Ordinal social ranking : simulations for CP-majority rule Nicolas - PowerPoint PPT Presentation

Ordinal social ranking : simulations for CP-majority rule Ordinal social ranking : simulations for CP-majority rule Nicolas Fayard 1 and Meltem Öztürk 1 1 LAMSADE, Université Paris-Dauphine DA2PL 2018- Poznan 1/25

Ordinal social ranking : simulations for CP-majority rule Social ranking problem 1 CP-Majority rule 2 CP-majority and transitive social ranking 3 Learning a CP-majority with maximum coalitions 4 2/25

Ordinal social ranking : simulations for CP-majority rule Social ranking problem Social ranking problem : example A company with three employees 1 , 2 and 3 working in the same department. According to the opinion of the manager of the company, the job performance of the different teams S ⊆ N = { 1 , 2 , 3 } is as follows : { 1 , 2 , 3 } ≻ { 3 } ≻ { 1 , 3 } ≻ { 2 , 3 } ≻ { 2 } ≻ { 1 , 2 } ≻ { 1 } ≻ ∅ 3/25

Ordinal social ranking : simulations for CP-majority rule Social ranking problem Social ranking problem : example { 1 , 2 , 3 } ≻ { 3 } ≻ { 1 , 3 } ≻ { 2 , 3 } ≻ { 2 } ≻ { 1 , 2 } ≻ { 1 } ≻ ∅ How to make a ranking over his three employees showing their attitude to work with others as a team or autonomously ? (the ability to cooperate is important) Remark : “Intuitively, 3 seems to be more influential than 1 and 2 ” . Can we state more precisely the reasons driving us to this conclusion? who between 1 and 2 is more “productive”? 4/25

Ordinal social ranking : simulations for CP-majority rule Social ranking problem Formal problem Input : A set of individuals : N = { 1 , . . . , n } A power relation on 2 N : S ≻ T : The “team” S is at least as strong as T . We suppose that ≻ is a complete preorder without any domain restriction. 5/25

Ordinal social ranking : simulations for CP-majority rule Social ranking problem Formal problem Output : A social ranking solution ρ ( ≻ ) , assigning to the power relation ≻ a total preorder on N . iP ≻ j : individual i is considered as more influential than j according to the social ranking ρ ( ≻ ) . iI ≻ j : individual i is considered as influential than j according to the social ranking ρ ( ≻ ) . Example : Power relation : { 1 , 2 , 3 } ≻ { 3 } ≻ { 1 , 3 } ≻ { 2 , 3 } ≻ { 2 } ≻ { 1 , 2 } ≻ { 1 } ≻ ∅ Social ranking : 3 P ≻ 1 I ≻ 2 6/25

Ordinal social ranking : simulations for CP-majority rule CP-Majority rule Ceteris Paribus Ceteris Paribus : all other things being equal To compare i and j , the only information that we use is the comparisons between S ∪ { i } and S ∪ { j } where S is a coalition containing neither i nor j example N = { 1 , 2 , 3 , 4 } with : 1234 ≻ 123 ≻ 124 ≻ 134 ≻ 12 ≻ 13 ≻ 234 ≻ 14 ≻ 2 ≻ 3 ≻ 1 ≻ 23 ≻ 24 ≻ 23 ≻ 4 1 vs. 2 S 1 ≺ 2 ∅ { 3 } 13 ≻ 23 { 4 } 14 ≻ 24 { 34 } 134 ≻ 234 7/25

Ordinal social ranking : simulations for CP-majority rule CP-Majority rule Ceteris Paribus Majority Definition (Ceteris Paribus Majority Rule) Let ≻∈ . The ceteris paribus majority relation (CP-majority) is the binary relation R � ⊆ N × N such that for all x , y ∈ N : xP ≻ y ⇔ d xy ( ≻ ) > d yx ( ≻ ) . xI ≻ y ⇔ d xy ( ≻ ) = d yx ( ≻ ) . D ij ( ≻ ) = { S ∈ 2 N \{ i , j } : S ∪ { i } ≻ S ∪ { j }} . We denote the cardinalities of D ij ( ≻ ) as d ij ( ≻ ) . 8/25

Ordinal social ranking : simulations for CP-majority rule CP-Majority rule Ceteris Paribus Majority Rule example N = { 1 , 2 , 3 , 4 } with : 1234 ≻ 123 ≻ 124 ≻ 134 ≻ 12 ≻ 13 ≻ 234 ≻ 14 ≻ 2 ≻ 3 ≻ 1 ≻ 23 ≻ 24 ≻ 23 ≻ 4 1 vs. 2 1 vs. 3 S S 1 ≺ 2 1 ≺ 3 ∅ ∅ { 3 } 13 ≻ 23 { 2 } 12 ≻ 23 { 4 } 14 ≻ 24 { 4 } 14 ≺ 34 { 34 } 134 ≻ 234 { 24 } 124 ≻ 234 1 P ≻ 2 1 I ≻ 3 Remark : Coalitions can be seen as voters Remark : The number of coalition voting for a unique binary relation is even 9/25

