***************
Proposition A.3
***************

-----------------------------------------
Phragmén's Sequential Rule (seq-Phragmén)
-----------------------------------------

profile with 52 voters and 8 candidates:
 10 x {a, b},
 3 x {c},
 12 x {d},
 21 x {a, b, c},
 6 x {c, d}

original winning committees:
 {a, b, d}

additional voter: {a, b}

winning committees after the modification:
 {a, c, d}

seq-Phragmén fails support monotonicity with additional voters

------------------------------------------
Phragmén's Minimax Rule (minimax-Phragmén)
------------------------------------------

profile with 18 voters and 8 candidates:
 13 x {b, c, d, e, f},
 2 x {a, g},
 2 x {a},
 1 x {g}

original winning committees:
 {a, b, c, d, e, f}
 {b, c, d, e, f, g}

additional voter: {a, b, c, d, e, f}

winning committees after the modification:
 {a, b, c, d, e, g}
 {a, b, c, d, f, g}
 {a, b, c, e, f, g}
 {a, b, d, e, f, g}
 {a, c, d, e, f, g}

minimax-Phragmén fails support monotonicity with additional voters

-------------------------------------------------
Sequential Proportional Approval Voting (seq-PAV)
-------------------------------------------------

profile with 19 voters and 8 candidates:
 7 x {a, b, d},
 4 x {a, b, e},
 3 x {a, c, d},
 5 x {a, c, e}

original winning committees:
 {a, b, c, d}

additional voter: {c, d}

winning committees after the modification:
 {a, b, d, e}

seq-PAV fails support monotonicity with additional voters

-----------------------------------------------
Sequential Approval Chamberlin-Courant (seq-CC)
-----------------------------------------------

profile with 12 voters and 8 candidates:
 3 x {a},
 1 x {a, c, d},
 1 x {b},
 2 x {b, c},
 1 x {b, d},
 2 x {c},
 2 x {d}

original winning committees:
 {a, c, d}

additional voter: {a, d}

winning committees after the modification:
 {a, b, c}

seq-CC fails support monotonicity with additional voters

-------------
Greedy Monroe
-------------

profile with 9 voters and 8 candidates:
 1 x {b, c, d},
 1 x {a, c, f},
 1 x {a, d, e},
 1 x {c, e},
 1 x {a, b},
 2 x {d, f},
 1 x {b, e},
 1 x {b, f}

original winning committees:
 {b, e, f}

additional voter: {e}

winning committees after the modification:
 {b, c, d}

Greedy Monroe fails candidate monotonicity with additional voters

------------------------------------------------------------
Method of Equal Shares (aka Rule X) with Phragmén completion
------------------------------------------------------------

profile with 9 voters and 8 candidates:
 1 x {b, d},
 1 x {a, b},
 1 x {b, d, e},
 1 x {a, e},
 2 x {c, d, e},
 1 x {c, e},
 1 x {a, c, e},
 1 x {b, c, d}

original winning committees:
 {a, d, e}

additional voter: {a}

winning committees after the modification:
 {b, c, e}

Equal Shares fails candidate monotonicity with additional voters

---------------------------------------------------------------------
Method of Equal Shares (aka Rule X) without completion (second phase)
---------------------------------------------------------------------

profile with 12 voters and 8 candidates:
 2 x {a, b, c},
 1 x {a, g},
 1 x {d, e},
 1 x {b, d, f},
 1 x {a, f},
 1 x {h},
 1 x {a, h},
 1 x {b, h},
 1 x {b, d},
 1 x {d, e, f},
 1 x {c, e, h}

original winning committees:
 {a, b, e}

additional voter: {e}

winning committees after the modification:
 {a, d, h}

Equal Shares without completion fails candidate monotonicity with additional voters

-------------------------------
Monroe's Approval Rule (Monroe)
-------------------------------

profile with 33 voters and 8 candidates:
 5 x {a, e},
 4 x {a, g},
 5 x {b, e},
 4 x {b, h},
 5 x {c, f},
 4 x {c, g},
 3 x {d, f},
 3 x {d, h}

original winning committees:
 {e, f, g, h}

additional voter: {e}

winning committees after the modification:
 {a, b, c, d}
 {e, f, g, h}

Monroe fails candidate monotonicity with additional voters

----------------------------------
Proportional Approval Voting (PAV)
----------------------------------

profile with 131 voters and 8 candidates:
 1 x {a, b},
 1 x {a, b, c},
 2 x {e, f},
 3 x {a, e},
 3 x {b, e},
 3 x {c, e},
 3 x {a, f},
 3 x {b, f},
 3 x {c, f},
 3 x {a, g},
 3 x {b, g},
 3 x {c, g},
 100 x {d}

original winning committees:
 {a, b, c, d}

change of voter 0: {a, b} --> {a, b, c, d}

winning committees after the modification:
 {d, e, f, g}

PAV fails support monotonicity without additional voters

--------------------------------
Approval Chamberlin-Courant (CC)
--------------------------------

profile with 13 voters and 8 candidates:
 2 x {a},
 2 x {a, d},
 2 x {a, e},
 2 x {c, d},
 2 x {c, e},
 2 x {b},
 1 x {d}

