The men who weren’t even there · 2010. 9. 9. · Power Extending the model Axiomatisation Application Conclusion The men who weren’t even there Legislative voting with absentees

Power Extending the model Axiomatisation Application Conclusion

The men who weren’t even thereLegislative 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.


Voting power

Voting situation

Voters cast votes for or againstLegislation rules determine whether notion is passed or not

Examples: National parliaments, shareholder’s meetings, UNSecurity Council, EU Council of Ministers, IMF

Voting power

(Share of) decision probability.


Voting power 2 – Example

EEC Council ofMinisters, 1958, QMV.

Country weightBelgium 2France 4Germany 4Italy 4Luxemburg 1Netherlands 2Quota 12

β = ( 521 ,521 ,

321 ,

321 ,

521 ,0)

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


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 approval2 may be a Yes vote – when veto is not used

Yes Abstain No

Abstain

Yes NoModel 1 (Machover and Felsenthal, 1997) Model 2 (Braham and Steffen, 2002)

Two models3 options: Yes/No/AbstainVoters decide: Abstain/Vote. If latter: Yes/No


The model expanded: 2.

The voting statistics of theHungarian National Assembly listthose:

1 voting Yes2 voting No3 voting Abstain (F-M abstain)4 not voting (B-S abstain)

5 not present

The numbers do not add upBeing absent is a “non-strategicabstention.”


The model expanded: 2.

The voting statistics of theHungarian National Assembly listthose:

1 voting Yes2 voting No3 voting Abstain (F-M abstain)4 not voting (B-S abstain)5 not present

The numbers do not add upBeing absent is a “non-strategicabstention.”


Absenteeism elaborated

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


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) ∈ Γ̃ is a triple:

a vector of maximum weights wa quota function q : Nn0 −→ N0a probability distribution on all w ′ : 0 ≤ w ′ ≤ w vectors

A power measure κ : Γ −→ Rn

A power measure with absenteeism:

κ̃ : Γ̃ −→ Rn

κ̃i(p,q,w) =∑

∀i: 0≤w ′


Absenteeism: Properties

Power index=power measure normalised to 1

PropositionThe power index of i is 1 iff i is the sole player.The power index of i is 0 iff i has weight 0.

CorollaryA minority has a positive power.

Claim (proof pending)

For a fixed q, assume that the expected size of the majoritycoalition exceeds the quota. Ceteris paribus the larger theassembly the smaller the minority power

A large assembly is but a voting machine.


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 , 1vT indicates T wins in v .

v =∨

T∈Wv

uT =∨

T⊆N1vT uT

Let pv : pv ≥ 0 for all v ∈ Γ and∑

v∈Γ pv = 1.

Generalised voting game

ṽ =∑v∈Γ

pv v .


Axiomatisation – Axioms

The value κ : Γ̃→ RN satisfies

Efficiency: if ∀ṽ ∈ Γ̃: ṽ(N) =n∑

i=1κi(ṽ),

Symmetry: if ∀ṽ ∈ Γ̃ such that i ∼ṽ j : κi(ṽ) = κj(ṽ),Marginality: if ∀ṽ , w̃ ∈ Γ̃, ∀i ∈ N such that ṽ ′i = w̃ ′i :κi(ṽ) = κi(w̃).

The Shapley value meets Efficiency, Symmetry and Marginality.


Axiomatisation – Result

Theorem

On the class Γ̃ solution κ satisfies Efficiency, Symmetry andMarginality if and only if κ = φ.

Fact1: ṽ =∑

w∈Γ

∨T⊆N

pw1wT uT =∨

T⊆N

∑w∈Γ

pw1wT uT

1 Write generalised voting games as max of generalisedunanimity games

2 Proof by induction: Divide N into N1 and N2, where in N1players are dummy in some winning coaition

3 Remove coalition, result is weighted voting game, withsame marginality for all other players.

4 Use Marginality and inductive assumption to determinevalue.

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


The National Assembly, Hungary

Simple model: all MPs are present with probability p, nocorrelation. The value of a coalition can be given:

v(S) =wS∑q

(wSi

)pi(1− p)wS−i

where wS =∑

i:Ni∈S wi .In 2009 p = 91.55%

2009 2006 2005 1994party seats seats seats seatsFidesz 139 141 168 20FKGP - - - 26KDNP 22 23 - 22MDF 9 11 9 38MSzP 189 190 177 209SzDSz 18 20 20 70Indep’t 6 1 11 -


National Assembly – Power of the minority

2009 2005 1994party seats S-S new seats S-S new seats S-S newFidesz 36.3 3.3 11.6 43.5 23.3 31.6 5.2 0 2.3FKGP - - - - - - 6.8 0 2.3KDNP 5.7 3.3 8.6 - - - 5.7 0 2.3MDF 2.4 3.3 2.8 2.3 6.7 4.8 9.9 0 2.3MSzP 49.3 83.3 69.5 45.9 40.0 51.0 54.3 100 88.4SzDSz 4.7 3.3 5.7 5.2 23.3 6.7 18.2 0 2.3Indep’t 1.6 3.3 1.9 2.8 6.7 5.8 - - -

Values closer to size.


National Assembly – RobustnessPlay with the value of p.

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


Conclusion

What we haveAbsenteeism leads to generalised voting gamesGood news! Parliamentary democracy 6= periodic electeddictatorship

if your parliament is small enough and yourMPs lack discipline.

The Shapley value is axiomatised for generalised votinggames.

Things to doAxiomatise Banzhaf index/measure (seems difficult)Examples from a smaller parliament (with lazy MPs)More accurate predictions with differences in partydisciplinePartisan voting


Conclusion

What we haveAbsenteeism leads to generalised voting gamesGood news! Parliamentary democracy 6= periodic electeddictatorship if your parliament is small enough

and yourMPs lack discipline.

The Shapley value is axiomatised for generalised votinggames.



Conclusion

What we haveAbsenteeism leads to generalised voting gamesGood news! Parliamentary democracy 6= periodic electeddictatorship if your parliament is small enough and yourMPs lack discipline.The Shapley value is axiomatised for generalised votinggames.



SPAM! RePEc and other things

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PowerPreliminaries

Extending the modelAbstentionAbsenteeism

AxiomatisationAxiomatisation

ApplicationNational Assembly, Hungary

ConclusionConclusion

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The men who weren’t even there · 2010. 9. 9. · Power Extending the model Axiomatisation Application Conclusion The men who weren’t even there Legislative voting with absentees