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 ---- Election-methods mailing list - see http://electorama.com/em for list info