Karol yczkowski Jagiellonian University (Cracow) (joint work with - - PowerPoint PPT Presentation

Jagiellonian Compromise

vs Cambridge Compromise:

On square root and proportional representations Karol Życzkowski

Jagiellonian University (Cracow) (joint work with Wojciech Słomczyński) Cost meeting, Istambul, November 3, 2015

Mathematical problems related to governance in European Union

a) How to vote in the European Council ? b) How to allocate seats in the European Parliament ?

 a two-tier decision-making system:

 the Member States at the lower level  the European Union at the upper level

Voting systems for the Council of the European Union

Figure by Annick Laruelle

 a two-tier decision-making system:

 the Member States at the lower level  the European Union at the upper level

Voting systems for the Council of the European Union

The Council of the EU votes by a qualified majority voting: a decision of the Council is taken, if it is approved by a qualified majority

 A representative of each country has to

vote yes or no and cannot split his vote

 example: if 30 millions of Italians support a

decision, and 29 M are against, an Italian minister says yes (on behalf of 59 millions).

 Thus 30 M Italians can overrule 45 M Spa-

niards (+29 millions of opposing Italians)

 One person-one vote system would be

perfect ... if all citizens of each country had the same opinion…

Indirect voting in the Council

 A representative of a member state with a

population N goes to Brussels and says yes according to the will of the majority of his co-patriots...

 How many of them are satisfied, N or N/2 ?

(since the representative followed their will). Indirect voting in the Council

 A representative of a member state with a

population N goes to Brussels and says yes according to the will of the majority of his co-patriots...

 How many of them are satisfied, N or N/2 ?

(since the representative followed their will).

 We do not know! These numbers will be

different in each cases. Statistical approach is needed to compute the average and to prove that the difference satisfied - dissatisfied scales as ... Sqrt (N) ! Indirect voting in the Council

 power index - probability that the vote of a

country will be decisive in a hypothetical ballot measures the potential voting power natural statistical assumption: all potential coalitions are equally likely Penrose-Banzhaf index

How to measure voting power?

How to compute the Banzhaf index ?

(Banzhaf, 1965):number of players n # of coalitions

2n

# of winning coalitions w # of coalitions with i-th player 2n-1 # of wining coalitions with i-th player Xi wi # of coalitions, for which the vote of Xi is critical ci := wi – ( w – wi

) = 2·wi – w

Banzhaf index = ci / 2n-1

probability that vote of Xi will be decisive

Penrose-Banzhaf index (normalised)

βi = ci / i ci (Penrose, 1946): pi = (1+ βi)/2 probability, that player Xi is going to winn

345 votes are distributed among 27 member states on a degressively proportional basis, e.g.:

 DE, FR, IT, UK

– 29 votes (weight)

 ES, PL

– 27 votes (weight), etc.



a) the sum of the weights of the Member States voting in favour is at least 255 (~73.9% of 345)



b) a majority of Member States (i.e. at least 14 out of 27) vote in favour



c) the Member States forming the qualified majority represent at least 62% of the overall population of the European Union

• ‘triple majority’

Treaty of Nice, 2001

a) ------



b) at least 55% of Member States (i.e. at least 15 out of 27) vote in favour



c) the Member States forming the qualified majority represent at least 65% of the overall population of the European Union c’) a blocking minority must include at least four Council members

• ‘double majority’

Treaty of Lisbon, 2007

(...) if two votings were required for every decision, one on a per capita basis and the other upon the basis of a single vote for each country, the system would be inaccurate in that it would tend to favour large countries. [Penrose, 1952]

• 2%
• 1%

0% 1% 2% 3% 4% 5%

Germany France U.K. Italy Spain Poland Romania Netherlands Greece Portugal Belgium Czech Republic Hungary Sweden Austria Bulgaria Denmark Slovakia Finland Irleland Lithuania Latvia Slovenia Estonia Cyprus Luxembourg Malta Differences in the Banzhaf index Constitution vs. NIce

Treaty of Nice vs treaty of Lisbon

(...) if two votings were required for every decision, one on a per capita basis and the other upon the basis of a single vote for each country, the system would be inaccurate in that it would tend to favour large countries. [L. Penrose, 1952]

