Profile

class poisson_approval.Profile(voting_rule='Approval', symbolic=False)[source]

A profile of preference (abstract class).

Parameters

voting_rule (str) – The voting rule. Possible values are APPROVAL, PLURALITY and ANTI_PLURALITY.
symbolic (bool) – Whether the computations are symbolic or numeric.

property abc

Share of voters with this ranking.

Type: Number

property acb

Share of voters with this ranking.

Type: Number

analyzed_strategies(strategies)[source]

Analyze a list of strategies for the profile.

Parameters: strategies (iterable) – An iterator of strategies, such as a list of strategies.
Returns: The analyzed strategies of the profile.
Return type: AnalyzedStrategies

Examples

Cf. ProfileOrdinal.analyzed_strategies_ordinal().

property analyzed_strategies_group

Analyzed group strategies.

Cf. analyzed_strategies() and strategies_group. This is implemented only for profiles where we consider that there is a natural notion of group, such as ProfileNoisyDiscrete.

Type: AnalyzedStrategies

property analyzed_strategies_ordinal

Analyzed ordinal strategies.

Cf. analyzed_strategies() and strategies_ordinal.

Type: AnalyzedStrategies

property analyzed_strategies_pure

Analyzed pure strategies.

Cf. analyzed_strategies() and strategies_pure. This is implemented only for discrete profiles such as ProfileTwelve or ProfileDiscrete.

Type: AnalyzedStrategies

property bac

Share of voters with this ranking.

Type: Number

property bca

Share of voters with this ranking.

Type: Number

best_responses_to_strategy(tau, ratio_optimistic=Fraction(1, 2))[source]

Convert best responses to a StrategyThreshold.

Parameters

tau (TauVector) – Tau-vector.
ratio_optimistic – The value of ratio_optimistic to use. Default: 1/2.

Returns

The conversion of the best responses into a strategy. Only the rankings present in this profile are mentioned in the strategy.

Return type

StrategyThreshold

property cab

Share of voters with this ranking.

Type: Number

property cba

Share of voters with this ranking.

Type: Number

property condorcet_winners

Condorcet winner(s).

Type: Winners

property contains_rankings

Whether the profile contains some rankings.

Type: bool

property contains_weak_orders

Whether the profile contains some weak orders.

Type: bool

property d_ballot_share_weak_voters_fanatic

Ballot shares due to the weak orders if they vote fanatically.

Voters of type 'a>b~c' (lovers):

In Approval or Plurality, they vote for a.
In Anti-plurality, half of them vote for ab (i.e. against c) and half of them vote for ac (i.e. against b).

Voters of type 'a~b>c' (haters):

In Approval or Plurality, half of them vote for a and half of them vote for b.
In Anti-plurality, they vote for ab (i.e. against c).

Type: dict

property d_ballot_share_weak_voters_sincere

Ballot shares due to the weak orders if they vote sincerely.

Voters of type 'a>b~c' (lovers):

In Approval or Plurality, they vote for a.
In Anti-plurality, half of them vote for ab (i.e. against c) and half of them vote for ac (i.e. against b).

Voters of type 'a~b>c' (haters):

In Approval or Anti-plurality, they vote for ab (i.e. against c).
In Plurality, half of them vote for a and half of them vote for b.

Type: dict

d_ballot_share_weak_voters_strategic(strategy)[source]

dict : Ballot shares due to the weak orders if they vote strategically.

For voters with a weak order, strategic voting is the same as sincere voting, except in two cases:

For voters of type 'a~b>c' (haters)`in Plurality, who have two dominant strategies: vote for a or b.
For voters of type 'a>b~c' (lovers) in Anti-Plurality, who have two dominant strategies: vote against b or c (i.e. respectively for ac or ab).

property d_candidate_anti_plurality_welfare

Anti-plurality welfare of each candidate, i.e. share of voters with utility > 0.

Type: DictPrintingInOrder

property d_candidate_plurality_welfare

Plurality welfare of each candidate, i.e. share of voters with utility 1.

Type: DictPrintingInOrder

property d_candidate_relative_anti_plurality_welfare

Relative anti-plurality welfare of each candidate. This is similar to d_candidate_anti_plurality_welfare, but renormalized so that the candidate with best welfare has 1 and the one with worst welfare has 0. In math: relative_welfare = (welfare - min_welfare) / (max_welfare - min_welfare). In the case where all candidates have the same welfare, by convention, the relative welfare is 1 for all of them.

