Introduction
In the literature of cooperative games, the notion of power
index [1–3] has been widely studied
to analyze the ``influence” of individuals taking into account their
ability to force a decision within groups or coalitions. In practical
situations, however, the information concerning the strength of
coalitions is hardly quantifiable. So, any attempt to numerically
represent the influence of groups and individuals clashes with the
complex and multi-attribute nature of the problem and it seems more
realistic to represent collective decision-making mechanisms using an
ordinal coalitional framework based on two main ingredients: a binary
relation over groups or coalitions and a ranking over the
individuals.
The main objective of the package socialranking is to
provide answers for the general problem of how to compare the elements
of a finite set \(N\) given a ranking
over the elements of its power-set (the set of all possible subsets of
\(N\)). To do this, the package
socialranking implements a portfolio of solutions from the
recent literature on social rankings [4–10].
Quick Start
A power relation (i.e, a ranking over subsets of a finite
set \(N\); see the Section on PowerRelation objects for a formal definition)
can be constructed using the newPowerRelation() or
newPowerRelationFromString() functions.
library(socialranking)
newPowerRelation(c(1,2), ">", 1, "~", c(), ">", 2)
## Elements: 1 2
## 12 > (1 ~ {}) > 2
newPowerRelationFromString("ab > a ~ {} > b")
## Elements: a b
## ab > (a ~ {}) > b
newPowerRelationFromString("12 > 1 ~ {} > 2", asWhat = as.numeric)
## Elements: 1 2
## 12 > (1 ~ {}) > 2
Functions used to analyze a given PowerRelation object
can be grouped into three main categories:
- Comparison functions, only comparing two elements;
- Score functions, calculating the scores for each
element;
- Ranking functions, creating
SocialRankingSolution objects.
Comparison and score functions are often used to evaluate a social
ranking solution (see section on PowerRelation
objects for a formal definition). Listed below are some of the most
prominent functions and solutions introduced in the aforementioned
papers.
| Comparison functions |
Score functions |
Ranking functions |
dominates() |
|
|
cumulativelyDominates() |
cumulativeScores() |
|
cpMajorityComparison()
cpMajorityComparisonScore() |
copelandScores()
kramerSimpsonScores() |
copelandRanking()
kramerSimpsonRanking() |
| |
lexcelScores() |
lexcelRanking()
dualLexcelRanking() |
| |
ordinalBanzhafScores() |
ordinalBanzhafRanking() |
These functions may be called as follows.
pr <- newPowerRelationFromString("ab > ac ~ bc > a ~ c > {} > b")
# a dominates b -> TRUE
dominates(pr, "a", "b")
## [1] TRUE
# b does not dominate a -> FALSE
dominates(pr, "b", "a")
## [1] FALSE
# calculate cumulative scores
scores <- cumulativeScores(pr)
# show score of element a
scores$a
## [1] 1 2 3 3 3
# performing a bunch of rankings
lexcelRanking(pr)
## a > b > c
## a > c > b
## a > b ~ c
## a > b ~ c
ordinalBanzhafRanking(pr)
## a ~ c > b
Finally an incidence matrix for all given coalitions can be
constructed using powerRelationMatrix(pr) or
as.relation(pr) from the relations package
[11]. The incidence matrix may be
displayed using relations::relation_incidence().
rel <- relations::as.relation(pr)
rel
## A binary relation of size 7 x 7.
relations::relation_incidence(rel)
## Incidences:
## ab ac bc a c {} b
## ab 1 1 1 1 1 1 1
## ac 0 1 1 1 1 1 1
## bc 0 1 1 1 1 1 1
## a 0 0 0 1 1 1 1
## c 0 0 0 1 1 1 1
## {} 0 0 0 0 0 1 1
## b 0 0 0 0 0 0 1
PowerRelation Objects
We first introduce some basic definitions on binary relations. Let
\(X\) be a set. A set \(R \subseteq X \times X\) is said a
binary relation on \(X\). For
two elements \(x, y \in X\), \(xRy\) refers to their relation, more
formally it means that \((x,y) \in R\).
A binary relation \((x,y) \in R\) is
said to be:
- reflexive, if for each \(x \in X,
xRx\)
- transitive, if for each \(x, y, z
\in X, xRy\) and \(yRz \Rightarrow
xRz\)
- total, if for each \(x,y \in X, x
\neq y \Rightarrow xRy\) or \(yRx\)
- symmetric, if for each \(x,y \in
X,xRy \Leftrightarrow yRx\)
- asymmetric, if for each \(x,y \in
X,(x,y) \in R \Rightarrow (y,x) \notin R\)
- antisymmetric, if for each \(x,y
\in X,xRy \cap yRx \Rightarrow x=y\)
A preorder is defined as a reflexive and transitive
relation. If it is total, it is called a total preorder.
Additionally if it is antisymmetric, it is called a linear
order.
Let \(N = \{1, 2, \dots, n\}\) be a
finite set of elements, sometimes also called players.
For some \(p \in \{1, \ldots, 2^n\}\),
let \(\mathcal{P} = \{S_1, S_2, \dots,
S_{p}\}\) be a set of coalitions such that \(S_i \subseteq N\) for all \(i \in \{1, \ldots, p\}\). Thus \(\mathcal{P} \subseteq 2^N\), where \(2^N\) denotes the power set of \(N\) (i.e., the set of all subsets or
coalitions of \(N\)).
\(\mathcal{T}(N)\) denotes the set
of all total preorders on \(N\), \(\mathcal{T}(\mathcal{P})\) the set of all
total preorders on \(\mathcal{P}\). A
single total preorder \(\succeq \in
\mathcal{T}(\mathcal{P})\) is said a power relation.
In a given power relation \(\succeq \in
\mathcal{T}(\mathcal{P})\) on \(\mathcal{P} \subseteq 2^N\), its symmetric
part is denoted by \(\sim\) (i.e.,
\(S \sim T\) if \(S \succeq T\) and \(T \succeq S\)), whereas its asymmetric part
is denoted by \(\succ\) (i.e., \(S \succ T\) if \(S \succeq T\) and not \(T \succeq S\)). In other terms, for \(S \sim T\) we say that \(S\) is indifferent to \(T\), whereas for \(S \succ T\) we say that \(S\) is strictly better than \(T\).
Lastly for a given power relation in the form of \(S_1 \succeq S_2 \succeq \ldots \succeq
S_m\), coalitions that are indifferent to one another can be
grouped into equivalence classes \(\sum_i\) such that we get the quotient
order \(\sum_1 \succ \sum_2 \succ \ldots
\succ \sum_m\).
Let \(N=\{1,2\}\) be two players
with its corresponding power set \(2^N =
\{\{1,2\}, \{1\}, \{2\}, \emptyset\}\). The following power
relation is given: \(\succeq =
\{(\{1,2\},\{2\}), (\{2\}, \emptyset), (\emptyset, \{2\}), (\emptyset,
\{1\})\}\). This power relation can be rewritten in a consecutive
order as: \(\{1,2\} \succ \{2\} \sim \emptyset
\succ \{1\}\). Its quotient order is formed by three equivalence
classes \(\sum_1 = \{\{1,2\}\}, \sum_2 =
\{\{2\}, \emptyset\},\) and \(\sum_3 =
\{\{1\}\}\); so the quotient order of \(\succeq\) is such that \(\{\{1,2\}\} \succ \{\{2\}, \emptyset\} \succ
\{\{1\}\}\).
