Poisson Approval studies the Poisson Game of Approval Voting.

## Poisson Approval

Poisson Approval studies the Poisson Game of Approval Voting.

### Features

• Implement only the case of 3 candidates.
• Deal with ordinal or cardinal profiles.
• Compute the asymptotic developments of the probability of pivot events when the number of players tends to infinity.
• Compute the best response to a given tau-vector.
• Explore automatically a grid of ordinal profiles or a grid of tau-vectors.
• Perform Monte-Carlo experiments on profiles or tau-vectors.

## History

### 0.8.1 (2020-02-04)

• BestResponse: the focus of this release is to correct rare bugs that used to happen when some offsets are very close to 1.
• API change: BestResponse now takes as parameters the tau-vector and the ranking, instead of all the events that are used for the computation.
• Exchanged the justifications 'Easy vs difficult pivot' and 'Difficult vs easy pivot' (their usages were switched, even if the result itself was correct).
• Use the asymptotic method only when there are two consecutive zeros in the “compass diagram” of the tau-vector (instead of: whenever it gives a result). The motivation is that the asymptotic method may rely on events that rely more on numerical approximation than the limit pivot theorem approach.
• To determine whether pivots are easy or difficult, we rely on expected scores in the duo events, instead of the pseudo-offsets of the trio. The motivation is that in some cases, the trio is computed with a numerical optimizer that relies more on numerical approximation than the duo events, which use only basic operations like addition, multiplication, etc. In the rare cases where the two methods differ, the latter is thus more reliable.
• Add a sub-algorithm of the “Offset method”, called “Offset method with trio approximation correction”. This is used in some rare cases where both pivots are difficult, but the numeric approximations of the trio event lead to an offset that is equal or even slightly greater than 1 (which is abnormal and leads to infinite geometric sums). In those cases, we now consider that the offset is lower and infinitely close to 1.
• Corrected a bug in the asymptotic method that could happen when the two personalized pivots had very close magnitudes. This uses the correction of Asymptotic.limit mentioned below.
• TauVector: added the attribute has_two_consecutive_zeros.
• Event: now computes the pseudo-offsets, e.g. psi_a, psi_ab, etc.
• Asymptotic: handles some edge cases more nicely.
• __str__ displays a coefficient as 0, 1 or -1 only if it is equal to that value. Close is not enough.
• limit does not use closeness to 0. It is not its role to decide what coefficients are negligible in the context. Only operations like multiplication are allowed to use closeness: for example, if mu_1 and - mu_2 are relatively close, the multiplication operator is allowed to decide that mu_1 + mu_2 is equal to 0.
• In multiplication, when the two magnitudes are close, the resulting magnitude is now always equal to the maximum. The same applies for the resulting nu when the nu’s are also equal.
• cached_property: corrected a bug. In the case of nested cached properties, the inner one was sometimes not recorded in cache. It did not lead to incorrect results but slowed down the program.

### 0.8.0 (2020-01-30)

• Implement the notion of fanatic voting, a variant of sincere voting: a given ratio of voters vote for their top candidate only. This is implemented for all subclasses of Profile.
• The utility barycenter now accepts iterables.
• Corrected bug: Profile.standardized_version now takes into account the auxiliary parameters like ratio_sincere, well_informed_voters, etc.

### 0.7.0 (2020-01-30)

• Add ProfileDiscrete: a profile with a discrete distribution of voters.
• Subclasses of Profile: better handling of the additional parameters like well_informed_voters or ratio_sincere. In the conversions to string (str or repr), they are now mentioned. They are also used in the equality tests between two profiles.

### 0.6.0 (2020-01-29)

• Implement ProfileCardinal.fictitious_play, where the update ratios of the perceived tau-vector and the actual tau-vector can be functions of the time. It is also faster that ProfileCardinal.iterated_voting, but can not detect cycles (only convergence).
• ProfileCardinal.iterated_voting_taus is renamed to ProfileCardinal.iterated_voting. It has been generalized by implementing a notion of perceived tau-vector, like for ProfileCardinal.fictitious_play. The syntax has been modified in consequence.
• ProfileCardinal.iterated_voting_strategies is deprecated and suppressed.
• Iterated voting and fictitious play do not need a StrategyThreshold as initial strategy, but any strategy that is consistent with the profile subclass. For example, with ProfileTwelve, you can use a StrategyTwelve.
• Strategy.profile is now a property that can be reassigned after the creation of the object.
• Add Strategy.deepcopy_with_attached_profile: make a deep copy and attach a given profile.
• Add the utility to_callable: convert an object to a callable (making it a constant function if it is not callable already).

