. 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

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