Tie-breaking approaches for collective decision making . r c - - PowerPoint PPT Presentation

Gabriella Pigozzi Marija Slavkovik Leon van der Torre

h t t p : \ \ i c r .

Tie-breaking approaches for collective decision making

Workshop and Meeting

• f the COST Action ICO602

Université Paris Dauphine 28, 29, 30 and 31 October 2008

Thursday, October 30, 2008

• 1. The problem & framework:reaching collective

supported decisions

• 2. Model-based fusion approach
• 3. Ties in model-based fusion results
• 4. Resolving ties: role of context in decision making
• 5. Characterization: Sensitive/robust decisions
• 6. Characterization:Skeptical and credulous

decisions

• 7. Conclusions & future work

Tie-breaking in collective DM

Overview

Thursday, October 30, 2008

how can a panel reach a supported decision whether to hire an employee? rule:

• a candidate is hired if and only if
• the candidate has an adequate CV

and

• the candidate made a good interview

method: each member submits his decision and the justifications for it according to the rule.

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

1/21 d p q

The problem

Tie-breaking in collective DM

Thursday, October 30, 2008

2/21 p q d 1 1 1 1 1 1 1

p ∧ q ↔ d

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Trivial ?

member1 member2 member3 panel

Tie-breaking in collective DM

Thursday, October 30, 2008

a panel of n members (designated by i in [1,n])

Formalization

3/21

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language

Formalization

3/21

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

L

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language a set of rules (propositional formulas)

Formalization

3/21

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R

Tie-breaking in collective DM

L

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language a set of rules (propositional formulas) a set of judgments submitted by i (complete and consistent with ):

Formalization

3/21

Si Si = {σi,1, . . . σi,m−1, δi}

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R R

Tie-breaking in collective DM

L

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language a set of rules (propositional formulas) a set of judgment submitted by i (complete and consistent with ):

• for : or

Formalization

3/21

Si Si = {σi,1, . . . σi,m−1, δi}

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R σ1 to σm−1 σ6 = ϕ5 → ϕ7 σ4 = ϕ5 R

Tie-breaking in collective DM

L

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language a set of rules (propositional formulas) a set of information submittes by i (complete and consistent with ):

• for : or

all sets submitted by the members (profile)

Formalization

3/21

Si Si = {σi,1, . . . σi,m−1, δi}

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R P = {S1, . . . Sn} σ1 to σm−1 σ6 = ϕ5 → ϕ7 σ4 = ϕ5 R

Tie-breaking in collective DM

L

Thursday, October 30, 2008

a panel of n members (designated by i from [1,n]) propositional language a set of rules (propositional formulas) a set of information submitted by i (complete and consistent with ):

• for : or

all sets submitted by an agent (profile) set of all complete alternatives

Formalization

3/21

Si Si = {σi,1, . . . σi,m−1, δi}

The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R P = {S1, . . . Sn} Ω = {ω1, . . . ωl} σ1 to σm−1 σ6 = ϕ5 → ϕ7 σ4 = ϕ5 P ⊆ Ω R {ωi ∪ R} ⊥

Tie-breaking in collective DM

L

Thursday, October 30, 2008

a method to merge sets of consistent beliefs ( ) under integrity constraints ( ) . distance function between alternatives ( ) intuition: how close is each alternative in to “winner” is the alternative closest to

Model-based fusion

4/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

R

ωi

d d(ω1, ω2) = d(ω2, ω1) P

D(ωi, P) = f(d(ωi, S1), . . . , d(ωi, Sn))

D(ω, P) = min{D(ωi, P)}

ω P d(ω1, ω1) = 0 Ω P

Tie-breaking in collective DM

Thursday, October 30, 2008

distance function - Hamming distance the judgment sets : the integrity constraint

• ther possible models

distance to the profile

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

d

P = {S1, S2, S3}

S1 = {¬ϕ1, ϕ2, ¬δ} S2 = {ϕ1, ϕ2, δ} S3 = {¬ϕ1, ϕ2, ¬δ}

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

ω5 = {¬ϕ1, ¬ϕ2, ¬δ} ω4 = {ϕ1, ¬ϕ2, ¬δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