Penrose square root law:

Voting power of a citizen in a country with population N is proportional to N -1/2

Bernoulli scheme for k=N/2 and p=q=1/2 + Stirling expansion gives probability

Square root weights - Penrose law

 this degressive system is distinguished by the

Penrose square root law (1952)

N

N 1

each citizen

• f each

country has the same potential voting power

 the ‘square root’ weights attributed to

Member States are proportional to the sides of the squares representing their populations

Square root weights - example

 The choice of an appropriate decision-taking

quota (threshold) affects both the distribution of voting power in the Council (and thus also the representativeness of the system) and the voting system’s effectiveness and transparency.

 Different authors have proposed different quotas

for a square root voting systems, usually varying from 60% to 74%.

 The optimal quota enables the computed voting

power of each country to be practically equal to the attributed voting weight.

Qualified majority threshold

Optimal threshold

EU-25: 62,0% EU-27: 61,6%

Jagiellonian Compromise W.S. & K.Z. 2004 square root weights

+

• ptimal quota

(mimimizing deviations from the Penrose rule that voting power of each state scales as square root of its population)

‘Nice’ / Jagiellonian Compromise / Lisabon

 it is extremely simple since it is based on a single criterion, and

thus it could be called a ‘single majority’ system;

 it is objective (no arbitrary weights or thresholds), hence

cannot a priori handicap any member of the European Union;

 it is representative: every citizen of each Member State has

the same potential voting power;

 it is transparent: the voting power of each Member State is

(approximately) proportional to its voting weight;

 it is easily extendible: if the number of Member States

changes, all that needs to be done is to set the voting weights according to the square root law and adjust the quota accordingly;

 it is moderately efficient: as the number of Member States

grows, the efficiency of the system does not decrease;

 it is also moderately conservative - it does not lead to a large

transfer of voting power relative to the existing arrangements.

Jagiellonian Compromise

 advocated or analysed by Laruelle, Widgrén (1998),

Baldwin, Berglöf, Giavazzi, Widgrén (2000), Felsenthal, Machover (2000-2004), Hosli (2000), Sutter (2000), Tiilikainen, Widgrén (2000), Kandogan (2001), Leech (2002), Moberg (2002), Hosli, Machover (2002), Leech, Machover (2003), Widgrén (2003), Baldwin, Widgrén (2004), Bilbao (2004), Bobay (2004), Kirsch (2004), Lindner (2004), Lindner, Machover (2004), Plechanovová (2004, 2006), Sozański (2004), Ade (2005), Koornwinder (2005), Pajala (2005), Maaser, Napel (2006), Taagepera, Hosli (2006)

 prior to the European Union summit in Brussels in June

2004, an open letter in support of square-root voting weights in the Council of Ministers endorsed by more than 40 scientists from 10 European countries

Square root weights

• support from academics

 Göran Persson (2000): Our formula has the advantage

• f being easy to understand by public opinion and

practical to use in an enlarged Europe […] it is transparent, logical and loyal.

 John Bruton (2004): Instead of double majority, we

could put in the Treaty a new, clear and automatic mathematical formula for allocating voting weights. Such a formula has been proposed by researchers in the London School of Economics. Their formula would allocate voting weights to countries on the basis of the square root of their population, rather than the number of population itself.

Square root weights

• support from politicians

Treaties are like roses and young girls. They last while they last.

Charles de Gaulle, Time, 12th July, 1963

Treaty of Lisbon (2007):

Double Majority voting system accepted for EU

however

(in Polish) English edition in preparation... The story described in

• ur popular

book

Every vote counts!

A walk through the world of elections

 How the optimal threshold q depends on

a) the number M of the members

• f the union?

b) the distribution of their population?

 W. Słomczyński and K. Życzkowski:

From a Toy Model to

the ‘Double Square Root’ Voting System Homo Oeconomicus 24, 381 (2007).

Qualified majority threshold II

EU-M 6 9 10 12 15 25 27 Ropt 73.0% 67.4% 65.5% 64.5% 64.4% 62,0% 61.4%

The value of the optimal quota qopt as a function of the number M of members of the EU.

