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 Fayard1 and Meltem Öztürk1

1LAMSADE, Université Paris-Dauphine

DA2PL 2018- Poznan

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Ordinal social ranking : simulations for CP-majority rule

1

Social ranking problem

2

CP-Majority rule

3

CP-majority and transitive social ranking

4

Learning a CP-majority with maximum coalitions

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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} ≻ ∅

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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”?

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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 2N : S ≻ T : The “team” S is at least as strong as T. We suppose that ≻ is a complete preorder without any domain restriction.

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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 : 3P≻1I≻2

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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 S 1 vs. 2 ∅ 1 ≺ 2 {3} 13 ≻ 23 {4} 14 ≻ 24 {34} 134 ≻ 234

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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 ⇔ dxy(≻) > dyx(≻). xI≻y ⇔ dxy(≻) = dyx(≻). Dij(≻) = {S ∈ 2N\{i,j} : S ∪ {i} ≻ S ∪ {j}}. We denote the cardinalities of Dij(≻) as dij(≻).

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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 S 1 vs. 2 S 1 vs. 3 ∅ 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

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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 2P≻1 3P≻2 1P≻3

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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

Nbr of voters

2 4 8 16

Nbr of different

36 414 720 100 000 100 000

CP-information tables

(all) (all)

% of Condorcet winner

33.33 26.66 13.93 9.5

% of transitive solution

66.66 40 19.5 10.5

TABLE – Probability to have Condorcet winner and a transitive social ranking

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Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking

Simulations

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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 Ss

s.t.

s PSij × Ss ≥ −M(1 − Rij)

∀i, j Ra1a2 + Ra2a3 + .... + Ran−1an − Rakak−1 < n − 1 * For all a1, a2, ..., an forming a cycle, for all k ∈ {a1, a2, ..., an} and for all n. with : (i.) Power relation : PSij =

• 1

if S ∪ {i} ≻ S ∪ {j} −1 if S ∪ {j} ≻ S ∪ {i} (ii.) Social ranking : Rij =

• 1

if iP≻j

• therwise

(iii.) Decision variables for coalition : Ss =

• 1

if the coalition Ss is kept

• therwise

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Ordinal social ranking : simulations for CP-majority rule CP-majority and transitive social ranking

Removing some coalitions for a transitive socialranking using CP-majority

FIGURE – Probability of Condorcet-like cycles and Condorcet winner

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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

TABLE – Probability for a coalition to be removed (in %), for N = 4 (10 000 simulations)

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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 S 1 vs. 2 S 1 vs. 3 ∅ 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

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Data sharing the same rule

FIGURE – Learning rule

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Max

s Ss

s.t. S PSijk × Ss ≥ −M(1 − Rijk) ∀i, j, k S PSijk × Ss ≤ −1 + MRijk ∀i, j, k With : i) Power relation : PSijk =      1 if S ∪ {i} ≻ S ∪ {j} for the power relation k −1 if S ∪ {j} ≻ S ∪ {i} for the power relation k

• therwise

ii) Social ranking : Rijk =

• 1

if iP≻j in the social ranking k

• therwise

iii) Decision Variables (common to all power relations!) : Ss = 1 if Ss is kept

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

FIGURE – Probability to find the CP-majority rule with exact subset of voting coalition for n = 4 for 500 symulation

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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 PR1 1, 2, 3, 12, 34 SR1 PR2 1, 3, 12, 23, 34 SR2 PR3 ∅,1, 3, 12, 34 SR3 Common CP-subset : 1, 3, 12, 34

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Min

ijk Vijk

s.t.               

• s PSijk × Ss ≥ −M(1 − Rijk) ∀ijk
• S PSijk × Ss ≤ −1 + MRijk ∀ijk

Rijk + Rjik = 1 ∀ijk Oijk − Rijk − Vijk ≤ 0 ∀ijk Oijk − Rijk + Vijk ≥ 0 ∀ijk With : i) Power relation : PSijk =

• 1

if S ∪ {i} ≻ S ∪ {j} for the power relation k −1 if S ∪ {j} ≻ S ∪ {i} for the power relation k ii) Objective social ranking : Oijk = 1 if iP≻jin the social ranking Ok iii) Decision Variables social ranking : Rijk = 1 if iP ≻ j in the social ranking k iv) Decision variables for coalition : SS = 1 if SS is kept v) Decision variables Kemeny distance : Vijk = 1 if Rijk = Oijk

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Data sharing the same subset of coalitions

FIGURE – Learning rules with noise for n = 4, Y being the number of coalitions shared by all power relations, with 1000 simulations

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Data sharing the same subset of coalitions

Number of power Y = 2 Y = 4 Y = 6 Y = 8 relation used to learn

2 3.278 4.326 4.481 4.544 3 3.736 5.427 6.619 7.567 4 3.519 5.328 6.918 8.247 5 3.219 5.22 7.05 8.605 6 3.258 5.173 7.055 8.788 7 3.127 5.112 7.016 8.84 8 2.95 5.098 6.94 8.896 9 2.661 4.913 6.973 8.896 10 2.705 4.862 6.936 8.943 11 2.505 4.765 6.847 8.887 12 2.56 4.691 6.831 8.918 13 2.406 4.61 6.806 8.885

TABLE – Average number of coalition in learning rules with noise, Y : the nbr of coalitions shared by all power relations, tested on 1000 simulations

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Conclusion

In this paper we introduced and studied the problem of how to rank the objects of a set N according to their ability to influence the ranking over the subsets of N, using the CP-majority rule. We have shown that this approach can lead to Condorcet paradox. We have shown that we can approximate the CP-majority by removing coalition. We have studied how to learn a rule that only use a subset

• f coalition

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Ordinal social ranking : simulations for CP-majority rule Learning a CP-majority with maximum coalitions

Conclusion

Current and alternative future work related to this problem : What happens if the power relation is not complete? Introducing Coopland’s method

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