+ + =

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 2 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 2 2 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 2 2 6 0 0 0

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

+ + =

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 2 2 6 0 0 0 1 3 1 5

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

Fusion example

5/21

• 1. The problem

Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

2 2 2 2 4 2 2 1 0 0 2 2 2 6 0 0 0 1 3 1 5

P = {S1, S2, S3}

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

1 1 1 0 1 0 0 1 0

ω4 ω5

R = {ϕ1 ∧ ϕ2 ↔ δ}

D(ωi, P) = n

j=1 d(ωi, Sj)

Hamming distance

Tie-breaking in collective DM

Thursday, October 30, 2008

in the general case the is no single “winner”:

Fusion & ties

6/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)} ω1 = S1 ω2 = S2 ω3 = S3

1 2 2 4 1 2 2 4 1 1 1 2 2 4 1 1 3 5

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

ω4 τ1 τ2 τ3

set of tied alternatives

R = {ϕ1 ∧ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative choose randomly from the tied alternatives?

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative choose randomly from the tied alternatives?

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)} ω1 = S1 ω2 = S2 ω3 = S3

1 2 2 4 1 2 2 4 1 1 1 2 2 4 1 1 3 5

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative choose randomly from the tied alternatives?

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)} ω1 = S1 ω2 = S2 ω3 = S3

1 2 2 4 1 2 2 4 1 1 1 2 2 4 1 1 3 5

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative choose randomly from the tied alternatives?

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)} ω1 = S1 ω2 = S2 ω3 = S3

1 2 2 4 1 2 2 4 1 1 1 2 2 4 1 1 3 5

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

all elements in are equally good to be the collective alternative choose randomly from the tied alternatives? revise: all tied alternatives are equally good to be the collective decision under the information considered in the fusion!

What ties mean?

7/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

T = {τi, |D(τi, P) = min{D(ωi, P)} ω1 = S1 ω2 = S2 ω3 = S3

1 2 2 4 1 2 2 4 1 1 1 2 2 4 1 1 3 5

ϕ1 ϕ2 δ

d(ωi, S1) d(ωi, S2) d(ωi, S3) D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

elicit preferential information.

More information

8/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

elicit preferential information. Which?

More information

8/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

elicit preferential information. Which? what information do we already have?

More information

8/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

elicit preferential information. Which? what information do we already have? idea: consider the context to resolve the ties!

More information

8/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

elicit preferential information. Which? what information do we already have? idea: consider the context to resolve the ties!

• ur proposal: to break ties by taking into

consideration the type of decision more desirable in a given context

More information

8/21

• 1. The problem

2.Model-based fusion approach Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

Example 1-2

9/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Case1: δ1: enforce tax on cigarettes ϕ1: smoking should be reduced among population ϕ2: higher cost of cigarettes reduces number of smokers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case2: δ1: enforce death penalty for drug trafficking ϕ1: less drugs available accounts for less drug abusers ϕ2: threat of death penalty reduces the number of drug dealers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 1-2

9/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Reversible decision

Case1: δ1: enforce tax on cigarettes ϕ1: smoking should be reduced among population ϕ2: higher cost of cigarettes reduces number of smokers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case2: δ1: enforce death penalty for drug trafficking ϕ1: less drugs available accounts for less drug abusers ϕ2: threat of death penalty reduces the number of drug dealers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 1-2

9/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Irreversible decision

Case1: δ1: enforce tax on cigarettes ϕ1: smoking should be reduced among population ϕ2: higher cost of cigarettes reduces number of smokers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case2: δ1: enforce death penalty for drug trafficking ϕ1: less drugs available accounts for less drug abusers ϕ2: threat of death penalty reduces the number of drug dealers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 1-2

