***************
Proposition A.4
***************

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

profile with 5 voters and 6 candidates:
 1 x {a, b},
 3 x {a},
 1 x {c}

original winning committee:
 {a, c}

Manipulation by voter 0: {a, b} --> {b}

winning committee after manipulation:
 {a, b}

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

profile with 6 voters and 6 candidates:
 1 x {c, d, e},
 1 x {a, b},
 1 x {b, f},
 1 x {a, c, d},
 1 x {b, c, f},
 1 x {c, e, f}

original winning committee:
 {b, c, f}

Manipulation by voter 0: {c, d, e} --> {e}

winning committee after manipulation:
 {b, c, e}

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

profile with 6 voters and 6 candidates:
 1 x {a, b},
 1 x {b, d},
 1 x {c, f},
 1 x {a, b, f},
 1 x {b, f},
 1 x {b, c}

original winning committee:
 {b, c, f}

Manipulation by voter 0: {a, b} --> {a}

winning committee after manipulation:
 {a, b, f}

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

profile with 12 voters and 6 candidates:
 1 x {b, e, f},
 1 x {a, b},
 1 x {d, e, f},
 1 x {d, e},
 1 x {b, f},
 2 x {c, d},
 1 x {a, b, c},
 1 x {a, c},
 1 x {a, b, e},
 1 x {a, e, f},
 1 x {b, c, d}

original winning committee:
 {a, b, d}

Manipulation by voter 0: {b, e, f} --> {c}
 (not a subset!)

winning committee after manipulation:
 {b, c, e}

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

profile with 5 voters and 6 candidates:
 1 x {a, b, c},
 1 x {b, d},
 1 x {b, c},
 1 x {a, d, e},
 1 x {b, e}

original winning committee:
 {b, d}

Manipulation by voter 0: {a, b, c} --> {a}

winning committee after manipulation:
 {a, b}

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

profile with 12 voters and 6 candidates:
 1 x {b, d},
 1 x {a, b, c},
 1 x {b, e},
 1 x {d, e},
 1 x {e, f},
 1 x {b, c, e},
 1 x {c, d, e},
 1 x {b, c},
 2 x {a, f},
 1 x {b, c, d},
 1 x {a, d}

original winning committee:
 {a, b, e}

Manipulation by voter 0: {b, d} --> {f}
 (not a subset!)

winning committee after manipulation:
 {b, d, f}

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

profile with 4 voters and 6 candidates:
 1 x {a, b},
 1 x {a, c, f},
 1 x {a, c, d},
 1 x {e, f}

original winning committee:
 {a, c}

Manipulation by voter 0: {a, b} --> {b}

winning committee after manipulation:
 {a, b}

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

profile with 6 voters and 6 candidates:
 1 x {a, b, c},
 1 x {a, b},
 1 x {b, f},
 1 x {c, e},
 1 x {b, e, f},
 1 x {b, d, f}

original winning committee:
 {b, f}

Manipulation by voter 0: {a, b, c} --> {c}

winning committee after manipulation:
 {b, c}

------------------------------------------
Phragmén's Leximax Rule (leximax-Phragmén)
------------------------------------------

profile with 4 voters and 6 candidates:
 1 x {a, b},
 3 x {b, c, d}

original winning committee:
 {b, c, d}

Manipulation by voter 0: {a, b} --> {a}

winning committee after manipulation:
 {a, b, c}

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

profile with 6 voters and 6 candidates:
 1 x {b, c, d},
 1 x {a, b},
 1 x {b, d},
 1 x {c, d},
 2 x {d, e}

original winning committee:
 {b, d, e}

Manipulation by voter 0: {b, c, d} --> {c}

winning committee after manipulation:
 {b, c, d}

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

profile with 6 voters and 6 candidates:
 1 x {a, b, c},
 1 x {b, d},
 2 x {a, b, e},
 1 x {a, b, d},
 1 x {a, b}

original winning committee:
 {a, b, d}

Manipulation by voter 0: {a, b, c} --> {c}

winning committee after manipulation:
 {a, b, c}

----------------------------------
Satisfaction Approval Voting (SAV)
----------------------------------

profile with 2 voters and 6 candidates:
 1 x {a, b, c},
 1 x {d, e}

original winning committee:
 {d}

Manipulation by voter 0: {a, b, c} --> {a}

winning committee after manipulation:
 {a}

