Options#

>>> RuleIteratedBucklin.print_options_parameters()
cm_option: ['lazy', 'exact']. Default: 'lazy'.
icm_option: ['lazy']. Default: 'lazy'.
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 candidate with least adjusted median Borda score (cf. below) is eliminated. Then the new Borda scores are computed. Etc. Ties are broken in favor of lower-index candidates: in case of a tie, the candidate with highest index is eliminated.

Adjusted median Borda score:

Let med_c be the median Borda score for candidate c. Let x_c the number of voters who put a lower Borda score to c. Then c’s adjusted median is med_c - x_c / (n_v + 1).

If med_c > med_d, then it is also true for the adjusted median. If med_c = med_d, then c has a better adjusted median iff x_c < x_d, i.e. if more voters give to c the Borda score med_c or higher.

So, the best candidate by adjusted median is the Bucklin winner. Here, at each round, we eliminate the candidate with lowest adjusted median Borda score, which justifies the name of “Iterated Bucklin method”.

Unlike Baldwin method (= Iterated Borda), Iterated Bucklin does not meet the Condorcet criterion. Indeed, a Condorcet winner may have the (strictly) worst median ranking.

is_cm_(): Non-polynomial or non-exact algorithms from superclass Rule.
is_icm_(): The algorithm from superclass Rule is polynomial and has a window of error of 1 manipulator.
is_im_(): Non-polynomial or non-exact algorithms from superclass Rule.
is_iia(): Non-polynomial or non-exact algorithms from superclass Rule.
is_tm_(): Exact in polynomial time.
is_um_(): Non-polynomial or non-exact algorithms from superclass Rule.

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 = RuleIteratedBucklin()(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 =
[[0.83333333 0.66666667 0.66666667]
 [1.66666667 1.                inf]]
candidates_by_scores_best_to_worst
[0, 1, 2]
scores_best_to_worst
[[0.83333333 0.66666667 0.66666667]
 [1.66666667 1.                inf]]
w = 0
score_w = [0.83333333 1.66666667]
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 (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 (True)        ||
 ||           ||                                       ||         ||
 V            V                                        V          V
majority_favorite_c_ut_ctb (False) ==> majority_favorite_c_ut (True)
 ||                                                               ||
 V                                                                V
IgnMC_c_ctb (False)                ==>                IgnMC_c (True)
 ||                                                               ||
 V                                                                V
InfMC_c_ctb (False)                ==>                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 = lazy
candidates_icm =
[0. 0. 0.]
necessary_coalition_size_icm =
[0. 5. 3.]
sufficient_coalition_size_icm =
[0. 6. 4.]

***********************************
*   Coalition Manipulation (CM)   *
***********************************
is_cm = nan
log_cm: cm_option = lazy, um_option = lazy, icm_option = lazy, tm_option = exact
candidates_cm =
[ 0.  0. nan]
necessary_coalition_size_cm =
[0. 0. 1.]
sufficient_coalition_size_cm =
[0. 2. 3.]

property candidates_by_scores_best_to_worst_#: 1d array of integers. Candidates are sorted according to their order of elimination.

property scores_#: 2d array of integers. scores[r, c] is the adjusted median Borda score of candidate c at elimination round r.

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.