9/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

All things being equal, be credulous

Reversible decision

Case1: δ1: enforce tax on cigarettes ϕ1: smoking should be reduced among population ϕ2: higher cost of cigarettes reduces number of smokers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case2: δ1: enforce death penalty for drug trafficking ϕ1: less drugs available accounts for less drug abusers ϕ2: threat of death penalty reduces the number of drug dealers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 1-2

9/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

All things being equal, be skeptical

Irreversible decision

Case1: δ1: enforce tax on cigarettes ϕ1: smoking should be reduced among population ϕ2: higher cost of cigarettes reduces number of smokers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case2: δ1: enforce death penalty for drug trafficking ϕ1: less drugs available accounts for less drug abusers ϕ2: threat of death penalty reduces the number of drug dealers The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 3-4

10/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Case 3: δ1: hire contractor ϕ1: there is enough money in the budget ϕ2: contractor appears reliable The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case 4: δ1: loan to friendly bank ϕ1: we possess enough liquid asset ϕ2: shares of own bank are expected to maintain market value The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 3-4

10/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Revisable decision

Case 3: δ1: hire contractor ϕ1: there is enough money in the budget ϕ2: contractor appears reliable The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case 4: δ1: loan to friendly bank ϕ1: we possess enough liquid asset ϕ2: shares of own bank are expected to maintain market value The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 3-4

10/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Panel will be held responsible

Case 3: δ1: hire contractor ϕ1: there is enough money in the budget ϕ2: contractor appears reliable The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case 4: δ1: loan to friendly bank ϕ1: we possess enough liquid asset ϕ2: shares of own bank are expected to maintain market value The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 3-4

10/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Revisable decision

All things being equal, go for sensitive alternative

Case 3: δ1: hire contractor ϕ1: there is enough money in the budget ϕ2: contractor appears reliable The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case 4: δ1: loan to friendly bank ϕ1: we possess enough liquid asset ϕ2: shares of own bank are expected to maintain market value The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

Example 3-4

10/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions

Panel will be held responsible

All things being equal, go for robust alternative

Case 3: δ1: hire contractor ϕ1: there is enough money in the budget ϕ2: contractor appears reliable The rule: R : δ1 ↔ ϕ1 ∧ ϕ2 Case 4: δ1: loan to friendly bank ϕ1: we possess enough liquid asset ϕ2: shares of own bank are expected to maintain market value The rule: R : δ1 ↔ ϕ1 ∧ ϕ2

Tie-breaking in collective DM

Thursday, October 30, 2008

in all examples, the logic formalization of the problem is the same

Context & propositions

11/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

in all examples, the logic formalization of the problem is the same how to capture decision characteristics in the formalization?

Context & propositions

11/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

Sensitive/robust

12/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

robust alternative - decision will change with many swaps in supporting reasons

Sensitive/robust

12/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

robust alternative - decision will change with many swaps in supporting reasons sensitive alternative - decision will change with few swaps in supporting reasons

Sensitive/robust

12/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

robust alternative - decision will change with many swaps in supporting reasons sensitive alternative - decision will change with few swaps in supporting reasons sensitive/robust: measure how changing an

• pinion on a decision or a justification would

affect a judgment set

Sensitive/robust

12/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

impact factor of shows how much will have to change if is replaced by

Impact factors

13/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

σj ∈ ωi ωi σj ¬σj uΩ(σj, ωi) = min{d(ωi, ωk)}

where ωi, ωk ∈ Ω ; σj ∈ ωi and ¬σj ∈ ωk

Tie-breaking in collective DM

Thursday, October 30, 2008

impact factor of shows how much will have to change if is replaced by

Impact factors

13/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

σj ∈ ωi ωi σj ¬σj

ω1 = S1 ω2 = S2

ω3 = S3

1(1) 0(2) 0(2) 4 0(2) 1 (1) 0(2) 4 1 (2) 1 (2) 1 (2) 4 0 (1) 0 (1) 0 (3) 5

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

uΩ(σj, ωi) = min{d(ωi, ωk)}

where ωi, ωk ∈ Ω ; σj ∈ ωi and ¬σj ∈ ωk

Tie-breaking in collective DM

Thursday, October 30, 2008

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons.

