ProfileCardinalContinuous

class poisson_approval.ProfileCardinalContinuous(ratio_sincere=0, ratio_fanatic=0, voting_rule='Approval', symbolic=False)[source]

A cardinal profile of preference with no atom (abstract class).

Notes

For parameters and attributes, cf. ProfileCardinal.

property abc

Share of voters with this ranking.

Type: Number

property acb

Share of voters with this ranking.

Type: Number

analyzed_strategies(strategies)

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=None)[source]

Convert best responses to a StrategyThreshold.

Parameters

tau (TauVector) – Tau-vector.
ratio_optimistic – The value of ratio_optimistic to use. Default: None (since it is a ProfileCardinalContinuous, we do not care about this value).

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)

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_candidate_relative_welfare

Relative welfare of each candidate. This is similar to d_candidate_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_welfare

Welfare of each candidate. E.g. 'a': 0.7 means that candidate 'a' has a welfare (average utility) equal to 0.7 for the voters. Since utilities are in [0, 1], so is the welfare.

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

fictitious_play(init, n_max_episodes, perception_update_ratio=<function one_over_t>, ballot_update_ratio=1, winning_frequency_update_ratio=<function one_over_t>, other_statistics_update_ratio=<function one_over_t>, other_statistics_tau=None, other_statistics_strategy=None, verbose=False)

Seek for convergence by fictitious play.

Parameters

init (Strategy or TauVector or str) –
The initialization.
- If it is a strategy, it must be an argument accepted by tau(), i.e. by tau_strategic().
- If it is a tau-vector, it is used directly.
- If it is a string:
  - 'sincere' or 'fanatic': tau_sincere or tau_fanatic is respectively used.
  - 'random_tau': use RandTauVectorUniform to draw a tau-vector uniformly at random that is consistent with the voting rule.
  - 'random_tau_undominated': use random_tau_undominated() to draw a tau-vector where all voters cast an undominated ballot at random.
n_max_episodes (int) – Maximal number of iterations.
perception_update_ratio (callable or Number) – The coefficient when updating the perceived tau: tau_perceived = (1 - perception_update_ratio(t)) * tau_perceived + perception_update_ratio(t) * tau_actual. For any t from 2 to n_max_episodes included, the update ratio must be in [0, 1]. The default function is one_over_t(), which leads to an arithmetic average. However, the recommended function is one_over_log_t_plus_one(), which accelerates the convergence. If perception_update_ratio is a Number, it is considered as a constant function.
ballot_update_ratio (callable or Number) – The ratio of voters who update their ballot: tau_actual = (1 - ballot_update_ratio(t)) * tau_actual + ballot_update_ratio(t) * tau_response. For any t from 2 to n_max_episodes included, the update ratio must be in [0, 1]. The default function is the constant 1, which corresponds to a full update. If ballot_update_ratio is a Number, it is considered as a constant function.
winning_frequency_update_ratio (callable or Number) – The coefficient when updating the winning frequency of each candidate: d_candidate_winning_frequency[c] = (1 - winning_frequency_update_ratio(t)) * d_candidate_winning_frequency[c] + winning_frequency_update_ratio(t) * winning_probability[c]. The default function is one_over_t(), which leads to an arithmetic average. Note that this parameters has an influence only in case of non-convergence.
other_statistics_update_ratio (callable or Number) – The coefficient when updating the other statistics (cf. below).
other_statistics_tau (dict) – Key: name of the statistic (different from converges, tau, strategy, tau_init, n_episodes, and d_candidate_winning_frequency). Value: a function whose input is a tau-vector, and whose output is a number or a numpy array.
other_statistics_strategy (dict) – Key: name of the statistic (different from converges, tau, strategy, tau_init, n_episodes, d_candidate_winning_frequency and the names in other_statistics_tau). Value: a function whose input is a strategy, and whose output is a number or a numpy array.
verbose (bool) – If True, print all intermediate steps.

Returns

Key converges: bool. True if the process converges.
Key tau: TauVector or None. The limit tau-vector. If None, it means that the process did not converge.
Key strategy: StrategyThreshold or None. The limit strategy. If None, it means that the process did not converge.
Key tau_init: the tau-vector at initialization.
Key n_episodes: the number of episodes until convergence. If the process did not converge, by convention, this value is n_max_episodes.
Key d_candidate_winning_frequency: dict. Key: candidate. Value: winning frequency. If the process reached a limit, the winning frequencies are computed in the limit only. If the process did not converge, the frequency is computed on the whole history.
Others keys are those of other_statistics_tau and other_statistics_strategy. Similarly to d_candidate_winning_frequency, they give the long-run average of the corresponding statistics.