A social ranking solution (also called social
ranking or, simply, solution) on \(N\), is a function \(R: \mathcal{T}(\mathcal{P}) \longrightarrow
\mathcal{T}(N)\) associating to each power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) a
total preorder \(R(\succeq)\) (or \(R^\succeq\)) over the elements of \(N\). By this definition, the notion \(i R^\succeq j\) means that applying the
social ranking solution to the power relation \(\succeq\) gives the result that \(i\) is ranked higher than or equal to \(j\).
Creating
PowerRelation Objects
A power relation in the socialranking package is defined
to be reflexive, transitive and total. In designing the package it was
deemed logical to have the coalitions specified in a consecutive order,
as seen in Example 1. Each coalition in that order
is split either by a ">" (left side strictly better) or
a "~" (two coalitions indifferent to one another). The
following code chunk shows the power relation from Example 1 and how a correlating PowerRelation
object can be constructed.
library(socialranking)
pr <- newPowerRelation(c(1,2), ">", 2, "~", c(), ">", 1)
pr
## Elements: 1 2
## 12 > (2 ~ {}) > 1
## [1] "PowerRelation" "SingleCharElements"
Notice how coalitions such as \(\{1,2\}\) are written as 12 to
improve readability. Similarly the function
newPowerRelationFromString saves some typing on the user’s
end by interpreting each character of a coalition as a separate player.
Note that spaces in that function are ignored.
newPowerRelationFromString("12 > 2~{} > 1", asWhat = as.numeric)
## Elements: 1 2
## 12 > (2 ~ {}) > 1
The compact notation is only done in PowerRelation
objects where every player is one digit or one character long. If this
is not the case, curly braces and commas are added where needed.
prLong <- newPowerRelation(
c("Alice", "Bob"), ">", "Bob", "~", c(), ">", "Alice"
)
prLong
## Elements: Alice Bob
## {Alice, Bob} > ({Bob} ~ {}) > {Alice}
## [1] "PowerRelation"
Some may have spotted a "SingleCharElements" class
missing in class(prLong) that has been there in
class(pr). "SingleCharElements" influences the
way coalitions are printed. If it is removed from
class(pr), the output will include the same curly braces
and commas displayed in prLong.
class(pr) <- class(pr)[-which(class(pr) == "SingleCharElements")]
pr
## Elements: 1 2
## {1, 2} > ({2} ~ {}) > {1}
Internally a PowerRelation is a list with four
attributes (see table below). Notice that every coalition vector is
turned into a set object from the sets
package[12].
| Attribute |
Description |
Value in pr |
elements |
Sorted vector of elements |
c(1,2) |
rankingCoalitions |
Coalitions in power relation |
list(set(1,2),set(2),set(),set(1)) |
equivalenceClasses |
List containing lists, each
containing coalitions in the
same equivalence class |
list(list(set(1,2)),
list(set(2), set()),
list(set(1))) |
Since each coalition vector is turned into a set,
coalitions such as c(1,2), c(2,1) and
c(1,1,2,2) are equivalent.
prAtts <- newPowerRelation(c(2,2,1,1,2), ">", c(1,1,1), "~", c())
prAtts
## Elements: 1 2
## 12 > (1 ~ {})
## [1] 1 2
## [[1]]
## {1, 2}
##
## [[2]]
## {1}
##
## [[3]]
## {}
prAtts$rankingComparators
## [1] ">" "~"
prAtts$equivalenceClasses
## [[1]]
## [[1]][[1]]
## {1, 2}
##
##
## [[2]]
## [[2]][[1]]
## {1}
##
## [[2]][[2]]
## {}
equivalenceClassIndex() determines at which index \(i\) a coalition \(S \in \sum_i\).
equivalenceClassIndex(prAtts, c(2,1))
## [1] 1
equivalenceClassIndex(prAtts, 1)
## [1] 2
equivalenceClassIndex(prAtts, c())
## [1] 2
# are the given coalitions in the same equivalence class?
coalitionsAreIndifferent(prAtts, 1, c())
## [1] TRUE
coalitionsAreIndifferent(prAtts, 1, c(1,2))
## [1] FALSE
Manipulating
PowerRelation Objects
It is strongly discouraged to directly manipulate
PowerRelation objects, since modifying one list or vector
entry would require updates on all attributes. Instead
newPowerRelation offers two parameters
rankingCoalitions and rankingComparators, each
corresponding to the same named attributes of a
PowerRelation object.
## Elements: 1 2
## {1, 2} > ({2} ~ {}) > {1}
# reverse power ranking
newPowerRelation(
rankingCoalitions = rev(pr$rankingCoalitions),
rankingComparators = pr$rankingComparators
)
## Elements: 1 2
## 1 > ({} ~ 2) > 12
Note that rankingComparators is optional. By default it
assumes rankingCoalitions to be a linear order.
newPowerRelation(rankingCoalitions = rev(pr$rankingCoalitions))
## Elements: 1 2
## 1 > {} > 2 > 12
If the length of the rankingComparators vector is
smaller or larger than the length of rankingCoalitions, the
function silently fills in any gaps.
# if too short -> comparator values are repeated
newPowerRelation(
rankingCoalitions = as.list(1:9),
rankingComparators = "~"
)
## Elements: 1 2 3 4 5 6 7 8 9
## (1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 ~ 7 ~ 8 ~ 9)
newPowerRelation(
rankingCoalitions = as.list(letters[1:9]),
rankingComparators = c(">", "~", "~")
)
## Elements: a b c d e f g h i
## a > (b ~ c ~ d) > (e ~ f ~ g) > (h ~ i)
# if too long -> ignore excessive comparators
newPowerRelation(
rankingCoalitions = pr$rankingCoalitions,
rankingComparators = c("~", ">", "~", ">", ">", "~")
)
## Elements: 1 2
## (12 ~ 2) > ({} ~ 1)
Creating Power
sets
As the number of elements \(n\)
increases, the number of possible coalitions increases to \(|2^N| = 2^n\). createPowerset
is a convenient function that not only creates a power set \(2^N\) which can be used to call
newPowerRelation, but also formats the function call in
such a way that makes it easy to rearrange the ordering of the
coalitions. RStudio offers shortcuts Alt+Up or Alt+Down (Option+Up or
Option+Down on MacOS) to move one or multiple lines of code up or down
(see fig. below).
createPowerset(
c("a", "b", "c"),
writeLines = TRUE,
copyToClipboard = FALSE
)
## newPowerRelation(
## c("a", "b", "c"),
## ">", c("a", "b"),
## ">", c("a", "c"),
## ">", c("b", "c"),
## ">", c("a"),
## ">", c("b"),
## ">", c("c"),
## ">", c(),
## )
If writeLines and copyToClipboard are both
set to FALSE, the function instead returns a list only
containing coalition vectors. This list can be passed directly as
rankingCoalitions parameter to
newPowerRelation.
ps <- createPowerset(1:2, includeEmptySet = FALSE)
ps
## [[1]]
## [1] 1 2
##
## [[2]]
## [1] 1
##
## [[3]]
## [1] 2
newPowerRelation(rankingCoalitions = ps)
## Elements: 1 2
## 12 > 1 > 2
newPowerRelation(rankingCoalitions = createPowerset(letters[1:4]))
## Elements: a b c d
## abcd > abc > abd > acd > bcd > ab > ac > ad > bc > bd > cd > a > b > c > d > {}
SocialRankingSolution Objects
The main goal of the socialranking package is to rank
elements based on a given power ranking. More formally we try to map
\(R: \mathcal{T}(\mathcal{P}) \rightarrow
\mathcal{T}(N)\), associating to each power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) a
total preorder \(R(\succeq)\) (or \(R^\succeq\)) over the elements of \(N\).