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons.

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons. define

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

M(ωi) = min{uΩ(σj, ωi)}, j ∈ [1, m − 1]

Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons. define

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

M(ωi) = min{uΩ(σj, ωi)}, j ∈ [1, m − 1]

M(ωi) = {σj|uΩ(σj, ωi) = M(ωi)}

Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons. define

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

M(ωi) = min{uΩ(σj, ωi)}, j ∈ [1, m − 1]

ωi u ωi ⇒ (uΩ(δ, ωi) − M(ωi)) ≤ (uΩ(δ, ωj) − M(ωj) ωi u ωj ⇒ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| ≥ |M(ωj)| ωi ∼u ωj ⇔ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| = |M(ωj)|

M(ωi) = {σj|uΩ(σj, ωi) = M(ωi)}

Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons. define

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

M(ωi) = min{uΩ(σj, ωi)}, j ∈ [1, m − 1]

ωi u ωi ⇒ (uΩ(δ, ωi) − M(ωi)) ≤ (uΩ(δ, ωj) − M(ωj) ωi u ωj ⇒ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| ≥ |M(ωj)| ωi ∼u ωj ⇔ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| = |M(ωj)|

M(ωi) = {σj|uΩ(σj, ωi) = M(ωi)}

Tie-breaking in collective DM

Thursday, October 30, 2008

more robust alternative - its decision has higher impact factor then its supporting reasons. more sensitive alternative - its decision has lower impact factors than its supporting reasons. define

Sensitive to robust

14/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

M(ωi) = min{uΩ(σj, ωi)}, j ∈ [1, m − 1]

ωi u ωi ⇒ (uΩ(δ, ωi) − M(ωi)) ≤ (uΩ(δ, ωj) − M(ωj) ωi u ωj ⇒ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| ≥ |M(ωj)| ωi ∼u ωj ⇔ uΩ(δ, ωi) − M(ωi) = uΩ(δ, ωj) − M(ωj)and|M(ωi)| = |M(ωj)|

M(ωi) = {σj|uΩ(σj, ωi) = M(ωi)}

Tie-breaking in collective DM

Thursday, October 30, 2008

Sensitive to robust

15/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making Sensitive/robust decisions 5.Skeptical and credulous decisions 6.Conclusions

ω1 = S1 ω2 = S2 ω3 = S3

1(1) 0(2) 0(2) 4 2-1=1 0(2) 1 (1) 0(2) 4 2-1=1 1 (2) 1 (2) 1 (2) 4 2-2=0 0 (1) 0 (1) 0 (3) 5 3-1=2

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

u

from most sensitive to most robust:

S3 u S1 ∼u S2 u ω4

Tie-breaking in collective DM

Thursday, October 30, 2008

credulous alternative - maximizes input from the group on supporting reasons

Credulous, skeptical

16/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

credulous alternative - maximizes input from the group on supporting reasons skeptical alternative - also takes into account how often a decision appears in the set of all possible alternatives

Credulous, skeptical

16/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

frequency factor of counts in how many in a given set of alternatives , appear.

Frequency factors

17/21

σj ∈ ωi ωi G σj vG(σj) = |G| G = {ωi|ωi ⊆ G; σj ∈ ωi}.

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

frequency factor of counts in how many in a given set of alternatives , appear.

Frequency factors

17/21

σj ∈ ωi ωi

ω1 = S1 ω2 = S2

ω3 = S3

1(2) 0(1) 0 (3) 4 0(1) 1(2) 0 (3) 4 1(2) 1(2) 1 (1) 4 0(1) 0(1) 0 (3) 5

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

G σj vG(σj) = |G| G = {ωi|ωi ⊆ G; σj ∈ ωi}.

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

frequency factor of counts in how many in a given set of alternatives , appear.

