The men who werent even there Legislative voting with absentees Lszl - - PowerPoint PPT Presentation

Power Extending the model Axiomatisation Application Conclusion

The men who weren’t even there

Legislative voting with absentees László Á. Kóczy1 Miklós P . Pintér2

1IEHAS & Óbuda University, Budapest 2Corvinus University Budapest

Xth Meeting of the Society for Social Choice and Welfare, 21–24 July 2010.

Power Extending the model Axiomatisation Application Conclusion

Voting power

Voting situation Voters cast votes for or against Legislation rules determine whether notion is passed or not Examples: National parliaments, shareholder’s meetings, UN Security Council, EU Council of Ministers, IMF Voting power (Share of) decision probability.

Power Extending the model Axiomatisation Application Conclusion

Voting power 2 – Example

EEC Council of Ministers, 1958, QMV. Country weight Belgium 2 France 4 Germany 4 Italy 4 Luxemburg 1 Netherlands 2 Quota 12 β = ( 5

21, 5 21, 3 21, 3 21, 5 21, 0)

Power index (Normalised) a priori measure of voting power.

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 1. abstention

Yes/no models do not cover all situations → ternary voting: allowing abstention:

1

may be a No vote – reduces chance for approval

2

may be a Yes vote – when veto is not used Two models 3 options: Yes/No/Abstain Voters decide: Abstain/Vote. If latter: Yes/No

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 1. abstention

Yes/no models do not cover all situations → ternary voting: allowing abstention:

1

may be a No vote – reduces chance for approval

2

may be a Yes vote – when veto is not used Two models 3 options: Yes/No/Abstain Voters decide: Abstain/Vote. If latter: Yes/No

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 1. abstention

Yes/no models do not cover all situations → ternary voting: allowing abstention:

1

may be a No vote – reduces chance for approval

2

may be a Yes vote – when veto is not used

Yes Abstain No Abstain Yes No Model 1 (Machover and Felsenthal, 1997) Model 2 (Braham and Steffen, 2002)

Two models 3 options: Yes/No/Abstain Voters decide: Abstain/Vote. If latter: Yes/No

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 2.

The voting statistics of the Hungarian National Assembly list those:

1

voting Yes

2

voting No

3

voting Abstain (F-M abstain)

4

not voting (B-S abstain) The numbers do not add up Being absent is a “non-strategic abstention.”

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 2.

The voting statistics of the Hungarian National Assembly list those:

1

voting Yes

2

voting No

3

voting Abstain (F-M abstain)

4

not voting (B-S abstain) The numbers do not add up Being absent is a “non-strategic abstention.”

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 2.

The voting statistics of the Hungarian National Assembly list those:

1

voting Yes

2

voting No

3

voting Abstain (F-M abstain)

4

not voting (B-S abstain)

5

not present The numbers do not add up Being absent is a “non-strategic abstention.”

Power Extending the model Axiomatisation Application Conclusion

The model expanded: 2.

The voting statistics of the Hungarian National Assembly list those:

1

voting Yes

2

voting No

3

voting Abstain (F-M abstain)

4

not voting (B-S abstain)

5

not present The numbers do not add up Being absent is a “non-strategic abstention.”

Power Extending the model Axiomatisation Application Conclusion

Absenteeism elaborated

Some points Absent = ill, busy elsewhere. Non-strategic decision. Absent voter (party) = abstaining voter (party) In legislative voting a party’s weight is the nr of MP’s

• present. Here:

absent (individual) voter = smaller weight for party Weighted voting game [q, w1, . . . , wn] or [q, w] With absent voters [q′, w′

1, . . . , w′ n] or [q′, w′], where

q′ = q′(w′

1, . . . , w′ n).

E.g. q′ = q, or q′ : q′/ w′

i = q/ wi.

