mathematics, power, and democracy - fernuniversität hagen · each voter may vote ‘yes’ or...

Mathematics, Power, and

Democracy

Werner Kirsch

Fakultat fur Mathematik and Informatik

FernUniversitat Hagen

IMW; Bielefeld, November 2010

Mathematics, Power, and Democracy Bielefeld, November 2010

Part of this talk is based joint work with Jessica Langner.

The well being of democracies regardless of their type and statusdepends on a small technical detail: the voting system.

Everything else is secondary.

Jose Ortega y Gasset

Voting systems (Parliaments)

Voting Systems (Parliaments)

A parliament is a voting system. The members of theparliament are the voters of this voting system.

Each voter may vote ‘Yes’ or ‘No’.Here and in what follows we neglect the possibility ofabstentions.

‘Normally’ a proposal is adopted if the number of votes infavor of the proposal is bigger than the number of votesagainst it (simple majority).

In some cases a qualified majority is required (e.g. two thirds),for example to make an amendment to the constitution.

Each voter may vote ‘Yes’ or ‘No’.

Here and in what follows we neglect the possibility ofabstentions.

Voting systems (Councils)

Bundesrat – different voting weights for the members

Voting Rules

The members of the Bundesrates represent their states(Lander).

According to their population the states get a number of threeto six votes. ‘voting weights’ wi ∈ 3, 4, 5, 6The votes of a state have to be cast in a block.

As a rule, a proposal is approved if more than one half of thevotes are in favor. (Quota: 1/2)

For certain proposals a majority of two thirds is required.(Quota: 2/3)

Voting Rules

According to their population the states get a number of threeto six votes.

‘voting weights’ wi ∈ 3, 4, 5, 6The votes of a state have to be cast in a block.

Voting Rules

According to their population the states get a number of threeto six votes. ‘voting weights’ wi ∈ 3, 4, 5, 6

The votes of a state have to be cast in a block.

Voting Rules

Voting Systems (More Examples)

UN security council

Five permanent members(China, France, Russia, UK, USA)

Ten non permanent members(Austria, Bosnia-Herzegovina, Brasil, Gabon, Japan,Lebanon, Mexico, Nigeria, Turkey, Uganda)

A proposal is approved if both

All permanent members affirm it andAt least four non permanent members do.

International Monetary Fund (cont’d)

The number of votes of a country in the Board of Governors isconnected with the country’s economic strength (through a rathersophisticated formula). Here are the voting weights for a fewcountries:

United States 371743Japan 133378Germany 130332France 107635UK 107635China 81151

......

Palau 281

For many important decisions of the IMF a majority of 85 % isrequired. The US-share is 16.74 % . . .

......

Palau 281

For many important decisions of the IMF a majority of 85 % isrequired.

The US-share is 16.74 % . . .

......

Palau 281

For many important decisions of the IMF a majority of 85 % isrequired. The US-share is 16.74 % . . .

The Council of the EU

The Council of the EU is one of the law making institutions ofthe EU.

The Council of Ministers consists of representatives of thegovernments of the member states, one representative foreach state.

The members of the Council used to have a voting weightdepending on the population of the country they represent.

Since 2004 the voting procedure in the Council is morecomplicated:

A three step procedure after the Nice Treaty (currentprocedure).‘Double Majority’ after the Lisbon Treaty (in effect after 2014).

Example:Council until 2004

Germany 10 Belgium 5

France 10 Sweden 4

UK 10 Austria 4

Italy 10 Denmark 3

Spain 8 Finland 3

Netherlands 5 Ireland 3

Greece 5 Luxembourg 2

Portugal 5

Quota 71,2%at least 62 out of 87 votes

A three step procedure after the Nice Treaty (currentprocedure).

‘Double Majority’ after the Lisbon Treaty (in effect after 2014).

Questions

How much power does a voter have in a given voting system ?For example: Is the voting weight a good measure for power?And what about more complicated voting systems?

What is a ‘fair’ distribution of power?. . . for example among the member states of the EU.

