. Convex,

, % Volume computation of the region describing the IAC region, vol.1

&. P1, = intersection

&. Vol, = volume ( maximal ( P1 ) ), vol.1

, P1 := POLYTOPE, vol.24, p.24

, % Volume computation of the region describing the event '' the committee { abc } is selected by both the k -Plurality rule and the k -Negative Plurality rule, vol.2

&. P2, = intersection

&. Vol, = volume ( maximal ( P2 ) ), vol.2

, the k -Negative Plurality rule both select {a ,b , c } under IAC > Pr := Vol, vol.2

, Pe = M (97) + M (98) + M (99) + M (100) + M (101) + M (102) + M (103) + M (104) + M (105) + M (106) + M (107) + M (108) + M (109) + M (110) + M (111) + M (112) + M (113) + M (114) + M (115) + M (116) + M (117) + M (118) + M (119) + M

, % The score of candidate a under the k -Borda rule

. Ba-=4*-m, ) +4* M (2) +4* M (3) +4* M (4) +4* M (5) +4* M (6) +4* M (7) +4* M (8) +4* M, p.4

*. , ) +4* M (11) +4* M (12) +4* M (13) +4* M (14) +4* M (15) +4* M (16) +4* M (17) +4* M (18) +4* M (19) +4* M (20) +4* M (21) +4* M (22) +4* M (23) +4* M

, * M (25) +3* M (26) +3* M (27) +3* M (28) +3* M (29) +3* M (30) +2* M (31) +2* M (32) + M (33) + M (35) +2* M (37) +2* M (38) + M (39) + M (41) +2* M (43) +2* M (44) + M (45) + M (47) +3* M (49) +3* M (50) +3* M (51) +3* M (52) +3* M (53) +3* M (54) +2* M (55) +2* M (56) + M (57) + M (59) +2* M (61) +2* M (62) + M (63) + M (65) +2* M

, * M (68) + M (69) + M (71) +3* M (73) +3* M (74) +3* M (75) +3* M (76) +3* M

, +3* M (78) +2* M (79) +2* M (80) + M (81) + M (83) +2* M (85) +2* M (86) + M (87) + M

, +2* M (91) +2* M (92) + M (93) + M (95) +3* M (97) +3* M (98) +3* M (99) +3* M (100) +3* M (101) +3* M (102) +2* M (103) +2* M (104) + M (105) + M (107) +2* M (109) +2* M (110) + M (111) + M (113) +2* M (115) +2* M (116) + M (117) + M

, % The score of candidate b under the k -Borda rule

. Bb-=3*-m, ) +3* M (2) +3* M (3) +3* M (4) +3* M (5) +3* M (6) +2* M (7) +2* M (8) + M (9) + M (11) +2* M (13) +2* M (14) + M (15) + M (17) +2* M (19) +2* M (20) + M (21) + M

, * M (25) +4* M (26) +4* M (27) +4* M (28) +4* M (29) +4* M (30) +4* M (31) +4* M (32) +4* M (33) +4* M (34) +4* M (35) +4* M (36) +4* M (37) +4* M (38) +4* M

, * M (40) +4* M (41) +4* M (42) +4* M (43) +4* M (44) +4* M (45) +4* M (46) +4* M (47) +4* M (48) +2* M (49) +2* M (50) + M (51) + M (53) +3* M (55) +3* M (56) +3* M (57) +3* M (58) +3* M (59) +3* M (60) + M (61) +2* M (63) +2* M (64) + M (66) + M

, +2* M (69) +2* M (70) + M (72) +2* M (73) +2* M (74) + M (75) + M (77) +3* M (79) +3* M (80) +3* M (81) +3* M (82) +3* M (83) +3* M (84) + M (85) +2* M (87) +2* M (88) + M (90) + M (91) +2* M (93) +2* M (94) + M (96) +2* M (97) +2* M (98) + M (99) + M

, * M (103) +3* M (104) +3* M (105) +3* M (106) +3* M (107) +3* M (108) + M

*. , ) +2* M (112) + M (114) + M (115) +2* M (117) +2* M (118) + M

, % The score of candidate c under the k -Borda rule

. Bc-=2*-m, ) +2* M (2) + M (3) + M (5) +3* M (7) +3* M (8) +3* M (9) +3* M (10) +3* M (11) +3* M (12) + M (13) +2* M (15) +2* M (16) + M (18) + M (19) +2* M (21) +2* M (22) + M

