Using voting probabilistic models for the European Union case
15-16-17 mars 2007
Summary:
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
Using voting probabilistic models
for the European Union case
J.L. Rouet, M.R. Feix, V. Merlin, D. Lepelley
ISTO, MAPMO, CREM, CERESUR
15-16-17 mars 2007
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
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Frame
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
•
N voters
•
Binary issue elections (‘Yes’, ‘No’)
•
Application to UE27
Treaty of Nice
European Constitution
•
1 key vote : Mandates ai , i = 1, . . . , N and Quota Q
2 key vote
•
Conclusion
•
•
2 kinds of mandates ai and bi i = 1, . . . , N
2 Quotas QA and QB
Qualified Majority Vote
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
Summary
Probabilistic models
Introduction
Generalization
Voting weight
1 Probabilistic models of vote
Introduction
Generalization
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
2 Voting configuration weight
3 Monte Carlo simulations
Algorithm
IC model
IAC model
4 Application to UE27
Treaty of Nice (1 key vote)
European Constitution (2 key vote)
5 Conclusion
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
4 de 24
Impartial Culture Model (IC)
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
•
Algorithm
IC model
IAC model
•
Application to UE27
Treaty of Nice
European Constitution
•
Conclusion
Voters votes independently from each other ‘yes’ or ‘no’
with probability 1/2.
2N vote configurations of equal weight
The Banzhaf index power is given by the decisive position
of a voter
•
Simple model
•
Allow easy computations
Using voting probabilistic models for the European Union case
15-16-17 mars 2007
5 de 24
Summary:
Probabilistic models
Introduction
Generalization
Voting weight
but
•
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
•
•
•
Described tied vote (108 voters produce a difference
‘yes/no’ ∼ 104 )
If N → ∞ the distribution of
votes is more an more
Pthe
N
A
piqued around 2 with A = i=1 ai
Do not allow to describe a consensus
The probability of acceptation decreases rapidly with N if
A
Q 6=
2
→ UE27 with Q = 75% A give only 2% of approval
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
6 de 24
Impartial Anonymous Culture Model (IAC)
Probabilistic models
Introduction
Generalization
•
Voting weight
•
Historical introduction : the vote has apparently an order
The Polya Urn : example with N = 4
MC simulations
1/2
Algorithm
IC model
IAC model
2/3
Application to UE27
1/2
1/3
1/3
2/3
Treaty of Nice
European Constitution
3/4
1/4
1/2
1/2
1/2
1/2
1/4
3/4
Conclusion
4/5
1/5
: 1/5
1/5
1/20
3/5
1/20
2/5
1/30
3/5
2/5
1/20 1/30
: 4 × 1/20 = 1/5
2/5
1/30
3/5
1/20
3/5
1/20
2/5
1/30
: 6 × 1/30 = 1/5
2/5
1/30
3/5
1/20
2/5
3/5
1/30 1/20
1/5
1/20
: 4 × 1/20 = 1/5
⇒ N + 1 voting configurations of equal probability
4/5
1/5
: 1/5
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
7 de 24
Generalization
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
(p1 , p2 , . . . pN ) : probability vector with pi probability that
voter i votes for the bill.
fN (p1 , p2 , . . . pN ) : probability distribution function of the
electorate body such that :
fN (p1 , p2 , . . . pN ) dp1 dp2 . . . dpN is the probability that
the probability of
•
•
•
•
voter 1 to vote for the bill belongs to [p1 , p1 + dp1 ] and
voter 2 to vote for the bill belongs to [p2 , p2 + dp2 ] and
...
voter N to vote for the bill belongs to [pN , pN + dpN ].
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
8 de 24
Reduced distribution function
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
fK (p1 , p2 , . . . pK ) =
Z
Z
fN (p1 , p2 , . . . , pK , . . . pN )dpK+1 . . . dpN
...
pK+1
pN
where fK (p1 , . . . pK )dp1 . . . dpK is the probability that the
probability of
•
voter 1 to vote for the bill belongs to [p1 , p1 + dp1 ] and
•
voter 2 to vote for the bill belongs to [p2 , p2 + dp2 ] and
•
...
•
voter K to vote for the bill belongs to [pK , pK + dpK ],
whatever the decisions of the voters K + 1 to N are.
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
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The one voter d.f.
Probabilistic models
Introduction
Generalization
The one voter d.f. reads
Voting weight
f (p) = f1 (p)
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
where f (p) dp is the probability that the probability of one
voter to vote for the bill belongs to [p, p + dp], whatever the
decisions of the other voters are.
Remarks :
Conclusion
•
•
•
For f (p) there is no correlation between voters,
The behavior of each voter is picked into f (p) (Straffin’s
independence assumption),
Pair correlations are taken into account defining f (p1 , p2 ).
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
Voting configuration weight
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
The weight of a given ‘n yes’ configuration is given by
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
Pc (n) =
Z
0
1
f (p) pn (1 − p)N −n dp
n P (n) is the probability of any ‘n’ yes
→ P (n) = CN
c
configuration.
