Workshop and Meeting of the COST Action ICO602 Université Paris Dauphine 28, 29, 30 and 31 October 2008 Tie-breaking approaches for collective decision making . r c Gabriella Pigozzi i \ \ : p Marija Slavkovik t t h Leon van der Torre Thursday, October 30, 2008
Overview Tie-breaking in collective DM 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 Thursday, October 30, 2008
The problem Tie-breaking in collective DM The problem how can a panel reach a supported decision 2.Model-based whether to hire an employee? fusion approach 3.Ties in model- rule: based results d - a candidate is hired if and only if 4.Resolving ties: role of context in p decision making -the candidate has an adequate CV 5.Sensitive/robust and decisions q -the candidate made a good interview 6.Skeptical and credulous decisions 7.Conclusions method: each member submits his decision and the justifications for it according to the rule. 1/21 Thursday, October 30, 2008
Trivial ? Tie-breaking in collective DM The problem 2.Model-based p ∧ q ↔ d fusion approach 3.Ties in model- p q d based results 4.Resolving ties: 1 1 1 member 1 role of context in decision making 1 0 0 member 2 5.Sensitive/robust decisions 0 1 0 member 3 6.Skeptical and credulous 1 1 0 decisions panel 7.Conclusions 2/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in decision making a panel of n members (designated by i in [1,n]) 5.Sensitive/robust decisions 6.Skeptical and credulous decisions 7.Conclusions 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: role of context in a panel of n members (designated by i from [1,n]) decision making 5.Sensitive/robust propositional language L decisions 6.Skeptical and credulous decisions 7.Conclusions 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem 2.Model-based fusion approach 3.Ties in model- based results 4.Resolving ties: a panel of n members (designated by i from [1,n]) role of context in decision making propositional language L 5.Sensitive/robust decisions a set of rules (propositional formulas) R 6.Skeptical and credulous decisions 7.Conclusions 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem 2.Model-based fusion approach 3.Ties in model- a panel of n members (designated by i from [1,n]) based results 4.Resolving ties: propositional language L role of context in decision making a set of rules (propositional formulas) R 5.Sensitive/robust decisions a set of judgments submitted by i (complete and S i 6.Skeptical and consistent with ): credulous S i = { σ i, 1 , . . . σ i,m − 1 , δ i } R decisions 7.Conclusions 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem 2.Model-based fusion approach a panel of n members (designated by i from [1,n]) 3.Ties in model- based results propositional language L 4.Resolving ties: role of context in a set of rules (propositional formulas) decision making R 5.Sensitive/robust a set of judgment submitted by i (complete and S i decisions consistent with ): S i = { σ i, 1 , . . . σ i,m − 1 , δ i } 6.Skeptical and R credulous - for : or decisions σ 1 to σ m − 1 σ 4 = ϕ 5 σ 6 = ϕ 5 → ϕ 7 7.Conclusions 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem a panel of n members (designated by i from [1,n]) 2.Model-based fusion approach propositional language L 3.Ties in model- based results a set of rules (propositional formulas) R 4.Resolving ties: role of context in a set of information submittes by i (complete decision making S i and consistent with ): 5.Sensitive/robust S i = { σ i, 1 , . . . σ i,m − 1 , δ i } R decisions - for : or 6.Skeptical and σ 1 to σ m − 1 σ 4 = ϕ 5 σ 6 = ϕ 5 → ϕ 7 credulous decisions all sets submitted by the members (profile) 7.Conclusions P = {S 1 , . . . S n } 3/21 Thursday, October 30, 2008
