ProfileNoisyDiscrete¶

class
poisson_approval.
ProfileNoisyDiscrete
(d, noise=None, 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, with noise.
Parameters:  d (dict) – Cf. examples below for the different types of input syntax.
 noise (Number, optional) – Cf. examples below for the different types of input syntax.
 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 candidatesa
andb
equally and prefer them to candidatec
.  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
andANTI_PLURALITY
.  symbolic (bool) – Whether the computations are symbolic or numeric.
It maps a ranking to a dict that maps a tuple (utility, noise) to a share of voters. It means that this share of voters have this ranking and with a utility for their second candidate that is uniformly distributed in an interval [umin, umax], where umin = max(utility  noise, 0) and umax = min(utility + noise, 1).
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 four following examples illustrate different ways to define the same profile, where:
 26/100 of the voters have ranking abc and a utility for b that is uniformly distributed between 0.30.01 and 0.3+0.01,
 53/100 of the voters have ranking abc and a utility for b that is uniformly distributed between 0.80.01 and 0.8+0.01,
 21/100 of the voters have ranking bac and a utility for a that is uniformly distributed between 0.30.01 and 0.1+0.01,
The first possible type of input syntax is a dict that maps a tuple (ranking, utility) to a share of voters. All groups of voters have the same noise about their utility, given by the parameter noise:
>>> from fractions import Fraction >>> profile = ProfileNoisyDiscrete({ ... ('abc', 0.3): Fraction(26, 100), ... ('abc', 0.8): Fraction(53, 100), ... ('bac', 0.1): Fraction(21, 100) ... }, noise=0.01) >>> print(profile) <abc 0.3 ± 0.01: 13/50, abc 0.8 ± 0.01: 53/100, bac 0.1 ± 0.01: 21/100> (Condorcet winner: a)
The second type of input syntax has the advantage of not repeating a ranking if is it used for several groups of voters. It is a dict that maps a ranking to a nested dictionary, itself mapping a utility to a share of voters. Like above, all groups of voters have the same noise, given by the parameter noise:
>>> from fractions import Fraction >>> profile = ProfileNoisyDiscrete({ ... 'abc': {0.3: Fraction(26, 100), 0.8: Fraction(53, 100)}, ... 'bac': {0.1: Fraction(21, 100)} ... }, noise=0.01) >>> print(profile) <abc 0.3 ± 0.01: 13/50, abc 0.8 ± 0.01: 53/100, bac 0.1 ± 0.01: 21/100> (Condorcet winner: a)
The two remaining types of input syntax are variants of the two types presented above. They are a bit more verbose but they enable to specify a specific noise for each group of voters.
Here is the variant of the first type of syntax. It is a dict that maps tuple (ranking, utility, noise) to a share of voters. It corresponds exactly to the attribute
d_ranking_utility_noise_share
:>>> from fractions import Fraction >>> profile = ProfileNoisyDiscrete({ ... ('abc', 0.3, 0.01): Fraction(26, 100), ... ('abc', 0.8, 0.01): Fraction(53, 100), ... ('bac', 0.1, 0.01): Fraction(21, 100) ... }) >>> print(profile) <abc 0.3 ± 0.01: 13/50, abc 0.8 ± 0.01: 53/100, bac 0.1 ± 0.01: 21/100> (Condorcet winner: a)
And here is the variant of the second type of syntax. It is a dict that maps a ranking to a nested dict, itself mapping a tuple (utility, noise) to a share of voters:
>>> from fractions import Fraction >>> profile = ProfileNoisyDiscrete({ ... 'abc': {(0.3, 0.01): Fraction(26, 100), (0.8, 0.01): Fraction(53, 100)}, ... 'bac': {(0.1, 0.01): Fraction(21, 100)} ... }) >>> print(profile) <abc 0.3 ± 0.01: 13/50, abc 0.8 ± 0.01: 53/100, bac 0.1 ± 0.01: 21/100> (Condorcet winner: a)
Some examples of operations on the profile:
>>> profile ProfileNoisyDiscrete({'abc': {(0.3, 0.01): Fraction(13, 50), (0.8, 0.01): Fraction(53, 100)}, 'bac': {(0.1, 0.01): 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) 0 >>> profile.analyzed_strategies_group Equilibrium: <abc: a, bac: b> ==> a (FF) <BLANKLINE> Nonequilibria: <abc: ab, bac: ab> ==> a, b (FF) <abc: ab, bac: b> ==> b (FF) <abc: utilitydependent (0.55), bac: ab> ==> a (FF) <abc: utilitydependent (0.55), bac: b> ==> a (FF) <abc: a, bac: ab> ==> a (FF) >>> print(profile.analyzed_strategies_group.winners_at_equilibrium) a
The profile can include weak orders:
>>> profile = ProfileNoisyDiscrete( ... {('abc', 0.3, 0.01): Fraction(26, 100), ('bac', 0.1, 0.01): Fraction(21, 100)}, ... d_weak_order_share={'a~b>c': Fraction(53, 100)}) >>> profile ProfileNoisyDiscrete({'abc': {(0.3, 0.01): Fraction(13, 50)}, 'bac': {(0.1, 0.01): Fraction(21, 100)}}, d_weak_order_share={'a~b>c': Fraction(53, 100)}) >>> print(profile) <abc 0.3 ± 0.01: 13/50, bac 0.1 ± 0.01: 21/100, a~b>c: 53/100> (Condorcet winner: a)
Please note that the voters with a weak order are not “noisy”.