Type: DictPrintingInOrder

property d_candidate_relative_plurality_welfare

Relative plurality welfare of each candidate. This is similar to d_candidate_plurality_welfare, but renormalized so that the candidate with best welfare has 1 and the one with worst welfare has 0. In math: relative_welfare = (welfare - min_welfare) / (max_welfare - min_welfare). In the case where all candidates have the same welfare, by convention, the relative welfare is 1 for all of them.

Type: DictPrintingInOrder

property d_ranking_share

Shares of rankings. E.g. 'abc': 0.3 means that a ratio 0.3 of the voters have ranking 'abc'.

Type: dict

property d_weak_order_share

Shares of weak orders. E.g. 'a~b>c': 0.3 means that a ratio 0.3 of the voters have weak order 'a~b>c'.

Type: dict

property has_majority_favorite

Whether there is a majority favorite (a candidate ranked first by strictly more than half of the voters).

Type: bool

property has_majority_ranking

Whether there is a majority ranking (a ranking shared by strictly more than half of the voters).

Type: bool

is_equilibrium(strategy)[source]

Whether a strategy is an equilibrium in this profile.

Parameters: strategy (Strategy) – A strategy that specifies at least all the rankings that are present in the profile.
Returns: Whether strategy is an equilibrium in this profile.
Return type: EquilibriumStatus

property is_generic_in_rankings

Whether the profile is generic in rankings (contains all rankings).

Type: bool

property is_profile_condorcet

Whether the profile is Condorcet. By convention, 1. means there is a strict Condorcet winner, 0.5 means there are one or more weak Condorcet winner(s), 0. means there is no Condorcet winner.

Type: float

property is_single_peaked

Whether the profile is single-peaked.

Type: bool

property is_standardized

Whether the profile is standardized. Cf. standardized_version().

Type: bool

classmethod order_and_label(t)[source]

Order and label of a discrete type.

Helper method for the ternary plots.

Parameters

t (object) – A type. Any type that is accepted as a key in the initialization dictionary.

Returns

order (str) – The ranking or weak order.
label (str) – The label to be used for the corner of the triangle.

Examples

Cf. ProfileNoisyDiscrete.order_and_label().

classmethod order_and_label_weak(t)[source]

Auxiliary function for order_and_label(), specialized for weak orders.

Parameters

t (object) – A weak order of the form 'a>b~c' or 'a~b>c'.

Returns

order (str) – The weak order itself.
label (str) – The label to be used for the corner of the triangle.

Examples

>>> Profile.order_and_label_weak('a~b>c')
('a~b>c', '$r(a\\sim b>c)$')

random_tau_undominated()[source]

Random tau based on undominated ballots.

This is used, for example, in iterated_voting().

Returns: A random tau-vector. Independently for each ranking, a proportion uniformly drawn in [0, 1] of voters use one undominated ballot, and the rest use the other undominated ballot. For example, in Approval voting, voters with ranking abc are randomly split between ballots a and ab.
Return type: TauVector

property standardized_version

Standardized version of the profile (makes it unique, up to permutations of the candidates).

Type: Profile

property strategies_group

Group strategies of the profile.

Yields: Strategy – All possible group strategies of the profile. This is implemented only for profiles where we consider that there is a natural notion of group, such as ProfileNoisyDiscrete.

Examples

Cf. ProfileNoisyDiscrete.

Type: Iterator

property strategies_ordinal

Ordinal strategies of the profile.

Yields: StrategyOrdinal – All possible ordinal strategies for this profile.

Examples

Cf. ProfileOrdinal.

Type: Iterator

property strategies_pure

Pure strategies of the profile.

Yields: Strategy – All possible pure strategies of the profile. This is implemented only for discrete profiles such as ProfileTwelve or ProfileDiscrete.

Examples

Cf. ProfileDiscrete.

Type: Iterator

property support_in_rankings

Support of the profile (in terms of rankings).

Type: SetPrintingInOrder of str

property support_in_weak_orders

Support of the profile (in terms of weak orders).

Type: SetPrintingInOrder of str

tau(strategy)[source]

Tau-vector associated to this profile and a given strategy.

Parameters: strategy (Strategy) – A strategy that specifies at least all the rankings that are present in the profile.
Returns: Tau-vector associated to this profile and strategy strategy.
Return type: TauVector

property weighted_maj_graph

Weighted majority graph.

Type: numpy.ndarray

τ(strategy)[source]

Tau-vector (alternate notation).

Parameters: strategy (Strategy) – A strategy that specifies at least all the rankings that are present in the profile.
Returns: Tau-vector associated to this profile and strategy strategy.
Return type: TauVector