Optimal quota – the normal approximation

     

 

   

k 1 i 2 i k k 4 k 1 i 2 i 2 2 2 2 k 1 i 2 i 2 k 1 i i 1 k 1 i i

) ( Assumption 1 w w : v , 2 ) ( w 1 2 1 : M) 1, (k 4 / 2 / 4 / 2 / ) ( function

• n

distributi cumulative normal standard

• ,

w 4 1 : , 2 1 w 2 1 : ) ( : ) ( , 2 : , , 1 : ) ( , , w ; , 1 w weights, voting

• ,

, 1 w

k n k k k n k n k k k k k q z M I i i M k

w q v

• v

e q m q w w m q w w m q q m m q z n q N z w M I card z n w q M k                                                                  

      

      

              

general result for arbitrary weights Voting power proportional to voting weight ! Feix, Lapelley, Merlin, Rouet 2007

Optimal quota –solution of the problem

   

% 9 . 15 ) 1 ( 1 ) ( ) ( increases M players

• f

number when the decrease, not does system the

• f

efficiency The N N 1 2 1 ) , , N ( get we ) ~ (w system voting Penrose for particular In (1979)) Taagepera (Laakso, players

• f

number effective

• w

1 : M where ), , , w ( 1 1 2 1 1 1 2 1 ) ( w 1 2 1 ) , , w ( , , w ; , 1 w weights, voting

• ,

, 1 w

k 1 i i k 1 i i 1 k k 1 i 2 i eff 1 k 1 i 2 i 1 * 1 k 1 i i

                                               

    

     n s M

• pt

k M n eff s M n M k

q A q A N q N w q M M M q w q q w q M k     

Special case: square root weights

• f Penrose

             

  

   M 1 i i M 1 i i M 1 i i k k

N N q M ..., , k , N N w 1 2 1 1 Optimal threshold for the Penrose weights

special case: square root weights wk

• f Penrose

and the

• ptimal quota

Optimal quota for random unions

• random choice of population
• critical quota are realisation dependent, but the

fluctuations around the mean are small

• average depends only on the number M of states
• the larger M the smaller the optimal quota qopt
• in limiting case M

 

the quota qopt tends to ½

M 10 12 14 16 18 20 22 24 26

<qopt >) (%)

65.5 65.2 63.6 63.2 62.9 62.2 61.7 61.2 60.6

Optimal threshold for a set of M random states

• M states, each with a random share xi of

the total population of the union, i Ni

• flat distribution in the probability simplex:

P(xi) ~  (1- i

xi), where xi =Ni / i Ni

• Jones 1991: average moments < i

(xi)s>

taken with s=1/2 & Stirling expansion imply the final result: optimal quota for a union of M random (typical) states:

• qopt

(M)= 1/2 + 1/(M1/2

Assumptions:

Optimal threshold for M random states

.

Why  in a voting problem?

Assumption of Gaussianhood and statistical approach as in the Buffon needle problem!

Result for in the case of M states with unknown (or varying) populations

Voting systems for the Council of the European Union

• C. Alvner

Brussels: European Parliament

(next elections in spring 2019)

Allocation of seats in the EU Parliament:

.How many seats for each state?

PM from a given state can split their votes (to represent opinion of the electorate) => In principle the seats should be distributed proportionally to the population of each state but …

Allocation of seats in the EU parliament:

.How many seats for each state?

PM from a given state can split their votes (to represent opinion of the electorate) => In principle the seats should be distributed proportionally to the population of each state but … population of Germany (82M) is 200 times larger than Malta (0.4M)...

Allocation of seats in the EU Parliament:

.LLegal rules: (Lisabon Treaty)

a) total number of seats equals S=751 (750) b) the largest state gets ≤ M=96 seats c) the smallest state gets ≥ m= 6 seats d) the system is… „degressively proportional”

