class svvamp.RuleApproval(approval_comparator='>', approval_threshold=0.0, **kwargs)[source]#

Approval voting.

Parameters:

  • approval_comparator (str) – Can be '>' (default) or '>='. When approval_comparator is '>', sincere voter v approves of candidates c iff preferences_ut[ v, c] > approval_threshold. When approval_comparator is '>=', previous relation is modified accordingly.

  • approval_threshold (number) – Number (default 0). Utility above which a sincere voter approves of a candidate.

  • Options

  • ------- – 

    >>> RuleApproval.print_options_parameters()
approval_comparator: ['>', '>=']. Default: '>'.
approval_threshold: is_number. Default: 0.
cm_option: ['exact']. Default: 'exact'.
icm_option: ['exact']. Default: 'exact'.
iia_subset_maximum_size: is_number. Default: 2.
im_option: ['exact']. Default: 'exact'.
tm_option: ['exact']. Default: 'exact'.
um_option: ['exact']. Default: 'exact'.

Notes

Each voter may vote for any number of candidates. The candidate with most votes is declared the winner. In case of a tie, the tied candidate with lowest index wins.

is_iia(): With our assumptions, Approval voting always meets IIA.

is_cm_(), is_icm_(), is_im_(), is_tm_(), is_um_(): Exact in polynomial time.

References

‘Approval voting’, Steven Brams and Peter Fishburn. In: American Political Science Review 72 (3 1978), pp. 831–847.

Examples

>>> profile = Profile(preferences_ut=[
...     [ 0. , -0.5, -1. ],
...     [ 1. , -1. ,  0.5],
...     [ 0.5,  0.5, -0.5],
...     [ 0.5,  0. ,  1. ],
...     [-1. , -1. ,  1. ],
... ], preferences_rk=[
...     [0, 1, 2],
...     [0, 2, 1],
...     [1, 0, 2],
...     [2, 0, 1],
...     [2, 1, 0],
... ])
>>> rule = RuleApproval()(profile)
>>> rule.demo_results_(log_depth=0)  

************************
*                      *
*   Election Results   *
*                      *
************************

***************
*   Results   *
***************
profile_.preferences_ut (reminder) =
[[ 0.  -0.5 -1. ]
 [ 1.  -1.   0.5]
 [ 0.5  0.5 -0.5]
 [ 0.5  0.   1. ]
 [-1.  -1.   1. ]]
profile_.preferences_rk (reminder) =
[[0 1 2]
 [0 2 1]
 [1 0 2]
 [2 0 1]
 [2 1 0]]
ballots =
[[False False False]
 [ True False  True]
 [ True  True False]
 [ True False  True]
 [False False  True]]
scores =
[3 1 3]
candidates_by_scores_best_to_worst
[0 2 1]
scores_best_to_worst
[3 3 1]
w = 0
score_w = 3
total_utility_w = 1.0

*********************************
*   Condorcet efficiency (rk)   *
*********************************
w (reminder) = 0

condorcet_winner_rk_ctb = 0
w_is_condorcet_winner_rk_ctb = True
w_is_not_condorcet_winner_rk_ctb = False
w_missed_condorcet_winner_rk_ctb = False

condorcet_winner_rk = 0
w_is_condorcet_winner_rk = True
w_is_not_condorcet_winner_rk = False
w_missed_condorcet_winner_rk = False

***************************************
*   Condorcet efficiency (relative)   *
***************************************
w (reminder) = 0

condorcet_winner_ut_rel_ctb = 0
w_is_condorcet_winner_ut_rel_ctb = True
w_is_not_condorcet_winner_ut_rel_ctb = False
w_missed_condorcet_winner_ut_rel_ctb = False

condorcet_winner_ut_rel = 0
w_is_condorcet_winner_ut_rel = True
w_is_not_condorcet_winner_ut_rel = False
w_missed_condorcet_winner_ut_rel = False