Frequency factors

17/21

σj ∈ ωi ωi

ω1 = S1 ω2 = S2

ω3 = S3

1(2) 0(1) 0 (3) 4 0(1) 1(2) 0 (3) 4 1(2) 1(2) 1 (1) 4 0(1) 0(1) 0 (3) 5

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

G σj vG(σj) = |G| G = {ωi|ωi ⊆ G; σj ∈ ωi}.

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions

• ver P
• ver Ω

Tie-breaking in collective DM

Thursday, October 30, 2008

more credulous alternative - the sum of the frequency factors over the reasons in the profile is higher .

Credulous & skeptical

18/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions

ωi c ωj ⇔ m−1

k=1 vP(σk) m−1 t=1 vP(σt)

σk ∈ ωi and σt ∈ ωj

Tie-breaking in collective DM

Thursday, October 30, 2008

more credulous alternative - the sum of the frequency factors over the reasons in the profile is higher .

18/21

more skeptical alternative - the sum of the f.factors of the reasons (over ) added to the f.factor of the decision (over ) is higher.

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions

ωi c ωj ⇔ m−1

k=1 vP(σk) m−1 t=1 vP(σt)

σk ∈ ωi and σt ∈ ωj

ωi c ωj ⇔ vΩ(δi) + m−1

k=1 vP(σk) vΩ(δj) + m−1 t=1 vP(σt)

σk ∈ ωi and σt ∈ ωj

P Ω

Credulous & skeptical

Tie-breaking in collective DM

Thursday, October 30, 2008

Ordering

19/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions Skeptical and credulous decisions 6.Conclusions

from least to most credulous: from least to most skeptical:

ω1 = S1 ω2 = S2

ω3 = S3

1(2) 0(1) 0 (3) 4(3,6) 0(1) 1(2) 0 (3) 4 (3,6) 1(2) 1(2) 1 (1) 4 (4,5) 0(1) 0(1) 0 (3) 5 (2,5)

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R = {ϕ1 ∧ ϕ2 ↔ δ}

ω4 c S1 ∼c S2 c S3 ω4 ∼s S3 s S1 ∼s S2

Tie-breaking in collective DM

Thursday, October 30, 2008

context helps to resolve between tied conflicting decisions

Conclusions

20/21

• 1. The problem -

framework 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

context helps to resolve between tied conflicting decisions but context does not always give an ordering:

Conclusions

20/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions Tie-breaking in collective DM

Thursday, October 30, 2008

contexts helps resolve between tied contradicting decisions but context does not always give an ordering:

Conclusions

20/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions

ω1 = S1 ω2 = S2

ω3 = S3

1(2/2) 0(2/1) 0 (2/2) 4(0,3,5) 0(2/1) 1(2/2) 0 (2/2) 4 (0,3,5) 1(2/2) 1(2/2) 1 (2/2) 4 (0,4,6) 0(2/1) 0(2/1) 1 (2/2) 5 (0,2,4)

ϕ1 ϕ2 δ

D(ωi, P)

ω4

R={ϕ1 ↔ ϕ2 ↔ δ}

Tie-breaking in collective DM

Thursday, October 30, 2008

Open questions

21/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions

how much can be resolved with context ?

Tie-breaking in collective DM

Thursday, October 30, 2008

Open questions

21/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions

how much can be resolved with context ? which other contexts can we distinguish?

Tie-breaking in collective DM

Thursday, October 30, 2008

Open questions

21/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions

how much can be resolved with context ? which other contexts can we distinguish? how to resolve what context can not ?

Tie-breaking in collective DM

Thursday, October 30, 2008

Open questions

21/21

• 1. The problem

2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making 5.Sensitive/robust decisions 6.Skeptical and credulous decisions Conclusions

how much can be resolved with context ? which other contexts-characterizations for decisions can we distinguish? how to resolve what context-characterizations can not ? idea: preferences over justifications based on context

Tie-breaking in collective DM

Thursday, October 30, 2008

gabriella.pigozzi@uni.lu marija.slavkovik@uni.lu leon.vandertorre@uni.lu

Thursday, October 30, 2008