In this context \(i R^\succeq j\)
tells us that, given a power relation \(\succeq\) and applying a social ranking
solution \(R(\succeq)\), \(i\) is ranked higher than or equal to \(j\). From here on out, >
and ~ also denote the asymmetric and the symmetric part of
a social ranking, respectively, \(i\)
> \(j\) indicating that
\(i\) is strictly better than \(j\), whereas in \(i\) ~ \(j\), \(i\)
is indifferent to \(j\).
In section 3.1 we show how a general
SocialRankingSolution object can be constructed using the
doRanking function. In the following sections, we will
introduce the notion of dominance[4],
cumulative dominance[13] and CP-Majority
comparison[6] that lets us compare two
elements before diving into the social ranking solutions of the Ordinal
Banzhaf Index[5], Copeland-like and
Kramer-Simpson-like methods[10], and
lastly the Lexicographical Excellence Solution[9] (Lexcel) and the Dual Lexicographical
Excellence solution[14] (Dual Lexcel).
Let \(\{a,b\} \succ (\{a,c\} \sim \{b,c\})
\succ (\{a\} \sim \{c\}) > \emptyset \succ \{b\}\) be a power
ranking. Using the following social ranking solutions, we get:
a > b > c for lexcelRankinga > c > b for dualLexcelRankinga > b ~ c for copelandRanking and
kramerSimpsonRankinga ~ c > b for
ordinalBanzhafRanking
Creating
SocialRankingSolution objects
A SocialRankingSolution object represents a total
preorder in \(\mathcal{T}(N)\) over the
elements of \(N\). Internally they are
saved as a list of vectors, each containing players that are indifferent
to one another. This is somewhat similar to the
equivalenceClasses attribute in PowerRelation
objects.
The function doRanking offers a generic way of creating
SocialRankingSolution objects. Given a
PowerRelation object and a sortable vector or list of
scores it determines the power relation between all players. Note that
length(scores) == length(powerRelation$elements) must be
TRUE. Additionally each index in scores
corresponds to the index in the sorted vector
powerRelation$elements.
pr <- newPowerRelationFromString("abc > ab ~ ac > bc")
# pr$elements == c("a", "b", "c")
# we define some arbitrary score vector where "a" scores highest
# "b" and "c" both score 1, thus they are indifferent
scores <- c(100, 1, 1)
doRanking(pr, scores)
## a > b ~ c
# we can also tell doRanking to punish higher scores
doRanking(pr, scores, decreasing = FALSE)
## b ~ c > a
By default elements are assumed to be indifferent to one another if
their scores are equal. Sometimes however other factors come into play
that make it non-obvious how to compare two scores. In those cases a
function comparing two scores can be passed that should return
TRUE if the two scores are considered equal.
scores <- c(0, 20, 21)
# b and c are considered to be indifferent,
# because their score difference is less than 2
doRanking(pr, scores, isIndifferent = function(a,b) abs(a-b) < 2)
## b ~ c > a
Comparison
Functions
Comparison functions only compare two elements in a given power
relation. They do not offer a social ranking solution. However in cases
such as CP-Majority comparison, those comparison functions may be used
to construct a social ranking solution in some particular cases.
Dominance
(Dominance [4]) Given a power relation
\(\succeq \in
\mathcal{T}(\mathcal{P})\) and two elements \(i,j \in N\), \(i\) dominates \(j\) in \(\succeq\) if \(S
\cup \{i\} \succeq S \cup \{j\}\) for each \(S \in 2^{N\setminus \{i,j\}}\). \(i\) also strictly dominates \(j\) if there exists \(S \in 2^{N\setminus \{i,j\}}\) such that
\(S \cup \{i\} \succ S \cup
\{j\}\).
The implication is that for every coalition \(i\) and \(j\) can join, \(i\) has at least the same positive impact
as \(j\).
The function dominates(pr, e1, e2) only returns a
logical value TRUE if e1 dominates
e2, else FALSE. Note that e1 not
dominating e2 does not indicate that
e2 dominates e1, nor does it imply that
e1 is indifferent to e2.
pr <- newPowerRelationFromString(
"3 > 1 > 2 > 12 > 13 > 23",
asWhat = as.numeric
)
# 1 clearly dominates 2
dominates(pr, 1, 2)
## [1] TRUE
## [1] FALSE
# 3 does not dominate 1, nor does 1 dominate 3, because
# {}u3 > {}u1, but 2u1 > 2u3
dominates(pr, 1, 3)
## [1] FALSE
## [1] FALSE
# an element i dominates itself, but it does not strictly dominate itself
# because there is no Sui > Sui
dominates(pr, 1, 1)
## [1] TRUE
dominates(pr, 1, 1, strictly = TRUE)
## [1] FALSE
For any \(S \in 2^{N \setminus
\{i,j\}}\), we can only compare \(S
\cup \{i\} \succeq S \cup \{j\}\) if both \(S \cup \{i\}\) and \(S \cup \{j\}\) take part in the power
relation.
Additionally, for \(S = \emptyset\),
we also want to compare \(\{i\} \succeq
\{j\}\). In some situations however a comparison between
singletons is not desired. For this reason the parameter
includeEmptySet can be set to FALSE such that
\(\emptyset \cup \{i\} \succeq \emptyset \cup
\{j\}\) is not considered in the CP-Majority comparison.
pr <- newPowerRelationFromString("ac > bc ~ b > a ~ abc > ab")
# FALSE because ac > bc, whereas b > a
dominates(pr, "a", "b")
## [1] FALSE
# TRUE because ac > bc, ignoring b > a comparison
dominates(pr, "a", "b", includeEmptySet = FALSE)
## [1] TRUE
Cumulative
Dominance
When comparing two players \(i,j \in
N\), instead of looking at particular coalitions \(S \in 2^{N \setminus \{i,j\}}\) they can
join, we look at how many stronger coalitions they can form at each
point. This property was originally introduced in [13] as a regular dominance axiom.
For a given power relation \(\succeq \in
\mathcal{T}(\mathcal{P})\) and its corresponding quotient order
\(\sum_1 \succ \dots \succ \sum_m\),
the power of a player \(i\) is given by
a vector \(\textrm{Score}_\textrm{Cumul}(i)
\in \mathbb{N}^m\) where we cumulatively sum the amount of times
\(i\) appears in \(\sum_k\) for each index \(k\).
(Cumulative Dominance Score) Given a power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) and
its quotient order \(\sum_1 \succ \dots \succ
\sum_m\), the cumulative score vector \(\textrm{Score}_\textrm{Cumul}(i) \in
\mathbb{N}^m\) of an element \(i \in
N\) is given by:
\[\begin{equation}
\textrm{Score}_\textrm{Cumul}(i) = \Big( \sum_{t=1}^k |\{S \in
\textstyle \sum_t : i \in S\}|\Big)_{k \in \{1, \dots, m\}}
\end{equation}\]
(Cumulative Dominance) Given two elements \(i,j \in N\), \(i\) cumulatively dominates \(j\) in \(\succeq\), if \(\textrm{Score}_\textrm{Cumul}(i)_k \geq
\textrm{Score}_\textrm{Cumul}(j)_k\) for each \(k \in \{1, \dots, m\}\). \(i\) also strictly cumulatively
dominates \(j\) if there exists a \(k\) such that \(\textrm{Score}_\textrm{Cumul}(i)_k >
\textrm{Score}_\textrm{Cumul}(j)_k\).