How effective is the voting system? For example: Are thereblocking minorities which can screw up the system?

Questions

Example

Council of Ministers of the EEC

Germany 4 votesFrance 4 votesItaly 4 votesNetherlands 2 votesBelgium 2 votesLuxembourg 1 vote

To approve a proposal 12 (out of 17) were needed. (quota 70%)

The voting weight is not (always) a good measure for power.

Example

Council of Ministers of the EEC

Germany 4 votesFrance 4 votesItaly 4 votesNetherlands 2 votesBelgium 2 votesLuxembourg 1 vote

To approve a proposal 12 (out of 17) were needed. (quota 70%)

The voting weight is not (always) a good measure for power.

Voting Systems (Mathematical Definition)

Definition

A voting system consists of a finite set V of voters and a subset Wof the power set P(V ). A subset M of V is called a coalition, thesets M in W are called winning coalitions.

We suppose:

V ∈ W (grand coalition)

∅ 6∈ W (empty coalition)

If A ⊂ B and A ∈ W then B ∈ W.

Weighted Voting Systems

A voting systems (V ,W) is called weighted, if there a functionw : V → R+ (weight) and a number q (quota), such that

A ∈ W ⇔∑v∈A

w(v) ≥ q∑v∈V

Definition

A voting system consists of a finite set V of voters and a subset Wof the power set P(V ). A subset M of V is called a coalition, thesets M in W are called winning coalitions.We suppose:

A ∈ W ⇔∑v∈A

w(v) ≥ q∑v∈V

Definition

A voting system consists of a finite set V of voters and a subset Wof the power set P(V ). A subset M of V is called a coalition, thesets M in W are called winning coalitions.We suppose:

A ∈ W ⇔∑v∈A

w(v) ≥ q∑v∈V

UN Security Council

Reminder: 5 permanent members, 10 nonpermanent membersWinning coalitions: All 5 permanent members and 4 (or more)nonpermanent members

The UN Security Council is a weighted voting system, as we mayset:

w(x) :=

7, for permanent members x ;1, for nonpermanent members.

andq := 39

Not every voting system can be written as a weighted votingsystem.

UN Security Council

Reminder: 5 permanent members, 10 nonpermanent membersWinning coalitions: All 5 permanent members and 4 (or more)nonpermanent membersThe UN Security Council is a weighted voting system, as we mayset:

w(x) :=

andq := 39

UN Security Council

Reminder: 5 permanent members, 10 nonpermanent membersWinning coalitions: All 5 permanent members and 4 (or more)nonpermanent membersThe UN Security Council is a weighted voting system, as we mayset:

w(x) :=

andq := 39

Power Indices

The Banzhaf-Index

If A ∈ W and v ∈ A then v is called decisive for the coalition A ifA \ v 6∈ W .

D(v) := A ∈ W | v is decisive for A

The Banzhaf-Power B(v) of a voter v is defined as

B(v) :=1

2#V−1 #D(v)

The Banzhaf-Index β(v) of a voter v is defined as

β(v) :=B(v)∑

u∈VB(u)

Example: Council of Ministers of the EEC

Country Votes Banzhaf IndexBelgium 11.8 % 14.3 %France 23.5 % 23.8 %Italy 23.5 % 23.8 %Germany 23.5 % 23.8 %Luxembourg 5.9 % 0.0 %Netherlands 11.8 % 14.3 %

Belgium 2 votesFrance 4 votesItaly 4 votesGermany 4 votesLuxembourg 1 voteNetherlands 2 votes

Quota 12

Power Indices II

The Shapley-Shubik-Index

For the Shapley-Shubik-Index we consider permutations π ∈ SV(rather than subsets of V ).We say that a voter v is decisive for a permutationπ = (π1, π2, . . . , πN) ∈ SV , (We set N := #V ), if v = πν and

πj | j ≤ ν ∈ Wbut πj | j < ν 6∈ W

We define the Shapley-Shubik-Index σ(v) of a voter v by

σ(v) :=1

#(SV )#π ∈ SV | v is decisive for π

Power-Indices

Power Indices measure ‘how frequently’ it happens that a givenvoter is decisive.