***************************************
*   Condorcet efficiency (absolute)   *
***************************************
w (reminder) = 0

condorcet_admissible_candidates =
[ True False False]
w_is_condorcet_admissible = True
w_is_not_condorcet_admissible = False
w_missed_condorcet_admissible = False

weak_condorcet_winners =
[ True False False]
w_is_weak_condorcet_winner = True
w_is_not_weak_condorcet_winner = False
w_missed_weak_condorcet_winner = False

condorcet_winner_ut_abs_ctb = 0
w_is_condorcet_winner_ut_abs_ctb = True
w_is_not_condorcet_winner_ut_abs_ctb = False
w_missed_condorcet_winner_ut_abs_ctb = False

condorcet_winner_ut_abs = 0
w_is_condorcet_winner_ut_abs = True
w_is_not_condorcet_winner_ut_abs = False
w_missed_condorcet_winner_ut_abs = False

resistant_condorcet_winner = nan
w_is_resistant_condorcet_winner = False
w_is_not_resistant_condorcet_winner = True
w_missed_resistant_condorcet_winner = False
>>> rule.demo_manipulation_(log_depth=0)  

*****************************
*                           *
*   Election Manipulation   *
*                           *
*****************************

*********************************************
*   Basic properties of the voting system   *
*********************************************
with_two_candidates_reduces_to_plurality =  False
is_based_on_rk =  False
is_based_on_ut_minus1_1 =  True
meets_iia =  True

****************************************************
*   Manipulation properties of the voting system   *
****************************************************
Condorcet_c_ut_rel_ctb (False)     ==>     Condorcet_c_ut_rel (False)
 ||                                                               ||
 ||     Condorcet_c_rk_ctb (False) ==> Condorcet_c_rk (False)     ||
 ||           ||               ||       ||             ||         ||
 V            V                ||       ||             V          V
Condorcet_c_ut_abs_ctb (False)     ==>     Condorcet_ut_abs_c (False)
 ||                            ||       ||                        ||
 ||                            V        V                         ||
 ||       maj_fav_c_rk_ctb (False) ==> maj_fav_c_rk (False)       ||
 ||           ||                                       ||         ||
 V            V                                        V          V
majority_favorite_c_ut_ctb (False) ==> majority_favorite_c_ut (False)
 ||                                                               ||
 V                                                                V
IgnMC_c_ctb (True)                 ==>                IgnMC_c (True)
 ||                                                               ||
 V                                                                V
InfMC_c_ctb (True)                 ==>                InfMC_c (True)

*****************************************************
*   Independence of Irrelevant Alternatives (IIA)   *
*****************************************************
w (reminder) = 0
is_iia = True
log_iia: iia_subset_maximum_size = 2.0
example_winner_iia = nan
example_subset_iia = nan

**********************
*   c-Manipulators   *
**********************
w (reminder) = 0
preferences_ut (reminder) =
[[ 0.  -0.5 -1. ]
 [ 1.  -1.   0.5]
 [ 0.5  0.5 -0.5]
 [ 0.5  0.   1. ]
 [-1.  -1.   1. ]]
v_wants_to_help_c =
[[False False False]
 [False False False]
 [False False False]
 [False False  True]
 [False False  True]]

************************************
*   Individual Manipulation (IM)   *
************************************
is_im = True
log_im: im_option = exact
candidates_im =
[0. 0. 1.]

*********************************
*   Trivial Manipulation (TM)   *
*********************************
is_tm = True
log_tm: tm_option = exact
candidates_tm =
[0. 0. 1.]

********************************
*   Unison Manipulation (UM)   *
********************************
is_um = True
log_um: um_option = exact
candidates_um =
[0. 0. 1.]

*********************************************
*   Ignorant-Coalition Manipulation (ICM)   *
*********************************************
is_icm = False
log_icm: icm_option = exact
candidates_icm =
[0. 0. 0.]
necessary_coalition_size_icm =
[0. 6. 4.]
sufficient_coalition_size_icm =
[0. 6. 4.]

***********************************
*   Coalition Manipulation (CM)   *
***********************************
is_cm = True
log_cm: cm_option = exact
candidates_cm =
[0. 0. 1.]
necessary_coalition_size_cm =
[0. 3. 2.]
sufficient_coalition_size_cm =
[0. 3. 2.]
property ballots_#

2d array of values in {0, 1}. ballots_[v, c] = 1 iff voter v votes for candidates c.

See also

approval_comparator, approval_threshold

property scores_#

1d array of integers. scores_[c] is the number of voters who vote for candidate c.

property w_#

Integer (winning candidate).

Default behavior: the candidate with highest value in vector scores_ is declared the winner. In case of a tie, the tied candidate with lowest index wins.