For a given PowerRelation object pr and two
elements e1 and e2,
cumulativeScores(pr) returns the vectors described in
definition 2 for each element,
cumulativelyDominates(pr, e1, e2) returns TRUE
or FALSE based on definition 3.
pr <- newPowerRelationFromString("ab > (ac ~ bc) > (a ~ c) > {} > b")
cumulativeScores(pr)
## $a
## [1] 1 2 3 3 3
##
## $b
## [1] 1 2 2 2 3
##
## $c
## [1] 0 2 3 3 3
##
## attr(,"class")
## [1] "CumulativeScores"
# for each index k, $a[k] >= $b[k]
cumulativelyDominates(pr, "a", "b")
## [1] TRUE
# $a[3] > $b[3], therefore a also strictly dominates b
cumulativelyDominates(pr, "a", "b", strictly = TRUE)
## [1] TRUE
# $b[1] > $c[1], but $c[3] > $b[3]
# therefore neither b nor c dominate each other
cumulativelyDominates(pr, "b", "c")
## [1] FALSE
cumulativelyDominates(pr, "c", "b")
## [1] FALSE
Similar to the dominance property from our previous section, two
elements not dominating one or the other does not indicate that they are
indifferent.
CP-Majority
comparison
The Ceteris Paribus Majority (CP-Majority) relation is a somewhat
relaxed version of the dominance property. Instead of checking if \(S \cup \{i\} \succeq S \cup \{j\}\) for all
\(S \in 2^{N \setminus \{i,j\}}\), the
CP-Majority relation \(iR^\succeq_\textrm{CP}j\) holds if the
number of times \(S \cup \{i\} \succeq S \cup
\{j\}\) is greater than or equal to the number of times \(S \cup \{j\} \succeq S \cup \{i\}\).
(CP-Majority [6]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The
Ceteris Paribus majority relation is the binary relation \(R^\succeq_\textrm{CP} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{CP}j \Leftrightarrow d_{ij}(\succeq) \geq
d_{ji}(\succeq)
\end{equation}\]
where \(d_{ij}(\succeq)\) represents
the cardinality of the set \(D_{ij}(\succeq)\), the set of all
coalitions \(S \in 2^{N \setminus
\{i,j\}}\) for which \(S \cup \{i\}
\succeq S \cup \{j\}\).
cpMajorityComparisonScore(pr, e1, e2) calculates the two
scores \(d_{ij}(\succeq)\) and \(-d_{ji}(\succeq)\). Notice the minus sign -
that way we can use the sum of both values to determine the relation
between e1 and e2.
pr <- newPowerRelationFromString("ab > (ac ~ bc) > (a ~ c) > {} > b")
cpMajorityComparisonScore(pr, "a", "b")
## [1] 2 -1
cpMajorityComparisonScore(pr, "b", "a")
## [1] 1 -2
if(sum(cpMajorityComparisonScore(pr, "a", "b")) >= 0) {
print("a >= b")
} else {
print("b > a")
}
## [1] "a >= b"
As a slight variation the logical parameter strictly
calculates \(d_{ij}(\succ)\) and \(-d_{ji}(\succ)\), the number of coalitions
\(S \in 2^{N\setminus \{i,j\}}\) where
\(S\cup\{i\}\succ S\cup\{j\}\).
# Now (ac ~ bc) is not counted
cpMajorityComparisonScore(pr, "a", "b", strictly = TRUE)
## [1] 1 0
# Notice that the sum is still the same
sum(cpMajorityComparisonScore(pr, "a", "b", strictly = FALSE)) ==
sum(cpMajorityComparisonScore(pr, "a", "b", strictly = TRUE))
## [1] TRUE
Coincidentally, cpMajorityComparisonScore with
strictly = TRUE can be used to determine if e1
(strictly) dominates e2.
cpMajorityComparisonScore should be used for simple and
quick calculations. The more comprehensive function
cpMajorityComparison(pr, e1, e2) does the same
calculations, but in the process retains more information about all the
comparisons that might be interesting to a user, i.e., the set \(D_{ij}(\succeq)\) and \(D_{ji}(\succeq)\) as well as the relation
\(iR^\succeq_\textrm{CP}j\). See the
documentation for a full list of available data.
# extract more information in cpMajorityComparison
cpMajorityComparison(pr, "a", "b")
## a > b
## D_ab = {c, {}}
## D_ba = {c}
## Score of a = 2
## Score of b = 1
# with strictly set to TRUE, coalition c does
# neither appear in D_ab nor in D_ba
cpMajorityComparison(pr, "a", "b", strictly = TRUE)
## a > b
## D_ab = {{}}
## D_ba = {}
## Score of a = 1
## Score of b = 0
The CP-Majority relation can generate cycles, which is the reason
that it is not offered as a social ranking solution. Instead we will
introduce the Copeland-like method and Kramer-Simpson-like method that make use of
the CP-Majority functions to determine a power relation between
elements. For further readings on CP-Majority, see [7] and [10].
Social Ranking
Solutions
Ordinal
Banzhaf
The Ordinal Banzhaf Score is a vector defined by the principle of
marginal contributions. Intuitively speaking, if a player joining a
coalition causes it to move up in the ranking, it can be interpreted as
a positive contribution. On the contrary a negative contribution means
that participating causes the coalition to go down in the ranking.
(Ordinal marginal contribution [5]) Let
\(\succeq \in
\mathcal{T}(\mathcal{P})\). For a given element \(i \in N\), its ordinal marginal
contribution \(m_i^S(\succeq)\)
with right to a coalition \(S \in
\mathcal{P}\) is defined as:
\[\begin{equation}
m_i^S(\succeq) = \begin{cases}
\hphantom{-}1 & \textrm{if } S \cup \{i\} \succ S\\
-1 & \textrm{if } S \succ S \cup \{i\}\\
\hphantom{-}0 & \textrm{otherwise}
\end{cases}
\end{equation}\]
(Ordinal Banzhaf relation) Let \(\succeq
\in \mathcal{T}(\mathcal{P})\). The Ordinal Banzhaf
relation is the binary relation \(R^\succeq_\textrm{Banz} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{Banz}j \Leftrightarrow
\text{Score}_\text{Banz}(i) \geq \text{Score}_\text{Banz}(j),
\end{equation}\]
where \(\text{Score}_\text{Banz}(i) =
\sum_{S} m^S_i(\succeq)\) for all \(S
\in N\setminus\{i\}\).
Note that if \(S \cup \{i\} \notin
\mathcal{P}\), \(m_i^S(\succeq) =
0\).
