Listing the families of Sufficient Coalitions of criteria involved in Sorting procedures

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1 Listing the families of Sufficient Coalitions of criteria involved in Sorting procedures Eda Ersek Uyanık 1, Olivier Sobrie 1,2, Vincent Mousseau 2, Marc Pirlot 1 (1) Université de Mons (2) École Centrale Paris DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 1- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 1 / 30

2 Sorting methods à la Electre Tri Bouyssou et Marchant (2007a,b) have characterized the Non Compensatory Sorting models/methods (NCS) Basically, an object is assigned to a category if it is better than the lower limit profile of the category on a sufficiently strong subset of criteria while this is not the case when comparing the object to the upper limit profile of the category E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 2- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 2 / 30

3 The non-compensatory sorting model The ingredients are : 2 categories (simplest case) : Good (G), Bad (B) objects to be sorted : X = i [n] X i with [n] = {1,..., n} i total preorder on X i limit profile b = (b 1,..., b n ) F = family of sufficient coalitions, which is a subset of 2 [n] up-closed by inclusion Assignment rule : For all x = (x 1,..., x n ) X x G iff {i [n] : x i i b i } F E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 3- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 3 / 30

4 A particular case : MR-Sort Case in which F can be determined by weights and a threshold weights w i with w i 0 for i [n] and i [n] w i = 1 threshold λ R i.e. x B iff w i λ i [n]:x i i b i A F iff w i λ i A E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 4- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 4 / 30

5 Examples Example 1. Conditions for successfull completion of examinations 4 subjects : French (F), English (E), Maths (µ), Physics (ϕ) at least 12/20 in each subject with at most one exception E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 5- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 5 / 30

6 Examples Example 1. Conditions for successfull completion of examinations 4 subjects : French (F), English (E), Maths (µ), Physics (ϕ) at least 12/20 in each subject with at most one exception This assignment rule to the category G = passed can be represented in the MR-Sort model Weights : w i = 1/4 for i = 1,... 4 Threshold : λ = 3/4 mark 12 for at least 3 subjects, w i 3/4 λ i:x i 12 mark 12 for at most 2 subjects, w i 2/4 < λ i:x i 12 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 5- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 5 / 30

7 Examples Example 2. Conditions for successfull completion of examinations 4 subjects : French (F), English (E), Maths (µ), Physics (ϕ) at least one mark 12/20 in the literary subjects (F, E) and at least one mark 12/20 in the scientific subjects (µ, ϕ) Such an assignment rule cannot be represented by weights and a threshold E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 6- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 6 / 30

8 Examples Example 2. Conditions for successfull completion of examinations 4 subjects : French (F), English (E), Maths (µ), Physics (ϕ) at least one mark 12/20 in the literary subjects (F, E) and at least one mark 12/20 in the scientific subjects (µ, ϕ) Such an assignment rule cannot be represented by weights and a threshold w 1 + w 3 λ w 1 + w 4 λ w 2 + w 3 λ w 2 + w 4 λ w 1 + w 2 < λ w 3 + w 4 < λ This system of inequalities has no solution : summing up the first 4 yields λ 1/2 while summing up the last 2 implies λ > 1/2. E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 6- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 6 / 30

9 Examples This rule can be represented in the NCS model The sufficient coalitions are F = {{F, µ}, {E, µ}, {F, ϕ}, {E, ϕ},...}... stands for all coalitions containing one of the latter. E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 7- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 7 / 30

10 Question To what extent is the NCS model more expressive than MR-Sort? Question : Which upper sets of 2 [n] can be represented by weights and threshold? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 8- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 8 / 30

11 Question To what extent is the NCS model more expressive than MR-Sort? Question : Which upper sets of 2 [n] can be represented by weights and threshold? Method : Use Linear programming to test this property E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 8- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 8 / 30