, * M (25) +2* M (26) + M (27) + M (29) +3* M (31) +3* M (32) +3* M (33) +3* M

, +3* M (35) +3* M (36) + M (37) +2* M (39) +2* M (40) + M (42) + M (43) +2* M (45) +2* M (46) + M (48) +4* M (49) +4* M (50) +4* M (51) +4* M (52) +4* M (53) +4* M (54) +4* M (55) +4* M (56) +4* M (57) +4* M (58) +4* M (59) +4* M (60) +4* M (61) +4* M

, * M (63) +4* M (64) +4* M (65) +4* M (66) +4* M (67) +4* M (68) +4* M (69) +4* M (70) +4* M (71) +4* M (72) + M (73) +2* M (75) +2* M (76) + M (78) + M (79) +2* M

, +2* M (82) + M (84) +3* M (85) +3* M (86) +3* M (87) +3* M (88) +3* M (89) +3* M (90)

, + M (92) + M (94) +2* M (95) +2* M (96) + M (97) +2* M (99) +2* M (100) + M (102) + M

, +2* M (105) +2* M (106) + M (108) +3* M (109) +3* M (110) +3* M (111) +3* M (112) +3* M (113) +3* M (114) + M (116) + M (118) +2* M (119) +2* M

, % The score of candidate d under the k -Borda rule

=. Bd, ) +2* M (3) +2* M (4) + M (6) + M (7) +2* M (9) +2* M (10) + M (12) +3* M (13) +3* M (14) +3* M (15) +3* M (16) +3* M (17) +3* M (18) + M (20) + M (22) +2* M (23) +2* M (24) + M (25) +2* M (27) +2* M (28) + M (30) + M (31) +2* M (33) +2* M (34) + M

, * M (37) +3* M (38) +3* M (39) +3* M (40) +3* M (41) +3* M (42) + M (44) + M

, +2* M (47) +2* M (48) + M (49) +2* M (51) +2* M (52) + M (54) + M (55) +2* M (57) +2* M (58) + M (60) +3* M (61) +3* M (62) +3* M (63) +3* M (64) +3* M (65) +3* M (66) + M (68) + M (70) +2* M (71) +2* M (72) +4* M (73) +4* M (74) +4* M (75) +4* M (76) +4* M (77) +4* M (78) +4* M (79) +4* M (80) +4* M (81) +4* M (82) +4* M (83) +4* M

, * M (85) +4* M (86) +4* M (87) +4* M (88) +4* M (89) +4* M (90) +4* M (91) +4* M (92) +4* M (93) +4* M (94) +4* M (95) +4* M (96) + M (98) + M (100) +2* M (101) +2* M

, + M (104) + M (106) +2* M (107) +2* M (108) + M (110) + M (112) +2* M

, * M (114) +3* M (115) +3* M (116) +3* M (117) +3* M (118) +3* M (119) +3* M

, % The score of candidate e under the k -Borda rule Be = M (2) + M (4) +2* M (5) +2* M (6) + M (8) + M (10) +2* M (11) +2* M (12) + M (14) + M, p.2

, * M (17) +2* M (18) +3* M (19) +3* M (20) +3* M (21) +3* M (22) +3* M (23) +3* M (24) + M (26) + M (28) +2* M (29) +2* M (30) + M (32) + M (34) +2* M (35) +2* M (36) + M (38) + M (40) +2* M (41) +2* M (42) +3* M (43) +3* M (44) +3* M (45) +3* M (46) +3* M (47) +3* M (48) + M (50) + M (52) +2* M (53) +2* M (54) + M (56) + M (58) +2* M

, * M (60) + M (62) + M (64) +2* M (65) +2* M (66) +3* M (67) +3* M (68) +3* M

, +3* M (70) +3* M (71) +3* M (72) + M (74) + M (76) +2* M (77) +2* M (78) + M (80) + M

, +2* M (83) +2* M (84) + M (86) + M (88) +2* M (89) +2* M (90) +3* M (91) +3* M (92) +3* M (93) +3* M (94) +3* M (95) +3* M (96) +4* M (97) +4* M (98) +4* M (99)

, * M (100) +4* M (101) +4* M (102) +4* M (103) +4* M (104) +4* M (105) +4* M (106) +4* M (107) +4* M (108) +4* M (109) +4* M (110) +4* M (111) +4* M (112) +4* M (113) +4* M (114) +4* M (115) +4* M (116) +4* M (117) +4* M (118) +4* M (119) +4* M

H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman et al., Justified representation in approval-based committee voting, % The conditions under which the k -Plurality rule and the k -Borda References, vol.48, pp.461-485, 2017.