Power Extending the model Axiomatisation Application Conclusion

Absenteeism continued

Consider game [q, w] ∈ Γ (Γ = n-player weighted voting games) Induced game [q′, w′] with probability p(w′). A generalised weighted voting game (p, q, w) ∈ ˜ Γ is a triple: a vector of maximum weights w a quota function q : Nn

0 −

→ N0 a probability distribution on all w′ : 0 ≤ w′ ≤ w vectors A power measure κ : Γ − → Rn A power measure with absenteeism: ˜ κ : ˜ Γ − → Rn ˜ κi(p, q, w) =

• ∀i: 0≤w′<w

p(w′)κi(q′(w′), w′)

Power Extending the model Axiomatisation Application Conclusion

Absenteeism: Properties

Power index=power measure normalised to 1 Proposition The power index of i is 1 iff i is the sole player. The power index of i is 0 iff i has weight 0. Corollary A minority has a positive power. Claim (proof pending) For a fixed q, assume that the expected size of the majority coalition exceeds the quota. Ceteris paribus the larger the assembly the smaller the minority power A large assembly is but a voting machine.

Power Extending the model Axiomatisation Application Conclusion

Absenteeism: Properties

Power index=power measure normalised to 1 Proposition The power index of i is 1 iff i is the sole player. The power index of i is 0 iff i has weight 0. Corollary A minority has a positive power. Claim (proof pending) For a fixed q, assume that the expected size of the majority coalition exceeds the quota. Ceteris paribus the larger the assembly the smaller the minority power A large assembly is but a voting machine.

Power Extending the model Axiomatisation Application Conclusion

Absenteeism: Properties

Power index=power measure normalised to 1 Proposition The power index of i is 1 iff i is the sole player. The power index of i is 0 iff i has weight 0. Corollary A minority has a positive power. Claim (proof pending) For a fixed q, assume that the expected size of the majority coalition exceeds the quota. Ceteris paribus the larger the assembly the smaller the minority power A large assembly is but a voting machine.

Power Extending the model Axiomatisation Application Conclusion

Axiomatisation – Setup

Based on Dubey (IJGT, 1975) and Young (IJGT, 1985). Notation: v ∨ w = max {v, w} if v, w ∈ Γ. uT is the unanimity game on T, 1v

T indicates T wins in v.

v =

• T∈Wv

uT =

• T⊆N

1v

TuT

Let pv: pv ≥ 0 for all v ∈ Γ and

v∈Γ pv = 1.

Generalised voting game ˜ v =

• v∈Γ

pvv .

Power Extending the model Axiomatisation Application Conclusion

Axiomatisation – Axioms

The value κ : ˜ Γ → RN satisfies Efficiency: if ∀˜ v ∈ ˜ Γ: ˜ v(N) =

n

• i=1

κi(˜ v), Symmetry: if ∀˜ v ∈ ˜ Γ such that i ∼˜

v j: κi(˜

v) = κj(˜ v), Marginality: if ∀˜ v, ˜ w ∈ ˜ Γ, ∀i ∈ N such that ˜ v′

i = ˜

w′

i :

κi(˜ v) = κi( ˜ w). The Shapley value meets Efficiency, Symmetry and Marginality.

Power Extending the model Axiomatisation Application Conclusion

Axiomatisation – Result

Theorem On the class ˜ Γ solution κ satisfies Efficiency, Symmetry and Marginality if and only if κ = φ. Fact1: ˜ v =

w∈Γ

• T⊆N

pw1w

T uT = T⊆N

• w∈Γ

pw1w

T uT

1

Write generalised voting games as max of generalised unanimity games

2

Proof by induction: Divide N into N1 and N2, where in N1 players are dummy in some winning coaition

3

Remove coalition, result is weighted voting game, with same marginality for all other players.

4

Use Marginality and inductive assumption to determine value.

5

In N2 veto players get the same value by Symmetry, which, by Efficiency is the Shapley value.

Power Extending the model Axiomatisation Application Conclusion

Axiomatisation – Result

Theorem On the class ˜ Γ solution κ satisfies Efficiency, Symmetry and Marginality if and only if κ = φ. Fact1: ˜ v =

w∈Γ

• T⊆N

pw1w

T uT = T⊆N

• w∈Γ

pw1w

T uT

1

Write generalised voting games as max of generalised unanimity games

2

Proof by induction: Divide N into N1 and N2, where in N1 players are dummy in some winning coaition

3

Remove coalition, result is weighted voting game, with same marginality for all other players.

4

Use Marginality and inductive assumption to determine value.

5

In N2 veto players get the same value by Symmetry, which, by Efficiency is the Shapley value.

