Malta,with400,000inhabitants,which representative mustdealwiththesituation systems. AninstitutionsuchastheEU compromise betweentheNiceandLisbon The systemproposedwouldbeafair of Ministers,regardlesscountryorigin. same influenceondecisionsoftheCouncil more suitabletogiveeverycitizenthe Penrose, consideringthatthissystemis suggested wasbasedonthesquarerootof Alternative votingsystemthatthey the TreatyofNiceandLisbon. to theshortcomingsofvotingsystem Council ofMinisters.Theydrawattention Penrose’s methodofallocatingvotesinthe recommending theuseofsquareroot letter toEUMemberStatesgovernments, Democratic Europe),andhavesentajoint formed agroupcalledSDE(Scientistfor Criticism treatiesofNiceandLisbon ehrad 63444 42 35 -17 2.80 Belgium 10.511-17 3.50 3242 2.80 -17 Portugal 10.5693.38 3251 3.39 Greece 11.125 4.21 2.88 -17 Netherlands3335 4042 16.334 -14 3.48 4.15 Romania 21.610 4649 4.85 Poland 38.157 5.71 -11 Spain 43.758 6177 6.44 Italy 58.751 6615 6.55 -5 UK 60.4216.90 7665 France 62.886 11.66 +23 8.49 9.02 7.99 7773 +9 Germany 82.438 7930 +6 8.27 9.46 9080 8.69 8.10 +7 Member States Qualified majorityvotingsystemfortheCouncilofMinisters:Citizen-basedJagiellonian In 2007,researchersfrom10countries 1. JagiellonianCompromise. Population (mil.) 2007 2007 (mil.) Compromise “JC”andnegotiatedDoubleMajority ”DM” Weight JC Jagiellonian compromise. compared tothesystemadoptedincase population oftheMemberStates. proportional tothesquarerootof the numberofvotesallocatedtobe root lawofPenrosewithonemajority,and countries. Suchasystemisbasedonsquare to ensureequalityinrepresentationforall decision-making systemwithintheCouncil, systematic campaigninfavorofanother accepted thenewsystem,itbegana Although, finally,in2004,Polandhas has generatedmuchtensionintheadoption. power forsomecountries(Poland),which the population.Thissystemleadstolossof criteria: thenumberofMemberStatesand qualified majoritydoublethattohavetwo should bechangedinasystemwith Council membersagreedthatthedecision at theUnitedNations. higher. ItisasifwecompareChinatoMalta has 82,000 Power JC Table 1showsthepowerofvote Following criticismofNice’system, (1)
,000 inhabitants,thatis211times Double Majority Power Deviation Table 1 % Population Power Power Deviation Member States Weight JC (mil.) 2007 JC(1) Double Majority % Czech 10.251 3202 3.34 2.77 -17 Hungary 10.076 3174 3.31 2.74 -17 Sweden 9.047 3008 3.14 2.63 -16 Austria 8.265 2875 3.00 2.53 -16 Bulgaria 7.718 2778 2.90 2.47 -15 Denmark 5.427 2330 2.43 2.19 -10 Slovakia 5.389 2321 2.42 2.18 -10 Finland 5.255 2293 2.39 2.17 -9 Ireland 4.209 2052 2.14 2.04 -5 Lithuania 3.403 1845 1.92 1.95 +2 Latvia 2.294 1515 1.58 1.81 +15 Slovenia 2.003 1415 1.47 1.78 +21 Estonia 1.344 1160 1.21 1.69 +40 Cyprus 0.766 875 0.91 1.63 +79 Luxembourg 0.459 678 0.71 1.59 +124 Malta 0.404 636 0.66 1.58 +139 TOTAL 492.881 95.916 100 100 Quota 59.058 61.57 Source: Pukelsheim, 2007, p. 15.