H. Aziz, S. Gaspers, N. Mattei, N. Narodytska, W. et al., Ties matter: Complexity of manipulation when tie-breaking with a random vote, Proc. of 27th AAAI Conference, pp.74-80, 2013.

S. Barberà and D. Coelho, How to choose a non-controversial list with k names, Social Choice and Welfare, vol.31, pp.79-96, 2008.

N. Betzler, A. Slinko, and J. Uhlmann, On the computation of fully proportional representation, Journal of Artificial Intelligence Research, vol.47, pp.475-519, 2013.

H. Bock, W. Dayb, and F. Mcmorris, Consensus rules for committee elections, Mathematical Social Sciences, vol.35, issue.3, pp.219-232, 1998.

S. J. Brams, Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures, 2008.

S. J. Brams and M. Brill, The excess method: A multiwinner approval voting procedure to allocate wasted votes, 2018.

S. J. Brams, D. M. Kilgour, and R. F. Potthoff, Multiwinner approval voting: an apportionment approach, Public Choice, vol.178, issue.1-2, pp.67-93, 2019.

S. J. Brams, D. M. Kilgour, and M. R. Sanver, A minimax procedure for electing committees, Public Choice, vol.132, issue.34, pp.401-420, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00119026

S. J. Brams, D. M. Kilgour, and M. R. Sanver, How to elect a representative committee using approval balloting, Mathematics and Democracy, pp.83-95, 2006.

M. Brill, J. Laslier, and P. Skowron, Multiwinner approval rules as apportionment methods, Journal of Theoretical Politics, vol.30, issue.3, pp.358-382, 2018.
URL : https://hal.archives-ouvertes.fr/halshs-02087610

W. Bruns, B. Ichim, T. Römer, R. Sieg, and C. Söger, Normaliz: Algorithms for rational cones and affine monoids, 2017.

W. Bruns, B. Ichim, and C. Söger, Computations of volumes and Ehrhart series in four candidates elections, Annals of Operations Research. Forthcoming, 2019.

D. Bubboloni, M. Diss, and M. Gori, Extensions of the Simpson voting rule to the committee selection setting, WP, vol.1813, 2018.
URL : https://hal.archives-ouvertes.fr/halshs-01827668

D. Cervone, W. V. Gehrlein, and W. Zwicker, Which scoring rule maximizes Condorcet efficiency under IAC?, Theory and Decision, vol.58, pp.145-185, 2005.

J. R. Chamberlin and P. N. Courant, Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Rule, The American Political Science Review, vol.77, issue.3, pp.718-733, 1983.

B. Debord, Prudent k-choice functions: Properties and algorithms, Mathematical SocialSciences, vol.26, pp.63-77, 1993.

M. Diss and A. Doghmi, Multi-winner scoring election methods: Condorcet consistency and paradoxes, Public Choice, vol.169, pp.97-116, 2016.
URL : https://hal.archives-ouvertes.fr/halshs-01285526

M. Diss and W. V. Gehrlein, The true impact of voting rule selection on Condorcet efficiency, Economics Bulletin, vol.35, pp.2418-2426, 2015.
URL : https://hal.archives-ouvertes.fr/halshs-01231013

M. Diss and W. V. Gehrlein, Borda's paradox with weighted scoring rules, Social Choice and Welfare, vol.38, pp.121-136, 2012.

C. Dodgson, L (1884) The principles of parliamentary representation

C. Dodgson, L (1876) A Method of Taking Votes on More than Two Issues

H. Droop, On methods of electing representatives, Journal of the Statistical Society of London, vol.44, issue.2, pp.141-202, 1881.

M. Dummett, Voting Procedures, 1984.

E. Elkind, P. Faliszewski, P. Skowron, and A. Slinko, Properties of multiwinner voting rules, Social Choice and Welfare, vol.48, issue.3, pp.599-632, 2017.

