The fate of the square root law for correlated voting · The fate of the square root law for correlated voting London School of Economics, March 2011 In the year 2000 presidential

The fate of the square root law

for correlated voting

Voting Power in Practice

London School of Economics

Symposium sponsored by the Leverhulme Trust

Werner Kirsch and Jessica Langner

Fakultat fur Mathematik and Informatik

FernUniversitat Hagen

The fate of the square root law for correlated voting London School of Economics, March 2011

talk based on:

W.K.: On Penrose’s Squareroot Law and Beyond.Homo Oeconomicus 2007

Jessica Langner: PhD-thesis, in preparation

W.K. and Jessica Langner: Mathematical Theory of CorrelatedVoting, in preparation.

strongly influenced by:

Felsenthal, D.S., Machover, M.:The measurement of voting power: Theory and practice, problemsand paradoxes.


talk based on:

W.K.: On Penrose’s Squareroot Law and Beyond.Homo Oeconomicus 2007

Jessica Langner: PhD-thesis, in preparation

W.K. and Jessica Langner: Mathematical Theory of CorrelatedVoting, in preparation.

strongly influenced by:

Felsenthal, D.S., Machover, M.:The measurement of voting power: Theory and practice, problemsand paradoxes.


Some Two-Tier Voting Systems (Councils)


Two-Tier Voting Systems

In a Two-Tier Voting Systems a union is formed by memberstates (or member organizations).

The voters in the member states are represented by a delegate.The delegates of the states are the member of a Council.

The delegates in the council are given a certain voting weightdepending on the size of the country they represent (or itseconomic strength, or . . . ).

The delegate’s vote cannot be split.


Two-Tier Voting Systems

In a Two-Tier Voting Systems a union is formed by memberstates (or member organizations).

The voters in the member states are represented by a delegate.The delegates of the states are the member of a Council.

The delegates in the council are given a certain voting weightdepending on the size of the country they represent (or itseconomic strength, or . . . ).

The delegate’s vote cannot be split.


Question

What is a ‘fair’ distribution of votes in a council?

Platz

We regard the process of decision making in a council as atwo-step voting system (two-tier system).

In a first step all voters in the different states decide upon thevoting of their respective representative in the council (oneperson, one vote).

The delegates cast their votes in the council according to themajority of voters in her/his country. The voting weight of adelegate depends on the population of his/her country.

A reasonable criterion for ‘fair’ voting weights:The decision of the council should agree with the public vote!


Question


Platz






Question


Platz






In the year 2000 presidential election in the USA Al Gore won themajority in the public vote but Georg W. Bush was elected by theElectoral College.

No distribution of voting weights can guarantee that the publicvote and the council vote coincide.

The best we can do is to choose the voting weights in such a waythat in most cases public vote and council vote agree.










A Mathematical Formulation

States and Voters

There are M states, labeled by Greek characters ν, κ, . . . .

There are Nν voters in the state ν . The votes of the citizensin state ν are denoted by Xνi with i = 1, . . . ,Nν and

Xνi =

{1, for ‘yes’;−1, for ‘no’.

The voting result in state ν is represented by

Sν =Nν∑i=1

Xνi

If Sν > 0 the majority in state ν votes affirmatively.

The popular vote in the union is given by:

P =M∑ν=1

Sν =M∑ν=1

Nν∑i=1

Xνi


A Mathematical Formulation

States and Voters

There are M states, labeled by Greek characters ν, κ, . . . .

There are Nν voters in the state ν . The votes of the citizensin state ν are denoted by Xνi with i = 1, . . . ,Nν and

Xνi =

{1, for ‘yes’;−1, for ‘no’.

The voting result in state ν is represented by

Sν =Nν∑i=1

Xνi

If Sν > 0 the majority in state ν votes affirmatively.

The popular vote in the union is given by:

P =M∑ν=1

Sν =M∑ν=1

Nν∑i=1

Xνi


We set

χ(x) :=

{1, for x > 0;−1, otherwise.

Thus

χν := χ(Sν) =

{1, if the majority in state ν votes ‘yes’;−1, if not.

If the state ν has voting weight wν , then the vote in thecouncil will be represented by

C =M∑ν=1

wν χ(Sν)

We define the democracy deficit by∆ :=

∣∣C − P |

The democracy deficit ∆(w1, . . . ,wM) is a function of theweights wν , ν = 1 . . . ,M .