Power Extending the model Axiomatisation Application Conclusion

The National Assembly, Hungary

Simple model: all MPs are present with probability p, no

• correlation. The value of a coalition can be given:

v(S) =

wS

• q

wS i

• pi(1 − p)wS−i

where wS =

i:Ni∈S wi.

In 2009 p = 91.55%

2009 2006 2005 1994 party seats seats seats seats Fidesz 139 141 168 20 FKGP

• 26

KDNP 22 23

• 22

MDF 9 11 9 38 MSzP 189 190 177 209 SzDSz 18 20 20 70 Indep’t 6 1 11

Power Extending the model Axiomatisation Application Conclusion

National Assembly – Power of the minority

2009 2005 1994 party seats S-S new seats S-S new seats S-S new Fidesz 36.3 3.3 11.6 43.5 23.3 31.6 5.2 2.3 FKGP

• 6.8

2.3 KDNP 5.7 3.3 8.6

• 5.7

2.3 MDF 2.4 3.3 2.8 2.3 6.7 4.8 9.9 2.3 MSzP 49.3 83.3 69.5 45.9 40.0 51.0 54.3 100 88.4 SzDSz 4.7 3.3 5.7 5.2 23.3 6.7 18.2 2.3 Indep’t 1.6 3.3 1.9 2.8 6.7 5.8

• Values closer to size.

Power Extending the model Axiomatisation Application Conclusion

National Assembly – Robustness

Play with the value of p. p = seats 1 0.95 0.9 0.85 0.8 0.75 0.7 MSzP 49.3 83.3 75.4 66.4 58.6 53.0 50.2 50.0 Fidesz 36.3 3.3 7.7 14.2 24.2 38.0 49.0 50.0 KDNP 5.7 3.3 7.6 8.5 7.6 3.0 0.2 0.0 SzDSz 4.7 3.3 6.9 5.4 6.1 2.9 0.2 0.0 MDF 2.4 3.3 1.3 3.5 2.1 1.9 0.2 0.0 indep’t 1.6 3.3 1.2 2.1 1.5 1.3 0.2 0.0

Power Extending the model Axiomatisation Application Conclusion

Conclusion

What we have Absenteeism leads to generalised voting games Good news! Parliamentary democracy = periodic elected dictatorship The Shapley value is axiomatised for generalised voting games. Things to do Axiomatise Banzhaf index/measure (seems difficult) Examples from a smaller parliament (with lazy MPs) More accurate predictions with differences in party discipline Partisan voting

Power Extending the model Axiomatisation Application Conclusion

Conclusion

What we have Absenteeism leads to generalised voting games Good news! Parliamentary democracy = periodic elected dictatorship The Shapley value is axiomatised for generalised voting games. Things to do Axiomatise Banzhaf index/measure (seems difficult) Examples from a smaller parliament (with lazy MPs) More accurate predictions with differences in party discipline Partisan voting

Power Extending the model Axiomatisation Application Conclusion

Conclusion

What we have Absenteeism leads to generalised voting games Good news! Parliamentary democracy = periodic elected dictatorship if your parliament is small enough The Shapley value is axiomatised for generalised voting games. Things to do Axiomatise Banzhaf index/measure (seems difficult) Examples from a smaller parliament (with lazy MPs) More accurate predictions with differences in party discipline Partisan voting

Power Extending the model Axiomatisation Application Conclusion

Conclusion

What we have Absenteeism leads to generalised voting games Good news! Parliamentary democracy = periodic elected dictatorship if your parliament is small enough and your MPs lack discipline. The Shapley value is axiomatised for generalised voting games. Things to do Axiomatise Banzhaf index/measure (seems difficult) Examples from a smaller parliament (with lazy MPs) More accurate predictions with differences in party discipline Partisan voting

Power Extending the model Axiomatisation Application Conclusion

Conclusion

What we have Absenteeism leads to generalised voting games Good news! Parliamentary democracy = periodic elected dictatorship if your parliament is small enough and your MPs lack discipline. The Shapley value is axiomatised for generalised voting games. Things to do Axiomatise Banzhaf index/measure (seems difficult) Examples from a smaller parliament (with lazy MPs) More accurate predictions with differences in party discipline Partisan voting

Power Extending the model Axiomatisation Application Conclusion

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