Non-interference and social welfare orderings satisfying strong Pareto

and anonymityTsuyoshi Adachi

Waseda University

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)

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Non-interferenceLet be as follows: There exists such thati. andii. andiii. Then, .

’s utility is only changed.

x (x’) is better than y (y’) for in both of the pairs.

The changes have the same sign

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The left vector is better than the right vectorDecreasing“Non-interference” requires

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Previous results (difficulty)Complete Non-Interference (CNI):

For all such that , , , and , .

SP and CNI is dictatorial. (MV, 2009a)

( SP, AN, and CNI are not compatible.)

DecreasingIncreasing

Previous results (characterization)Individual Damege Principle (IDP):

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For all such that , , , , and , .

SP,AN,IDP (MV, 2009ab)

: the leximin ordering

Decreasing

Previous results (characterization)Individual Benefit Principle (IBP):

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For all such that , , , , and , .

SP,AN,IBP (MV, 2009ab)

: the leximax ordering

Increasing

Previous results (characterization)Uniform Non-Interference (UNI):

1. SP,AN,UNI implies .2. is SP,AN,UNI (MV, 2009a)

For all such that , , , and , .

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Let

The same value

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

Non-interference on D

For all such that , , , and , .

Non-interference on (NI on ):

i. ii. and

is the set of possible :

• CNI NI on • IDP NI on • IBP NI on • UNI NI on

The existing axiomsThe first quadrant

The third quadrant

• Lower-Half Noninterference (LNI) denotes NI on

Let

Lower-Half Noninterference

Remark : For all SP , LNI UNI Proof

i. [LNI UNI] is clearii. UNI LNI

• By Remark,

– SP, AN, LNI implies

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LNI and UNI

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The same value

By UNIBy SP

NI on NI on

Existing results (recosideration)• SP, AN, UNI implies .

NI on

• SP and CNI is dictarorial SP, AN, and CNI are not compatible• SP, AN, IDP

NI on

• SP, AN, IBP

NI on

Other examples

• SP, AN, and NI on ?

The key lemma

Lemma: Let be SP, AN. Then, i.[ is NI on with ]ii.[ is NI on with ]

Let

Leximin and Leximax orderings

Proposition 1: i. is NI on ii.( )

Theorem 1: i.[SP, AN, NI on ]

[ and ] ii.( )

○ 　　　　×

Compatibility with SP and AN

By Proposition 1, ×

By Lemma,

　　　 or non-existence

Theorem 2: NI on is compatible with SP and AN , , or 　 .

Incompatible

NI on is implied by IBP, IDP, or LNI

I. [SP, AN, NI on ]

II. [SP, AN, NI on ]

III. [ satisfies SP, AN, NI on ]

Characterization of SWOs

Theorem 3: The combination of SP, AN, and NI oni. characterizes , ii. characterizes , iii. is satisfied by , oriv. is incompatible.

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.

Individual 2

Individual 1

NI on implies [ ]

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

Individual 2

Individual 1

Step 1: For all in ,

By SP and AN,

By NI on ,

The same length

1

By AN,

Individual 2

Individual 11

1

By Step 1, for all in ,

By Step 1, For all in ,