Under the Nice Treaty, each Member randomly, and is equal to the share of State is assigned a number of votes that possible winning coalitions. reflect the population. A legislative For the Lisbon Treaty, approximately proposal passes the Council, in the Treaty 12.9% out of all possible coalitions are of Nice, if three conditions are leading to a condition regarding the accomplished: number of votes, the number approval of randomly chosen legislative and percentage of state population, the proposals, and for the Treaty of Nice this system being called triple majority. index is equal to 2.1%. Calculations show Lisbon Treaty states that a legislative that by introducing the double majority proposal pass if only two criteria are met: decision-making reduce the danger of 55% of Member States “for” representing blocking process from the EU levels. On 65% of the EU population. Additionally, a the other hand, the figures show that the minority block includes at least four Nice system, which is currently, extension members who could block a proposal that to the EU-27 was not fear and confirming a majority may agree. Most experts all that the decision system will be blocked believed that the main drawback of the (Kurpas, Schonlau, 2006, pp. 2-5). Another Treaty of Nice was the low efficiency of shortcoming of the system from Nice is that decision making. This performance, it is necessary to apply simultaneously the measured by Coleman power to act index, three criteria in calculating the qualified calculate the probability that the Council majority, replaced by the Treaty of Lisbon, approved various issues, selected which uses only two criteria. The Penrose’s Law and Decision-Making Processes in the Council of European Union
19 20 Theoretical and Applied Economics in atransparentwayeasily implemented decision-making processes? Canbedone to havethesamepower toinfluence in whicheverycitizenofanyMemberState equal value,isviolatedinbothsystems. voting powerofacitizenanystateis principle ofdemocracy,throughwhichthe These calculationsshowthatthebasic being requiredasincasetheTreatyofNice. State, complexmathematicalcalculations potential votingpowerofeachMember a simplecitizenisnotabletocalculatethe the systemisnotmoretransparentbecause only twocriteriafixesanotherbasicflaw: of Lisbon.Thefactthatthissystemuses regarded asavisionarycritiqueoftheTreaty system in1952.Hisfindingscanbe deficiency byanalyzingdoublemajority drawn byPenrose,whodiscoveredthis manner, thesameconclusionwasalso medium sizedMemberStates.Inanironic consequences leadtodecreasedpowerof increase. Thecombinedeffectsofthese number ofstatessupportingaproposalto a disproportionatepowerprovidedthatthe population, whilethesmallstateswillhave will gainmorefromdirectcontactwiththe two desires.Inthissystemthelargestates Lisbon Treatyapparentlyreflectsonlythose citizens. Doublemajoritybroughtbythe Member Statesandtheprincipleofequal principles: theprincipleofequality compromise betweentwofundamental of Ministers? mal decision-makingfortheCouncil One maybedesignedavotingsystem 2. Issquarerootsolutionforopti- All votingsystemsintheEUwerea of degressiveproportionality a mathematicalapplicationoftheprinciple tionate tothesquarerootofpopulation,is from theacademicworld. The proposalreceivedconsiderablesupport the Councilregardlessofcountryorigin. that everycitizenisequallyrepresentedin can beshownthatthesquarerootruleensures (Kauppi, Widgren,2004,pp.221-229).It scheme maybejustifiedintermsoffairness however, usingitasabasisforvoting although thislawmayseemstrange, (Penrose’s law). be proportionaltothesquarerootofN if thevotingpowerofeachcountryshould system ofCouncilwouldberepresentative square rootrule.Thustheindirectvote the votingpowerofacitizendrops Penrose hasshownthatinsuchelections, a citizenfromsmallsizedcountry. on theoutcomeofelectionprocessthan with alargepopulationhaslessinfluence +1, thenthesimplycitizenfromacountry can begainedbyasimplemajorityof50% elections inastatewithNpopulation,which voting powerpotential.Ifweconsiderdirect every citizenofanystatehasthesame system ofanidealrepresentationinwhich who developedtheprincipleofavoting enlargements? efficiently andtomatchinfutureEU be asinglestate).Note that adegressive tion tothepopulation”(as ifEuropewould association ofstates)and “votesinpropor- vote” (asifEuropewouldbeasimple between twoextremities:„acountry-one The solutionofsharingvotes,propor- Kauppi andWidgrenshowedthat A partialanswerwasgivenbyPenrose, (2) andis system is used in German Parliament, by The problem is how votes are allocated which is designated a number of and how to set the threshold to achieve a representatives for each Land. But the certain distribution of power. One solution assertion that “each country’s voting power would be that the share of votes to be should be proportional to the square root proportional to the square root and then to of the population” does not solve the found the optimal threshold that will problem entirely. Kirsh showed that the produce maximum transparency of the square root tells us just how power should system, i.e. the system in which voting be distributed between countries (2004, power of each Member State will be pp. 2-3). It is not clear how this law should approximately equal to the share of votes be implemented in practice in terms of share as is observed in Figure 1. (Slomczynski, of the vote. Zyczkowski).
18 16 14 equal w eights w eights prop to the square root of population 12 w eights prop to population 10
8 6 4 2 0 UK Italy Malta Spain Latvia Ireland Poland Finland France Estonia Cyprus Austria Greece Belgium Portugal Bulgaria Slovakia Slovenia Romania Czech R Hungary Sweden Denmark Lithuania Germany Luxemburg Netherlands
Figure 1. The weights attributed to the Member States: proportionally to population, proportionally to the square root of population, and uniformly
Source: Slomczynski, Zyczkowski, 2008, p. 11.
