1 9/11/2011 Aggregation of Binary - PowerPoint Presentation

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9/11/2011

Aggregation of Binary Evaluations without Manipulations

Dvir

Falik

Elad

Dokow

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9/11/2011

“Doctrinal paradox”

Majority rule is not consistent!

The defendant is guilty

The defendant was sane at the time

The defendant

killed the victim

001Judge 1010Judge 2111Judge 3

011Majority

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“Doctrinal paradox”

Assume that for solving this paradox the society decide only on p and q.

The defendant is guilty

The defendant was sane at the time

The defendant

killed the victim

001Judge 1010Judge 2111Judge 3

111Majority

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“Doctrinal paradox”

Judge 1 can declare 0 on p and manipulate the result of the third column .

The defendant is guilty

The defendant was sane at the time

The defendant

killed the victim

000Judge 1010Judge 2111Judge 3

010Majority

Slide5

Linear classification

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“Condorcet paradox” (1785)

Majority rule is not consistent!

IS c>a

IS b>c

IS a>b

0

11Judge 1101Judge 2110Judge 3

111Majority

Arrow Theorem: There is no function

which is IIA

paretian

and not dictatorial.

a>b>c

c>a>b

b>c>a

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9/11/2011

Gibbard

Satterhwaite theorem:

Social choice function:

Social welfare function:

GS theorem

: For any , there is no Social choice function which is onto A, and not

manipulatable

. 7

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Example:

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011

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010My opinionSocial aggregatorFacility location

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Example:

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010My opinionSocial aggregatorFull Manipulation

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Example:

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010My opinionSocial aggregatorFull ManipulationPartial Manipulation

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Example:

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010My opinionSocial aggregatorFull ManipulationPartial ManipulationHamming manipulation

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Example:

GS theorem

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010My opinion: c>a>bSocial aggregatorabc

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The modelA finite, non-empty set of issues K={

1

,…

,k}

A vector

is

an evaluation.The evaluations in are called feasible, the others are infeasible.In our example, (1,1,0) is feasible ; but (1,1,1) is infeasible.

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A society is a finite set .

A

profile

of feasible evaluations

is

an matrix all of whose rows lie in X.An aggregator for N over X is a mapping .

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Different definitions of Manipulation

Manipulation:

An aggregator f is

manipulatable

if there exists a judge

i,

an opinion , an evaluation , coordinate j, and a profile such that: partialPartial

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Different definitions of Manipulation

Manipulation:

An aggregator f is

manipulatable

if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that: fullFullAnd:We denote by and say that c is between a and b if . We denote by the set .

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Different definitions of Manipulation

Manipulation:

An aggregator f is

manipulatable

if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that: fullFull

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Different definitions of ManipulationAny other definition of manipulation should be between the

partial

and the

full

manipulation.

If is not partial manipulable then f is not full manipulable

.

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Hamming ManipulationHamming manipulation: An aggregator f is Hamming

manipulatable

if there exists a judge

i,

an

opinion , an evaluation , and a profile such that:

Hamming distance:

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Theorem (Nehiring

and Puppe, 2002): Social aggregator f is

PMF (partial manipulation free) if and only if f is IIA and monotonic.Theorem (Nehiring and Puppe, 2002):Every Social aggregator which is IIA, paretian and monotonic is dictatorial if and only if X is Totally Blocked. 9/11/2011

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Partial Manipulation

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Corollary (Nehiring

and Puppe, 2002):Every Social aggregator which is

PMF and paretian is dictatorial if and only if X is Totally Blocked. 9/11/201121

Partial Manipulation

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IIAAn aggregator is independent of irrelevant alternatives (IIA) if for every and any two profiles and satisfying

for all , we have

3

2

1

Judge 1Judge 2

Judge 3

aggregator

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Paretian

An aggregator is

Paretian

if we have whenever the profile is such that for all .

3

211Judge 11Judge 21

Judge 31aggregator

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Monotonic

An aggregator

is

IIA and

Monotonic

if for every coordinate j, if then for every we have . 3211Judge 10Judge 20

Judge 31aggregator

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Almost dictator function:

Fact: For any set is not

Hamming/full manipulatable.9/11/201125

Almost dictator

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Close to PMF (C-PMF)

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26An aggregator is C-PMF

if there exist a PMF

function

s.t

for every profile in which we have that .

is an IIA and monotonic function.

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Question: what are the conditions on such that there exists a C-PMF, Hamming\full non-manipulatable

social

aggregator?

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Let

be an IIA and Monotonic function. Let be a function with the following property: there isn’t any between and .

Let be a function with the following property: for every , .The sets of those function will be denoted byEasy to notice that 9/11/201128

Nearest

Neighbor

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Social welfare

maximizer

(SWM)9/11/201129One special function which is C-PMF and

depend

not only in the outcome of

but

on the whole profile

is the SWM aggregator.

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Full Manipulation Free aggregator

Theorem:

For any set

is not full

manipulatable

. Furthermore, if is annonymous, then is annonymous. Remark: This proposition doesn’t hold for in which . Theorem: For any set the SWM aggregator is not full manipulatable.

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Theorem 1:

For any set

9/11/201131 Hamming Nearest Neighbor

If

then judge

i

can’t manipulate by choosing instead of .

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Conclusions:

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32 Hamming Nearest Neighbor

1. An Hamming Nearest Neighbor function is not

manipulatable

on .

2. Manipulation can’t be too ‘far’.

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Hamming Nearest Neighbor

If then judge

i

can’t manipulate by choosing instead of .

Theorem 2:

For any set

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MIPE-minimally infeasiblepartial evaluation

Let , a vector

with

entries for issues in

J only

is a

J-evaluation.A MIPE is a J-evaluationfor some which is infeasible, but such that every restriction of x to a proper subset of J is feasible.

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Hamming Nearest Neighbor What happens in intermediate cases?

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Example

(P

or q)s

s

q

p

00000010000100110100110111101111

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Example

(p or q)s

3

s

4

q

2

p200000010000100110100110111101111Weighted columns:My opinion: 1 0 1 0684623751 1 1 086644553

5200110100110

1

0

1

0

1

Maj

:

0

0

1

1

0

1

0

0

1

1

1

1

0

1

1

1

Maj

:

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Lines, Cycles

Joint work with: Michal Feldman,

Reshef Mair, Ilan Nehama. 38

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Main Theorem

:

An onto aggregator f on the line is HMF if and only if it is monotonic and 1-SSI.

Main Theorem

: For sufficiently large cycles, any onto HMF aggregator is 1-dictatorial.