Options#

>>> RuleKimRoush.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

At each round, all candidates with a Veto score strictly lower than average are simultaneously eliminated. When all remaining candidates have the same Veto score, the candidate with lowest index is declared the winner.

Kim-Roush method does not meets InfMC.

is_cm_(): Non-polynomial or non-exact algorithms from superclass Rule.
is_icm_(): Non-exact algorithm from superclass Rule.
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.

References

‘Statistical Manipulability of Social Choice Functions’, K.H. Kim and F.W. Roush, 1996.

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 = RuleKimRoush()(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. -2. -2.]
 [-5. inf inf]]
candidates_by_scores_best_to_worst
[0 1 2]
scores_best_to_worst
[[-1. -2. -2.]
 [-5. inf inf]]
w = 0
score_w = [-1. -5.]
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 (False)       ||
 ||           ||                                       ||         ||
 V            V                                        V          V
majority_favorite_c_ut_ctb (False) ==> majority_favorite_c_ut (False)
 ||                                                               ||
 V                                                                V
IgnMC_c_ctb (False)                ==>                IgnMC_c (False)
 ||                                                               ||
 V                                                                V
InfMC_c_ctb (False)                ==>                InfMC_c (False)

*****************************************************
*   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 = nan
log_icm: icm_option = lazy
candidates_icm =
[ 0.  0. nan]
necessary_coalition_size_icm =
[0. 0. 0.]
sufficient_coalition_size_icm =
[ 0. inf inf]

***********************************
*   Coalition Manipulation (CM)   *
***********************************
is_cm = False
log_cm: cm_option = lazy, um_option = lazy, icm_option = lazy, tm_option = exact
candidates_cm =
[0. 0. 0.]
necessary_coalition_size_cm =
[0. 1. 3.]
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. When several candidates are eliminated during the same round, they are sorted by Veto score at that round (more vetos are eliminated first) and, in case of a tie, by their index ( highest indexes are eliminated first).

property scores_#

2d array of integers. scores[r, c] is minus the Veto score of candidate c at elimination round r.

By convention, if candidate c does not participate to round r, then scores[r, c] = numpy.inf.

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.