The function ordinalBanzhafScores returns two numbers
for each element: the number of coalitions \(S\) where a player’s contribution has a
positive impact, and the number of coalitions \(S\) where a player’s contribution has a
negative impact. These two numbers are added and elements are ranked
highest to lowest.
pr <- newPowerRelation(
c(1,2),
">", c(1),
">", c(2)
)
# both players 1 and 2 have an Ordinal Banzhaf Score of 1
# therefore they are indifferent to one another
ordinalBanzhafScores(pr)
## $`1`
## [1] 1 0
##
## $`2`
## [1] 1 0
##
## attr(,"class")
## [1] "OrdinalBanzhafScores"
ordinalBanzhafRanking(pr)
## 1 ~ 2
pr <- newPowerRelationFromString("ab > a > {} > b")
# player b has a negative impact on the empty set
# -> player b's score is 1 - 1 = 0
# -> player a's score is 2 - 0 = 2
sapply(ordinalBanzhafScores(pr), function(score) sum(score))
## a b
## 2 0
ordinalBanzhafRanking(pr)
## a > b
Copeland-like
method
The Copeland-like method of ranking elements based on the CP-Majority
rule is strongly inspired by the Copeland score from social choice
theory[15]. The score of an element \(i \in N\) is determined by the amount of
the pairwise CP-Majority winning comparisons \(i R^\succeq_\textrm{CP} j\), minus the
number of all losing comparisons \(j
R^\succeq_\textrm{CP} i\) against all other elements \(j \in N \setminus \{i\}\).
(Copeland-like relation [10]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The
Copeland-like relation is the binary relation \(R^\succeq_\textrm{Cop} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{Cop}j \Leftrightarrow \text{Score}_\text{Cop}(i)
\geq \text{Score}_\text{Cop}(j),
\end{equation}\]
where \(\text{Score}_\text{Cop}(i) = |\{j
\in N \setminus \{i\}: d_{ij}(\succeq) \geq d_{ji}(\succeq)\}| - |\{j
\in N \setminus \{i\}: d_{ij}(\succeq) \leq
d_{ji}(\succeq)\}|\)
copelandScores(pr) returns two numerical values for each
element, a positive number for the winning comparisons (shown in \(\text{Score}_\text{Cop}(i)\) on the left)
and a negative number for the losing comparisons (in \(\text{Score}_\text{Cop}(i)\) on the
right).
pr <- newPowerRelationFromString("(abc ~ ab ~ c ~ a) > (b ~ bc) > ac")
scores <- copelandScores(pr)
# Based on CP-Majority, a>=b and a>=c (+2), but b>=a (-1)
scores$a
## [1] 2 -1
sapply(copelandScores(pr), sum)
## a b c
## 1 0 -1
## a > b > c
Kramer-Simpson-like
method
Strongly inspired by the Kramer-Simpson method of social choice
theory[16, 17], elements are ranked
inversely to their greatest pairwise defeat over all possible
CP-Majority comparisons.
(Kramer-Simpson-like relation [10])
Let \(\succeq \in
\mathcal{T}(\mathcal{P})\). The Kramer-Simpson-like
relation is the binary relation \(R^\succeq_\textrm{KS} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{KS}j \Leftrightarrow \text{Score}_\text{KS}(i)
\leq \text{Score}_\text{KS}(j),
\end{equation}\]
where \(\text{Score}_\text{KS}(i) = \max_j
d_{ji}(\succeq)\) for all \(j \in N
\setminus \{i\}\).
kramerSimpsonScores(pr) returns a single numerical value
for each element, which sorted lowest to highest gives us the ranking
solution.
pr <- newPowerRelationFromString("(abc ~ ab ~ c ~ a) > (b ~ bc) > ac")
unlist(kramerSimpsonScores(pr))
## a b c
## 0 0 1
## a ~ b > c
There is a small caveat to Definition 8. By
default this function does not compare \(d_{ii}(\succeq)\). In other terms, the
score of every element is the maximum CP-Majority comparison score
against all other elements.
This is slightly different from the definition found in [10], where the CP-Majority comparison \(d_{ii}(\succeq)\) is also considered. Since
by definition \(d_{ii}(\succeq) = 0\),
the Kramer-Simpson scores in those cases will never be negative,
possibly discarding valuable information.
To still account for the original definition in [10], the functions
kramerSimpsonScores and kramerSimpsonRanking
offer a compIvsI parameter that can be set to
TRUE if one wishes for \(d_{ii}(\succeq)\) to be included in the
comparisons.
pr <- newPowerRelationFromString(
"b > (a ~ c) > ab > (ac ~ bc) > {} > abc"
)
kramerSimpsonRanking(pr)
## b > a > c
# notice how b's score is negative
unlist(kramerSimpsonScores(pr))
## a b c
## 1 -1 2
kramerSimpsonScores(pr, elements = "b", compIvsI = TRUE)
## $b
## [1] 0
##
## attr(,"class")
## [1] "KramerSimpsonScores"
Lexicographical
Excellence Solution
The idea behind the lexicographical excellence solution (Lexcel) is
to reward elements appearing more frequently in higher ranked
equivalence classes.
For a given power relation \(\succeq\) and its quotient order \(\sum_1 \succ \dots \succ \sum_m\), we
denote by \(i_k\) the number of
coalitions in \(\sum_k\) containing
\(i\):
\[\begin{equation}
i_k = |\{S \in \textstyle \sum_k: i \in S\}|
\end{equation}\]
for \(k \in \{1, \dots, m\}\). Now,
let \(\text{Score}_\text{Lex}(i)\) be
the \(m\)-dimensional vector \(\text{Score}_\text{Lex}(i) = (i_1, \dots,
i_m)\) associated to \(\succeq\). Consider the lexicographic order
\(\geq_\textrm{Lex}\) among vectors
\(\mathbf{i}\) and \(\mathbf{j}\): \(\mathbf{i} \geq_\textrm{Lex} \mathbf{j}\)
if either \(\mathbf{i} = \mathbf{j}\)
or there exists \(t : i_r = j_r, r \in
\{1,\dots,t-1\}\), and \(i_t >
j_t\).
(Lexicographic-Excellence relation [8]) Let \(\succeq \in
\mathcal{T}(\mathcal{P})\) with its corresponding quotient order
\(\sum_1 \succ \dots \succ \sum_m\).
The Lexicographic-Excellence relation is the binary relation
\(R^\succeq_\textrm{Lex} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{Lex}j \Leftrightarrow \text{Score}_\text{Lex}(i)
\geq_{\textrm{Lex}} \text{Score}_\text{Lex}(j)
\end{equation}\]
pr <- newPowerRelationFromString(
"12 > (123 ~ 23 ~ 3) > (1 ~ 2) > 13",
asWhat = as.numeric
)
# show the number of times an element appears in each equivalence class
# e.g. 3 appears 3 times in [[2]] and 1 time in [[4]]
lapply(pr$equivalenceClasses, unlist)
## [[1]]
## [1] 1 2
##
## [[2]]
## [1] 1 2 3 2 3 3
##
## [[3]]
## [1] 1 2
##
## [[4]]
## [1] 1 3
lexScores <- lexcelScores(pr)
for(i in names(lexScores))
paste0("Lexcel score of element ", i, ": ", lexScores[i])
# at index 1, element 2 ranks higher than 3
lexScores['2'] > lexScores['3']
## [1] TRUE
# at index 2, element 2 ranks higher than 1
lexScores['2'] > lexScores['1']
## [1] TRUE
## 2 > 1 > 3
For some generalizations of the Lexcel solution see also [9].