We set

χ(x) :=


Thus

χν := χ(Sν) =



C =M∑ν=1

wν χ(Sν)


∣∣C − P |



We set

χ(x) :=


Thus

χν := χ(Sν) =



C =M∑ν=1

wν χ(Sν)


∣∣C − P |



Goal

Choose the weights wν in such a way that ∆(w1, . . . ,wM) isminimal!

More precisely we want the average square error

E(

∆2)

= E( ( M∑

ν=1

wν χ( Nν∑

i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )

to be as small as possible.

?But . . .

. . . what is E (=average) supposed to mean?


Goal

Choose the weights wν in such a way that ∆(w1, . . . ,wM) isminimal!More precisely we want the average square error

E(

∆2)

= E( ( M∑

ν=1

wν χ( Nν∑

i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )


?But . . .



Goal


E(

∆2)

= E( ( M∑

ν=1

wν χ( Nν∑

i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )


?But . . .



Goal


E(

∆2)

= E( ( M∑

ν=1

wν χ( Nν∑

i=1

Xνi)−

M∑ν=1

Nν∑i=1

Xνi)2 )


?But . . .



Voting Measures

The voting system, represented by X1, . . . ,XN , is fed withproposals in a completely random way.The voters react to the (random) proposal in a deterministicrational way.

The randomness of the proposals induce a probability measure Pon the space {−1,+1}N , the possible voting outcomes.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

2


Voting Measures

The voting system, represented by X1, . . . ,XN , is fed withproposals in a completely random way.The voters react to the (random) proposal in a deterministicrational way.

The randomness of the proposals induce a probability measure Pon the space {−1,+1}N , the possible voting outcomes.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

2


Voting Measures

A measure P on {−1, 1}N is called a voting measure if

P(X1 = ξ1, . . . ,XN = ξN) = P(X1 = −ξ1, . . . ,XN = −ξN)

Example: 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) , that is

P(X1 = ξ1, . . . ,XN = ξN) =1

2N


Voting Measures

A measure P on {−1, 1}N is called a voting measure if

P(X1 = ξ1, . . . ,XN = ξN) = P(X1 = −ξ1, . . . ,XN = −ξN)

Example: 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) , that is

P(X1 = ξ1, . . . ,XN = ξN) =1

2N


Optimal Weights for Independent States

We assume now that Xνi and Xκj are independent for ν 6= κWe keep the notation Sν =

∑i Xνi

and χν = ±1 for Sν > 0 (resp.Sν ≤ 0 ).

We want to minimize the function

D(w1, . . . ,wM) = E(

∆(w1, . . . ,wM)2)

=M∑

ν,κ=1

(wνwκ E

(χνχκ

)− 2wν E

(χνSκ

)+ E

(SνSκ

))

We have (in general) χ2ν = 1 and χνSν = |Sν | .

Moreover, due to independence, we know for ν 6= κ

E(χν χκ

)= 0

E(χν Sκ

)= 0

E(Sν Sκ

)= 0


Optimal Weights for Independent States

We assume now that Xνi and Xκj are independent for ν 6= κWe keep the notation Sν =

∑i Xνi

and χν = ±1 for Sν > 0 (resp.Sν ≤ 0 ).

We want to minimize the function

D(w1, . . . ,wM) = E(

∆(w1, . . . ,wM)2)

=M∑

ν,κ=1

(wνwκ E

(χνχκ

)− 2wν E

(χνSκ

)+ E

(SνSκ

))

We have (in general) χ2ν = 1 and χνSν = |Sν | .

Moreover, due to independence, we know for ν 6= κ

E(χν χκ

)= 0

E(χν Sκ

)= 0

E(Sν Sκ

)= 0


Independent States

The D simplifies to:

D(w1, . . . ,wM) =M∑ν=1

(w2ν − 2wνE

(|Sν |

)+ E

(S2ν

)).

This is minimized by:

gν := E(|Sν |

)The quantity E

(|Sν |

)describes the margin of the voting outcome.

Democracy Deficit

The minimal value of D is given by:

D0 := D(g1, . . . , gM) =M∑ν=1

(E(∣∣Sν∣∣2) − E

(∣∣Sν∣∣)2)


Independent States

The D simplifies to:

D(w1, . . . ,wM) =M∑ν=1

(w2ν − 2wνE

(|Sν |

)+ E

(S2ν

)).

This is minimized by:

gν := E(|Sν |

)The quantity E

(|Sν |

)describes the margin of the voting outcome.

Democracy Deficit

The minimal value of D is given by:

D0 := D(g1, . . . , gM) =M∑ν=1

(E(∣∣Sν∣∣2) − E

(∣∣Sν∣∣)2)


Now we assume that all voters are independent.