original winning committees:
 {a, b, c}

change of voter 0: {a} --> {a, b, c}

winning committees after the modification:
 {a, b, c}
 {b, d, e}

CC fails support monotonicity without additional voters

-------------------------------
Monroe's Approval Rule (Monroe)
-------------------------------

profile with 18 voters and 8 candidates:
 2 x {a},
 2 x {a, d},
 2 x {a, e},
 4 x {b},
 4 x {c, d},
 1 x {b, e},
 3 x {c, e}

original winning committees:
 {a, b, c}

change of voter 0: {a} --> {a, b, c}

winning committees after the modification:
 {a, b, c}
 {b, d, e}

Monroe fails support monotonicity without additional voters

------------------------------------------
Phragmén's Minimax Rule (minimax-Phragmén)
------------------------------------------

profile with 19 voters and 8 candidates:
 1 x {h},
 13 x {b, c, d, e, f},
 2 x {a, g},
 2 x {a},
 1 x {g}

original winning committees:
 {a, b, c, d, e, f}
 {b, c, d, e, f, g}

change of voter 0: {h} --> {a, b, c, d, e, f, h}

winning committees after the modification:
 {a, b, c, d, e, g}
 {a, b, c, d, f, g}
 {a, b, c, e, f, g}
 {a, b, d, e, f, g}
 {a, c, d, e, f, g}

minimax-Phragmén fails support monotonicity without additional voters

-----------------------------
Minimax Approval Voting (MAV)
-----------------------------

profile with 9 voters and 8 candidates:
 1 x {f},
 1 x {b},
 1 x {c},
 1 x {d},
 1 x {b, e, g},
 1 x {b, e, h},
 1 x {b, f, g},
 1 x {b, f, h},
 1 x {b, g}

original winning committees:
 {b, c, d, e, f}

change of voter 0: {f} --> {b, c, d, e, f}

winning committees after the modification:
 {b, c, d, g, h}

minimaxav fails support monotonicity without additional voters

-----------------------------------------
Phragmén's Sequential Rule (seq-Phragmén)
-----------------------------------------

profile with 32 voters and 8 candidates:
 7 x {c},
 4 x {a, b},
 1 x {a, b, c},
 16 x {a, b, d},
 4 x {c, d}

original winning committees:
 {a, b, c}

change of voter 0: {c} --> {a, b, c}

winning committees after the modification:
 {a, c, d}

seq-Phragmén fails support monotonicity without additional voters

-------------------------------------------------
Sequential Proportional Approval Voting (seq-PAV)
-------------------------------------------------

profile with 8 voters and 8 candidates:
 1 x {c, d},
 1 x {a, c},
 1 x {a, d},
 1 x {a, f},
 1 x {b, c},
 2 x {b, f},
 1 x {c, e}

original winning committees:
 {a, c, f}

change of voter 0: {c, d} --> {a, c, d, f}

winning committees after the modification:
 {a, b, c}

seq-PAV fails support monotonicity without additional voters

------------------------------------------------------------
Reverse Sequential Proportional Approval Voting (revseq-PAV)
------------------------------------------------------------

profile with 18 voters and 8 candidates:
 2 x {a, e},
 2 x {b, c, d},
 1 x {d, e},
 3 x {c, e},
 1 x {b, d, e},
 1 x {a, b, c},
 1 x {c, d, e},
 2 x {a, d, e},
 1 x {b, d},
 1 x {a, b},
 1 x {a, d},
 1 x {a, b, d},
 1 x {b, c}

original winning committees:
 {c, d, e}

change of voter 0: {a, e} --> {a, c, d, e}

winning committees after the modification:
 {b, d, e}

revseq-PAV fails support monotonicity without additional voters

-----------------------------------------------
Sequential Approval Chamberlin-Courant (seq-CC)
-----------------------------------------------

profile with 13 voters and 8 candidates:
 1 x {e},
 1 x {a},
 1 x {a, d},
 3 x {b},
 2 x {a, c},
 1 x {b, c, d},
 2 x {c},
 2 x {d}

original winning committees:
 {b, c, d}

change of voter 0: {e} --> {b, d, e}

winning committees after the modification:
 {a, b, c}

seq-CC fails support monotonicity without additional voters

------------------------------------------------------------
Method of Equal Shares (aka Rule X) with Phragmén completion
------------------------------------------------------------

profile with 7 voters and 8 candidates:
 1 x {b},
 1 x {a, b, e},
 2 x {b, e},
 1 x {c},
 1 x {a, c},
 1 x {a}

original winning committees:
 {a, b, e}

change of voter 0: {b} --> {a, b, e}

winning committees after the modification:
 {a, b, c}

Equal Shares fails support monotonicity without additional voters

-------------
Greedy Monroe
-------------

profile with 4 voters and 8 candidates:
 1 x {d},
 1 x {c},
 1 x {b},
 1 x {a, c}

original winning committees:
 {b, c}

change of voter 0: {d} --> {b, c, d}

winning committees after the modification:
 {a, c}

Greedy Monroe fails support monotonicity without additional voters

