class svvamp.RuleWoodall(**kwargs)[source]#

Woodall Rule.

Options#

>>> RuleWoodall.print_options_parameters()
cm_option: ['fast', 'slow', 'very_slow', 'exact']. Default: 'fast'.
icm_option: ['exact']. Default: 'exact'.
iia_subset_maximum_size: is_number. Default: 2.
im_option: ['lazy', 'exact']. Default: 'lazy'.
tm_option: ['exact']. Default: 'exact'.
um_option: ['lazy', 'exact']. Default: 'lazy'.

Notes

Each voter must provide a strict total order. Among the candidates of the Smith set (in the sense of smith_set_rk), elect the one that is eliminated latest by RuleIRV.

  • is_cm_():

    • cm_option = 'fast': Rely on RuleIRV’s fast algorithm. Polynomial heuristic. Can prove CM but unable to decide non-CM (except in rare obvious cases).

    • cm_option = 'slow': Rely on RuleExhaustiveBallot’s exact algorithm. Non-polynomial heuristic (\(2^{n_c}\)). Quite efficient to prove CM or non-CM.

    • cm_option = 'very_slow': Rely on RuleIRV’s exact algorithm. Non-polynomial heuristic (\(n_c!\)). Very efficient to prove CM or non-CM.

    • cm_option = 'exact': Non-polynomial algorithm from superclass Rule.

    Each algorithm above exploits the faster ones. For example, if cm_option = 'very_slow', SVVAMP tries the fast algorithm first, then the slow one, then the ‘very slow’ one. As soon as it reaches a decision, computation stops.

Woodall does not meets_condorcet_c_ut_abs_ctb:

>>> profile = Profile(preferences_ut=[
...     [ 0. ,  0.5, -0.5],
...     [ 0.5, -0.5,  1. ],
... ], preferences_rk=[
...     [1, 0, 2],
...     [2, 0, 1],
... ])
>>> RuleWoodall()(profile).w_
1
>>> profile.condorcet_winner_ut_abs_ctb
0

Woodall does not meets_condorcet_c_ut_rel:

>>> profile = Profile(preferences_ut=[
...     [ 1. ,  1. ,  1. ],
...     [ 0.5, -0.5,  1. ],
... ], preferences_rk=[
...     [0, 1, 2],
...     [2, 0, 1],
... ])
>>> RuleWoodall()(profile).w_
0
>>> profile.condorcet_winner_ut_rel
2

References

‘Four Condorcet-Hare Hybrid Methods for Single-Winner Elections’, James Green-Armytage, 2011.

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 = RuleWoodall()(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 =
[[0 1 2]
 [0 2 1]
 [1 0 2]
 [2 0 1]
 [2 1 0]]
scores =
[[1 0 0]
 [2 0 1]]
candidates_by_scores_best_to_worst
[0, 2, 1]
scores_best_to_worst
[[1 0 0]
 [2 1 0]]
w = 0
score_w = [1 2]
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 =  True
is_based_on_rk =  True
is_based_on_ut_minus1_1 =  False
meets_iia =  False

****************************************************
*   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 (True)      ||
 ||           ||               ||       ||             ||         ||
 V            V                ||       ||             V          V
Condorcet_c_ut_abs_ctb (False)     ==>     Condorcet_ut_abs_c (True)
 ||                            ||       ||                        ||
 ||                            V        V                         ||
 ||       maj_fav_c_rk_ctb (True)  ==> maj_fav_c_rk (True)        ||
 ||           ||                                       ||         ||
 V            V                                        V          V
majority_favorite_c_ut_ctb (True)  ==> majority_favorite_c_ut (True)
 ||                                                               ||
 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 = nan
log_im: im_option = lazy
candidates_im =
[ 0.  0. nan]

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

********************************
*   Unison Manipulation (UM)   *
********************************
is_um = nan
log_um: um_option = lazy
candidates_um =
[ 0.  0. nan]

*********************************************
*   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 = nan
log_cm: cm_option = fast, um_option = lazy, tm_option = exact
candidates_cm =
[ 0.  0. nan]
necessary_coalition_size_cm =
[0. 1. 2.]
sufficient_coalition_size_cm =
[0. 2. 4.]
property candidates_by_scores_best_to_worst_#

1d array of integers. All candidates, sorted from the winner to the last candidate in the election’s result.

Default behavior: candidates_by_scores_best_to_worst[k] is the candidate with k-th highest value in scores_. By definition, candidates_by_scores_best_to_worst[0] = w_.

property scores_#

Scores of the candidates in the election.

See specific documentation for each voting rule. Typical type in most subclasses: 1d or 2d array. Typical behavior in most subclasses:

  • If scores_ is a 1d array, then scores_[c] is the numerical score for candidate c.

  • If scores_ is a 2d array, then scores_[:, c] is the score vector for candidate c.

It is not mandatory to follow this behavior.

property w_#
>>> profile = Profile(preferences_ut=[
...     [3, 0, 1, 2],
...     [0, 1, 2, 3],
...     [0, 3, 2, 1],
...     [0, 3, 2, 1],
...     [3, 0, 2, 1],
...     [1, 0, 2, 3],
... ], preferences_rk=[
...     [0, 3, 2, 1],
...     [3, 2, 1, 0],
...     [1, 2, 3, 0],
...     [1, 2, 3, 0],
...     [0, 2, 3, 1],
...     [3, 2, 0, 1],
... ])
>>> rule = RuleWoodall()(profile)
>>> rule.w_
3