[EM] Examples with 4 options for immune methods

[Jobst Heitzig]

This is a comprehensive (at least I hope so) list of examples with 4
options of which more than one is immune, with all defeats of different
strength.

An option is *immune* here when each of its defeats is countered by a
chain of stronger defeats leading back. When there are at least two
immune options, then there is no Condorcet winner or loser, exactly 2
options are immune, and the options can be renamed so that the defeats
are A>B>C>D>A,A>C,B>D without loss of generality.

Interesting sets:
Smith = {ABCD}
Undominated = Banks = {ABD}
Copeland = {AB}

The immune set varies and is denoted by {..} in the list.

The strength of defeat X>Y is abbreviated by XY in the list.

The examples are ordered by largest defeat, 2nd largest defeat, etc.,
considering AB before AC, BC, BD, CD, and DA.
Those of the 6!=720 orders which have only one immune option are not
listed. Note that, for the remaining examples, the immune set is thus
determined by the 3 largest defeats!

The winners of the three major immune methods and of Plain Condorcet are
listed beneath.


COMPREHENSIVE (?) LIST OF 104 EXAMPLES:

                     Tideman
                     | River
                     | | Beatpath
                     | | |  Plain Condorcet
                     | | |  |
AB>BD>... {AC}
 CD>DA>AC>BC         A A C  C
 CD>DA>BC>AC         A A C  C
 DA>CD>AC>BC         A A C  C
 DA>CD>BC>AC         A A C  C
AB>CD>... {AC}
 BD>DA>AC>BC         A C C  C
 BD>DA>BC>AC         A C C  C

AC>BC>BD>... {AB}
 CD>DA>AB            A B B  B
AC>BC>CD>... {AB}
 BD>DA>AB            A A B  B
 DA>AB>BD            A A B  B
 DA>BD>AB            A A B  B
AC>BC>DA>... {BD}
 CD>AB>BD            D D B  B
AC>BD>... {AB}
 BC>CD>DA>AB         A B B  B
 CD>BC>DA>AB         A B B  B
 CD>DA>AB>BC         A B B  B
 CD>DA>BC>AB         A B B  B
AC>CD>... {AB}
 BC>BD>DA>AB         A A B  B
 BC>DA>AB>BD         A A B  B
 BC>DA>BD>AB         A A B  B
 BD>BC>DA>AB         A A B  B
 BD>DA>AB>BC         A A B  B
 BD>DA>BC>AB         A A B  B
 DA>BC>AB>BD         A A B  B
 DA>BC>BD>AB         A A B  B
 DA>BD>AB>BC         A A B  B
 DA>BD>BC>AB         A A B  B
AC>DA>... {BD}
 BC>CD>AB>BD         D D B  B
 CD>BC>AB>BD         D D B  B

BC>AC>BD>... {AB}
 CD>DA>AB            A B B  B
BC>AC>CD>... {AB}
 BD>DA>AB            A B B  B
 DA>AB>BD            A B B  B
 DA>BD>AB            A B B  B
BC>AC>DA>... {BD}
 CD>AB>BD            D B B  B
BC>BD>... {AB}
 AC>CD>DA>AB         A B B  B
 CD>AC>DA>AB         A B B  B
BC>CD>... {AB}
 AC>BD>DA>AB         A B B  B
 AC>DA>AB>BD         A B B  B
 AC>DA>BD>AB         A B B  B
 BD>AC>DA>AB         A B B  B
BC>DA>... {BD}
 AC>CD>AB>BD         D B B  B

