Non-interference and social welfare orderings satisfying strong Pareto and anonymity

1. Non-interference and social welfare orderings satisfying strong Pareto and anonymity Tsuyoshi Adachi Waseda University

2. A social welfare ordering : a reflexive, complete and transitive binary relation on : utility vectors • Axioms • Efficiency • Strong Pareto (SP): • Impartiality • Anonymity (AN): • If for some permutation , then . • Noninterference (Mariotti & Veneziani, 2009a)

3. Non-interference ’s utility is only changed. Let be as follows: There exists such that • and • and Then, . x (x’) is better than y (y’) for in both of the pairs. The changes have the same sign “Non-interference” requires Decreasing The left vectoris better than the right vector

4. Previous results (difficulty) Complete Non-Interference (CNI): For all such that , , , and , . Decreasing Increasing SP and CNI is dictatorial. (MV, 2009a) ( SP, AN, and CNI are not compatible.)

5. Previous results (characterization) Individual Damege Principle (IDP): For all such that , , , , and , . Decreasing : the leximin ordering SP,AN,IDP(MV, 2009ab)

6. Previous results (characterization) Individual Benefit Principle (IBP): For all such that , , , , and , . Increasing : the leximax ordering SP,AN,IBP(MV, 2009ab)

7. Previous results (characterization) Uniform Non-Interference (UNI): For all such that , , , and , . The same value Let SP,AN,UNI implies . is SP,AN,UNI (MV, 2009a)

8. MV (2009ab) • SP, AN and CNI are not compatible. • IDP, IBP, and UNI (as restrictions of CNI) • Characterization of (resp. ) without equity axioms (resp. inequity axioms) Hammond (1976), D’Aspremont and Gevers (1977) • Our questions: • Restrictions of CNI other than IDP, IBP, and UNI. • SWOs characterized by the combination of such axioms Introduction of a general class of non-interference axioms

9. Non-interference on D Non-interference on (NI on ): For all such that , , , and , . is the set of possible : and

10. The existing axioms The first quadrant • CNINI on • IDPNI on • IBPNI on • UNINI on The third quadrant

11. Lower-Half Noninterference Let • Lower-Half Noninterference (LNI) denotes NI on

12. LNI and UNI Remark : For all SP , LNIUNI Proof • [LNIUNI] is clear • UNILNI • By Remark, • SP, AN, LNIimplies is smaller then NI on NI on By SP By UNI The same value

13. Existing results (recosideration) • SP, AN, IDP • SP, AN, UNI implies . • SP and CNIis dictarorial • SP, AN, and CNI are not compatible NI on • SP, AN, IBP NI on NI on NI on

14. Other examples • SP, AN, and NI on ?

15. The key lemma Let Lemma: Let be SP, AN. Then, [isNI on with ] [ is NI on with ]

16. Leximin and Leximax orderings Proposition 1: is NI on ( ) • Theorem 1: • [SP, AN, NI on ] • [ and ] • ( ) × ○

17. Compatibility with SP and AN By Lemma, or non-existence By Proposition 1, × • Theorem 2:NI on is compatible with SP and AN • , , or . Incompatible NI on is implied by IBP, IDP, or LNI

18. Characterization of SWOs • [SP, AN, NI on ] • [SP, AN, NI on ] • [ satisfies SP, AN, NI on ] • Theorem 3: The combination of SP, AN, and NI on • characterizes , • characterizes , • is satisfied by , or • is incompatible.

19. Conclusions • Generalized class of NI axioms • Characterization of NI axioms compatible with SP and AN. • A NI axiom is compatible iff it is implied by IBP, IDP, or LNI • SWOs characterized by SP, AN, and a NI axiom. • The leximin, leximax, and utilitarian orderings have an important role.

20. Proof of the Lemma • The two person case : Individual 1 and 2. • Let : i.e., there exists such that . • (Note that for all ) • Let be SP, AN, NI on NI on implies [ ] Individual 2 Individual 1

21. Step 1: For all in , Individual 2 By SP and AN, By NI on , By AN, The same length Individual 1 1

22. By Step 1, By Step 1, for all in , For all in , Individual 2 1 Individual 1 1