Thus Penrose’s law should be respected Ministers 27 optimal threshold would be and the potential influence of each citizen 61.57%. For an arbitrary number of in the Council would be the same, which Member States the best threshold q can be would lead to compliance with the principle calculated according to the formula: of representativeness. Such a system should + + also be transparent, simple and objective 1 N1 N 2... N m q = 1+ + + and would not create any advantage or 2 N1 N 2 ... N m disadvantage for any country. where N represents the population of This system was called the Jagiellonian i country i.
Compromise. Thus for the EU Council of The Penrose’s Law and Decision-Making Processes in the Council of European Union
21 22 Theoretical and Applied Economics sum ofsquaresbetween dependence ofthesquarerootresidual a doublemajority.Figure2 illustrates the Member Statesasincaseofthesystemwith does notdecreasewiththenumberof as thesolution conditions isincreasedefficiency. threshold thatchangedependingonfuture electoral bodysize.Thisshowsthata the optimalthresholddecreaseswith optimal threshold.Calculationsshowthat increase itwillresultinachangeofthe the thresholdR: Banzhaf indexandtheshare ofthevote, Jagiellonian Compromise.DistributionofvotesandvotingpowerinCMwiththreshold61.5% uebug049678 0.709 Malta 0.404 Luxembourg 0.459 677.86 Cyprus 0.766635.88 0.659 Estonia 1.344 0.909 875.45 Slovenia 2.003 1.208 1159.61 1.478 Latvia 2.294 1415.40 Lithuania 3.403 1.918 Ireland 4.209 1844.81514.79 1.578 Finland 5.255 2051.59 2.138 Slovakia 5.389 2.388 2292.51 2.418 Denmark 5.427 2321.46 2.428 2.898Bulgaria 7.718 2329.69 2.77 Austria 8.265 Sweden 9.047 2.9982875.05 Hungary 10.076 3.138 3007.95 3.308 Czech 10.251 3174.36 Belgium 10.511 3.378Portugal3.3383201.73 10.569 3242.13 4.210 3.388 Greece 11.125 3251.09 Netherlands 3.479 16.334 4041.56 3335.4 Romania 21.610 4.851 Poland 38.157 4648.68 Spain 43.758 6.445 6177.14 Italy 58.751 6.906 6615 UK 60.421 France 62.886 7664.97 Germany 82.438 7.994 9.441 8.273 7937.18 7771.30 9079.54 8.104 Effectiveness ofthesquarerootrule Jagiellonian Compromiseisalsoknown Since thenumberofMemberStateswill tt ouain qaero fpplto Banzhaf index % Square root of population Population State Pop.-q% Distribution of votes and the voting power in the Council of Ministers. . Basedonthe σ Jagiellonian Compromise –threshold of 61.5 % normalized the shareofvoteand votesareequal. optimal valueforthreshold isreachedwhen population mention theoneinTreatyofNice. is higherthantheTreatyofLisbon,notto measure byBanzhafindex(seeTable2),which 16.43%. Thevotingshareandpower Werner, 2009).Inadditiontheefficiencyis between showed thatthemaximumrelativedeviation with anerrorlessthan0.00005‰.Analysis is reachedfor Since theminimumvalueofthisfunction β 0i σ (3) and 2 wegetathresholdofq=61.5%, = R ∑ β x N 27 = qoi 1 opt islessthan0.14%(Langner, β , thenonecansaythatthe x ( R ) − ∑ y N = 1 N N x y 2 Table 2 0.016 0.014 0.012 0.01 0.008 0.006 27 0.004 Ropt = 0,615 0.002 0 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.615
Figure 2. The cumulative residual σ between the voting weight and power for EU-27 countries as dependent on the value of the threshold R= 0.615
Source: Slomczynski, Zyczkowski, 2007, p. 11.
