White Manipulation in Judgment Aggregation Gabriella Pigozzi - - PowerPoint PPT Presentation

University of Tsukuba, LGS’09, 26 - 29 August 2009

White Manipulation in Judgment Aggregation

Gabriella Pigozzi Marija Slavkovik Davide Grossi

ILLC Amsterdam

judgment aggregation (JA) has two problems: aggregation functions that satisfy a desirable set of properties do not exist aggregation operators that exist are manipulable the question is: is lying, cheating and manipulation really that bad ?

2

What is this all about

LGS’09, University of Tsukuba , 26 - 29 August 2009

the colloquial term “white lies”

3

White Manipulability

LGS’09, University of Tsukuba , 26 - 29 August 2009

the colloquial term “white lies”

3

White Manipulability

LGS’09, University of Tsukuba , 26 - 29 August 2009

the colloquial term “white lies”

3

White Manipulability

LGS’09, University of Tsukuba , 26 - 29 August 2009

the colloquial term “white lies”

3

White Manipulability

manipulation - lying with the intent to improve the

• utcome for the agent who lies

LGS’09, University of Tsukuba , 26 - 29 August 2009

white manipulation - lying with the intent to improve the outcome for all the agents involved the colloquial term “white lies”

3

White Manipulability

manipulation - lying with the intent to improve the

• utcome for the agent who lies

LGS’09, University of Tsukuba , 26 - 29 August 2009

4

In the rest of the talk

introduce the basic concepts of judgment aggregation redefine the judgment aggregation function introduce in JA: scoring functions, social welfare notions define white manipulation initial results conclusions

LGS’09, University of Tsukuba , 26 - 29 August 2009

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) f : profiles − → judgment sets

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) f : profiles − → judgment sets

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) f : profiles − → judgment sets

5

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) f : profiles − → judgment sets impasse

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are not manipulable if they satisfy independence and (weak) monotonicity[1]

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

6

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are not manipulable if they satisfy independence and (weak) monotonicity[1]

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

6

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are not manipulable if they satisfy independence and (weak) monotonicity[1]

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

6

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are not manipulable if they satisfy independence and (weak) monotonicity[1]

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

6

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are manipulable if they satisfy independence and (weak) monotonicity*

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the premise based procedure

7

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are manipulable if they satisfy independence and (weak) monotonicity*

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the premise based procedure

7

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are manipulable if they satisfy independence and (weak) monotonicity*

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the premise based procedure

7

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are manipulable if they satisfy independence and (weak) monotonicity*

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the premise based procedure

7

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

judgment aggregation functions are manipulable if they satisfy independence and (weak) monotonicity*

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the premise based procedure the premise based procedure is manipulable [2]

7

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

the conclusion based procedure

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

8

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

the conclusion based procedure

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

8

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

the conclusion based procedure

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

8

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

the conclusion based procedure

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) the conclusion based procedure is manipulable [3]

8

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

distance based merging

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

9

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

distance based merging

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b)

9

how individual judgments on logically connected issues can be aggregated into a collective judgment on the same issues

LGS’09, University of Tsukuba , 26 - 29 August 2009

Judgment Aggregation

distance based merging

a = X is good at teaching b = X is good at research x = hire X

• prof. A

yes no no

• prof. B

yes yes yes

• prof. C

no yes no Majority yes yes no

hiring committee example with rule : x ↔ (a ∧ b) distance based merging is manipulable [4]

9

10

The ideas are ...

impasse is in the set of possible outputs of the aggregation function, but not part of any profile assume that agents have preferences over outputs and neither of the agents prefers the output scoring functions determine preference ordering over the elements of the set of possible outputs of the aggregation function

↓ ↓

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

JA function we defined as example we work with - quota rule score function we define as a function that, given a judgment set, scores all other possible outcomes based

• n that judgment set

we work with examples of distance based scoring functions:

11

JA function & scoring functions

f : Ω − → Φ ∪ {↓} f q HSe : Φ − → (Φ↓ − → N) V Sk : Φ − → (Φ↓ − → N)

LGS’09, University of Tsukuba , 26 - 29 August 2009

• Definition. [Manipulation] Let s be a scoring function.

An aggregation function f is manipulable if and only if there exists a judgment profile ω ∈ Ω and an agent i such that f(ω) ≺s

i f(ω′), where ω′ ∈ Ω is some i-variant

• f ω.

Utilitarian social welfare Egalitarian social welfare

12

Social welfare notions in JA

LGS’09, University of Tsukuba , 26 - 29 August 2009

USW s(ω)(ϕ) = n

i=1 s(ϕ)(ϕi)

ESW s(ω)(ϕ) = max{s(ϕ)(ϕi) | ϕi ∈ Φ}

using a scoring function, a preference profile can be built from a judgment profile having a preference profile, social welfare notions can be applied to JA

Utilitarian social welfare Egalitarian social welfare

12

Social welfare notions in JA

LGS’09, University of Tsukuba , 26 - 29 August 2009

USW s(ω)(ϕ) = n

i=1 s(ϕ)(ϕi)

ESW s(ω)(ϕ) = max{s(ϕ)(ϕi) | ϕi ∈ Φ}

using a scoring function, a preference profile can be built from a judgment profile having a preference profile, social welfare notions can be applied to JA