We will, in deed, give it a probabilistic interpretation later.

Power-Indices

Power Indices measure ‘how frequently’ it happens that a givenvoter is decisive.

We will, in deed, give it a probabilistic interpretation later.

Multi-Layer Voting Systems

Typical Examples

Each member state of the EU (or the IMF) is represented byone delegate in the Council of Ministers (or the board ofGovernors).

The delegate of a country has a given voting weightdepending on the population (or the economic strength) of hiscountry. This vote cannot be split.

We assume that the delegates follow the majority vote in theirrespective country.

This constitutes a two-layer (or two-step) voting system forthe citizens in these countries.

Typical Examples

Mathematical Formulation

The set of voters consists of M disjoint sets Sν , ν = 1, . . . ,M(states). The set Sν contains Nν elements.The vote cast by the voter number i in state Sν is denoted by Xνi .

Xνi =

+1, if he/she votes ‘YES’;−1, if she/he votes ‘NO’.

The popular vote Xν in state Sν is given by:

Xν =Nν∑i=1

With χ(x) =

+1, if x > 0;−1, if x ≤ 0.

the voting behavior of the

delegate of Sν is given by

Yν = χ(Xν)

If we give the state Sν a weight wν then the voting outcome in theCouncil is

M∑ν=1

wν Yν =M∑ν=1

wν χ( Nν∑i=1

We consider a two-layer voting system as ‘fair’ if the voting resultin the council is as close as possible to the popular voteM∑ν=1

Nν∑i=1

Xνi , i.e. we want to minimize (with respect to the wν ):

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣The quantity ∆ is called the democracy deficit of the votingsystem.Note, that it is impossible to make ∆ identically equal to zero!

M∑ν=1

wν Yν =M∑ν=1

wν χ( Nν∑i=1

Nν∑i=1

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣

The quantity ∆ is called the democracy deficit of the votingsystem.Note, that it is impossible to make ∆ identically equal to zero!

M∑ν=1

wν Yν =M∑ν=1

wν χ( Nν∑i=1

Nν∑i=1

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣The quantity ∆ is called the democracy deficit of the votingsystem.

Note, that it is impossible to make ∆ identically equal to zero!

M∑ν=1

wν Yν =M∑ν=1

wν χ( Nν∑i=1

Nν∑i=1

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣The quantity ∆ is called the democracy deficit of the votingsystem.Note, that it is impossible to make ∆ identically equal to zero!

It seems therefore reasonable to minimize

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

which is some kind of mean square error.

But . . .

. . . what is E supposed to mean?

It seems therefore reasonable to minimize

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

which is some kind of mean square error.

But . . .

. . . what is E supposed to mean?

General Framework

The voting system is fed with proposals in a completely randomway.

The voters react to the (random) proposal in a deterministicrational way.

The randomness of the proposals induce a probability measure Pon the space −1,+1V .A proposal ω and its counter-proposal ¬ω should have the sameprobability.The rationality of the voting implies that Xv (¬ω) = −Xv (ω)) .Consequently, the probability P is invariant under inversion

(X1, . . . ,XN) −→ (−X1, . . . ,−XN)

In particular:

P(Xi = 1) = P(Xi = −1) =1

General Framework

The voting system is fed with proposals in a completely randomway.The voters react to the (random) proposal in a deterministicrational way.

(X1, . . . ,XN) −→ (−X1, . . . ,−XN)

In particular:

P(Xi = 1) = P(Xi = −1) =1

General Framework

The voting system is fed with proposals in a completely randomway.The voters react to the (random) proposal in a deterministicrational way.

(X1, . . . ,XN) −→ (−X1, . . . ,−XN)

In particular:

P(Xi = 1) = P(Xi = −1) =1

Examples

Independent Voters

If the voting behavior Xi of the voter i is independent from theother voters, then P is just a product measures P0 withP0 = 1

2(δ−1 + δ+1) .

