RuleCopeland#

class svvamp.RuleCopeland(tie_break_rule='lexico', **kwargs)[source]#

Copeland rule

Parameters:

  • tie_break_rule (str) – ‘lexico’ of ‘random’. Default: ‘lexico’. If tie_break_rule is ‘lexico’, then the candidate with the lowest index is selected in case of a tie (usual behavior of SVVAMP for the other voting rules). If tie_break_rule is random, then each time a profile is loaded, a tie-break order over the candidates is drawn at random. This tie-break is used for the result of the sincere election, but also in case of manipulation.

  • Options

  • ------- – 

    >>> RuleCopeland.print_options_parameters()
cm_option: ['fast', 'exact']. Default: 'fast'.
icm_option: ['exact']. Default: 'exact'.
iia_subset_maximum_size: is_number. Default: 2.
im_option: ['lazy', 'exact']. Default: 'lazy'.
precheck_heuristic: is_bool. Default: True.
tie_break_rule: ['lexico', 'random']. Default: 'lexico'.
tm_option: ['lazy', 'exact']. Default: 'exact'.
um_option: ['lazy', 'exact']. Default: 'lazy'.

Notes

Select the candidate with most victories in the matrix of victories.

Copeland does not meets_condorcet_c_ut_rel:

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

Copeland does not meets_majority_favorite_c_ut_ctb:

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

  • is_im_(): Deciding CM is in P (but a dedicated algorithm is not implemented yet).

  • is_cm_(): Deciding CM is NP-complete.

References

‘The Computational Difficulty of Manipulating an Election’, J. J. Bartholdi III, C. A. Tovey and M. A. Trick, 1989.

‘Manipulation of copeland elections’, Piotr Faliszewski, Edith Hemaspaandra and Henning Schnoor, 2010.

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 = RuleCopeland()(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 = 2.0
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 (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 (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 = False
log_cm: cm_option = fast, um_option = lazy, tm_option = exact
candidates_cm =
[0. 0. 0.]
necessary_coalition_size_cm =
[0. 2. 3.]
sufficient_coalition_size_cm =
[0. 2. 3.]
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.