Features Rule

class embedded_voting.RuleFeatures(score_components=1, embeddings_from_ratings=None)

Voting rule in which the aggregated score of a candidate is the norm of the feature vector of this candidate.

Intuitively, for each candidate, her feature on embedding dimension d is the ideal rating that a voter of group d should put to that candidate. In this model, the actual rating of a voter for this candidate would be a mean of the features, weighted by the voter’s embedding: embeddings[voter, :] @ features[candidate, :]. Considering all the voters and all the candidates, we then obtain ratings = embeddings @ features.T, i.e. features = (inv(embeddings) @ ratings).T.

Since embeddings is not always invertible, we consider in practice features = (pinv(embeddings) @ ratings).T. This can be seen as a least-square approximation of the inital model.

Finally, the score of a candidate is the Euclidean norm of her vector of features.

Examples

>>> ratings = Ratings(np.array([[.5, .6, .3], [.7, 0, .2], [.2, 1, .8]]))
>>> embeddings = Embeddings(np.array([[1, 1], [1, 0], [0, 1]]), norm=True)
>>> election = RuleFeatures()(ratings, embeddings)
>>> election.scores_
[0.669..., 0.962..., 0.658...]
>>> election.ranking_
[1, 0, 2]
>>> election.winner_
1
>>> election.welfare_
[0.0353..., 1.0, 0.0]

features_

This function return the feature vector of all candidates.

Returns:The matrix of features. Its shape is n_candidates, n_dim. Return type:np.ndarray

plot_features(plot_kind='3D', dim: list = None, row_size=5, show=True)

This function plot the features vector of all candidates in the given dimensions.

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].
• row_size (int) – The number of subplots by row. By default, it is set to 5 plots by row.
• show (bool) – If True, plot the figure at the end of the function.