Remark

The Banzhaf B(v) can be expressed through this measure, namely:

B(v) = 2P(D(v)

Reminder:

Examples

Independent Voters

If the voting behavior Xi of the voter i is independent from theother voters, then P is just a product measures P0 withP0 = 1

2(δ−1 + δ+1) .

Remark

The Banzhaf B(v) can be expressed through this measure, namely:

B(v) = 2P(D(v)

Reminder:

Examples

Notation

We set:

Q = Q(N) :=N⊗i=1

2δ1 +

2δ−1)

and (to be used later):

Qp = Q(N)p :=

N⊗i=1

(pδ1 + (1− p)δ−1

Examples

The CBM

For the Common Believe Model (CBM) (or Collective Bias Model(CBM) ) we suppose there is a strong common value system, forexample there may be a strong church or religious group or astate-wide ideology or the media may be under the control of asmall group of people or . . .

This common believe (or collective bias) is expressed through arandom variable Z with values in the interval [−1, 1] anddistribution µ .We define

Pµ(A) :=

∫[−1,1]

Q 12(1+z)(A) dµ(z)

If µ = δ0 we get back to the independent voters model.

Examples

The CBM

For the Common Believe Model (CBM) (or Collective Bias Model(CBM) ) we suppose there is a strong common value system, forexample there may be a strong church or religious group or astate-wide ideology or the media may be under the control of asmall group of people or . . .This common believe (or collective bias) is expressed through arandom variable Z with values in the interval [−1, 1] anddistribution µ .

We define

Pµ(A) :=

∫[−1,1]

Q 12(1+z)(A) dµ(z)

Examples

The CBM

For the Common Believe Model (CBM) (or Collective Bias Model(CBM) ) we suppose there is a strong common value system, forexample there may be a strong church or religious group or astate-wide ideology or the media may be under the control of asmall group of people or . . .This common believe (or collective bias) is expressed through arandom variable Z with values in the interval [−1, 1] anddistribution µ .We define

Pµ(A) :=

∫[−1,1]

Q 12(1+z)(A) dµ(z)

Examples

The CBM

For the Common Believe Model (CBM) (or Collective Bias Model(CBM) ) we suppose there is a strong common value system, forexample there may be a strong church or religious group or astate-wide ideology or the media may be under the control of asmall group of people or . . .This common believe (or collective bias) is expressed through arandom variable Z with values in the interval [−1, 1] anddistribution µ .We define

Pµ(A) :=

∫[−1,1]

Q 12(1+z)(A) dµ(z)

Examples

Remarks

The measure Pµ is spin-flip invariant, if µ is symmetric (at 0).

If µ is the uniform distribution on [−1, 1], then theShapley-Shubik-Index σ(v) can be expressed as:

Pµ(D(v)

Examples

Remarks

The measure Pµ is spin-flip invariant, if µ is symmetric (at 0).

If µ is the uniform distribution on [−1, 1], then theShapley-Shubik-Index σ(v) can be expressed as:

Pµ(D(v)

Fair representation

We return to the question of fair voting weights.

We recall that we want to chose the weights wν in such a way,that the democracy deficit

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣is as small as possible,

in the sense, that we try to minimize

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

Let us assume, that voters in different states act in an independentway of each others, i.e. Xνi and Xρj with ν 6= ρ are independent ofeach others.

But Xνi and Xνj may be dependent.

Fair representation

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

∣∣∣is as small as possible,in the sense, that we try to minimize

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

Fair representation

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

Fair representation

∆ :=∣∣∣ M∑ν=1

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

E(∆2) = E( ( M∑

wν χ( Nν∑i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

Fair Representation

Result

The quantity E(∆2) is minimized by the choice

wν = E(∣∣ Nν∑

Xνi∣∣)

Independent Voters

For independent Xνi we have (for large Nν):

E(∣∣ Nν∑

∣∣) ' √Nν

by the central limit theorem.