Return type

dict

Notes

Comparison between iterated_voting() and fictitious_play():

iterated_voting() can detect cycles (whereas fictitious_play() only looks for a limit).
fictitious_play() accepts update ratios that are functions of the time (whereas iterated_voting() only accepts constants).
fictitious_play() is faster and uses less memory, because it only looks for a limit and not for a cycle.

In general, you should use iterated_voting() only if you care about cycles, with the constraint that it implies having constant update ratios.

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

have_ranking_with_utility_above_u(ranking, u)

Share of voters who have a given ranking and a utility for their middle candidate that is strictly above a given value.

Parameters

ranking (str) – A ranking, e.g. 'abc'.
u (Number) – A utility between 0 and 1 (included).

Returns

The share of voters who have ranking ranking and a utility for their middle candidate strictly greater than u. This does not include the voters who have a weak order of preference.

Return type

Number

have_ranking_with_utility_below_u(ranking, u)

Share of voters who have a given ranking and a utility for their middle candidate that is strictly below a given value.

Parameters

ranking (str) – A ranking, e.g. 'abc'.
u (Number) – A utility between 0 and 1 (included).

Returns

The share of voters who have ranking ranking and a utility for their middle candidate strictly lower than u. This does not include the voters who have a weak order of preference.

Return type

Number

have_ranking_with_utility_u(ranking, u)[source]

Share of voters who have a given ranking and a utility for their middle candidate that is equal to a given value.

Since it is a continuous profile, this method always returns 0.

is_equilibrium(strategy)

Whether a strategy is an equilibrium.

Parameters

strategy (StrategyThreshold) – A strategy that specifies at least all the rankings that are present in the profile. If some voters have a utility for their second candidate that is equal to the utility threshold of the strategy, then the ratio of optimistic voters must be specified.

Returns

Whether strategy is an equilibrium in this profile. This is based on the assumption that:

A proportion ratio_sincere of voters cast their ballot sincerely (in the sense of tau_sincere),
A proportion ratio_fanatic of voters vote for their top candidate only,
And the rest of the voters use strategy.

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

iterated_voting(init, n_max_episodes, perception_update_ratio=1, ballot_update_ratio=1, winning_frequency_update_ratio=<function one_over_t>, other_statistics_update_ratio=<function one_over_t>, other_statistics_tau=None, other_statistics_strategy=None, verbose=False)

Seek for convergence by iterated voting.

Parameters

init (Strategy or TauVector or str) –
The initialization.
- If it is a strategy, it must be an argument accepted by tau(), i.e. by tau_strategic().
- If it is a tau-vector, it is used directly.
- If it is a string:
  - 'sincere' or 'fanatic': tau_sincere or tau_fanatic is respectively used.
  - 'random_tau': use RandTauVectorUniform to draw a tau-vector uniformly at random that is consistent with the voting rule.
  - 'random_tau_undominated': use random_tau_undominated() to draw a tau-vector where all voters cast an undominated ballot at random.
n_max_episodes (int) – Maximal number of iterations.
perception_update_ratio (Number) – Number in [0, 1]. The coefficient when updating the perceived tau: tau_perceived = (1 - perception_update_ratio) * tau_perceived + perception_update_ratio * tau_actual.
ballot_update_ratio (Number) – Number in [0, 1]. The ratio of voters who update their ballot: tau_actual = (1 - ballot_update_ratio) * tau_actual + ballot_update_ratio * tau_response.
winning_frequency_update_ratio (callable or Number) – The coefficient when updating the winning frequency of each candidate: d_candidate_winning_frequency[c] = (1 - winning_frequency_update_ratio(t)) * d_candidate_winning_frequency[c] + winning_frequency_update_ratio(t) * winning_probability[c]. The default function is one_over_t(), which leads to an arithmetic average. Note that this parameters has an influence only in case of non-convergence.
other_statistics_update_ratio (callable or Number) – The coefficient when updating the other statistics (cf. below).
other_statistics_tau (dict) – Key: name of the statistic (different from converges, tau, strategy, tau_init, n_episodes, and d_candidate_winning_frequency). Value: a function whose input is a tau-vector, and whose output is a number or a numpy array.
other_statistics_strategy (dict) – Key: name of the statistic (different from converges, tau, strategy, tau_init, n_episodes, d_candidate_winning_frequency and the names in other_statistics_tau). Value: a function whose input is a strategy, and whose output is a number or a numpy array.
verbose (bool) – If True, print all intermediate steps.