→ ∞ = (−1/ 2) When N then the value of critical P(x1, x2...xN ) CN (x1, x2..xN ) for any threshold tends to 50% in accordance with N = ≥ ∑ xi 1 Penrose’s law. Table 3 shows the average critical xi 0, and where the normalization i=1 threshold in terms of N, the number of electoral constant is expressed by the Euler gamma body obtained as a result of a symmetric Dirichlet N N (− ) distribution, with density given by the formula function C = Γ( )n 2 . N 2
N Average optimal threshold Ropt as a function of the number of states N Table 3 No. of Member 10 12 14 16 18 20 22 24 26 27 States N Ropt N 66.0% 65.8% 64.6% 64.4% 63.4% 63.1% 62.6% 62.0% 61.4% 61.57%
The results above indicate that for a proportion equal to yes and no. This given number of states N, the choice of the approach requires that the people who voted weighting in proportion to square root of to be in an even number, otherwise the voter the population and close to the threshold will not be in such a position. The 27 R opt ensures optimum representativeness probability of compliance leaves the system of voting power as each country hypothesis that each member of the becomes proportional to the square root of population are equally likely to vote in two population and voting power of individual ways (yes/no). a citizen is almost identical. − N −1 N 1 1 Square root is derived from the P = N −1 = t ie 2 probability that an individual voter to 2 ()− influence the outcome of a binary vote (yes/ = N 1 ! ≈ 2 1 2 π ()− N −1 − N 1 N −1 no). A voter has some direct influence only 2 N 1 2 if the rest of the people N voted in a The Penrose’s Law and Decision-Making Processes in the Council of European Union
23 24 Theoretical and Applied Economics advantages: different countries; hence theabsenceofdisadvantages for the thresholdarenotset arbitrarily)and may becalledsinglemajoritysystem; Figure 3. Ease ofdecision-makingprocessforvarious Majority threshold We concludetheJagiellonian Source: Source: % population % Members - is objective(theshareofthevoteand is simply,basedononecriterionand 50-61.5 12.9 50-61.5 11.7 70-70 1.6 1.1 70-60 2.2 1.6 70-50 2.5 1.7 60-70 4.8 5.6 60-60 8.5 11.0 50-70 9.2 8.3 60-50 11.1 15.1 55-65 12.9 12.5 50-60 21.9 19.8 55-55 23.0 19.9 50-50 35.8 31.9 Nisa 2.8 Nisa 1.8 EffectivenessoftheNiceTreaty,TreatyLisbonandJagiellonianCompromise 10 15 20 25 30 Langner,Werner,2009,p. 15. 0 5 Baldwin, Widgren,2007,p.6. 51
53 thresholds
55 UE-27
57
59
61 UE-27+Turkey Probability Probability Passage Passage 61,5% 63
65 Table 4 65%
67
69 share ofvotes(Ratzer,2006). possibility ofgivingeachstateanequal proportional distributioncancelsthe allocated toastateis the numberofstatechanges, allyouhave share ofvotes; each stateisroughlyproportionaltothe power potential; the MemberStateshassamevoting height of a fixedarea:thepointismaximum represents anormaldistributionandithas Formally, theapproachtovalueof standard deviationvariation considering abinomialdistributionwith 71 The resultcanbeviewedby 73 is easilyadaptableandextensible: if transparency: thevotingpowerof representativeness: everycitizenof 75
77 N 1
79 − 1
81 . Iftheshareofvotes
83 JC DC17 DC15 263 Lisabona 255 a Nis N , sucha N 4 − N-1 1 . to do is to reallocate votes in accordance proportional to the square root of the with law and as the square root threshold; country’s population according to the
is more effective than the treaties of N v = i Nice and Lisbon; i n formula N for all i countries. moderate conservative: it does not ∑ j j=1 lead to a dramatic transfer of power from The number of votes obtained will not the current situation. be rational numbers, but some 3. Approximation by rounding approximations have to be made. These when applying Penrose’s law in deci- approximations, in the absence of precise sion-making processes of the Council rules, will be given to the accuracy of of Ministers and the impact on coali- floating point representation, in which the tions formation of Romania performance of computer use. Zyczkowski and Slomczynski were The application of square root law in rounded voting weights to four decimal the EU Council decision-making processes places obtained. has consequences for the coalitions Case study will show that this rounding formation of a Member State. We show that may have effects on the optimal threshold the approximation by rounding the share R and the minimum value of σ, which is of votes, and no exact calculation, which produced by the voting system of the must return a Member State, may have a square root of Penrose. major effect on the optimal threshold R and Approximation of k-digit rounding the minimum value of σ can be reached means that the share of votes is done by k by the voting system of Penrose’s square decimals, where the share of votes is root law. n = ∑vi 1 To make a voting system “fair” in the normalized such , where vi is the i=1 Council of Ministers, Banzhaf index should share of votes if the country i. To show calculated according to the formula data rounding errors, the minimization of N β = i σ, we consider four countries as follows: i n
