ProfileDiscrete

class poisson_approval.ProfileDiscrete(d, d_weak_order_share=None, normalization_warning=True, ratio_sincere=0, ratio_fanatic=0, voting_rule='Approval', symbolic=False)[source]

Profile with a discrete distribution of voters.

Parameters:
  • d (dict) – The first possible format is a dict that maps a tuple (ranking, utility) to the share of voters who have this ranking, and this utility for their second candidate. The second possible format is a dict of dict that maps a ranking (first key) and a utility (second key) to the corresponding share of voters; it corresponds more closely to the attribute d_ranking_utility_share mentioned below. Cf. examples below.
  • d_weak_order_share (dict) – E.g. {'a~b>c': 0.2, 'a>b~c': 0.1}. d_weak_order_share['a~b>c'] is the probability that a voter likes candidates a and b equally and prefer them to candidate c.
  • normalization_warning (bool) – Whether a warning should be issued if the input distribution is not normalized.
  • ratio_sincere (Number) – The ratio of sincere voters, in the interval [0, 1]. This is used for tau().
  • ratio_fanatic (Number) – The ratio of fanatic voters, in the interval [0, 1]. This is used for tau(). The sum of ratio_sincere and ratio_fanatic must not exceed 1.
  • voting_rule (str) – The voting rule. Possible values are APPROVAL, PLURALITY and ANTI_PLURALITY.
  • symbolic (bool) – Whether the computations are symbolic or numeric.
d_ranking_utility_share

To a ranking (first key) and a utility (second key), it associates the share of voters who have this ranking and this utility for their second candidate.

Type:dict of dict

Notes

If the input distribution is not normalized, the profile will be normalized anyway and a warning is issued (unless normalization_warning is False).

Examples

The first possible input syntax is a dict that maps a tuple (ranking, utility) to a share of voters:

>>> from fractions import Fraction
>>> profile = ProfileDiscrete({
...     ('abc', 0.3): Fraction(26, 100),
...     ('abc', 0.8): Fraction(53, 100),
...     ('bac', 0.1): Fraction(21, 100)
... })
>>> print(profile)
<abc 0.3: 13/50, abc 0.8: 53/100, bac 0.1: 21/100> (Condorcet winner: a)

The second possible input syntax is a dict that maps a ranking to a nested dict, itself mapping a utility to a share of voters:

>>> from fractions import Fraction
>>> profile = ProfileDiscrete({
...     'abc': {0.3: Fraction(26, 100), 0.8: Fraction(53, 100)},
...     'bac': {0.1: Fraction(21, 100)}
... })
>>> print(profile)
<abc 0.3: 13/50, abc 0.8: 53/100, bac 0.1: 21/100> (Condorcet winner: a)

Some examples of operations on the profile:

>>> profile
ProfileDiscrete({'abc': {0.3: Fraction(13, 50), 0.8: Fraction(53, 100)}, 'bac': {0.1: Fraction(21, 100)}})
>>> profile.d_ranking_share
{'abc': Fraction(79, 100), 'bac': Fraction(21, 100)}
>>> profile.abc
Fraction(79, 100)
>>> profile.have_ranking_with_utility_above_u('abc', 0.5)
Fraction(53, 100)
>>> profile.have_ranking_with_utility_below_u('abc', 0.5)
Fraction(13, 50)
>>> profile.have_ranking_with_utility_u('abc', 0.3)
Fraction(13, 50)
>>> profile.analyzed_strategies_pure
Equilibrium:
<abc: a, bac: b> ==> a (FF)
<BLANKLINE>
Non-equilibria:
<abc: ab, bac: ab> ==> a, b (FF)
<abc: ab, bac: b> ==> b (FF)
<abc: utility-dependent (0.55), bac: ab> ==> a (FF)
<abc: utility-dependent (0.55), bac: b> ==> a (FF)
<abc: a, bac: ab> ==> a (FF)
>>> print(profile.analyzed_strategies_pure.winners_at_equilibrium)
a

The profile can include weak orders:

>>> profile = ProfileDiscrete({('abc', 0.3): Fraction(26, 100), ('bac', 0.1): Fraction(21, 100)},
...                           d_weak_order_share={'a~b>c': Fraction(53, 100)})
>>> profile
ProfileDiscrete({'abc': {0.3: Fraction(13, 50)}, 'bac': {0.1: Fraction(21, 100)}}, d_weak_order_share={'a~b>c': Fraction(53, 100)})
>>> print(profile)
<abc 0.3: 13/50, bac 0.1: 21/100, a~b>c: 53/100> (Condorcet winner: a)
abc

Share of voters with this ranking.

