Majority.Merit

Type Choices

type Choices = HashSet

choices :: (Bounded choice, Enum choice, Eq choice, Hashable choice) => Choices choice

Return a set of Choices by enumerating the alternatives of its type. Useful on sum types.

Type Grades

type Grades = Set

How many grades should be used? A greater number of grades permits a finer distinction but demands a higher degree of expertise and discernment. The optimal number is the highest number of grades that constitutes a common language, that is, that allows Judges to make absolute judgments. Too little grades may lead too often to ties.

Note, however, that if the inputs or grades depend on the set of choices, i.e. if judges change their grades when choices are added or dropped, then the Arrow paradox cannot be avoided. To avoid this the scale of grades must be fixed and absolute so that more or fewer choices does not change the inputs or messages of other choices.

grades :: [grade] -> Grades (Ranked grade)

Type Ranked

newtype Ranked a

Helper type to rank data without a good Ord instance.

Constructors

Ranked (Integer, a)

Instances

Functor Ranked

Defined in Majority.Merit

Methods

fmap :: (a -> b) -> Ranked a -> Ranked b

(<$) :: a -> Ranked b -> Ranked a

Eq (Ranked a)

Defined in Majority.Merit

Methods

(==) :: Ranked a -> Ranked a -> Bool

(/=) :: Ranked a -> Ranked a -> Bool

Ord (Ranked a)

Defined in Majority.Merit

Methods

compare :: Ranked a -> Ranked a -> Ordering

(<) :: Ranked a -> Ranked a -> Bool

(<=) :: Ranked a -> Ranked a -> Bool

(>) :: Ranked a -> Ranked a -> Bool

(>=) :: Ranked a -> Ranked a -> Bool

max :: Ranked a -> Ranked a -> Ranked a

min :: Ranked a -> Ranked a -> Ranked a

Show a => Show (Ranked a)

Defined in Majority.Merit

Methods

showsPrec :: Int -> Ranked a -> ShowS

show :: Ranked a -> String

showList :: [Ranked a] -> ShowS

zipRank :: [a] -> [Ranked a]

zipRank xs returns a list with the items of xs wrapped so that they are compareable according to their position into xs.

rankKey :: [(k, a)] -> [(Ranked k, a)]

rank :: Ranked a -> Integer

unRank :: Ranked a -> a

enum :: (Bounded a, Enum a, Ord a) => Set a

Return the Set enumerating the alternatives of its type parameter. Useful on sum types.

Type Judges

type Judges = HashMap

Map each judge to its default grade (usually the same for all judges but not necessarily).

For instance, when a judge gives no grade or has “no opinion”, this default grade could mean that the judge chooses “To Reject” the choice: the rationale being that a judge having “no opinion” concerning a choice has not even taken the time to evaluate it and thus has implicitly rejected it.

judges :: Eq judge => Hashable judge => [judge] -> grade -> Judges judge grade

Type Opinions

type Opinions judge grade = HashMap judge (Distribution grade)

Profile of opinions of some judges about a single choice.

opinions :: Eq judge => Hashable judge => Judges judge grade -> Opinions judge grade -> (Opinions judge grade, HashSet judge)

(ok, ko) = opinions js os returns:

• in ok the opinions of the judges js updated by those in os,
• in ko the opinions of judges not in js.

Type Distribution

type Distribution grade = Map grade Share

Usually, a judge gives a singleGrade to a given choice. However, when applying the Majority Judgment to a Tree of Sections, what a judge gives to a parent Section is composed by the grades he or she has given to the sub-Sections, and those can be different. In that case, each grade given to a sub-Section contributes to a Share of the parent Section which therefore is not necessarily a singleGrade, but more generally a Distribution of grades. And the sub-Sections can actually themselves have sub-Sections, hence not being given a grade, but a Distribution of grades too.

singleGrade :: grade -> Distribution grade

Type Share

type Share = Rational

Usually a judge attributes a singleGrade to a given choice, and then the Share of this grade is 1. However, when introducing vote colleges (giving more power to some judges), or when introducing Sections (decomposing a judgment into several sub-judgments), it becomes possible that only a percentage of grade is attributed by a judge to a given choice. This is what a Share is used for.

Type OpinionsByChoice

type OpinionsByChoice choice judge grade = HashMap choice (Opinions judge grade)

Profile of opinions of some Judges about some choices.

opinionsByChoice :: Eq choice => Hashable choice => [(choice, Opinions judge grade)] -> OpinionsByChoice choice judge grade

Type Merit

newtype Merit grade

Profile of merit about a single choice.

