Options#

>>> RuleRankedPairs.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

In the matrix of duels matrix_duels_rk, victories (and ties) are sorted by decreasing amplitude. If two duels have the same score, we take first the one where the winner has the smallest index; if there is still a choice to make, we take first the duel where the loser has the highest index.

Starting with the largest victory, we build a directed graph whose nodes are the candidates and edges are victories. But if a victory creates a cycle in the graph, it is not validated and the edge is not added.

At the end, we have a transitive connected directed graph, whose adjacency relation is included in the relation of victories (with ties broken), matrix_victories_rk_ctb. The maximal node of this graph (by topological order) is declared the winner.

This method meets the Condorcet criterion.

is_cm_(): Deciding CM is NP-complete. Non-polynomial or non-exact algorithms from superclass Rule.
is_icm_(): Exact in polynomial time.
is_im_(): Deciding IM is NP-complete. Non-polynomial or non-exact algorithms from superclass Rule.
is_iia_(): Exact in polynomial time.
is_tm_(): Exact in polynomial time.
is_um_(): Non-polynomial or non-exact algorithms from superclass Rule.

References

‘Independence of clones as a criterion for voting rules’, Nicolaus Tideman, 1987.

‘Complexity of Unweighted Coalitional Manipulation under Some Common Voting Rules’, Lirong Xia et al., 2009.

‘Schulze and Ranked-Pairs Voting are Fixed-Parameter Tractable to Bribe, Manipulate, and Control’, Lane A. Hemaspaandra, Rahman Lavaee and Curtis Menton, 2012.

‘A Complexity-of-Strategic-Behavior Comparison between Schulze’s Rule and Ranked Pairs’, David Parkes and Lirong Xia, 2012.

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 = RuleRankedPairs()(profile)
>>> rule.meets_condorcet_c_rk_ctb
True
>>> 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. 3. 3.]
 [0. 0. 0.]
 [0. 3. 0.]]
candidates_by_scores_best_to_worst
[0, 2, 1]
scores_best_to_worst
[[0. 3. 3.]
 [0. 0. 3.]
 [0. 0. 0.]]
w = 0
score_w = [0. 3. 3.]
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 = False
log_cm: cm_option = lazy, 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 candidates_by_scores_best_to_worst_#: 1d array of integers. candidates_by_scores_best_to_worst[k] is the k-th candidate by topological order on the graph generated by Ranked Pairs.

property score_w_#

score_w_ = scores_[w_, :].

Type:: 1d array. score_w_ is w_’s score vector

property scores_#: 2d array of integers. scores[c, d] is equal to matrix_duels_rk[c, d] iff this duel was validated in Ranked Pairs, 0 otherwise.

Note

Unlike for most other voting systems, scores matrix must be read in rows, in order to comply with our convention for the matrix of duels: c’s score vector is scores[c, :].

property scores_best_to_worst_#

2d array. scores_best_to_worst_ is the scores of the candidates, from the winner to the last candidate of the election.

scores_best_to_worst[k, j] is the score of the k-th best candidate of the election against the j-th. It is the result in matrix_duels_rk if this duels was validated by Ranked Pairs, 0 otherwise.

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.