12 Question To what extent is the NCS model more expressive than MR-Sort? Question : Which upper sets of 2 [n] can be represented by weights and threshold? Method : Use Linear programming to test this property In example 2, solve, e.g., the following : subject to min λ w 1 + w 3 λ w 1 + w 4 λ w 2 + w 3 λ w 2 + w 4 λ w 1 + w 2 λ + ɛ w 3 + w 4 λ + ɛ To formulate the constraints of the LP, we only need the minimal sufficient coalitions (MSC) and the maximal insufficient coalitions (MIC) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 8- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 8 / 30

13 Minimal sufficient coalitions The MSC form an antichain in 2 [n],. Antichain in 2 [n] : Set of subsets of [n] such that none of them is included in another There is a one-to-one correspondence between upper sets and antichains of 2 [n] Reformulation : For each antichain of 2 [n], test whether it can be represented by an MR-Sort rule E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot 9- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 9 / 30

14 Number of antichains of 2 [n] D(n), the sequence of Dedekind numbers, is A in the On-Line Encyclopedia of Integer Sequences D(1) = 3, D(2) = 6, D(3) = 20, D(4) = 168, D(5) = 7581, D(6) = , D(7) = , D(8) = D(9) is not known If testing whether an antichain is representable by weights and threshold requires 1 second CPU time, answering this question for n = 7 would require D(7) seconds, i.e years. E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot10- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 10 / 30

15 Same question in other contexts Boolean functions : D(n) is the number of monotone Boolean functions in n variables Cooperative games : D(n) 2 is the number of simple games Correspondences NCS model monotone Boolean function simple game sufficient coalitions true points winning coalitions MR-Sort threshold function weighted majority game threshold threshold quota E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot11- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 11 / 30

16 Same question in other contexts Question : Which antichains are representable by weights and threshold? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot12- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 12 / 30

17 Same question in other contexts Question : Which antichains are representable by weights and threshold? Which monotone boolean functions are threshold functions? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot12- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 12 / 30

18 Same question in other contexts Question : Which antichains are representable by weights and threshold? Which monotone boolean functions are threshold functions? Which simple games are weighted majority games? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot12- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 12 / 30

19 Same question in other contexts Question : Which antichains are representable by weights and threshold? Which monotone boolean functions are threshold functions? Which simple games are weighted majority games? Many papers study this question in the different contexts For instance : characterization results such as the asummability property in the context of Boolean functions Study of subclasses of simple games, e.g. invariant-trade robust games Listing all weighted majority games up to n = 7 Do we need more? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot12- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 12 / 30

20 Additional objectives study other classes in NCS list all antichains in order to be able to simulate the random choice of a NCS model E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot13- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 13 / 30

21 Representation of a NCS rule by a k-additive capacity Any upper set of sufficient coalitions can be represented by a capacity µ and a threshold λ The capacity of a coalition A is computed in terms of its interaction function m (Möbius transform) : µ(a) = S A m(s) = i A m i + i,j A, i j m ij + i,j,l A, i,j,l The capacity µ is k-additive if m(s) = 0 for all S > k m ijl +... Linear Programming permits to test whether a family of sufficient coalitions can be represented by a 2-additive, a 3-additive,..., capacity E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot14- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 14 / 30

22 Simulation Context : Assume we devised a method for learning a NCS model We want to test whether the method works A basic requirement is the following : if the learning set has been assigned by a NCS rule, the learning method should be able to retrieve this rule (approximately) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot15- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 15 / 30

23 Simulation Context : Assume we devised a method for learning a NCS model We want to test whether the method works A basic requirement is the following : if the learning set has been assigned by a NCS rule, the learning method should be able to retrieve this rule (approximately) Procedure to test this requirement : select a NCS model at random generate objects that will constitute a learning set assign these objects using the NCS model test the learning method E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot15- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 15 / 30

24 Additional objectives study other classes in NCS : 1-additive, 2-additive, 3-additive,... list all antichains in order to be able to simulate the random choice of a NCS model We want to list all antichains and classify them up to n = 6 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot16- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 16 / 30