∑ N j j=1 Population and share of votes for four countries for all countries, where Ni is the population of country i. Practical implementation of Table 5
Current νi this index, thus calculated, is difficult and Country Ni νi no. (3 decimals) represents a problem of multi-dimensional 1 România 21610213 4648.678 0.4648 0.465 optimization. To estimate an ideal 2 Denmark 5427459 2329.6907 0.2329 0.233 3 Ireland 4209019 2051.589 0.2051 0.205 distribution of voting power, Penrose 4 Cyprus 766414 875.450 0.0875 0.088 suggested that the share of votes to be The Penrose’s Law and Decision-Making Processes in the Council of European Union
25 26 Theoretical and Applied Economics where thecountry’svote is computedaccordingtoformulas: threshold coalitions this tableitcouldbereadthewinning are formedinaccordancewithTable6.From possible coalitionscouldbeobtainedwhich the shareofvotes,votesforall Cyprus, i.e. formed fromRomania,Denmark,Irelandand only onewinningcoalitionofcountries the deviation and means thatforallthresholds0 all countries and couldbecalculatedwiththeformula coalitions are calculatedinTable7. Onlythresholds η Coalitions Coalitions i of states states of =2 ,,, .0 2,3 1,2,3,4 1.000 0.438128014 Coalitions andtheshareofvotes,without 2,3,4 0.525673088 4 0.087545074 0.087545074 4 2,3,4 0.525673088 0.292704012 3,4 1,3,4 0.757571875 0.320514149 2,4 0.464867863 1,2,4 0.785382012 1 1,2,3 0.902995877 1,4 0.552412937 1,4 0.552412937 Nici 0.205158938 3 1,3 0.670026801 0.000 0.232969075 una 2 1,2 0.697836938 Using fourdecimalapproximationfor For examplefor0.90299 σ β ∑ forallthethresholds j i n = i B 1 =1,2,3,4.Deviation 2 i B Coalitions Coalitions . of states states of j = i ∑ j n is critical, = η 1 i n ≤ j ≤ R 1thereis ≤ i Table 6 ∑ 1and . This ν i η i R σ , in Table8. coalitions andtheshareofvotesareshown rounding. Inthiscasethepossible σ and thishappensinthe0.5256< even iftheoppositionhasmorevotes. of 0.5meansthatacoalitioncanbesuccessful for R 0.087545074 0.205158938 2 2 2 0 0.108664 00.108664 2 2 0.087545074 2 0.205158938 10.082441 1 3 0.205158938 3 0.232969075 20.062765 2 2 0.232969075 4 0.292704012 10.060114 1 3 0.292704012 5 0.730832026 00.092898 2 2 0.730832026 6 0.464867863 10.030906 3 3 0.464867863 5 0.438128014 20.060498 2 2 0.438128014 6 0.552412937 10.030906 3 3 0.552412937 5 0.525673088 00.092898 2 2 0.525673088 6 0.670026801 10.060114 1 3 0.670026801 5 0.697836938 20.062765 2 2 0.697836938 4 0.757571875 10.082441 1 3 0.757571875 3 0.785382012 00.108664 2 2 0.785382012 2 0.902995877 1 1.000 0.902995877 1 1 1 0.126844 iscalculatedfromtheshareofvotes,with Coalitions Coalitions of states states of 0.000 0.087545074 1 1 1 1 0.126844 10.126844 1 1 1 0.087545074 0.000 ,,, .0 1 0.465 1 1,2,3,4 1.000 R > 2,3,4 0.526 None 0.000 3,4 1,3,4 0.758 0.293 2,4 1,2,4 0.785 0.321 2,3 1,2,3 0.903 0.438 Deviation fromthefairvoting,without The situationischangingverymuchif The lowestvalueof 1,4 0.552 4 1,4 0.552 0.088 3 1,3 0.670 0.205 2 1,2 0.698 0.233 ≥ Coalitions andtheshareofvotes, 0.5areshownasthelowerthreshold with 3decimalsrounding R ≤ ∑ ν rounding i η 1 Coalitions Coalitions of states states of η 2 σ min η 3 =0.030906 R η ≤ 4 0.5524. Table 7 Table 8 ∑ ν σ i σ Comparison with Table 6 shows that the The lowest value of min = 0.060114 coalition consisting of Denmark, Ireland and this happens in the 0.670 Notes (1) The share of the vote in the case of Jagiellonian 65% of its population are taken from Slomczynski Compromise is square root of population, rounded to and Zyczkowski, 2007, p. 12. Deviation from the nearest number. Thus we have in Malta = 635.9 Jagiellonian compromise is for the Malta (1.58-0.66)/ → 636. The threshold for qualified majority decisions 0.66 = +1.3939 → +139% respect the formula: (+95.916)/2=59058,47 → 59058. (2) See draft Constitution with regard to the European Since the share of Malta is 636/95916=0,006631 → Parliament. 0.66%, the relative index is 0,66. 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