Lexcel score vectors are very similar to the cumulative score vectors
(see section on Cumulative Dominance) in that
the number of times an element appears in a given equivalence class is
of interest. In fact, applying the base function cumsum on
an element’s lexcel score gives us its cumulative score.
lexcelCumulated <- lapply(lexScores, cumsum)
cumulScores <- cumulativeScores(pr)
paste0(names(lexcelCumulated), ": ", lexcelCumulated, collapse = ', ')
## [1] "1: 1:4, 2: c(1, 3, 4, 4), 3: c(0, 3, 3, 4)"
paste0(names(cumulScores), ": ", cumulScores, collapse = ', ')
## [1] "1: 1:4, 2: c(1, 3, 4, 4), 3: c(0, 3, 3, 4)"
Dual
Lexicographical Excellence Solution
Similar to the Lexcel ranking, the Dual Lexcel also uses the Lexcel
score vectors from definition 9 to establish a
ranking. However, instead of rewarding higher frequencies in high
ranking coalitions, it punishes higher frequencies in lower ranking
coalitions, or, it punishes mediocrity[14].
Take the values \(i_k\) for \(k \in \{1, \dots, m\}\) and the Lexcel
score vector \(\text{Score}_\text{Lex}(i)\) from section
Lexicographical Excellence Solution. Consider
the dual lexicographical order \(\geq_\textrm{DualLex}\) among vectors \(\mathbf{i}\) and \(\mathbf{j}\): \(\mathbf{i} \geq_\textrm{DualLex}
\mathbf{j}\) if either \(\mathbf{i} =
\mathbf{j}\) or there exists \(t: i_t
< j_t\) and \(i_r = j_r, r\in\{t+1,
\dots, m\}\).
(Dual Lexicographical-Excellence relation [14]) Let \(\succeq
\in \mathcal{T}(\mathcal{P})\). The Dual
Lexicographic-Excellence relation is the binary relation \(R^\succeq_\textrm{DualLex} \subseteq N \times
N\) such that for all \(i,j \in
N\):
\[\begin{equation}
iR^\succeq_\textrm{DualLex}j \Leftrightarrow
\text{Score}_\text{Lex}(i) \geq_\textrm{DualLex}
\text{Score}_\text{Lex}(j)
\end{equation}\]
The S3 class LexcelScores does not account for Dual
Lexcel comparisons. Instead -rev(x) is called on a Lexcel
score vector x such that the resulting comparisons produces
a Dual Lexcel ranking solution.
pr <- newPowerRelationFromString(
"12 > (123 ~ 23 ~ 3) > (1 ~ 2) > 13",
asWhat = as.numeric
)
lexScores <- lexcelScores(pr)
# in regular Lexcel, 1 scores higher than 3
lexScores['1'] > lexScores['3']
## [1] TRUE
# turn Lexcel score into Dual Lexcel score
dualLexScores <- structure(
lapply(lexcelScores(pr), function(r) -rev(r)),
class = 'LexcelScores'
)
# now 1 scores lower than 3
dualLexScores['1'] > dualLexScores['3']
## [1] FALSE
# element 2 comes out at the top in both Lexcel and Dual Lexcel
lexcelRanking(pr)
## 2 > 1 > 3
## 2 > 3 > 1
Relations
Incidence Matrix
In our vignette we focused more on the intuitive aspects of power
relations and social ranking solutions. To reiterate, a power relation
is a total preorder, or a reflexive and transitive relation \(\succeq \in \mathcal{P} \times
\mathcal{P}\), where \(\sim\)
denotes the symmetric part and \(\succ\) its asymmetric part.
A power relation can be viewed as an incidence matrix \((b_{ij}) = B \in \{0,1\}^{|\mathcal{P}| \times
|\mathcal{P}|}\). Given two coalitions \(i, j \in \mathcal{P}\), if \(iRj\) then \(b_{ij} = 1\), else \(0\).
With help of the relations package, the functions
relations::as.relation(pr) and
powerRelationMatrix(pr) turn a PowerRelation
object into a relation object. relations then
offers ways to display the relation object as an incidence
matrix with relation_incidence(rel) and to test basic
properties such relation_is_linear_order(rel),
relation_is_acyclic(rel) and
relation_is_antisymmetric(rel) (see relations
package for more [11]).
pr <- newPowerRelationFromString("ab > a > {} > b")
rel <- relations::as.relation(pr)
relations::relation_incidence(rel)
## Incidences:
## ab a {} b
## ab 1 1 1 1
## a 0 1 1 1
## {} 0 0 1 1
## b 0 0 0 1
c(
relations::relation_is_acyclic(rel),
relations::relation_is_antisymmetric(rel),
relations::relation_is_linear_order(rel),
relations::relation_is_complete(rel),
relations::relation_is_reflexive(rel),
relations::relation_is_transitive(rel)
)
## [1] TRUE TRUE TRUE TRUE TRUE TRUE
Note that the columns and rows are sorted by their names in
relation_domain(rel), hence why each name is preceded by
the ordering number.
# a power relation where coalitions {1} and {2} are indifferent
pr <- newPowerRelationFromString("12 > (1 ~ 2)", asWhat = as.numeric)
rel <- relations::as.relation(pr)
# we have both binary relations {1}R{2} as well as {2}R{1}
relations::relation_incidence(rel)
## Incidences:
## 12 1 2
## 12 1 1 1
## 1 0 1 1
## 2 0 1 1
# FALSE
c(
relations::relation_is_acyclic(rel),
relations::relation_is_antisymmetric(rel),
relations::relation_is_linear_order(rel),
relations::relation_is_complete(rel),
relations::relation_is_reflexive(rel),
relations::relation_is_transitive(rel)
)
## [1] FALSE FALSE FALSE TRUE TRUE TRUE
Cycles and Transitive
Closure
A cycle in a power relation exists, if there is one coalition \(S \in 2^N\) that appears twice. For
example, in \(\{1,2\} \succ (\{1\} \sim
\emptyset) \succ \{1,2\}\), the coalition \(\{1,2\}\) appears at the beginning and at
the end of the power relation.
Properly handling power relations and calculating social ranking
solutions with cycles is somewhat ill-defined, hence a warning message
is shown as soon as one is created.
newPowerRelation(c(1,2), ">", 2, ">", 1, "~", 2, ">", c(1,2))
## Warning in newPowerRelation(c(1, 2), ">", 2, ">", 1, "~", 2, ">", c(1, 2)): Found the following duplicates. Did you mean to introduce cycles?
## {2}
## {1, 2}
## Elements: 1 2
## 12 > 2 > (1 ~ 2) > 12
Recall that a power relation is transitive, meaning for three
coalitions \(x, y, z \in 2^N\), if
\(xRy\) and \(yRz\), then \(xRz\). If we introduce cycles, we pretty
much introduce symmetry. Assume we have the power relation \(x \succ y \succ x\). Then, even though
\(xRy\) and \(yRx\) are defined as the asymmetric part of
the power relation \(\succeq\),
together they form the symmetric power relation \(x \sim y\).
transitiveClosure(pr) is a function that turns a power
relation with cycles into one without one. In the process of removing
duplicate coalitions, it turns all asymmectric relations within a cycle
into symmetric relations.
pr <- suppressWarnings(newPowerRelation(1, '>', 2, '>', 1))
pr
## Elements: 1 2
## 1 > 2 > 1
## Elements: 1 2
## (1 ~ 2)
# two cycles, (1>3>1) and (2>23>2)
pr <- suppressWarnings(
newPowerRelationFromString(
"1 > 3 > 1 > 2 > 23 > 2",
asWhat = as.numeric
)
)
transitiveClosure(pr)
## Elements: 1 2 3
## (1 ~ 3) > (2 ~ 23)
# overlapping cycles
pr <- suppressWarnings(
newPowerRelationFromString("c > ac > b > ac > (a ~ b) > abc")
)
transitiveClosure(pr)
## Elements: a b c
## c > (ac ~ b ~ a) > abc
Bibliography
1. Shapley LS, Shubik M. A method for evaluating the distribution of
power in a committee system. American political science review.
