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Power Extending the model Axiomatisation Application Conclusion The men who werent even there Legislative voting with absentees Lszl . Kczy 1 . Pintr 2 Mikls P 1 IEHAS & buda University, Budapest 2 Corvinus University

  1. Power Extending the model Axiomatisation Application Conclusion The men who weren’t even there Legislative voting with absentees László Á. Kóczy 1 . Pintér 2 Miklós P 1 IEHAS & Óbuda University, Budapest 2 Corvinus University Budapest Xth Meeting of the Society for Social Choice and Welfare, 21–24 July 2010.

  2. 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.

  3. 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.

  4. Power Extending the model Axiomatisation Application Conclusion The model expanded: 1. abstention Yes/no models do not cover all situations → ternary voting : allowing abstention : may be a No vote – reduces chance for approval 1 may be a Yes vote – when veto is not used 2 Two models 3 options: Yes/No/Abstain Voters decide: Abstain/Vote. If latter: Yes/No

  5. Power Extending the model Axiomatisation Application Conclusion The model expanded: 1. abstention Yes/no models do not cover all situations → ternary voting : allowing abstention : may be a No vote – reduces chance for approval 1 may be a Yes vote – when veto is not used 2 Two models 3 options: Yes/No/Abstain Voters decide: Abstain/Vote. If latter: Yes/No

  6. Power Extending the model Axiomatisation Application Conclusion The model expanded: 1. abstention Yes/no models do not cover all situations → ternary voting : allowing abstention : may be a No vote – reduces chance for approval 1 may be a Yes vote – when veto is not used 2 Abstain Yes Abstain No 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

  7. Power Extending the model Axiomatisation Application Conclusion The model expanded: 2. The voting statistics of the Hungarian National Assembly list those: voting Yes 1 voting No 2 voting Abstain (F-M abstain) 3 not voting (B-S abstain) 4 The numbers do not add up Being absent is a “non-strategic abstention.”

  8. Power Extending the model Axiomatisation Application Conclusion The model expanded: 2. The voting statistics of the Hungarian National Assembly list those: voting Yes 1 voting No 2 voting Abstain (F-M abstain) 3 not voting (B-S abstain) 4 The numbers do not add up Being absent is a “non-strategic abstention.”

  9. Power Extending the model Axiomatisation Application Conclusion The model expanded: 2. The voting statistics of the Hungarian National Assembly list those: voting Yes 1 voting No 2 voting Abstain (F-M abstain) 3 not voting (B-S abstain) 4 not present 5 The numbers do not add up Being absent is a “non-strategic abstention.”

  10. Power Extending the model Axiomatisation Application Conclusion The model expanded: 2. The voting statistics of the Hungarian National Assembly list those: voting Yes 1 voting No 2 voting Abstain (F-M abstain) 3 not voting (B-S abstain) 4 not present 5 The numbers do not add up Being absent is a “non-strategic abstention.”

  11. 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 , w 1 , . . . , w n ] or [ q , w ] With absent voters [ q ′ , w ′ 1 , . . . , w ′ n ] or [ q ′ , w ′ ] , where q ′ = q ′ ( w ′ n ) . 1 , . . . , w ′ E.g. q ′ = q , or q ′ : q ′ / � w ′ i = q / � w i .

  12. 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 : N n 0 − → N 0 a probability distribution on all w ′ : 0 ≤ w ′ ≤ w vectors A power measure κ : Γ − → R n A power measure with absenteeism: κ : ˜ → R n ˜ Γ − � p ( w ′ ) κ i ( q ′ ( w ′ ) , w ′ ) ˜ κ i ( p , q , w ) = ∀ i : 0 ≤ w ′ < w

  13. 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.

  14. 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.

  15. 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.

  16. 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 . u T is the unanimity game on T , 1 v T indicates T wins in v . � � 1 v v = u T = T u T T ∈W v T ⊆ N Let p v : p v ≥ 0 for all v ∈ Γ and � v ∈ Γ p v = 1. Generalised voting game � p v v . ˜ v = v ∈ Γ

  17. Power Extending the model Axiomatisation Application Conclusion Axiomatisation – Axioms Γ → R N satisfies The value κ : ˜ n v ∈ ˜ Efficiency: if ∀ ˜ Γ : ˜ v ) , κ i (˜ v ( N ) = � i = 1 v j : κ i (˜ Symmetry: if ∀ ˜ v ∈ ˜ Γ such that i ∼ ˜ v ) , v ) = κ j (˜ w ∈ ˜ Marginality: if ∀ ˜ Γ , ∀ i ∈ N such that ˜ i : v , ˜ v ′ i = ˜ w ′ w ) . κ i (˜ v ) = κ i ( ˜ The Shapley value meets Efficiency, Symmetry and Marginality.

  18. 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: ˜ p w 1 w p w 1 w v = � � T u T = � � T u T w ∈ Γ T ⊆ N T ⊆ N w ∈ Γ Write generalised voting games as max of generalised 1 unanimity games Proof by induction: Divide N into N 1 and N 2 , where in N 1 2 players are dummy in some winning coaition Remove coalition, result is weighted voting game, with 3 same marginality for all other players. Use Marginality and inductive assumption to determine 4 value. In N 2 veto players get the same value by Symmetry, which, 5 by Efficiency is the Shapley value.

  19. 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: ˜ p w 1 w p w 1 w v = � � T u T = � � T u T w ∈ Γ T ⊆ N T ⊆ N w ∈ Γ Write generalised voting games as max of generalised 1 unanimity games Proof by induction: Divide N into N 1 and N 2 , where in N 1 2 players are dummy in some winning coaition Remove coalition, result is weighted voting game, with 3 same marginality for all other players. Use Marginality and inductive assumption to determine 4 value. In N 2 veto players get the same value by Symmetry, which, 5 by Efficiency is the Shapley value.

  20. 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: w S � w S � � p i ( 1 − p ) w S − i v ( S ) = i q where w S = � i : N i ∈ S w i . 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 -

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