25 Saving time : Equivalent families of MSC Example n = 4 These 3 families are equivalent (12)(34) (13)(24) (14)(23) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot17- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 17 / 30

26 Saving time : Equivalent families of MSC Example n = 4 These 3 families are equivalent (12)(34) (13)(24) (14)(23) Only test one representative of each class of equivalent antichains The number of inequivalent antichains is R(n) The sequence R(n) is A in the OEIS. R(7) was recently computed (Stephen and Yusun 2014) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot17- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 17 / 30

27 Number of inequivalent antichains n R(n) To answer the question for n = 7, R(7) seconds are needed i.e years The case n = 6 is within reach : 4.5 hours E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot18- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 18 / 30

28 Listing all inequivalent antichains We follow Stephen and Yusun (2014) Generate the different possible types of antichains Antichain type : (k 1,..., k n ) k i = number of a coalitions of size i in the antichain Examples : the type of {13, 14, 23, 24} is (0, 4, 0, 0) the type of {14, 123, 234} is (0, 1, 2, 0) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot19- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 19 / 30

29 Number of coalitions in an antichain Sperner s theorem : The maximal size of an antichain of 2 [n] is ( ) n n/2 Hence n i=1 k i ( ) n n/2 For n = 3, k 1 + k 2 + k 3 3 For n = 4, k 1 + k 2 + k 3 + k 4 6 For each type, generate all possible antichains and discard the equivalent ones (but count them) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot20- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 20 / 30

30 Case n = 3 Type Representative # equivalent (1,0,0) {1} 3 (2,0,0) {1, 2} 3 (3,0,0) {1, 2, 3} 1 (0,1,0) {12} 3 (1,1,0) {1, 23} 3 (0,2,0) {13, 23} 3 (0,3,0) {12, 13, 23} 1 (0,0,1) {123} 1 Total trivial cases : empty family (no sufficient coalition) and the family in which all coalitions are sufficient E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot21- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 21 / 30

31 Small savings The set of complementary coalitions forms an antichain For instance, for n = 4, the complementary coalitions of the antichain {12, 23, 134} constitute the antichain {34, 14, 2} When type (k 1, k 2,..., k n 1, 0) has been explored, no need to explore type (k n 1,..., k 2, k 1, 0) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot22- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 22 / 30

32 Small savings The set of complementary coalitions forms an antichain For instance, for n = 4, the complementary coalitions of the antichain {12, 23, 134} constitute the antichain {34, 14, 2} When type (k 1, k 2,..., k n 1, 0) has been explored, no need to explore type (k n 1,..., k 2, k 1, 0) The antichains involving a singleton can be built using the antichains of 2 [n 1] (counting the equivalent ones is not straightforward) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot22- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 22 / 30

33 Small savings The set of complementary coalitions forms an antichain For instance, for n = 4, the complementary coalitions of the antichain {12, 23, 134} constitute the antichain {34, 14, 2} When type (k 1, k 2,..., k n 1, 0) has been explored, no need to explore type (k n 1,..., k 2, k 1, 0) The antichains involving a singleton can be built using the antichains of 2 [n 1] (counting the equivalent ones is not straightforward) There is a one-to-one correspondence between families of type (0,..., 0, k i, 0,..., 0) and of type (0,..., 0, ( ) n i ki, 0,..., 0) for instance : {12} yields {13, 23} E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot22- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 22 / 30

34 List of inequivalent antichains for n = 4 Type Representative # eq. (0,0,0,0) {} 1 (0,0,0,1) {1234} 1 (0,0,1,0) {124} 4 (0,0,2,0) {234, 124} 6 (0,0,3,0) {134, 123, 124} 4 (0,0,4,0) {134, 123, 234, 124} 1 (0,1,0,0) {24} 6 (0,1,1,0) {14, 123} 12 (0,1,2,0) {24, 134, 123} 6 (0,2,0,0) {12, 23} 12 {23, 14} 3 (0,2,1,0) {24, 134, 23} 12 (0,3,0,0) {13, 12, 34} 12 {24, 12, 14} 4 {24, 34, 14} 4 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot23- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 23 / 30