E. Elkind, J. Lang, and A. Saffidine, Choosing collectively optimal sets of alternatives based on the Condorcet criterion, Proceedings IJCAI11, pp.186-191, 2011.

E. Elkind, J. Lang, and A. Saffidine, Condorcet winning sets, Social Choice and Welfare, vol.44, issue.3, pp.493-517, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01509956

A. El-ouafdi, D. Lepelley, and H. Smaoui, Probabilities of electoral outcomes in four-candidate elections, 2018.

M. Franz, Convex -a Maple package for convex geometry, 2016.

P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon, Multiwinner Rules on Paths From k-Borda to Chamberlin-Courant, Proceedings IJCAI17, pp.192-198, 2017.

P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon, Committee Scoring Rules: Axiomatic Classification and Hierarchy, Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16), 2016.

P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon, Multiwinner analogues of the Plurality rule: Axiomatic and algorithmic perspectives, Social Choice and Welfare, vol.51, issue.3, pp.513-550, 2018.

D. S. Felsenthal and Z. Maoz, A Comparative Analysis of Sincere and Sophisticated Voting Under the Plurality and Approval Voting Procedures, Behavioral Science, vol.33, pp.116-130, 1988.

P. C. Fishburn, An analysis of simple voting systems for electing committees, SIAM Journal on Applied Mathematics, vol.41, pp.499-502, 1981.

P. C. Fishburn, Condorcet social choice functions, SIAM Journal of Applied Mathematics, vol.33, pp.469-489, 1977.

P. C. Fishburn and W. V. Gehrlein, Borda's Rule, Positional Voting, and Condorcet's Simple Majority Principle, Public Choice, vol.28, pp.79-88, 1976.

W. V. Gehrlein, The Condorcet criterion and committee selection, Mathematical Social Sciences, vol.10, pp.199-209, 1985.

W. V. Gehrlein, On the Probability that all Weighted Scoring Rules Elect the Condorcet Winner, Quality and Quantity, vol.33, issue.1, pp.77-84, 1999.

W. V. Gehrlein and P. C. Fishburn, Coincidence Probabilities for Simple Majority and Positional Voting Rules, Social Science Research, vol.7, pp.272-283, 1978.

W. V. Gehrlein and P. C. Fishburn, Probabilities of elections outcomes for large electorates, Journal of Economic Theory, vol.19, pp.38-49, 1978.

W. V. Gehrlein and P. C. Fishburn, The probability of the paradox of voting: A computable solution, Journal of Economic Theory, vol.13, pp.14-25, 1976.

W. V. Gehrlein and D. Lepelley, Elections, Voting Rules and Paradoxical Outcomes, 2017.

W. V. Gehrlein and D. Lepelley, Voting paradoxes and group coherence, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01243452

W. V. Gehrlein and D. Lepelley, On the probability of observing Borda's Paradox, Social Choice and Welfare, vol.35, pp.1-23, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01243471

W. V. Gehrlein and D. Lepelley, The Condorcet efficiency of approval voting and the probability of electing the Condorcet loser, Journal of Mathematical Economics, vol.29, pp.271-283, 1998.

W. V. Gehrlein, D. Lepelley, and I. Moyouwou, Voters' preference diversity, concepts of agreement and Condorcet's paradox, Quality & Quantity, vol.49, issue.6, pp.2345-2368, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01452557

G. T. Guilbaud, Les théories de l'intérêt général et le problème logique de l'agrégation, Economie Appliquée, vol.5, pp.501-584, 1952.

E. Kamwa, On stable rules for selecting committees, Journal of Mathematical Economics, vol.70, pp.36-44, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01631177

E. Kamwa, M. , and V. , Some remarks on the Chamberlin-Count rule, 2014.

E. Kamwa, M. , and V. , Scoring rules over subsets of alternatives: Consistency and paradoxes, Journal of Mathematical Economics, vol.61, pp.130-138, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01702492

E. Kamwa, M. , and V. , Coincidence of Condorcet Committees. Social Choice and Welfare, vol.50, issue.1, pp.171-189, 2018.

E. Kamwa and F. Valognes, Scoring rules and preference restrictions: The Strong Borda Paradox revisited, Revue d'Economie Politique, vol.127, issue.3, pp.375-395, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01631180

B. Kaymak and M. R. Sanver, Sets of alternatives as Condorcet winners, Social Choice and Welfare, vol.20, pp.477-494, 2003.