Type:Number
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().

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
analyzed_strategies_ordinal

Analyzed ordinal strategies.

Cf. analyzed_strategies() and strategies_ordinal.

Type:AnalyzedStrategies
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
bac

Share of voters with this ranking.

Type:Number
bca

Share of voters with this ranking.

Type:Number
best_responses_to_strategy(d_ranking_best_response)

Convert best responses to a StrategyThreshold.

Parameters:d_ranking_best_response (dict) – Key: ranking. Value: BestResponse.
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
cab

Share of voters with this ranking.

Type:Number
cba

Share of voters with this ranking.

Type:Number
condorcet_winners

Condorcet winner(s).

Type:Winners
contains_rankings

Whether the profile contains some rankings.

Type:bool
contains_weak_orders

Whether the profile contains some weak orders.

Type:bool
d_ballot_share_weak_voters_fanatic

Ballot shares due to the weak orders if they vote fanatically

Voters of the type 'a>b~c':

  • 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 the type 'a~b>c':

  • 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
d_ballot_share_weak_voters_sincere

Ballot shares due to the weak orders if they vote sincerely

Voters of the type 'a>b~c':

  • 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 the type 'a~b>c':

  • 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
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>, 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:
  • 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 1 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 1 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.
  • verbose (bool) – If True, print all intermediate steps.
Returns:

  • 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 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.

Return type:

dict

Notes

Comparison between iterated_voting() and fictitious_play():

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

has_majority_favorite

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

Type:bool
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)[source]

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

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)[source]

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

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 given utility for their middle candidate.

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 equal to u. This does NOT include the voters who have a weak order of preference. I.e. if u`=0 or `u=1, then the share is 0.

Return type:

Number

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 threshold utility 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
is_generic_in_rankings

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

Type:bool
is_profile_condorcet

Whether the profile is Condorcet. 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
is_single_peaked

Whether the profile is single-peaked.

Type:bool
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>, 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:
  • n_max_episodes (int) – Maximal number of iterations.
  • perception_update_ratio (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 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.
  • verbose (bool) – If True, print all intermediate steps.
Returns:

  • 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 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.

cycle_taus_perceived, cycle_strategies and cycle_taus_actual have the same length. If it 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():

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)[source]

Order and label of a discrete type.

Cf. Profile.order_and_label().

Examples

>>> ProfileDiscrete.order_and_label(('abc', 0.5))
('abc', '$r(abc, u_b = 0.5)$')
>>> ProfileDiscrete.order_and_label('a~b>c')
('a~b>c', '$r(a\\sim b>c)$')
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 ProfileCardinal.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
standardized_version

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

Examples

>>> from fractions import Fraction
>>> profile = ProfileDiscrete({
...     ('abc', 0.3): Fraction(26, 100),
...     ('abc', 0.8): Fraction(53, 100),
...     ('bac', 0.1): Fraction(21, 100)
... })
>>> print(profile.standardized_version)
<abc 0.3: 13/50, abc 0.8: 53/100, bac 0.1: 21/100> (Condorcet winner: a)
>>> profile.is_standardized
True
Type:ProfileDiscrete
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
strategies_ordinal

ordinal strategies of the profile.

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

Examples

Cf. ProfileOrdinal.

Type:Iterator
strategies_pure

pure strategies of the profile.

Yields:StrategyThreshold – All possible pure strategies of the profile.
Type:Iterator
support_in_rankings

Support of the profile (in terms of rankings).

Type:SetPrintingInOrder of str
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, 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 self.ratio_sincere, self.ratio_fanatic and 1 - self.ratio_sincere - self.ratio_fanatic.
Return type:TauVector
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.

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 threshold utility 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
weighted_maj_graph

Weighted majority graph.

Type:np.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