35 Case n = 4 (suite) Type Representative # eq. (0,3,1,0) {13, 34, 23, 124} 4 (0,4,0,0) {24, 12, 13, 34} 3 {24, 12, 14, 23} 12 (0,5,0,0) {24, 12, 14, 13, 34} 6 (0,6,0,0) {24, 12, 14, 34, 23, 13} 1 (1,0,0,0) {1} 4 (1,0,1,0) {234, 1} 4 (1,1,0,0) {14, 2} 12 (1,2,0,0) {13, 34, 2} 12 (1,3,0,0) {24, 34, 23, 1} 4 (2,0,0,0) {4, 3} 6 (2,1,0,0) {4, 23, 1} 6 (3,0,0,0) {4, 2, 1} 4 (4,0,0,0) {4, 2, 3, 1} 1 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot24- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 24 / 30

36 Number of inequivalent families that cannot be represented by weights and threshold C 2 number and percentage of inequivalent families that can be represented by a 2-additive capacity and not by a 1-additive one C 3 number and percentage of inequivalent families that can be represented by a 3-additive capacity and not by a 2-additive one n R(n) C 2 C (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (10.00 %) 0 (00.00 %) (43.33 %) 0 (00.00 %) (93.19 %) 338 (02.07 %) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot25- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 25 / 30

37 Number of families that cannot be represented by weights and threshold n D(n) C 2 C (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (00.00 %) 0 (00.00 %) (10.71 %) 0 (00.00 %) (56.64 %) 0 (00.00 %) (96.88 %) (01.86 %) E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot26- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 26 / 30

38 Comments The expressivity gap between NCS and MR-Sort quickly grows with n A 2-additive capacity is enough up to n = 5 This work allows to sample NCS models in a representative way up to n = 6 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot27- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 27 / 30

39 Simulation Problem : How to draw a NCS model uniformly at random? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot28- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 28 / 30

40 Simulation Problem : How to draw a NCS model uniformly at random? Two possible answers : pick an antichain in the list of all inequivalent antichains (with equal probabilities) pick an antichain in the list of all inequivalent antichains (with probability proportional to the number of representatives) These answers are feasible up to n = 6 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot28- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 28 / 30

41 Simulation Problem : How to draw a NCS model uniformly at random? Two possible answers : pick an antichain in the list of all inequivalent antichains (with equal probabilities) pick an antichain in the list of all inequivalent antichains (with probability proportional to the number of representatives) These answers are feasible up to n = 6 Remark : This is also an issue for drawing an MR-Sort model "uniformly at random" Families of sufficient coalitions representable by weights and thresholds have been enumerated at least up to n = 7 E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot28- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 28 / 30

42 Further questions in guise of a conclusion Have the classes of 2-additively, 3-additively,... representable antichains been studied in another context? Is there a more subtle way of sampling antichains in a representative manner, up to larger n? Is the expressivity gap between NCS and MR-Sort really so large? E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot29- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 29 / 30

43 Expressivity gap between NCS and MR-Sort Generate all Boolean vectors for n Assign them using a NCS rule that is not a MR-Sort rule Learn a MR-Sort model minimizing the number of wrong assignments E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot30- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 30 / 30

44 Expressivity gap between NCS and MR-Sort Generate all Boolean vectors for n Assign them using a NCS rule that is not a MR-Sort rule Learn a MR-Sort model minimizing the number of wrong assignments n D(n) % non-additive 0/1 loss min. max. avg % 1/16 1/16 1/ % 1/32 3/ / % 1/64 8/ /64 Up to n = 6, MR Sort seem able to yield a good approximation of NCS rules E. Ersek, O. Sobrie, V. Mousseau, M. Pirlot30- DA2PL Workshop November 20-21, 2014 Ecole Centrale Paris 30 / 30