„degressive proportionality”: function A(t)/t is non increasing. (before rouding !) Examples include a) base + prop. a+ct (Pukelsheim 2007) b) piecewise linear c) parabolic a+bt-ct2 (Ramirez 2006) d) power like, a+ct

e) hiperbolic a+bt-c/t

but …

. Let t denote population of a state i

and A=A(t) the number of seats in EUP

„degressive proportionality”: function A(t)/t is non increasing. (before rouding !) Examples include a) base + prop. a+ct (Pukelsheim 2007) b) piecewise linear c) parabolic a+bt-ct2 (Ramirez 2006) d) power like, a+ct

e) hiperbolic a+bt-c/t

but none is mathematically distinguished

. Let t denote population of a state i

and A=A(t) the number of seats in EUP

1,50 2,00 2,50 3,00 3,50 4,00 4,50

• 1,00
• 0,50

0,00 0,50 1,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50

Ln (Seats) Ln (Population)

Seats (2009-2014) Parabolic Ramírez Fix + Prop. Pukelsheim Min Max

log – log scale

.

Cambridge meeting – Jan. 2011

Mathematics

• Prof. Geoffrey Grimmett (University of Cambridge), Director
• Prof. Friedrich Pukelsheim (University of Augsburg), co-Director
• Prof. Jean-François Laslier (École Polytechnique, Paris)
• Prof. Victoriano Ramírez González (University of Granada)
• Prof. Wojciech Słomczyński (Jagiellonian University, Kraków)
• Prof. Martin Zachariasen (University of Copenhagen)
• Prof. Karol Życzkowski (Jagiellonian University, Kraków)

Public Policy

• Prof. Richard Rose (University of Aberdeen)

AFCO Committee in attendance Mr Andrew Duff MEP (Rapporteur) Mr Rafał Trzaskowski MEP (Vice-President) Mr Guy Deregnaucourt (Administrator)

Cambridge meeting –Jan. 2011

adopted a simple solution:

Base + proportional system: A(t)=min{b+[t/d], M} where b=5, M=96, [...] rounding up,

and the divisor d is chosen such that the total number of seats for all 28 states agrees with the bound,

i=1 [A(ti)] =S=751

28

 *After the 2009 election, Andrew Duff MEP proposed a new version of his report, adopted by the parliamentary Committee on Constitutional Affairs (AFCO) in April 2011.

* Plenary session of the Parliament referred

the report back to AFCO in July 2011.

* Third version of the report

was published in September 2011 and adopted by AFCO in January 2012.

* It was withdrawn before being discussed

in plenary in March 2012 for fear that it would likely be turned down….

• 1. EU Council: Jagiellonian Compromise -
• ne criterion: voting weight is proportional to the

square root of the population; decision is taken

if the sum of the weights of members of a coalition exceeds the optimal threshold (e.g. 61.1% quota)

• 1b. An approximate expression for optimal quota

for M states:

qopt

(M)= 1/2 + 1/(M1/2

• 2. EU Parliament: Cambridge Compromise -

apportionment of the seats according to:

base + prop system.

Conclusions I

• 1. Even having a sound solution of a

mathematical problem related to politics it is rather difficult to catch the attention (and understanding !) of lawyers and politicians

2

But time goes on and some problems stay for good in the scientific literature Perhaps their time will come, some day…

Conclusions II

• 1. Even having a sound solution of a

mathematical problem related to politics it is rather difficult to catch the attention (and understanding !) of lawyers and politicians

2

But time goes on and some problems stay for good in the scientific literature Perhaps their time will come, some day…

3 My very best wishes to Turkey!! Welcome to the European Union !!!

Conclusions II

Teşekkür ederim !

.

Log-log coordinates

L(ln p) = ln A(p)

 A fulfifls Inc ↔ L′ ≥ 0  A fulfifls UDP ↔ L′ ≤ 1  A fulfifls Inc and UDP ↔ 0 ≤ L′ ≤ 1

Inc = increasing UDP = degressively proportional with unrounded allocations

Subproportionality

 L is convex iff A is subproportional  L is concave iff A is superproportional

the smaller a pair of states is, the larger is the gain

• f the small member in the pair over the large one
• the base+prop method is subproportional in the main ‘affine’

part of its domain, i.e. for all countries, but the largest one

• the parabolic method is subproportional for small and medium

countries and superproportional for the ‘large six’ (the graph of L is an S-shape curve).

Subproportionality

L1 - parabolic L2 - base+prop L3 - spline L4 - power

Subproportionality

L1 - parabolic L2 - base+prop L3 - spline L4 - power