M. Kilgour, Multi-winner voting, Estudios de Economia Applicada, vol.36, issue.1, pp.167-180, 2018.

M. Kilgour, Approval balloting for multi-winner elections, pp.105-124, 2010.

M. Kilgour, M. , and E. , Approval balloting for fixed-size committees, Electoral Systems: Paradoxes, Assumptions, and Procedures, pp.305-326, 2012.

C. Klamler, The Dodgson Ranking and Its Relation to Kemeny's Method and Slater's Rule, Social Choice and Welfare, vol.23, pp.91-102, 2004.

C. Klamler, A Comparison of the Dodgson Method and the Copeland Rule, Economics Bulletin, vol.4, issue.8, pp.1-7, 2003.

D. Lepelley, On the probability of electing the Condorcet loser, Mathematical Social Sciences, vol.25, pp.105-116, 1993.

T. Lu and C. Boutilier, Budgeted social choice: From consensus to personalized decision making, Proceedings of IJCAI-11, pp.280-286, 2011.

A. Mathur and A. Bhattacharyya, On the Gap between Outcomes of Voting Rules, Proc. of the 16th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2017, 2017.

N. Mattei, N. Narodytska, and W. T. , How hard is it to control an election by breaking ties?, Proc. of 21st ECAI, pp.1067-1068, 2014.

V. Merlin, M. Tataru, and F. Valognes, On the probability that all decision rules select the same winner, Journal of Mathematical Economics, vol.33, pp.183-207, 2000.

I. Moyouwou and H. Tchantcho, A note on Approval Voting and electing the Condorcet loser, Mathematical Social Sciences, vol.89, pp.70-82, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01452548

S. Obraztsova, E. Elkind, and N. Hazon, Ties matter: Complexity of voting manipulation revisited, Proc. of 10th AAMAS Conference, pp.71-78, 2011.

H. Nurmi, Discrepancies in the Outcomes Resulting From Different Voting Schemes, Theory and Decision, vol.25, pp.193-208, 1988.

H. Nurmi, Comparing Voting Systems, 1987.

F. Plassmann and N. Tideman, How Frequently Do Different Voting Rules Encounter Voting Paradoxes in Three-candidate elections?, Social Choice and Welfare, vol.42, pp.31-75, 2014.

R. F. Potthoff and S. J. Brams, Proportional representation: Broadening the options, Journal of Theoritical Politics, vol.10, issue.2, pp.147-178, 1998.

A. D. Procaccia, J. S. Rosenschein, and A. Zohar, On the complexity of achieving proportional representation, Social Choice and Welfare, vol.30, pp.353-362, 2008.

T. C. Ratliff, Selecting committees, Public Choice, vol.126, pp.343-355, 2006.

T. C. Ratliff, Some startling inconsistencies when electing committees, Social Choice and Welfare, vol.21, pp.433-454, 2003.

T. C. Ratliff, A Comparison of Dodgson's Method and the Borda Count, Economic Theory, vol.20, pp.357-372, 2002.

T. C. Ratliff, A Comparison of Dodgson's Method and Kemeny's Rule, Social Choice and Welfare, vol.18, pp.79-90, 2001.

M. Regenwetter and B. Grofman, Approval voting, Borda winners and Condorcet winners: evidence from seven elections, Management Science, vol.44, pp.520-533, 1998.

D. G. Saari, Mathematical structure of voting paradoxes I. Pairwise votes, Economic Theory, vol.15, pp.1-53, 2000.

P. Skowron, P. Faliszewski, and A. Slinko, Axiomatic Characterization of Committee Scoring Rules, 2016.

P. Skowron, P. Faliszewski, and A. Slinko, Achieving fully proportional representation, 2013.

P. Skowron, L. Yu, P. Faliszewski, and E. Elkind, The complexity of fully proportional representation for single-crossing electorates, Theoretical Computer Science, vol.569, pp.43-57, 2015.

S. Sterne, On representative government and personal representation, Philadelphia: J.B. Lippincott, 1871.

H. Young, An axiomatization of the Borda rule, Journal of Economic Theory, vol.9, pp.43-52, 1974.