1954;48:787–92.
2. Banzhaf III JF. Weighted voting doesn’t work: A mathematical
analysis. Rutgers L Rev. 1964;19:317.
3. Holler MJ, Nurmi H. Power, voting, and voting power: 30 years after.
Springer; 2013.
4. Moretti S, Öztürk M. Some axiomatic and algorithmic perspectives on
the social ranking problem. In: International conference on algorithmic
DecisionTheory. Springer; 2017. p. 166–81.
5. Khani H, Moretti S, Öztürk M. An ordinal banzhaf index for social
ranking. In: 28th international joint conference on artificial
intelligence (IJCAI 2019). 2019. p. 378–84.
6. Haret A, Khani H, Moretti S, Öztürk M. Ceteris paribus majority for
social ranking. In: 27th international joint conference on artificial
intelligence (IJCAI-ECAI-18). 2018. p. 303–9.
7. Fayard N, Öztürk-Escoffier M. Ordinal social ranking: Simulation for
CP-majority rule. In: DA2PL’2018 (from multiple criteria decision aid to
preference learning). 2018.
8. Bernardi G, Lucchetti R, Moretti S. Ranking objects from a preference
relation over their subsets. Social Choice and Welfare. 2019;52:589–606.
9. Algaba E, Moretti S, Rémila E, Solal P. Lexicographic solutions for
coalitional rankings. Social Choice and Welfare. 2021;1–33.
10. Allouche T, Escoffier B, Moretti S, Öztürk M. Social ranking
manipulability for the CP-majority, banzhaf and lexicographic excellence
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IJCAI-20. International Joint Conferences on Artificial
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10.24963/ijcai.2020/3.
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relations. 2022.
https://CRAN.R-project.org/package=relations.
12. Meyer D, Hornik K.
Generalized and
customizable sets in R. Journal of Statistical
Software. 2009;31:1–27.
13. Moretti S. An axiomatic approach to social ranking under coalitional
power relations. Homo Oeconomicus. 2015;32:183–208.
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dominant set selection problem and its application to value alignment.
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15. Copeland AH. A reasonable social welfare function. mimeo, 1951.
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Economic Theory. 1975;16:310–34.
3
SocialRankingSolutionObjectsThe main goal of the
socialrankingpackage is to rank elements based on a given power ranking. More formally we try to map \(R: \mathcal{T}(\mathcal{P}) \rightarrow \mathcal{T}(N)\), associating to each power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) a total preorder \(R(\succeq)\) (or \(R^\succeq\)) over the elements of \(N\).In this context \(i R^\succeq j\) tells us that, given a power relation \(\succeq\) and applying a social ranking solution \(R(\succeq)\), \(i\) is ranked higher than or equal to \(j\). From here on out,
>and~also denote the asymmetric and the symmetric part of a social ranking, respectively, \(i\)>\(j\) indicating that \(i\) is strictly better than \(j\), whereas in \(i\)~\(j\), \(i\) is indifferent to \(j\).In section 3.1 we show how a general
SocialRankingSolutionobject can be constructed using thedoRankingfunction. In the following sections, we will introduce the notion of dominance[4], cumulative dominance[13] and CP-Majority comparison[6] that lets us compare two elements before diving into the social ranking solutions of the Ordinal Banzhaf Index[5], Copeland-like and Kramer-Simpson-like methods[10], and lastly the Lexicographical Excellence Solution[9] (Lexcel) and the Dual Lexicographical Excellence solution[14] (Dual Lexcel).Let \(\{a,b\} \succ (\{a,c\} \sim \{b,c\}) \succ (\{a\} \sim \{c\}) > \emptyset \succ \{b\}\) be a power ranking. Using the following social ranking solutions, we get:
a > b > cforlexcelRankinga > c > bfordualLexcelRankinga > b ~ cforcopelandRankingandkramerSimpsonRankinga ~ c > bforordinalBanzhafRanking3.1 Creating
SocialRankingSolutionobjectsA
SocialRankingSolutionobject represents a total preorder in \(\mathcal{T}(N)\) over the elements of \(N\). Internally they are saved as a list of vectors, each containing players that are indifferent to one another. This is somewhat similar to theequivalenceClassesattribute inPowerRelationobjects.The function
doRankingoffers a generic way of creatingSocialRankingSolutionobjects. Given aPowerRelationobject and a sortable vector or list of scores it determines the power relation between all players. Note thatlength(scores) == length(powerRelation$elements)must beTRUE. Additionally each index inscorescorresponds to the index in the sorted vectorpowerRelation$elements.By default elements are assumed to be indifferent to one another if their scores are equal. Sometimes however other factors come into play that make it non-obvious how to compare two scores. In those cases a function comparing two scores can be passed that should return
TRUEif the two scores are considered equal.3.2 Comparison Functions
Comparison functions only compare two elements in a given power relation. They do not offer a social ranking solution. However in cases such as CP-Majority comparison, those comparison functions may be used to construct a social ranking solution in some particular cases.
3.2.1 Dominance
(Dominance [4]) Given a power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) and two elements \(i,j \in N\), \(i\) dominates \(j\) in \(\succeq\) if \(S \cup \{i\} \succeq S \cup \{j\}\) for each \(S \in 2^{N\setminus \{i,j\}}\). \(i\) also strictly dominates \(j\) if there exists \(S \in 2^{N\setminus \{i,j\}}\) such that \(S \cup \{i\} \succ S \cup \{j\}\).
The implication is that for every coalition \(i\) and \(j\) can join, \(i\) has at least the same positive impact as \(j\).
The function
dominates(pr, e1, e2)only returns a logical valueTRUEife1dominatese2, elseFALSE. Note thate1not dominatinge2does not indicate thate2dominatese1, nor does it imply thate1is indifferent toe2.For any \(S \in 2^{N \setminus \{i,j\}}\), we can only compare \(S \cup \{i\} \succeq S \cup \{j\}\) if both \(S \cup \{i\}\) and \(S \cup \{j\}\) take part in the power relation.
Additionally, for \(S = \emptyset\), we also want to compare \(\{i\} \succeq \{j\}\). In some situations however a comparison between singletons is not desired. For this reason the parameter
includeEmptySetcan be set toFALSEsuch that \(\emptyset \cup \{i\} \succeq \emptyset \cup \{j\}\) is not considered in the CP-Majority comparison.3.2.2 Cumulative Dominance
When comparing two players \(i,j \in N\), instead of looking at particular coalitions \(S \in 2^{N \setminus \{i,j\}}\) they can join, we look at how many stronger coalitions they can form at each point. This property was originally introduced in [13] as a regular dominance axiom.
For a given power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) and its corresponding quotient order \(\sum_1 \succ \dots \succ \sum_m\), the power of a player \(i\) is given by a vector \(\textrm{Score}_\textrm{Cumul}(i) \in \mathbb{N}^m\) where we cumulatively sum the amount of times \(i\) appears in \(\sum_k\) for each index \(k\).