Square root law by Penrose.

Fair Representation

Result

The quantity E(∆2) is minimized by the choice

wν = E(∣∣ Nν∑

Xνi∣∣)

Independent Voters

For independent Xνi we have (for large Nν):

E(∣∣ Nν∑

∣∣) ' √Nν

by the central limit theorem.

Square root law by Penrose.

Fair Representation

Result

Eµ(∣∣ Nν∑

∣∣) ' ∫|z | dµ(z) Nν

Voting models as spin systems

The space −1,+1N with the probability measure P can belooked upon as a spin system which is invariant under spin flips.Let Q(N) denote the uniform distribution on Ω = −1,+1N .Q(N) corresponds to independent, identically distributed (anduniformly distributed) Xi .Any probability measure P on Ω which is spin-flip invariant can bewritten as

P = Z−1 eβH(x1,...,xN)Q(N)

where Z is a normalizing constantand H(x1, . . . , xN) = H(−x1, . . . ,−xN)where β plays the role of an inverse temperature (or a couplingstrength).

Examples

H(X1, . . . ,XN) ≡ 1 gives the model with independent voters.

The Common Believe Model can be interpreted as a system ofnoninteracting spins in a random, but constant magnetic field.

The Curie-Weiss Model is given by the Hamiltonian

HCW (x1, . . . , xN) = − 1

( N∑i=1

In this model the spins (voters) tend to behave in the sameway as (the average of) the others. If the parameter β is verysmall, the spins are almost independent, if β is very largealmost all of the spins are aligned.

Examples

HCW (x1, . . . , xN) = − 1

( N∑i=1

Examples

HCW (x1, . . . , xN) = − 1

( N∑i=1

Optimal Weights for the Curie-Weiss Model

Results

For β < 1 we have

(∣∣ N∑i=1

∣∣) ' √NFor β > 1 we have

(∣∣ Nν∑i=1

∣∣) ' N

For β = 1 we have

(∣∣ Nν∑i=1

∣∣) ' N34

Optimal Weights for the Curie-Weiss Model

Results

For β < 1 we have

(∣∣ N∑i=1

∣∣) ' √NFor β > 1 we have

(∣∣ Nν∑i=1

∣∣) ' N

For β = 1 we have

(∣∣ Nν∑i=1

∣∣) ' N34

Reality Check

In the EU with 27 member states there exist currently two votingsystems for the Council of Ministers

The voting system of Nice which was decided in Nice in 2000.The system is currently used.

The voting system of Lisbon which was negotiated in Lisbonin 2007 and ratified last year. It will be effective in 2014 or so.

The Polish government tried to block the Lisbon treaty becausethey believed that according to this treaty Poland isunder-represented in the Council.

In fact, their slogan was:

Square root or death.

Reality Check

Reality Check: Nice and Lisbon vs. Square Root

State Nice Lisbon State Nice Lisbon

Germany -18,45 24,34 Austria 3,63 -15,50

France -4,05 7,67 Bulgaria 5,08 -15,15

UK -3,81 7,44 Denmark -10,52 -10,04

Italy -2,14 6,06 Slowakia -10,48 -10,00

Spain 9,58 -5,74 Finland -9,01 -9,54

Poland 14,26 -9,35 Ireland 4,32 -3,29

Romania -13,10 -13,89 Lithuania 11,57 0,30

Netherlands -5,98 -17,00 Latvia -22,05 13,07

Greece 5,66 -17,56 Slovenia -15,74 19,72

Portugal 8,69 -17,35 Estonia 2,21 39,22

Belgium 8,99 -17,12 Cyprus 40,26 81,38

Czechia 9,80 -17,23 Luxembourg 77,76 126,34

Hungary 10,15 -17,34 Malta 42,30 138,56

Sweden -1,54 -16,62

What is the right measure?

It depends!

To formulate a constitution: No preknowledge! Independenceis the ‘best guess’.

To describe a given system at a specific time: Do statisticalexperiment.

