Positional scoring rules

General class

class embedded_voting.RulePositional(points, rule=None)

This class enables to extend a voting rule to an ordinal input with a positional scoring rule.

Parameters:

• points (list) – The vector of the positional scoring rule. Should be of the same length than the number of candidates. In each ranking, candidate ranked at position i get points[i] points.
• rule (Rule) – The aggregation rule used to determine the aggregated scores of the candidates.

fake_ratings_

The modified ratings of voters (with ordinal scores) on which we run the election.

Type:ratings

points

The vector of the positional scoring rule. Should be of the same length than the number of candidates. In each ranking, candidate ranked at position i get points[i] points.

Type:np.ndarray

base_rule

The aggregation rule used to determine the aggregated scores of the candidates.

Type:Rule

_rule

The aggregation rule instantiated with the fake_ratings.

Type:Rule

_score_components

The number of components in the score of every candidate. If > 1, we perform a lexical sort to obtain the ranking.

Examples

>>> ratings = np.array([[.1, .2, .8, 1], [.7, .9, .8, .6], [1, .6, .1, .3]])
>>> embeddings = Embeddings([[1, 0], [1, 1], [0, 1]], norm=True)
>>> election = RuleSVDNash()(ratings, embeddings)
>>> election.ranking_
[3, 0, 1, 2]
>>> election_bis = RulePositional([2, 1, 1, 0])(ratings, embeddings)
>>> election_bis.fake_ratings_
Ratings([[0. , 0.5, 0.5, 1. ],
         [0.5, 1. , 0.5, 0. ],
         [1. , 0.5, 0. , 0.5]])
>>> election_bis.set_rule(RuleSVDNash())(ratings, embeddings).ranking_
[1, 3, 0, 2]

plot_fake_ratings(plot_kind='3D', dim=None, list_candidates=None, list_titles=None, row_size=5, show=True)

This function plot the candidates in the fake ratings, obtained using the scoring vector points.

Parameters:

• plot_kind (str) – The kind of plot we want to show. Can be '3D' or 'ternary'.
• dim (list) – The 3 dimensions we are using for our plot. By default, it is set to [0, 1, 2].
• list_candidates (int list) – The list of candidates we want to plot. Should contains integers lower than n_candidates. By default, we plot all candidates.
• list_titles (str list) – Contains the title of the plots. Should be the same length than list_candidates.
• row_size (int) – Number of subplots by row. By default, it is set to 5 plots by rows.
• show (bool) – If True, displays the figure at the end of the function.

set_rule(rule)

This function updates the base_rule used for the election.

Parameters:rule (Rule) – The new rule to use. Returns:The object itself. Return type:RulePositional

Particular cases

class embedded_voting.RulePositionalPlurality(n_candidates, rule=None)

This class enables to extend a voting rule to an ordinal input with Plurality rule (vector [1, 0, ..., 0]).

Parameters:rule (Rule) – The aggregation rule used to determine the aggregated scores of the candidates.

Examples

>>> ratings = np.array([[.1, .2, .8, 1], [.7, .9, .8, .6], [1, .6, .1, .3]])
>>> embeddings = Embeddings(np.array([[1, 0], [1, 1], [0, 1]]), norm=True)
>>> election = RulePositionalPlurality(4, rule=RuleSVDNash(use_rank=True))(ratings, embeddings)
>>> election.fake_ratings_
Ratings([[0., 0., 0., 1.],
         [0., 1., 0., 0.],
         [1., 0., 0., 0.]])
>>> election.ranking_
[0, 1, 3, 2]

class embedded_voting.RulePositionalVeto(n_candidates, rule=None)

This class enables to extend a voting rule to an ordinal input with Veto rule (vector [1, ..., 1, 0]).

Parameters:rule (Rule) – The aggregation rule used to determine the aggregated scores of the candidates.

Examples

>>> ratings = np.array([[.1, .2, .8, 1], [.7, .9, .8, .6], [1, .6, .1, .3]])
>>> embeddings = Embeddings(np.array([[1, 0], [1, 1], [0, 1]]), norm=True)
>>> election = RulePositionalVeto(n_candidates=4, rule=RuleSVDNash())(ratings, embeddings)
>>> election.fake_ratings_
Ratings([[0., 1., 1., 1.],
         [1., 1., 1., 0.],
         [1., 1., 0., 1.]])
>>> election.ranking_
[1, 3, 0, 2]

class embedded_voting.RulePositionalKApproval(n_candidates, k=2, rule=None)

This class enables to extend a voting rule to an ordinal input with k-Approval rule (vector [1, 1, ..., 0] with k ones).

Parameters:

• k (int) – The k parameter of the k-approval. By default, it is set to 2.
• rule (Rule) – The aggregation rule used to determine the aggregated scores of the candidates.

Examples

>>> ratings = np.array([[.1, .2, .8, 1], [.7, .9, .8, .6], [1, .6, .1, .3]])
>>> embeddings = Embeddings(np.array([[1, 0], [1, 1], [0, 1]]), norm=True)
>>> election = RulePositionalKApproval(n_candidates=4, k=2, rule=RuleSVDNash(use_rank=True))(
...     ratings, embeddings)
>>> election.fake_ratings_
Ratings([[0., 0., 1., 1.],
         [0., 1., 1., 0.],
         [1., 1., 0., 0.]])
>>> election.ranking_
[1, 2, 0, 3]

class embedded_voting.RulePositionalBorda(n_candidates, rule=None)

This class enables to extend a voting rule to an ordinal input with Borda rule (vector [m-1, m-2, ..., 1, 0]).

Parameters:rule (Rule) – The aggregation rule used to determine the aggregated scores of the candidates.

Examples

>>> ratings = np.array([[.1, .2, .8, 1], [.7, .9, .8, .6], [1, .6, .1, .3]])
>>> embeddings = Embeddings(np.array([[1, 0], [1, 1], [0, 1]]), norm=True)
>>> election = RulePositionalBorda(n_candidates=4, rule=RuleSVDNash())(ratings, embeddings)
>>> election.fake_ratings_
Ratings([[0.        , 0.33333333, 0.66666667, 1.        ],
         [0.33333333, 1.        , 0.66666667, 0.        ],
         [1.        , 0.66666667, 0.        , 0.33333333]])
>>> election.ranking_
[1, 3, 2, 0]