Independent Voters

For independent voters we have

gν = E(|Sν |

)≈√

2π

√N

This is the square root law by Penrose.

Democracy Deficit

The minimal (averaged) democracy deficit is given by:

D0 = D(g1, . . . , gM) ≈ π−2π N

The averaged democracy deficit per voter converges to 0 :

E((∆0

N

)2)≤ C

N



Independent Voters


gν = E(|Sν |

)≈√

2π

√N


Democracy Deficit


D0 = D(g1, . . . , gM) ≈ π−2π N


E((∆0

N

)2)≤ C

N



Independent Voters


gν = E(|Sν |

)≈√

2π

√N


Democracy Deficit


D0 = D(g1, . . . , gM) ≈ π−2π N


E((∆0

N

)2)≤ C

N


The Collective Bias Model

In this model the voters are influenced by a mainstreamopinion, e. g. a common believe due to the country’s traditionor the influence of opinion makers.

For a given proposal ω we model this ‘common believe’ by avalue ζ ∈ [−1, 1] which depends on the proposal at hand.

The value ζ = 1 means there is such a strong common believein favor of the proposal that all voters will vote ‘yes’, ζ = −1means all voters will vote ‘no’.

In general, ζ denotes the expected outcome of the voting, i. e.E(Xi ) . The individual voting results Xi fluctuate around thisvalue randomly.














CBM (mathematical definition)

For ζ ∈ [−1, 1] let Pζ be the probability measure on {−1, 1}with Pζ(1) = 1

2(1 + ζ) and Pζ(−1) = 12(1− ζ) . Then

Eζ(Xi ) = ζ .

We define Pζ on {−1, 1}N by

Pζ(X1 = a1,X2 = a2, . . . ,XN = aN)

= Pζ(a1) · Pζ(a2) · . . . · Pζ(aN)

Let µ be a measure on [−1, 1 with µ([a, b]) giving theprobability that ζ has a value between a and b.

We define the CBM-measure by

Pµ(X1 = a1,X2 = a2, . . . ,XN = aN)

=

∫ 1

−1Pζ(X1 = a1,X2 = a2, . . . ,XN = aN) dµ(ζ)


CBM (mathematical definition)

For ζ ∈ [−1, 1] let Pζ be the probability measure on {−1, 1}with Pζ(1) = 1

2(1 + ζ) and Pζ(−1) = 12(1− ζ) . Then

Eζ(Xi ) = ζ .

We define Pζ on {−1, 1}N by

Pζ(X1 = a1,X2 = a2, . . . ,XN = aN)

= Pζ(a1) · Pζ(a2) · . . . · Pζ(aN)

Let µ be a measure on [−1, 1 with µ([a, b]) giving theprobability that ζ has a value between a and b.

We define the CBM-measure by

Pµ(X1 = a1,X2 = a2, . . . ,XN = aN)

=

∫ 1

−1Pζ(X1 = a1,X2 = a2, . . . ,XN = aN) dµ(ζ)


Note, that Pζ is not a voting measure (unless ζ = 0).

However Pµ is a voting measure if µ is invariant under signchange. We call µ the bias measure.

If the measure µ is concentrated in 0, then Pµ makes thevoting results Xi independent, thus we are in the case ofindependent voters.

If µ is the uniform distribution on [−1, 1] , then thecorresponding measure was already considered by Straffin(1977) when he established an intimate connection to of thismodel to the Shapley-Shubik index.

In a similar way, the Penrose-Banzhaf measure is connectedwith the model of independent voters.


























CBM

Optimal weights

For the CBM we havegν = E

(|Sν |

)≈ µ1 N

where µ1 =∫|ζ| dµ(ζ) .

Democracy Deficit for the CBM


D0 ≈(µ2 − µ1

2)

N2

with µ2 =∫|ζ|2 dµ(ζ) .

The averaged democracy deficit per voter converges to a nonzerolimit:

E((∆0

N

)2)−→ µ2 − µ1

2


CBM

Optimal weights

For the CBM we havegν = E

(|Sν |

)≈ µ1 N

where µ1 =∫|ζ| dµ(ζ) .

Democracy Deficit for the CBM


D0 ≈(µ2 − µ1

2)

N2

with µ2 =∫|ζ|2 dµ(ζ) .

The averaged democracy deficit per voter converges to a nonzerolimit:

E((∆0

N

)2)−→ µ2 − µ1

2


The Cooperation Model

The Curie-Weiss model is a model of cooperative behavior fromstatistical physics. It models ferromagnetic system in whichelementary magnets prefer to align.