It depends!

Outlook

New results for correlation across borders.

Use for opinion polls (in particular for election night forecasts).

Outlook

New results for correlation across borders.

Use for opinion polls (in particular for election night forecasts).

Efficiency

Definition

Let (V ,W) be a voting system and P be a (voting) measure on−1, 1V . We may assume that V = 1, 2, . . . ,N and identify−1, 1V , −1, 1N and P(V ) .

The efficiency η of the voting system is defined as

η := P(W)

Example: Parliament

If the voting rule is simple majority and N is odd, then η = 12

independent of the choice of the measure P.

Efficiency

Definition

Let (V ,W) be a voting system and P be a (voting) measure on−1, 1V . We may assume that V = 1, 2, . . . ,N and identify−1, 1V , −1, 1N and P(V ) .The efficiency η of the voting system is defined as

η := P(W)

Example: Parliament

Efficiency

Definition

Let (V ,W) be a voting system and P be a (voting) measure on−1, 1V . We may assume that V = 1, 2, . . . ,N and identify−1, 1V , −1, 1N and P(V ) .The efficiency η of the voting system is defined as

η := P(W)

Example: Parliament

The meaning of efficiency

If the efficiency of a voting system is close to 12 it is relatively easy

to make decisions. In almost any case either the supporters of aproposal or the supporters of the counter-proposal can enforce adecision in their respective favor. If the efficiency is low astalemate will happen rather frequently.

On the other hand, sometimes a high efficiency is not alwaysdesired. For example, in most countries an amendment of theconstitution requires a large majority resulting in low efficiency.

For example in the European Union EU the member states tend tobe afraid of being overruled by the others. So, they prefer to keepthe efficiency low, thus strengthening their blocking power.

The meaning of efficiency

If the efficiency of a voting system is close to 12 it is relatively easy

to make decisions. In almost any case either the supporters of aproposal or the supporters of the counter-proposal can enforce adecision in their respective favor. If the efficiency is low astalemate will happen rather frequently.

On the other hand, sometimes a high efficiency is not alwaysdesired. For example, in most countries an amendment of theconstitution requires a large majority resulting in low efficiency.

For example in the European Union EU the member states tend tobe afraid of being overruled by the others. So, they prefer to keepthe efficiency low, thus strengthening their blocking power.

Efficiency and Enlargement

We are interested in the dependence of η on the number N ofvoters.

This plays a crucial role in enlargement processes. For example theEU was extended various times, in particular in 2004 and 2007from 15 members to 27 members.The voting mechanism is (a combination of instances of) weightedvoting with a quota kept fixed at about 70 %.It look like the politicians believed the kept also the efficiency ofthe system fixed. Is this true?

Efficiency for Independent Voters

Setting

Let wi be an infinite sequence of weights, q a quota and considerthe voting systems VN = 1, 2, . . . ,N with voting weightsw1,w2, . . . ,wN and quota q .

Let P denote the product measure on VN and set

∑Ni=1 wi

2(∑Ni=1 wi

)2 → 0

Result

Under these conditions

ηN := P(WN) = O(e−C aN

Efficiency for Independent Voters

Setting

Let P denote the product measure on VN and set

∑Ni=1 wi

2(∑Ni=1 wi

)2 → 0

Result

ηN := P(WN) = O(e−C aN

Efficiency for the Common Believe Model

Setting

Let Pµ denote the CBM measure on VN and set

∑Ni=1 wi

2(∑Ni=1 wi

)2 → 0

Result

ηN := P(WN) = ' µ(]q, 1]

2µ(q)

Efficiency for the Common Believe Model

Setting

Let Pµ denote the CBM measure on VN and set

∑Ni=1 wi

2(∑Ni=1 wi

)2 → 0

Result

ηN := P(WN) = ' µ(]q, 1]

2µ(q)

Efficiency for the EU-Council

The efficiency for the Nice-procedure is about 2%.

The efficiency for the Lisbon-procedure is about 15%

Thank you for your attention!

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