Constructors

Merit

Fields

• unMerit :: Map grade Share

Instances

(Ord grade, Show grade) => IsList (Merit grade)

Defined in Majority.Merit

Associated Types

type Item (Merit grade) :: Type

Methods

fromList :: [Item (Merit grade)] -> Merit grade

fromListN :: Int -> [Item (Merit grade)] -> Merit grade

toList :: Merit grade -> [Item (Merit grade)]

Eq grade => Eq (Merit grade)

Defined in Majority.Merit

Methods

(==) :: Merit grade -> Merit grade -> Bool

(/=) :: Merit grade -> Merit grade -> Bool

Ord grade => Ord (Merit grade)

Defined in Majority.Value

Methods

compare :: Merit grade -> Merit grade -> Ordering

(<) :: Merit grade -> Merit grade -> Bool

(<=) :: Merit grade -> Merit grade -> Bool

(>) :: Merit grade -> Merit grade -> Bool

(>=) :: Merit grade -> Merit grade -> Bool

max :: Merit grade -> Merit grade -> Merit grade

min :: Merit grade -> Merit grade -> Merit grade

Show grade => Show (Merit grade)

Defined in Majority.Merit

Methods

showsPrec :: Int -> Merit grade -> ShowS

show :: Merit grade -> String

showList :: [Merit grade] -> ShowS

Ord grade => Semigroup (Merit grade)

Defined in Majority.Merit

Methods

(<>) :: Merit grade -> Merit grade -> Merit grade

sconcat :: NonEmpty (Merit grade) -> Merit grade

stimes :: Integral b => b -> Merit grade -> Merit grade

type Item (Merit grade)

Defined in Majority.Merit

type Item (Merit grade) = (grade, Share)

merit :: Ord grade => Foldable opinions => opinions (Distribution grade) -> Merit grade

merit os returns the Merit given by opinions os

meritFromList :: Ord grade => Foldable opinions => Functor opinions => opinions grade -> Merit grade

normalizeMerit :: Merit grade -> Merit grade

'normalizeMerit m' multiply all Shares by their least common denominator to get integral Shares.

Type MeritByChoice

newtype MeritByChoice choice grade

Profile of merit about some choices.

Constructors

MeritByChoice

Fields

• unMeritByChoice :: HashMap choice (Merit grade)

Instances

(Eq choice, Hashable choice, Show choice) => IsList (MeritByChoice choice grade)

Defined in Majority.Merit

Associated Types

type Item (MeritByChoice choice grade) :: Type

Methods

fromList :: [Item (MeritByChoice choice grade)] -> MeritByChoice choice grade

fromListN :: Int -> [Item (MeritByChoice choice grade)] -> MeritByChoice choice grade

toList :: MeritByChoice choice grade -> [Item (MeritByChoice choice grade)]

(Eq choice, Eq grade) => Eq (MeritByChoice choice grade)

Defined in Majority.Merit

Methods

(==) :: MeritByChoice choice grade -> MeritByChoice choice grade -> Bool

(/=) :: MeritByChoice choice grade -> MeritByChoice choice grade -> Bool

(Show choice, Show grade) => Show (MeritByChoice choice grade)

Defined in Majority.Merit

Methods

showsPrec :: Int -> MeritByChoice choice grade -> ShowS

show :: MeritByChoice choice grade -> String

showList :: [MeritByChoice choice grade] -> ShowS

(Eq choice, Hashable choice, Ord grade) => Semigroup (MeritByChoice choice grade)

Defined in Majority.Merit

Methods

(<>) :: MeritByChoice choice grade -> MeritByChoice choice grade -> MeritByChoice choice grade

sconcat :: NonEmpty (MeritByChoice choice grade) -> MeritByChoice choice grade

stimes :: Integral b => b -> MeritByChoice choice grade -> MeritByChoice choice grade

type Item (MeritByChoice choice grade)

Defined in Majority.Merit

type Item (MeritByChoice choice grade) = (choice, Merit grade)

meritByChoice :: (Ord grade, Eq choice, Hashable choice) => OpinionsByChoice choice judge grade -> MeritByChoice choice grade

meritByChoice gs cs os returns the Merits given to choices cs by opinions os from the Judges js on the Grades gs.