Options#

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

Notes

The Slater method is defined similarly to the Kemeny method, but relying on the matrix of victories instead of the matrix of duels.

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 = RuleSlater()(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 =
[2 0 1]
candidates_by_scores_best_to_worst
[0 2 1]
scores_best_to_worst
[2 1 0]
w = 0
score_w = 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 (True)  ==> Condorcet_c_rk (True)      ||
 ||           ||               ||       ||             ||         ||
 V            V                ||       ||             V          V
Condorcet_c_ut_abs_ctb (True)      ==>     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 = lazy, um_option = lazy, icm_option = exact, tm_option = exact
candidates_cm =
[ 0.  0. nan]
necessary_coalition_size_cm =
[0. 1. 2.]
sufficient_coalition_size_cm =
[0. 2. 3.]

property candidates_by_scores_best_to_worst_#

1d array of integers. This is an optimal Kemeny order.

In case several orders are optimal, the first one by lexicographic order is given. This implies that if several winners are possible, the one with lowest index is declared the winner.

property scores_#: 1d array of integers. By convention, scores are integers from 1 to n_c, with n_c for the winner and 1 for the last candidate in Kemeny optimal order.

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.