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
11 de 24
Special cases
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
IC model
1/2N
f (p) = δ(p − 12 )
n
f (p) = 1
IAC model
1/[(N + 1)CN
]
1
1
1/2
• f (p) = 2 δ(p) + 2 δ(p − 1) Consensual Culture
•
•
Application to UE27
f(p)
Treaty of Nice
European Constitution
IC
UC
Conclusion
IAC
0
1/2
1
p
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
12 de 24
Remarks
Probabilistic models
Introduction
Generalization
Voting weight
•
IAC case : f (p) = 1
MC simulations
•
Algorithm
IC model
IAC model
•
Application to UE27
Treaty of Nice
European Constitution
•
Conclusion
•
1
1
: no bias for a large number of elections
2
Z0 1/2
Z 1
1
f (p)dp =
f (p)dp = : equal chance to vote
2
0
1/2
’yes’ or ’no’
Maximization of the uncertainty (Shannon entropy)
Z
pf (p)dp =
IC case : f (p) = δ(p − 1/2)
•
describe of an undecided electorate
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
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Meaning of f (p)
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Pc (n) =
R1
with γ =
0
f (p)pn (1 − p)N −n dp =
Conclusion
0
f (p)[pγ (1 − p)1−γ ]N dp
n
N
In the limit
Application to UE27
Treaty of Nice
European Constitution
R1
[pγ (1 − p)1−γ ]N
n
N → ∞
fixed,
but γ =
n → ∞
N
is more and more picked around p = γ,
then Pc (n) ∼ f (γ)
and P (n) =
R1
0
n
pn (1 − p)N −n dp = f ( N
) C1n
n
1
N f(N )
→ the percentage of ’yes’ votes gives f (p) itself
N
1
N
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
14 de 24
The asymptotic limit
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
When
N →∞
mandates scattered enough
we have
1 key vote P (x) =
1
A f (x/A)
with A =
Conclusion
2 key vote P (x, y) =
with A =
PN
i=1 ai
x
1
A B δ( A
and B =
PN
−
i=1 bi
PN
y
B )f (x/A
i=1 ai
= y/B)
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
15 de 24
Numerical Simulations : Monte Carlo method
Probabilistic models
Introduction
Generalization
•
N electors give 2N configurations
•
if N large enumeration is not possible
Voting weight
MC simulations
Algorithm
IC model
IAC model
→ Monte Carlo method
Algorithm
Application to UE27
Treaty of Nice
European Constitution
Conclusion
1. Pick a random number p accordingly to f (p)
2. for i = 1, N : pick a random number r ∈ [0, 1[ into a
uniform distribution function
. if r ≤ p voter i vote ‘yes’
. if r > p voter i vote ‘no’
3. go to point 1 for another election
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
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Simulations
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
•
Application to UE27
•
Treaty of Nice
European Constitution
•
Conclusion
N = 100 voters (1030 configurations), and N = 1, 000
voters
50, 000 elections
2 key vote case :
. Mandates a equal to 1
. Mandates b chosen at random in a ratio 1 to 5
Using voting probabilistic models for the European Union case
15-16-17 mars 2007
17 de 24
Summary:
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
N = 100
N = 1000
M = 50, 000 elections, Key A : equal mandates, Key B :
mandates taken at random in a uniform distribution with a
ratio 1 to 5, all mandates normalized to 100.
Using voting probabilistic models for the European Union case
15-16-17 mars 2007
18 de 24
Summary:
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
N = 100
N = 1000
M = 50, 000 elections, Key A : equal mandates, Key B :
mandates taken at random in a uniform distribution with a
ratio 1 to 5, all mandates normalized to 100.
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
19 de 24
Treaty of Nice (1 key vote)
Probabilistic models
Introduction
Generalization
Constitutionally : 3 keys
Voting weight
Population : 62 % of the population
MC simulations
State : state majority
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
Mandates : Q = 74.8 % of the mandates with ai ∼
√
pi
Citoyen
Pop. Mandates State Number Percentage
0
0
0 62380460
46.48%
30.90%
0
0
1 41479780
0
.00%
0
1
0
0
1
1
8
.00%
1
0
0
4728388
3.52%
17.07%
1
0
1 22910326
1
1
0
16
.00%
1
1
1
2718750
2.03%
Mandats
Etat
227configurations
→ In practice : one key vote
Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
Results
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
IC
IAC
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Using voting probabilistic models for the European Union case
Summary:
European Constitution (2 key vote)
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
15-16-17 mars 2007
Two key vote : 2 legitimacy
Citizen : 65 % of the population
• State
: 55 % of states
•
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
Results
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
IC
IAC
MC simulation with M = 2, 700 elections
X state legitimacy, Y citizen legitimacy (normalized to 100)
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
Results
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
IC (12%)
IAC (35%)
MC simulation with M = 2, 700 elections
X state legitimacy, Y citizen legitimacy (normalized to 100)
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Using voting probabilistic models for the European Union case
Summary:
15-16-17 mars 2007
24 de 24
Conclusion
Probabilistic models
Introduction
Generalization
Voting weight
MC simulations
Algorithm
IC model
IAC model
Application to UE27
Treaty of Nice
European Constitution
Conclusion
Data analysis (USA presidential election) suggests the IAC
model
The asymptotic limit (N → ∞) gives analytical results
Power index could be derived
Generalization for a M issue vote defining
fN (p1 , p2 , . . . pM −1 )