Formalization Tie-breaking in collective DM The problem a panel of n members (designated by i from [1,n]) 2.Model-based fusion approach propositional language L 3.Ties in model- a set of rules (propositional formulas) based results R 4.Resolving ties: a set of information submitted by i (complete S i role of context in decision making and consistent with ): S i = { σ i, 1 , . . . σ i,m − 1 , δ i } R 5.Sensitive/robust - for : or decisions σ 1 to σ m − 1 σ 4 = ϕ 5 σ 6 = ϕ 5 → ϕ 7 6.Skeptical and all sets submitted by an agent (profile) credulous decisions P = {S 1 , . . . S n } 7.Conclusions set of all complete alternatives Ω = { ω 1 , . . . ω l } 3/21 { ω i ∪ R} � ⊥ P ⊆ Ω Thursday, October 30, 2008
Model-based fusion Tie-breaking in collective DM 1. The problem a method to merge sets of consistent beliefs ( ) Model-based P fusion approach under integrity constraints ( ) . R 3.Ties in model- based results distance function between alternatives ( ) d ω i 4.Resolving ties: role of context in decision making d( ω 1 , ω 1 ) = 0 d ( ω 1 , ω 2 ) = d ( ω 2 , ω 1 ) 5.Sensitive/robust decisions intuition: how close is each alternative in to 6.Skeptical and Ω P credulous decisions D ( ω i , P ) = f ( d ( ω i , S 1 ) , . . . , d ( ω i , S n )) 7.Conclusions “winner” is the alternative closest to P ω D ( ω , P ) = min { D ( ω i , P ) } 4/21 Thursday, October 30, 2008
Fusion example Tie-breaking in collective DM 1. The problem distance function - Hamming distance d Model-based fusion approach the judgment sets : P = {S 1 , S 2 , S 3 } 3.Ties in model- based results S 3 = {¬ ϕ 1 , ϕ 2 , ¬ δ } S 1 = {¬ ϕ 1 , ϕ 2 , ¬ δ } S 2 = { ϕ 1 , ϕ 2 , δ } 4.Resolving ties: the integrity constraint R = { ϕ 1 ∧ ϕ 2 ↔ δ } role of context in decision making other possible models ω 4 = { ϕ 1 , ¬ ϕ 2 , ¬ δ } 5.Sensitive/robust decisions ω 5 = {¬ ϕ 1 , ¬ ϕ 2 , ¬ δ } 6.Skeptical and credulous distance to the profile D ( ω i , P ) = � n j =1 d ( ω i , S j ) decisions 7.Conclusions 5/21 Thursday, October 30, 2008
Fusion example Tie-breaking in collective DM Hamming R = { ϕ 1 ∧ ϕ 2 ↔ δ } 1. The problem distance Model-based ϕ 1 ϕ 2 δ d ( ω i , S 1 ) d ( ω i , S 2 ) d ( ω i , S 3 ) D ( ω i , P ) fusion approach P = {S 1 , S 2 , S 3 } 3.Ties in model- 0 1 0 based results ω 1 = S 1 4.Resolving ties: 1 1 1 role of context in ω 2 = S 2 decision making 5.Sensitive/robust 0 1 0 ω 3 = S 3 decisions 6.Skeptical and 1 0 0 ω 4 credulous decisions 0 0 0 7.Conclusions ω 5 D ( ω i , P ) = � n j =1 d ( ω i , S j ) 5/21 Thursday, October 30, 2008
Fusion example Tie-breaking in collective DM Hamming R = { ϕ 1 ∧ ϕ 2 ↔ δ } 1. The problem distance Model-based ϕ 1 ϕ 2 δ d ( ω i , S 1 ) d ( ω i , S 2 ) d ( ω i , S 3 ) D ( ω i , P ) fusion approach P = {S 1 , S 2 , S 3 } 3.Ties in model- 0 1 0 0 based results ω 1 = S 1 4.Resolving ties: 1 1 1 role of context in ω 2 = S 2 decision making 5.Sensitive/robust 0 1 0 ω 3 = S 3 decisions 6.Skeptical and 1 0 0 ω 4 credulous decisions 0 0 0 7.Conclusions ω 5 D ( ω i , P ) = � n j =1 d ( ω i , S j ) 5/21 Thursday, October 30, 2008
Fusion example Tie-breaking in collective DM Hamming R = { ϕ 1 ∧ ϕ 2 ↔ δ } 1. The problem distance Model-based ϕ 1 ϕ 2 δ d ( ω i , S 1 ) d ( ω i , S 2 ) d ( ω i , S 3 ) D ( ω i , P ) fusion approach P = {S 1 , S 2 , S 3 } 3.Ties in model- 0 1 0 0 2 based results ω 1 = S 1 4.Resolving ties: 1 1 1 role of context in ω 2 = S 2 decision making 5.Sensitive/robust 0 1 0 ω 3 = S 3 decisions 6.Skeptical and 1 0 0 ω 4 credulous decisions 0 0 0 7.Conclusions ω 5 D ( ω i , P ) = � n j =1 d ( ω i , S j ) 5/21 Thursday, October 30, 2008
Fusion example Tie-breaking in collective DM Hamming R = { ϕ 1 ∧ ϕ 2 ↔ δ } 1. The problem distance Model-based ϕ 1 ϕ 2 δ d ( ω i , S 1 ) d ( ω i , S 2 ) d ( ω i , S 3 ) D ( ω i , P ) fusion approach P = {S 1 , S 2 , S 3 } 3.Ties in model- 0 1 0 0 2 0 based results ω 1 = S 1 4.Resolving ties: 1 1 1 role of context in ω 2 = S 2 decision making 5.Sensitive/robust 0 1 0 ω 3 = S 3 decisions 6.Skeptical and 1 0 0 ω 4 credulous decisions 0 0 0 7.Conclusions ω 5 D ( ω i , P ) = � n j =1 d ( ω i , S j ) 5/21 Thursday, October 30, 2008
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