BD>AB>... {AC}
 CD>DA>AC>BC         A A C  C
 CD>DA>BC>AC         A A C  C
 DA>CD>AC>BC         A A C  C
 DA>CD>BC>AC         A A C  C
BD>AC>... {AB}
 BC>CD>DA>AB         A B B  B
 CD>BC>DA>AB         A B B  B
 CD>DA>AB>BC         A B B  B
 CD>DA>BC>AB         A B B  B
BD>BC>... {AB}
 AC>CD>DA>AB         A B B  B
 CD>AC>DA>AB         A B B  B
BD>CD>AB>... {AC}
 DA>AC>BC            A A C  C
 DA>BC>AC            A A C  C
BD>CD>AC>... {AB}
 BC>DA>AB            A B B  B
 DA>AB>BC            A B B  B
 DA>BC>AB            A B B  B
BD>CD>BC>... {AB}
 AC>DA>AB            A B B  B
BD>CD>DA>... {BC}
 AB>AC>BC            B B C  C
 AB>BC>AC            B B C  C
 AC>AB>BC            B B B  B
BD>DA>... {BC}
 AB>CD>AC>BC         B B C  C
 AB>CD>BC>AC         B B C  C
 CD>AB>AC>BC         B B C  C
 CD>AB>BC>AC         B B C  C
 CD>AC>AB>BC         B B B  B

CD>AB>... {AC}
 BD>DA>AC>BC         A C C  C
 BD>DA>BC>AC         A C C  C
CD>AC>... {AB}
 BC>BD>DA>AB         A A B  B
 BC>DA>AB>BD         A A B  B
 BC>DA>BD>AB         A A B  B
 BD>BC>DA>AB         A A B  B
 BD>DA>AB>BC         A A B  B
 BD>DA>BC>AB         A A B  B
 DA>BC>AB>BD         A A B  B
 DA>BC>BD>AB         A A B  B
 DA>BD>AB>BC         A A B  B
 DA>BD>BC>AB         A A B  B
CD>BC>... {AB}
 AC>BD>DA>AB         A B B  B
 AC>DA>AB>BD         A B B  B
 AC>DA>BD>AB         A B B  B
 BD>AC>DA>AB         A B B  B
CD>BD>AB>... {AC}
 DA>AC>BC            A C C  C
 DA>BC>AC            A C C  C
CD>BD>AC>... {AB}
 BC>DA>AB            A A B  B
 DA>AB>BC            A A B  B
 DA>BC>AB            A A B  B
CD>BD>BC>... {AB}
 AC>DA>AB            A B B  B
CD>BD>DA>... {BC}
 AB>AC>BC            B C C  C
 AB>BC>AC            B C C  C
 AC>AB>BC            B C B  B
CD>DA>... {BC}
 AC>BD>AB>BC         B C B  B
 BD>AB>AC>BC         B C C  C
 BD>AB>BC>AC         B C C  C
 BD>AC>AB>BC         B C B  B

DA>AC>... {BD}
 BC>CD>AB>BD         D D B  B
 CD>BC>AB>BD         D D B  B
DA>BC>... {BD}
 AC>CD>AB>BD         D B B  B
DA>BD>... {BC}
 AB>CD>AC>BC         B B C  C
 AB>CD>BC>AC         B B C  C
 CD>AB>AC>BC         B B C  C
 CD>AB>BC>AC         B B C  C
 CD>AC>AB>BC         B B B  B
DA>CD>... {BC}
 AC>BD>AB>BC         B C B  B
 BD>AB>AC>BC         B C C  C
 BD>AB>BC>AC         B C C  C
 BD>AC>AB>BC         B C B  B

(END OF LIST)


Note that Beatpath and Plain Condorcet are unanimous in all these examples!

Statistics (ignoring Plain Condorcet):

# possible orderings with Smith set {ABCD}: 720
# situations with non-unique immune (= # examples): 104 (ca. 14%)
# Tideman unique: 45 (ca. 6%)
# River unique: 5 (ca. 0.7%)
# Beatpath unique: 51 (ca. 7%)
# all equal: 3 (ca. 0.4%)
# Tideman and River different: 50 (ca. 7%)
# Tideman and Beatpath different: 96 (ca. 13%)
# River and Beatpath different: 56 (ca. 8%)


Jobst