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

Instant-Condorcet Runoff Voting (ICRV), also known as Benham rule.

Options#

>>> RuleICRV.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: ['lazy', 'exact']. Default: 'exact'.
um_option: ['lazy', 'exact']. Default: 'lazy'.

Notes

Principle: eliminate candidates as in IRV; stop as soon as there is a Condorcet winner.

  • Even round r (including round 0): if a candidate w has only victories against all other non-eliminated candidates (i.e. is a Condorcet winner in this subset, in the sense of matrix_victories_rk), then w is declared the winner.

  • Odd round r: the candidate who is ranked first (among non-eliminated candidates) by least voters is eliminated, like in RuleIRV.

This method meets the Condorcet criterion.

  • 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.

  • is_icm_(): Exact in polynomial time.

  • is_im_(): Non-polynomial or non-exact algorithms from superclass Rule.

  • not_iia(): Exact in polynomial time.

  • is_tm_(): Exact in polynomial time.

  • is_um_(): Non-polynomial or non-exact algorithms from superclass Rule.

References

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

See also

RuleExhaustiveBallot, RuleIRV, RuleIRVDuels, RuleCondorcetAbsIRV, RuleCondorcetVtbIRV

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 = RuleICRV()(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 (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.

Candidates that are not eliminated at the moment a Condorcet winner is detected are sorted by their number of victories in matrix_victories_rk (restricted to candidates that are not eliminated at that time).

Other candidates are sorted in the reverse order of their IRV elimination.

property scores_#

2d array.

For even rounds r (including round 0), scores[r, c] is the number of victories for c in matrix_victories_rk (restricted to non-eliminated candidates). Ties count for 0.5.

For odd rounds r, scores[r, c] is the number of voters who rank c first (among non-eliminated candidates).

Examples

>>> profile = Profile(preferences_rk=[
...     [0, 1, 2, 3],
...     [1, 2, 0, 3],
...     [2, 0, 1, 3],
... ])
>>> rule = RuleICRV()(profile)
>>> rule.scores_
array([[ 2.,  2.,  2.,  0.],
       [ 1.,  1.,  1.,  0.],
       [ 1.,  1.,  1., nan],
       [ 1.,  1.,  1., nan],
       [ 1.,  0., nan, nan]])
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.