Between Liberalism and Democracy D. Samet , D. Schmeidler

1. BetweenLiberalism and DemocracyD. Samet, D. Schmeidler www.tau.ac.il/~samet

2. Sen’s Liberalism Paradox Who should be allowed to read Lady Chatterley’s Lover ?

3. Sen’s Liberalism Paradox  ≻{R} ≻ {B} R’spreferences: {R} ≻{B} ≻ B’spreferences: Liberal social preferences: Social alternatives are subsetsof individuals – the qualified ones. {B}  {R} ≻ ≻ Pareto social preferences: {R}≻ {B}

4. The model N = {1,…,n} individuals i’s vote A profile – nxn matrix P of 0,1. j k i 1 . . . . . 0 idoes notqualifyk P = iqualifiesj A rule: f(P) = (f1(P), . . . , fn(P)), a 0,1 vector, designating the socially qualified.

5. Monotonicity Individuals qualify additional persons no one looses social qualification. P≥P’ f(P) ≥ f(P’)

6. Independence j’s qualification depends onlyon how individuals qualify her. Pij = P’ij for all i fj(P) ≥ fj(P’)

7. Symmetry Dale Pat Dale Pat I Pat is qualified Dale and are qualified Let  be a permutation of {1, . . . , n} The profile P is defined by (P)ij = P(i) (j) The rule f is defined by (f)j = f(j) f(P) = f(P)

8. Consent rules t . Fix integers s and t. s – the quota for qualification t – the quota for disqualification The quotas satisfy: s, t 1 s + t  n + 2 n+2 . n+1 . 1 . . . s 1 n+1 n+2 The consent rule fst : One’s self qualification wins social consent iffs individuals support it. One’s self disqualification wins social consent ifft individuals support it.

9. Consent rules The consent rule fst : One’s self qualification wins social consent iffs individuals support it. One’s self disqualification wins social consent ifft individuals support it. Pjj = 1: fj(P) = 1 |{i : Pij = 1}| ≥ s Pjj = 0: fj(P) = 0 |{i : Pij = 0}| ≥ t TheoremA rule satisfies monotonicity, independence, and symmetryiffit is a consent rule.

10. Duality Notation: for x ∈ {0,1}, ¯ = 1 – x For a profile P,¯ = ( ¯¯ ) For a rule f , ¯(P) = ( ¯¯ ) x P Pij f fi(P) f( ¯ ) = ¯(P) f P TheoremA rule satisfies monotonicity, independence, symmetry and dualityiffit is a consent rule of the form fss.

11. Special rules t . n+2 . n+1 . 1 . . . 1 n+1 n+2 • The size of the quotas s,t reflects the power of society • The difference |s-t| reflects the asymmetry of qualification and disqualification. s The size and the difference are the smallest for s = t = 1. The liberal rule f11 : fj11(P) = Pjj

12. Special rules t . n+2 . n+1 . 1 . . . 1 n+1 n+2 The choice between qualification and disqualification of j is determined by simple majority where j’s vote for herself counts like all other votes for her. • The size of the quotas s,t reflects the power of society • The difference |s-t| reflects the asymmetry of qualification and disqualification. These are the rules that bridge between liberalism and democracy (majoritarian rule) s Maximal societal influence for equal quotas ... The majoritarian rule fss for odd n and s = (n+1)/2. fjss(P) = x  |{i : Pij = x}| ≥ (n+1)/2

13. Special rules t . n+2 . n+1 . 1 . . . 1 n+1 n+2 • The size of the quotas s,t reflects the power of society • The difference |s-t| reflects the asymmetry of qualification and disqualification. The choice between qualification and disqualification of j is determined by simple majority of all the voters other than j. s Maximal societal influence for equal quotas ... The majoritarian rule fss for even n and s = n/2 + 1. fjss(P) = x  |{i j : Pij = x}| ≥ n/2 + 1

14. Special rules t . n+2 . n+1 . 1 . . . 1 n+1 n+2 • The size of the quotas s,t reflects the power of society • The difference |s-t| reflects the asymmetry of qualification and disqualification. s The most extreme case of asymmetry between qualification and disqualification is attained at (n+1,1) and (1,n+1). At (n+1,1) players are disqualified no matter how they vote. At (1,n+1) players are qualified no matter how they vote.

15. Self determination “Let Hobbits determine who is a Hobbit.” J.R.R. Tolkien Theorem: The only rule that satisfies monotonicity, independence, non-degeneracy, and self-determination is the liberal rule. Self determination If non-Hobbits change their votes about Hobbits, then the set of Hobbits does not change. Non-degeneracy For each j there are profiles P and Q such that fj(P) = 1 and fj(Q) = 0.