Curie Weiss Measure

H(X1, . . . ,XN) = − 1

N

( N∑i=1

Xi

)2

This is the energy function for the spin configuration (=votingoutcome) X1, . . . ,XN . We use this to define probability measures

Pβ(X1, . . . ,XN) =e−βH(X1,...,XN)

Z

The parameter β measures the strength of the interactionbetween the voters.


The Cooperation Model

The Curie-Weiss model is a model of cooperative behavior fromstatistical physics. It models ferromagnetic system in whichelementary magnets prefer to align.

Curie Weiss Measure

H(X1, . . . ,XN) = − 1

N

( N∑i=1

Xi

)2

This is the energy function for the spin configuration (=votingoutcome) X1, . . . ,XN . We use this to define probability measures

Pβ(X1, . . . ,XN) =e−βH(X1,...,XN)

Z

The parameter β measures the strength of the interactionbetween the voters.


Optimal Weights for the Curie-Weiss Model

gν = E(∣∣Sν∣∣) ≈

C1(β)√

Nν , for β < 1;

C2 Nν

34 , for β = 1;

C3(β) Nν , for β > 1.


A Model with Global Collective Behavior

So far we have always assumed that voter in different statesact independently.

In this section we consider the case of collective behavioracross country borders.

We assume that all voters act according to the Collective Biasmeasure Pµ .

This means there is a common believe, expressed through themeasure µ , for all voters in the union.



So far we have always assumed that voter in different statesact independently.

In this section we consider the case of collective behavioracross country borders.

We assume that all voters act according to the Collective Biasmeasure Pµ .

This means there is a common believe, expressed through themeasure µ , for all voters in the union.



We have again to minimize the function

D(w1, . . . ,wM) = E(

∆(w1, . . . ,wM)2)

=M∑

ν,κ=1

(wνwκ E

(χνχκ

)− 2wν E

(χνSκ

)+ E

(SνSκ

))But this time the function does not simplify immediately!

However, for large Nν and Nκ we have for any ν, κ :

Eµ(χνχκ

)≈ 1

Eµ(χνSκ

)≈ Eµ

(|Sκ∣∣) ≈ µ1 Nκ

and

E(SνSκ

)≈ µ2 Nν Nκ .



We have again to minimize the function

D(w1, . . . ,wM) = E(

∆(w1, . . . ,wM)2)

=M∑

ν,κ=1

(wνwκ E

(χνχκ

)− 2wν E

(χνSκ

)+ E

(SνSκ

))But this time the function does not simplify immediately!

However, for large Nν and Nκ we have for any ν, κ :

Eµ(χνχκ

)≈ 1

Eµ(χνSκ

)≈ Eµ

(|Sκ∣∣) ≈ µ1 Nκ

and

E(SνSκ

)≈ µ2 Nν Nκ .


Democracy Deficit for the Global CBM

For large N we obtain:

D(w1, . . . ,wM)

≈M∑

ν,κ=1

wνwκ + 2M∑ν=1

wν

M∑κ=1

µ1 Nκ +M∑

ν,κ=1

µ2 NνNκ

=( M∑ν=1

wν)2 − 2µ1

( M∑ν=1

wν)

N + µ2 N2

= W 2 − 2µ1 W + µ2 N2

This last expression depends only on the sum W =∑M

ν=1 wν ofthe voting weights and not on the individual weights wν .

For large Nν the asymptotic value of D does not depend on the waythe weights are distributed among the member states of the union.


Democracy Deficit for the Global CBM

For large N we obtain:

D(w1, . . . ,wM)

≈M∑

ν,κ=1

wνwκ + 2M∑ν=1

wν

M∑κ=1

µ1 Nκ +M∑

ν,κ=1

µ2 NνNκ

=( M∑ν=1

wν)2 − 2µ1

( M∑ν=1

wν)

N + µ2 N2

= W 2 − 2µ1 W + µ2 N2

This last expression depends only on the sum W =∑M

ν=1 wν ofthe voting weights and not on the individual weights wν .For large Nν the asymptotic value of D does not depend on the waythe weights are distributed among the member states of the union.


Conclusion

If the voters act independently or almost independently, thesquare root law applies.

If the voters inside a country are strongly correlated to eachother, but not to voters in other countries, then proportionalrepresentation is optimal.

If all voters are strongly correlated (also across borders), anydistribution of voting weights is as good as any other one.


Documents

The fate of the square root law for correlated voting · The fate of the square root law for correlated voting London School of Economics, March 2011 In the year 2000 presidential