### 0.5.1 (2020-01-18)

• Configure Codecov.
• Reach 100% coverage for this version.

### 0.5.0 (2020-01-11)

• In iterated voting, implement the possibility to move only progressively towards the best response:
• Add ProfileCardinal.iterated_voting_taus: at each iteration, a given ratio of voters update their ballot.
• Replace the former method ProfileCardinal.iterated_voting by ProfileCardinal.iterated_voting_strategies: as in former versions, at each iteration, the threshold utility of each ranking’s strategy is moved in the direction of the best response’s threshold utility. The method now returns a cycle of tau-vectors and the corresponding cycle of best response strategies, in order to be consistent with ProfileCardinal.iterated_voting_taus.
• Add the utility barycenter: compute a barycenter while respecting the type of one input if the other input has weight 0.
• Accelerate the algorithm used in iterated voting.
• In ProfileCardinal, add the possibility of partial sincere voting:
• Add parameter ratio_sincere: ratio of sincere voters.
• Add property tau_sincere: the tau-vector if all voters vote sincerely.
• The former method tau is renamed tau_strategic: the tau_vector if all voters vote strategically.
• The new method tau takes both sincere and strategic voting into account.
• The method is_equilibrium has a new implementation to take this feature into account.
• Add TauVector.isclose: whether the tau-vector is close to another tau-vector (in the sense of math.isclose). This method is used by the new version of ProfileCardinal.is_equilibrium.
• Add Profile.best_responses_to_strategy: convert a dictionary of best responses to a StrategyThreshold that mentions only the rankings that are present in the profile.
• In random generators of profiles (GeneratorProfileOrdinalUniform, GeneratorProfileOrdinalGridUniform, GeneratorProfileOrdinalVariations, GeneratorProfileHistogramUniform): instead of having explicit arguments like well_informed_voters or ratio_sincere, there are **kwargs that are directly passed to the __init__ of the relevant Profile subclass.
• Update the tutorials with these new features.

### 0.4.0 (2020-01-08)

• Add image_distribution: estimate the distribution of f(something) for a random something.
• Update the tutorial on mass simulations with this new feature.

### 0.3.0 (2020-01-08)

• GeneratorExamples: run another generator until the generated object meets a given test.
• GeneratorStrategyOrdinalUniform: draw a StrategyOrdinal uniformly.
• GeneratorProfileOrdinalGridUniform: draw a ProfileOrdinal uniformly on a grid of rational numbers.
• GeneratorTauVectorGridUniform: draw a TauVector uniformly on a grid of rational numbers.
• Utilities:
• Add rand_integers_fixed_sum: draw an array of integers with a given sum.
• Add rand_simplex_grid: draw a random point in the simplex, with rational coordinates of a given denominator.
• Update probability: allow for a tuple of generators.
• Tutorials:
• Add a tutorial on asymptotic developments.
• Update the tutorial on mass simulations with the new features.

### 0.2.1 (2020-01-05)

• Relaunch deployment.

### 0.2.0 (2020-01-05)

• Modify masks_distribution: remove the trailing zeros. This has the same impact on ProfileOrdinal.distribution_equilibria.
• Modify NiceStatsProfileOrdinal.plot_cutoff: center the textual indications.
• Replace all notations r with profile and sigma with strategy.

### 0.1.1 (2019-12-24)

• Convert all the documentation to NumPy format, making it more readable in plain text.

### 0.1.0 (2019-12-20)

• First release on PyPI.
• Implement only the case of 3 candidates.
• Deal with ordinal or cardinal profiles.
• Compute the asymptotic developments of the probability of pivot events when the number of players tends to infinity.
• Compute the best response to a given tau-vector.
• Explore automatically a grid of ordinal profiles or a grid of tau-vectors.
• Perform Monte-Carlo experiments on profiles or tau-vectors.