Returns

Key converges: bool. True if the process converges (i.e. cycle of length 1).
Key cycle_taus_perceived: list of TauVector. The limit cycle of perceived tau-vectors. cycle_taus_perceived[t] is a barycenter of cycle_taus_perceived[t - 1] with cycle_taus_actual[t - 1], parametrized by perception_update_ratio.
Key cycle_strategies: list of StrategyThreshold. The limit cycle of strategies. cycle_strategies[t] is the best response to cycle_taus_perceived[t].
Key cycle_taus_actual: list of TauVector. The limit cycle of actual tau-vectors. cycle_taus_actual[t] is a barycenter of cycle_taus_actual[t - 1] and the tau-vector resulting from strategies[t], parametrized by ballot_update_ratio.
Key tau_init: the tau-vector at initialization.
Key n_episodes: the number of episodes until convergence. If the process did not converge, by convention, this value is n_max_episodes.
Key d_candidate_winning_frequency: dict. Key: candidate. Value: winning frequency. If the process reached a limit or a periodical orbit, the winning frequencies are computed in the limit only. If the process did not converge, the frequency is computed on the whole history.
Others keys are those of other_statistics_tau and other_statistics_strategy. Similarly to d_candidate_winning_frequency, they give the long-run average of the corresponding statistics.

The return values cycle_taus_perceived, cycle_strategies and cycle_taus_actual are lists of the same length. If the length is 1, the process converges to this limit. If it is greater than 1, the process reaches a periodical orbit. If it is 0, by convention, it means that the process does not converge and does not reach a periodical orbit.

Return type

dict

Notes

Comparison between iterated_voting() and fictitious_play():

iterated_voting() can detect cycles (whereas fictitious_play() only looks for a limit).
fictitious_play() accepts update ratios that are functions of the time (whereas iterated_voting() only accepts constants).
fictitious_play() is faster and uses less memory, because it only looks for a limit and not for a cycle.

In general, you should use iterated_voting() only if you care about cycles, with the constraint that it implies having constant update ratios.

classmethod order_and_label(t)

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)

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()

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

share_sincere(strategy)

Share of voters that happen to cast a sincere ballot.

Parameters: strategy – An argument accepted by share_sincere_among_strategic_voters().
Returns: The ratio of voters that happen to cast a sincere ballot. This includes all kinds of voters (sincere, fanatic or strategic).
Return type: Number

property share_sincere_among_fanatic_voters

Share of fanatic voters that happen to cast a sincere ballot.

In Plurality or Anti-Plurality, this number is always 1 because fanatic voters are sincere by our definition.

In Approval, this is the proportion of fanatic voters whose ballot happen to be sincere, i.e. whose utility for their second candidate is lower or equal to 0.5.

Note that this share is relative to the share of fanatic voters: therefore, it is independent of ratio_fanatic and can be defined conventionally even if ratio_fanatic is 0, with the same computation.

Type: Number

share_sincere_among_strategic_voters(strategy)

Share of strategic voters that happen to cast a sincere ballot.

Parameters: strategy (StrategyThreshold) – A strategy that specifies at least all the rankings that are present in the profile. If some voters have a utility for their second candidate that is equal to the utility threshold of the strategy, then the ratio of optimistic voters must be specified.
Returns: The ratio of sincere voters among strategic voters. In particular, if all strategic voters happen to be sincere with the given strategy, then the ratio is 1, even if there are non-strategic voters (for example, in Approval, fanatic voters who have a utility > 0.5 for their second candidate and are therefore not “sincere”, according to our definition).
Return type: Number

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)

Tau-vector associated to a strategy, with partial sincere and fanatic voting.

Parameters: strategy – An argument accepted by tau_strategic().
Returns: A share ratio_sincere of the voters vote sincerely (in the sense of tau_sincere), a share ratio_fanatic vote only for their top candidate (cf. tau_fanatic), and the rest of the voters vote strategically (in the sense of tau_strategic()). In other words, this tau-vector is the barycenter of tau_sincere, tau_fanatic and tau_strategic(strategy), with respective weights ratio_sincere, ratio_fanatic and 1 - ratio_sincere - ratio_fanatic.
Return type: TauVector

property tau_fanatic

Tau-vector associated to fanatic voting.

Returns

In Approval or Plurality, all voters approve of their top candidate only,
In Anti-plurality, all voters vote against their bottom candidate (i.e. for the other two).

Return type

TauVector

Notes

In Plurality and Anti-plurality, sincere and fanatic voting are the same. They differ only in Approval.

property tau_sincere

Tau-vector associated to sincere voting.

Returns

In Approval, all voters approve of their top candidate, and voters approve of their middle candidate if and only if their utility for her is strictly greater than 0.5.
In Plurality, all voters vote for their top candidate.
In Anti-plurality, all voters vote against their bottom candidate (i.e. for the other two).

Return type

TauVector

Notes

In Plurality and Anti-plurality, sincere and fanatic voting are the same. They differ only in Approval.

tau_strategic(strategy)

Tau-vector associated to a strategy (fully strategic voting).

Parameters: strategy (StrategyThreshold) – A strategy that specifies at least all the rankings that are present in the profile. If some voters have a utility for their second candidate that is equal to the utility threshold of the strategy, then the ratio of optimistic voters must be specified.
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)

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