Utilitarian social welfare Egalitarian social welfare

12

Social welfare notions in JA

LGS’09, University of Tsukuba , 26 - 29 August 2009

USW s(ω)(ϕ) = n

i=1 s(ϕ)(ϕi)

ESW s(ω)(ϕ) = max{s(ϕ)(ϕi) | ϕi ∈ Φ}

using a scoring function, a preference profile can be built from a judgment profile having a preference profile, social welfare notions can be applied to JA

• Definition. [White manipulability] Let SW be a social welfare

function and s a scoring function. An aggregation function f is white manipulable if and only if there exists an agent i and a judgment profile ω ∈ Ω such that f(ω) ≺s

i f(ω′) and

SW(f(ω)) < SW(f(ω′)), where ω′ ∈ Ω is some i-variant of ω.

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

x x ( )

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

x x ( ) 20 possible

profiles

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

x x ( )

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

(1, 0, 0) ≻HSe

2

(0, 0, 0) ≻HSe

2

(0, 1, 0) ∼HSe

2

(1, 1, 1) ≻HSe

2

↓ (1, 1, 1) ≻HSe

1

(1, 0, 0) ∼HSe

1

(0, 1, 0) ≻HSe

1

(0, 0, 0) ≻HSe

1

↓ (0, 1, 0) ≻HSe

3

(0, 0, 0) ≻HSe

3

(1, 0, 0) ∼HSe

3

(1, 1, 1) ≻HSe

3

↓ x x ( )

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

(1, 0, 0) ≻HSe

2

(0, 0, 0) ≻HSe

2

(0, 1, 0) ∼HSe

2

(1, 1, 1) ≻HSe

2

↓ (1, 1, 1) ≻HSe

1

(1, 0, 0) ∼HSe

1

(0, 1, 0) ≻HSe

1

(0, 0, 0) ≻HSe

1

↓ (0, 1, 0) ≻HSe

3

(0, 0, 0) ≻HSe

3

(1, 0, 0) ∼HSe

3

(1, 1, 1) ≻HSe

3

↓ x x ( )

13 LGS’09, University of Tsukuba , 26 - 29 August 2009

Hiring example revisited

Φ

(a,b,X) X ↔ a ∧ b (prof. A) ϕ1 (1,0,0) (prof. B) ϕ2 (1,1,1) (prof. C) ϕ3 (0,1,0) ϕ4 (0,0,0)

(1, 0, 0) ≻HSe

2

(0, 0, 0) ≻HSe

2

(0, 1, 0) ∼HSe

2

(1, 1, 1) ≻HSe

2

↓ (1, 1, 1) ≻HSe

1

(1, 0, 0) ∼HSe

1

(0, 1, 0) ≻HSe

1

(0, 0, 0) ≻HSe

1

↓ (0, 1, 0) ≻HSe

3

(0, 0, 0) ≻HSe

3

(1, 0, 0) ∼HSe

3

(1, 1, 1) ≻HSe

3

↓ x x ( ) f(ω)

• ne agent can white manipulate alone and improve

the social welfare the group can agree on how to manipulate and this improve the social welfare idea: negotiate on how to lie example: fallback bargaining

14 LGS’09, University of Tsukuba , 26 - 29 August 2009

Coordinated white manipulation

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = n

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = n

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = n

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = 2

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = 2

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c  

consensus for r = 2

M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

r = n

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

r = n

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

r = n

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓  

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓   r = 2

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

introduced by S.J. Brams and D.M. Kilgour (1998)[5] bargainers “fallback” on less and less preferred alternatives

15 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining

M =   a b c d a c b d b a d c   M h =   100 000 111, 010 ↓ 111 100, 010 000 ↓ 010 000 111, 100 ↓   r = 2

d = 1 d = 2 d = 3 d = 4

hiring example:

d = 1 d = 2 d = 3 d = 4

if is the least preferred outcome, it will not be the result of the bargaining (for both scoring functions) for r=n, the utilitarian social welfare of the bargaining

• utput is the highest

16 LGS’09, University of Tsukuba , 26 - 29 August 2009

Fallback Bargaining & WM

↓

treat the inconsistency as an impasse and the impasse as a possible outcome introduce the idea of manipulability as a positive concept extend the judgment aggregation framework with an automatically built preference profile introduce social welfare concepts in the judgment aggregation framework

17 LGS’09, University of Tsukuba , 26 - 29 August 2009

Our contribution

analyze further the fallback bargaining for other social welfare functions analyze other agreement reaching protocols for the use

• f white manipulation

analyze profiles with different preferences regarding the impasse redefine manipulation concepts in terms of coalition manipulation concepts extend the JA framework towards game theory

18 LGS’09, University of Tsukuba , 26 - 29 August 2009

Future work

• 1. manipulability of JA functions:

19 LGS’09, University of Tsukuba , 26 - 29 August 2009

References

A complete conclusion-based procedure for judgment aggregation G.Pigozzi, M. Slavkovik and L.van der Torre. In Proceedings of 1rst Conference on Algorithmic Decision Theory (forthcoming)

• 2. manipulability of premise based JA:
• 3. manipulability of conclusion based JA:
• 4. manipulability of distance based merging:
• 5. fallback bargaining: