# [EM] preferences and approval with four candidates

Jobst Heitzig heitzig-j at web.de
Sat Jun 4 14:00:25 PDT 2005

```Hello folks!

I have thought some time about how to visualize voting situations
graphically and came about the following model:

Candidates and voters are represented by points in some metric
space, preferences are according to distance, and a candidate is
approved iff s/he is at most 1 unit apart.

For 3 candidates, we need only 2 dimensions: we use the plane and
place the candidates at the corners of a regular triangle with
side length 4/3. Then a voter who walks aroung in that plane will
come about all 44 possible complete and transitive preferences
except the pathological one where all candidates are ranked equal
and none is approved. This is a nice geometric exercise :-)

For 4 candidates, one can also use a 2-dim. space, but not the
plane but the sphere or the surface of a regular tetrahedron:
we put the candidates at the corners of a regular tetrahedron with
side length 4/3. Then a voter who walks aroung on its surface will
come about all complete and transitive preferences in which 1, 2,
or 3 of the four candidates are approved. The positions of these
preferences are sketched below.

I don't know whether this is of any use...

Yours, Jobst

D

. . . . . D>>A             D>>C

D>>A=C

. . . . . . . . . . . D>A    D>>A>C     D>>C>A    D>C

D>A>>C               D>C>>A
D>A=C
. . . . . . . . . . . . . . . . . . . A=D           D>A>C     D>C>A           C=D
.                                          A=D>>C                       C=D>>A
.                                                   A=D>C         C=D>A
.
.         . . . . . . . . . . . . . A>D    A>D>>C    A>D>C   A=C=D   C>D>D    C>D>>A    C>D
.         .
.         .                                             A>C=D         C>A=D
.         .                                 A>>D>C                                 C>>D>A
.         .                                                 A>C>D     C>A>D                      C>>D
.         .         . . . . . . . A>>D                            A=C>D
.         .         .                        A>>C=D                                     C>>A=D
.         .         .                                A>>C>D                       C>>A>D
.         .         .                                        A>C>>D  A=C>>D C>A>>D
.         .         .
.         .         .            A               A>>C         A>C       A=C       C>A         C>>A               C
.         .         .
.         .         .                                        A>C>>B A=C>>B  C>A>>B
.         .         .                                A>>C>B                       C>>A>B
.         .         .                        A>>B=C                                     C>>A=B
.         .                                                       A=C>B
.         .      A>>D             A>>B                      A>C>B     C>A>B                      C>>B             C>>D
.         .                                 A>>B>C                                 C>>B>A
.         .           A>>B=D                            A>B=C         C>A=B                           C>>B=D
.
.         .       A>D    A>>D>B     A>>B>D    A>B    A>B>>C    A>B>C   A=B=C   C>B>A    C>B>>A    C>B    C>>B>D     C>>D>B    C>D
.         .
.         .       A>D>>B               A>B>>D                 A=B>C         B=C>A                 C>B>>D               C>D>>B
.                           A>B=D                    A=B>>C                       B=C>>A                    C>B=D
.       A=D           A>D>B     A>B>D           A=B           B>A>C     B>C>A           B=C           C>B>D     C>D>B           C=D
.          A=D>>B                       A=B>>D                    B>A=C                    B=C>>D                       C=D>>B
.                 A=D>B         A=B>D                 B>A>>C               B>>C>A                 B=C>D         C=D>B

.       D>A    D>A>>B    D>A>B   A=B=D   B>A>D    B>A>>D    B>A    B>>A>C     B>>C>A    B>C    B>C>>D    B>C>D   B=C=D   D>C>B    D>C>>B    D>C
.
.                       D>A=B         B>A=D                            B>>A=C                           B>C=D         D>B=C
D>>A>B                                 B>>A>D                                   B>>C>D                                 D>>C>B
D>>A                      D>B>A     B>D>A                      B>>A             B>>C                      B>D>C     D>B>C                      D>>C
B=D>A                                                                           B=D>C
D>>A=B                                     B>>A=D                               B>>C=D                                     D>>B=C
D>>B>A                       B>>D>A                                             B>>D>C                       D>>B>C
D>B>>A  B=D>>A B>D>>A                                                           B>D>>C  B=D>>C D>B>>C

D               D>>B         D>B       B=D       B>D         B>>D               B               B>>D         B>D       B=D       D>B         D>>B               D

```