Ordinal social ranking : simulations for CP-majority rule CP-Majority rule Condorcet-like paradox 2 ≻ 1 ≻ 3 ≻ 4 ≻ 23 ≻ 13 ≻ 12 ≻ 14 ≻ 34 ≻ 24 ≻ 134 ≻ 124 ≻ 234 ≻ 123 A = { 1 , 2 , 3 } 1 vs. 2 2 vs. 3 1 vs. 3 1 ≺ 2 2 ≻ 3 1 ≻ 3 13 ≺ 23 12 ≺ 13 12 ≺ 23 14 ≻ 24 34 ≺ 24 14 ≻ 34 134 ≺ 234 124 ≺ 134 124 ≺ 234 2 P ≻ 1 3 P ≻ 2 1 P ≻ 3 10/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Simulations 1 ≻ 2 ≻ 3 ≻ 12 ≻ 13 ≻ 23 and 12 ⊐ 13 ⊐ 23 ⊐ 1 ⊐ 2 ⊐ 3 share the same CP-information table Transitive solution? Unique Condorcet winner? N 3 4 5 6 2 4 8 16 Nbr of voters 36 414 720 100 000 100 000 Nbr of different (all) (all) CP-information tables 33.33 26.66 13.93 9.5 % of Condorcet winner 66.66 40 19.5 10.5 % of transitive solution T ABLE – Probability to have Condorcet winner and a transitive social ranking 11/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Simulations 12/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Removing some coalitions for a transitive social ranking using CP-majority Max � s S s s P Sij × S s ≥ − M ( 1 − R ij ) � � ∀ i , j s . t . R a 1 a 2 + R a 2 a 3 + .... + R a n − 1 a n − R a k a k − 1 < n − 1 * For all a 1 , a 2 , ..., a n forming a cycle, for all k ∈ { a 1 , a 2 , ..., a n } and for all n . with : � 1 if S ∪ { i } ≻ S ∪ { j } ( i . ) Power relation : P Sij = − 1 if S ∪ { j } ≻ S ∪ { i } � 1 if iP ≻ j ( ii . ) Social ranking : R ij = 0 otherwise � 1 if the coalition S s is kept ( iii . ) Decision variables for coalition : S s = 0 otherwise 13/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Removing some coalitions for a transitive socialranking using CP-majority F IGURE – Probability of Condorcet-like cycles and Condorcet winner 14/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Probability for a coalition to be removed S 1 2 3 4 ∅ probability 40.35 11.13 10.99 11.09 15.18 S 12 13 14 23 24 34 probability 4.20 4.53 4.15 4.60 6.10 5.73 T ABLE – Probability for a coalition to be removed (in %), for N = 4 (10 000 simulations) 15/25

Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking Ceteris Paribus Majority Rule example N = { 1 , 2 , 3 , 4 } with : 1234 ≻ 123 ≻ 124 ≻ 134 ≻ 12 ≻ 13 ≻ 234 ≻ 14 ≻ 2 ≻ 3 ≻ 1 ≻ 23 ≻ 24 ≻ 23 ≻ 4 1 vs. 2 1 vs. 3 S S 1 ≺ 2 1 ≺ 3 ∅ ∅ { 3 } 13 ≻ 23 { 2 } 12 ≻ 23 { 4 } 14 ≻ 24 { 4 } 14 ≺ 34 { 34 } 134 ≻ 234 { 24 } 124 ≻ 234 1 P ≻ 2 1 I ≻ 3 16/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions Data sharing the same rule F IGURE – Learning rule 17/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions Max � s S s �� S P Sijk × S s ≥ − M ( 1 − R ijk ) ∀ i , j , k s . t . � S P Sijk × S s ≤ − 1 + MR ijk ∀ i , j , k With : i ) Power relation :  1 if S ∪ { i } ≻ S ∪ { j } for the power relation k   − 1 if S ∪ { j } ≻ S ∪ { i } for the power relation k P Sijk =  0 otherwise  ii ) Social ranking : � 1 if iP ≻ j in the social ranking k R ijk = 0 otherwise iii ) Decision Variables (common to all power relations!) : S s = 1 if S s is kept 18/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions F IGURE – Probability to find the CP-majority rule with exact subset of voting coalition for n = 4 for 500 symulation 19/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions Data sharing the same subset of coalitions : example Power Relation Sub-rule Social Ranking 1 , 2 , 3 , 12 , 34 PR 1 SR 1 1 , 3 , 12 , 23 , 34 PR 2 SR 2 ∅ , 1 , 3 , 12 , 34 PR 3 SR 3 Common CP-subset : 1 , 3 , 12 , 34 20/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions Min � ijk V ijk  s P Sijk × S s ≥ − M ( 1 − R ijk ) ∀ ijk �    S P Sijk × S s ≤ − 1 + MR ijk ∀ ijk �     R ijk + R jik = 1 ∀ ijk s . t .  O ijk − R ijk − V ijk ≤ 0 ∀ ijk     O ijk − R ijk + V ijk ≥ 0 ∀ ijk   With : i ) Power relation : � 1 if S ∪ { i } ≻ S ∪ { j } for the power relation k P Sijk = − 1 if S ∪ { j } ≻ S ∪ { i } for the power relation k ii ) Objective social ranking : O ijk = 1 if iP ≻ j in the social ranking O k iii ) Decision Variables social ranking : R ijk = 1 if iP ≻ j in the social ranking k iv ) Decision variables for coalition : S S = 1 if S S is kept v ) Decision variables Kemeny distance : V ijk = 1 if R ijk � = O ijk 21/25

Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions Data sharing the same subset of coalitions F IGURE – Learning rules with noise for n = 4, Y being the number of coalitions shared by all power relations, with 1000 simulations 22/25