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

analyzed_strategies_group
¶ Analyzed group strategies.
Cf.
analyzed_strategies()
andstrategies_group
. This is implemented only for profiles where we consider that there is a natural notion of group, such asProfileNoisyDiscrete
.Type: AnalyzedStrategies

analyzed_strategies_ordinal
¶ Analyzed ordinal strategies.
Cf.
analyzed_strategies()
andstrategies_ordinal
.Type: AnalyzedStrategies

analyzed_strategies_pure
¶ Analyzed pure strategies.
Cf.
analyzed_strategies()
andstrategies_pure
. This is implemented only for discrete profiles such asProfileTwelve
orProfileDiscrete
.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

contains_rankings
¶ Whether the profile contains some rankings.
Type: bool

contains_weak_orders
¶ Whether the profile contains some weak orders.
Type: bool
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 Antiplurality, 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 Antiplurality, they vote for ab (i.e. against c).
Type: dict
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 Antiplurality, 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 Antiplurality, 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
Dictionary that maps a ranking to a dictionary that maps a tuple (u_min, u_max) to a share of voters.
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. bytau_strategic()
.  If it is a tauvector, it is used directly.
 If it is a string:
'sincere'
or'fanatic'
:tau_sincere
ortau_fanatic
is respectively used.'random_tau'
: useRandTauVectorUniform
to draw a tauvector uniformly at random that is consistent with the voting rule.'random_tau_undominated'
: userandom_tau_undominated()
to draw a tauvector where all voters cast an undominated ballot at random.
 If it is a strategy, it must be an argument accepted by
 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 anyt
from 1 to n_max_episodes included, the update ratio must be in [0, 1]. The default function isone_over_t()
, which leads to an arithmetic average. However, the recommended function isone_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 anyt
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 isone_over_t()
, which leads to an arithmetic average. Note that this parameters has an influence only in case of nonconvergence.  verbose (bool) – If True, print all intermediate steps.
Returns:  Key
tau
:TauVector
or None. The limit tauvector. 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()
andfictitious_play()
:iterated_voting()
can detect cycles (whereasfictitious_play()
only looks for a limit).fictitious_play()
accepts update ratios that are functions of the time (whereasiterated_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. init (Strategy or TauVector or str) –

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
 ranking (str) – A ranking, e.g.

