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Proposition A.2
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1st profile:
------------

profile with 10 voters and 3 candidates:
 2 x {a},
 3 x {a, c},
 3 x {b, c},
 2 x {b}

winning committees for k=1 and k=2:
----------------------------------
Proportional Approval Voting (PAV)
----------------------------------

Computing only one winning committee (resolute=True)


Optimal PAV-score: 6

1 winning committee:
 {c}

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

Computing only one winning committee (resolute=True)


Optimal PAV-score: 10

1 winning committee:
 {a, b}

 PAV: {c} vs {a, b}
--------------------------------
Approval Chamberlin-Courant (CC)
--------------------------------

Computing only one winning committee (resolute=True)


Optimal CC-score: 6

1 winning committee:
 {c}

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

Computing only one winning committee (resolute=True)


Optimal CC-score: 10

1 winning committee:
 {a, b}

 CC: {c} vs {a, b}
-------------------------------
Monroe's Approval Rule (Monroe)
-------------------------------

Computing only one winning committee (resolute=True)


Optimal Monroe score: 6

1 winning committee:
 {c}

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

Computing only one winning committee (resolute=True)


Optimal Monroe score: 10

1 winning committee:
 {a, b}

 Monroe: {c} vs {a, b}
------------------------------------------
Phragmén's Minimax Rule (minimax-Phragmén)
------------------------------------------

Computing only one winning committee (resolute=True)


1 winning committee:
 {c}

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

Computing only one winning committee (resolute=True)


1 winning committee:
 {a, b}

 minimax-Phragmén: {c} vs {a, b}
-----------------------------
Minimax Approval Voting (MAV)
-----------------------------

Computing only one winning committee (resolute=True)


1 winning committee:
 {c}

Minimum maximal distance: 2
Corresponding distances to voters:
[2, 2, 1, 1, 1, 1, 1, 1, 2, 2]

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

Computing only one winning committee (resolute=True)


1 winning committee:
 {a, b}

Minimum maximal distance: 2
Corresponding distances to voters:
[1, 1, 2, 2, 2, 2, 2, 2, 1, 1]

 minimaxav: {c} vs {a, b}

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2nd profile:
------------

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

winning committees for k=2 and k=3:
-------------
Greedy Monroe
-------------


The Monroe assignment computed by Greedy Monroe
has a Monroe score of 14.
Assignment (unsatisfatied voters marked with *):

 candidate a assigned to: 0, 1, 2, 3, 4, 5, 6, 7, 8
 candidate b assigned to: 9*, 10, 11, 12*, 13*, 14*, 15, 16, 17

1 winning committee:
 {a, b}

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


The Monroe assignment computed by Greedy Monroe
has a Monroe score of 16.
Assignment (unsatisfatied voters marked with *):

 candidate a assigned to: 0, 1, 2, 3, 4, 5
 candidate c assigned to: 6, 7, 8, 9, 10, 11
 candidate d assigned to: 12*, 13*, 14, 15, 16, 17

1 winning committee:
 {a, c, d}

 Greedy Monroe: {a, b} vs {a, c, d}

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3rd profile:
------------

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

winning committees for k=3 and k=4:
------------------------------------------------------------
Method of Equal Shares (aka Rule X) with Phragmén completion
------------------------------------------------------------


Phase 1:

starting budget:
  (3/4, 3/4, 3/4, 3/4)

adding candidate number 1: a
 with maxmimum cost per voter q = 1/2
 remaining budget:
  (1/4, 1/4, 3/4, 3/4)
 tie broken in favor of a,
 candidates {a, c, d, e} are tied
 (all would impose a maximum cost of 1/2).

adding candidate number 2: c
 with maxmimum cost per voter q = 3/4
 remaining budget:
  (1/4, 0, 3/4, 0)
 tie broken in favor of c,
 candidates {c, d, e} are tied
 (all would impose a maximum cost of 3/4).

adding candidate number 3: e
 with maxmimum cost per voter q = 3/4
 remaining budget:
  (0, 0, 0, 0)

1 winning committee:
 {a, c, e}

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


Phase 1:

starting budget:
  (1, 1, 1, 1)

adding candidate number 1: a
 with maxmimum cost per voter q = 1/2
 remaining budget:
  (1/2, 1/2, 1, 1)
 tie broken in favor of a,
 candidates {a, c, d, e} are tied
 (all would impose a maximum cost of 1/2).

adding candidate number 2: c
 with maxmimum cost per voter q = 1/2
 remaining budget:
  (1/2, 0, 1, 1/2)
 tie broken in favor of c,
 candidates {c, d, e} are tied
 (all would impose a maximum cost of 1/2).

adding candidate number 3: d
 with maxmimum cost per voter q = 1/2
 remaining budget:
  (0, 0, 1, 0)
 tie broken in favor of d,
 candidates {d, e} are tied
 (all would impose a maximum cost of 1/2).

adding candidate number 4: b
 with maxmimum cost per voter q = 1
 remaining budget:
  (0, 0, 0, 0)
 tie broken in favor of b,
 candidates {b, e} are tied
 (all would impose a maximum cost of 1).

1 winning committee:
 {a, b, c, d}

 Equal Shares: {a, c, e} vs {a, b, c, d}