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
 ranking (str) – A ranking, e.g.

have_ranking_with_utility_u
(ranking, u)¶ Share of voters who have a given ranking and a given utility for their middle candidate.
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 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 oftau_sincere
),  A proportion
ratio_fanatic
of voters vote for their top candidate only,  And the rest of the voters use strategy.
Return type: EquilibriumStatus  A proportion

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 singlepeaked.
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. bytau_strategic()
.  If it is a tauvector, it is used directly.
 If it is a string:
'sincere'
or'fanatic'
:tau_sincere
ortau_fanatic
is respectively used.'random_tau'
: useRandTauVectorUniform
to draw a tauvector uniformly at random that is consistent with the voting rule.'random_tau_undominated'
: userandom_tau_undominated()
to draw a tauvector where all voters cast an undominated ballot at random.
 If it is a strategy, it must be an argument accepted by
 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 isone_over_t()
, which leads to an arithmetic average. Note that this parameters has an influence only in case of nonconvergence.  verbose (bool) – If True, print all intermediate steps.
Returns:  Key
cycle_taus_perceived
: list ofTauVector
. The limit cycle of perceived tauvectors.cycle_taus_perceived[t]
is a barycenter ofcycle_taus_perceived[t  1]
withcycle_taus_actual[t  1]
, parametrized by perception_update_ratio.  Key
cycle_strategies
: list ofStrategyThreshold
. The limit cycle of strategies.cycle_strategies[t]
is the best response tocycle_taus_perceived[t]
.  Key
cycle_taus_actual
: list ofTauVector
. The limit cycle of actual tauvectors.cycle_taus_actual[t]
is a barycenter ofcycle_taus_actual[t  1]
and the tauvector resulting fromstrategies[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()
andfictitious_play()
:iterated_voting()
can detect cycles (whereasfictitious_play()
only looks for a limit).fictitious_play()
accepts update ratios that are functions of the time (whereasiterated_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. init (Strategy or TauVector or str) –

classmethod
order_and_label
(t)[source]¶ Order and label of a discrete type.
Cf.
Profile.order_and_label()
.Examples
>>> ProfileNoisyDiscrete.order_and_label(('abc', 0.5, 0.01)) ('abc', '$r(abc, u_b = 0.5 ± 0.01)$') >>> ProfileNoisyDiscrete.order_and_label(('abc', 0.5)) ('abc', '$r(abc, u_b \\simeq 0.5)$') >>> ProfileNoisyDiscrete.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 tauvector. 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 = ProfileNoisyDiscrete({ ... ('abc', 0.3, 0.01): Fraction(26, 100), ... ('abc', 0.8, 0.01): Fraction(53, 100), ... ('bac', 0.1, 0.01): Fraction(21, 100) ... }) >>> print(profile.standardized_version) <abc 0.3 ± 0.01: 13/50, abc 0.8 ± 0.01: 53/100, bac 0.1 ± 0.01: 21/100> (Condorcet winner: a) >>> profile.is_standardized True
Type: ProfileNoisyDiscrete

strategies_group
¶ group strategies of the profile.
Yields: StrategyThreshold – All possible group strategies of the profile. 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: Strategy – All possible pure strategies of the profile. This is implemented only for discrete profiles such as ProfileTwelve
orProfileDiscrete
.Examples
Cf.
ProfileDiscrete
.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)¶ Tauvector 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 oftau_sincere
), a shareratio_fanatic
vote only for their top candidate, and the rest of the voters vote strategically (in the sense oftau_strategic()
). In other words, this tauvector is the barycenter oftau_sincere
,tau_fanatic
andtau_strategic(strategy)
, with respective weightsself.ratio_sincere
,self.ratio_fanatic
and1  self.ratio_sincere  self.ratio_fanatic
.Return type: TauVector

tau_fanatic
¶ Tauvector associated to fanatic voting.
Returns:  In Approval or Plurality, all voters approve of their top candidate only.,
 In Antiplurality, all voters vote against their bottom candidate (i.e. for the other two).
Return type: TauVector Notes
In Plurality and Antiplurality, sincere and fanatic voting are the same. They differ only in Approval.

tau_sincere
¶ Tauvector 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 Antiplurality, all voters vote against their bottom candidate (i.e. for the other two).
Return type: TauVector Notes
In Plurality and Antiplurality, sincere and fanatic voting are the same. They differ only in Approval.

tau_strategic
(strategy)¶ Tauvector 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: Tauvector associated to this profile and strategy strategy. Return type: TauVector

weighted_maj_graph
¶ Weighted majority graph.
Type: np.ndarray