(Cumulative Dominance Score) Given a power relation \(\succeq \in \mathcal{T}(\mathcal{P})\) and its quotient order \(\sum_1 \succ \dots \succ \sum_m\), the cumulative score vector \(\textrm{Score}_\textrm{Cumul}(i) \in \mathbb{N}^m\) of an element \(i \in N\) is given by:
\[\begin{equation} \textrm{Score}_\textrm{Cumul}(i) = \Big( \sum_{t=1}^k |\{S \in \textstyle \sum_t : i \in S\}|\Big)_{k \in \{1, \dots, m\}} \end{equation}\]
(Cumulative Dominance) Given two elements \(i,j \in N\), \(i\) cumulatively dominates \(j\) in \(\succeq\), if \(\textrm{Score}_\textrm{Cumul}(i)_k \geq \textrm{Score}_\textrm{Cumul}(j)_k\) for each \(k \in \{1, \dots, m\}\). \(i\) also strictly cumulatively dominates \(j\) if there exists a \(k\) such that \(\textrm{Score}_\textrm{Cumul}(i)_k > \textrm{Score}_\textrm{Cumul}(j)_k\).
For a given
PowerRelationobjectprand two elementse1ande2,cumulativeScores(pr)returns the vectors described in definition 2 for each element,cumulativelyDominates(pr, e1, e2)returnsTRUEorFALSEbased on definition 3.Similar to the dominance property from our previous section, two elements not dominating one or the other does not indicate that they are indifferent.
3.2.3 CP-Majority comparison
The Ceteris Paribus Majority (CP-Majority) relation is a somewhat relaxed version of the dominance property. Instead of checking if \(S \cup \{i\} \succeq S \cup \{j\}\) for all \(S \in 2^{N \setminus \{i,j\}}\), the CP-Majority relation \(iR^\succeq_\textrm{CP}j\) holds if the number of times \(S \cup \{i\} \succeq S \cup \{j\}\) is greater than or equal to the number of times \(S \cup \{j\} \succeq S \cup \{i\}\).
(CP-Majority [6]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The Ceteris Paribus majority relation is the binary relation \(R^\succeq_\textrm{CP} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{CP}j \Leftrightarrow d_{ij}(\succeq) \geq d_{ji}(\succeq) \end{equation}\]
where \(d_{ij}(\succeq)\) represents the cardinality of the set \(D_{ij}(\succeq)\), the set of all coalitions \(S \in 2^{N \setminus \{i,j\}}\) for which \(S \cup \{i\} \succeq S \cup \{j\}\).
cpMajorityComparisonScore(pr, e1, e2)calculates the two scores \(d_{ij}(\succeq)\) and \(-d_{ji}(\succeq)\). Notice the minus sign - that way we can use the sum of both values to determine the relation betweene1ande2.As a slight variation the logical parameter
strictlycalculates \(d_{ij}(\succ)\) and \(-d_{ji}(\succ)\), the number of coalitions \(S \in 2^{N\setminus \{i,j\}}\) where \(S\cup\{i\}\succ S\cup\{j\}\).Coincidentally,
cpMajorityComparisonScorewithstrictly = TRUEcan be used to determine ife1(strictly) dominatese2.cpMajorityComparisonScoreshould be used for simple and quick calculations. The more comprehensive functioncpMajorityComparison(pr, e1, e2)does the same calculations, but in the process retains more information about all the comparisons that might be interesting to a user, i.e., the set \(D_{ij}(\succeq)\) and \(D_{ji}(\succeq)\) as well as the relation \(iR^\succeq_\textrm{CP}j\). See the documentation for a full list of available data.The CP-Majority relation can generate cycles, which is the reason that it is not offered as a social ranking solution. Instead we will introduce the Copeland-like method and Kramer-Simpson-like method that make use of the CP-Majority functions to determine a power relation between elements. For further readings on CP-Majority, see [7] and [10].
3.3 Social Ranking Solutions
3.3.1 Ordinal Banzhaf
The Ordinal Banzhaf Score is a vector defined by the principle of marginal contributions. Intuitively speaking, if a player joining a coalition causes it to move up in the ranking, it can be interpreted as a positive contribution. On the contrary a negative contribution means that participating causes the coalition to go down in the ranking.
(Ordinal marginal contribution [5]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). For a given element \(i \in N\), its ordinal marginal contribution \(m_i^S(\succeq)\) with right to a coalition \(S \in \mathcal{P}\) is defined as:
\[\begin{equation} m_i^S(\succeq) = \begin{cases} \hphantom{-}1 & \textrm{if } S \cup \{i\} \succ S\\ -1 & \textrm{if } S \succ S \cup \{i\}\\ \hphantom{-}0 & \textrm{otherwise} \end{cases} \end{equation}\]
(Ordinal Banzhaf relation) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The Ordinal Banzhaf relation is the binary relation \(R^\succeq_\textrm{Banz} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{Banz}j \Leftrightarrow \text{Score}_\text{Banz}(i) \geq \text{Score}_\text{Banz}(j), \end{equation}\]
where \(\text{Score}_\text{Banz}(i) = \sum_{S} m^S_i(\succeq)\) for all \(S \in N\setminus\{i\}\).
Note that if \(S \cup \{i\} \notin \mathcal{P}\), \(m_i^S(\succeq) = 0\).
The function
ordinalBanzhafScoresreturns two numbers for each element: the number of coalitions \(S\) where a player’s contribution has a positive impact, and the number of coalitions \(S\) where a player’s contribution has a negative impact. These two numbers are added and elements are ranked highest to lowest.3.3.2 Copeland-like method
The Copeland-like method of ranking elements based on the CP-Majority rule is strongly inspired by the Copeland score from social choice theory[15]. The score of an element \(i \in N\) is determined by the amount of the pairwise CP-Majority winning comparisons \(i R^\succeq_\textrm{CP} j\), minus the number of all losing comparisons \(j R^\succeq_\textrm{CP} i\) against all other elements \(j \in N \setminus \{i\}\).
(Copeland-like relation [10]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The Copeland-like relation is the binary relation \(R^\succeq_\textrm{Cop} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{Cop}j \Leftrightarrow \text{Score}_\text{Cop}(i) \geq \text{Score}_\text{Cop}(j), \end{equation}\]
where \(\text{Score}_\text{Cop}(i) = |\{j \in N \setminus \{i\}: d_{ij}(\succeq) \geq d_{ji}(\succeq)\}| - |\{j \in N \setminus \{i\}: d_{ij}(\succeq) \leq d_{ji}(\succeq)\}|\)
copelandScores(pr)returns two numerical values for each element, a positive number for the winning comparisons (shown in \(\text{Score}_\text{Cop}(i)\) on the left) and a negative number for the losing comparisons (in \(\text{Score}_\text{Cop}(i)\) on the right).3.3.3 Kramer-Simpson-like method
Strongly inspired by the Kramer-Simpson method of social choice theory[16, 17], elements are ranked inversely to their greatest pairwise defeat over all possible CP-Majority comparisons.
(Kramer-Simpson-like relation [10]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The Kramer-Simpson-like relation is the binary relation \(R^\succeq_\textrm{KS} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{KS}j \Leftrightarrow \text{Score}_\text{KS}(i) \leq \text{Score}_\text{KS}(j), \end{equation}\]
where \(\text{Score}_\text{KS}(i) = \max_j d_{ji}(\succeq)\) for all \(j \in N \setminus \{i\}\).
kramerSimpsonScores(pr)returns a single numerical value for each element, which sorted lowest to highest gives us the ranking solution.There is a small caveat to Definition 8. By default this function does not compare \(d_{ii}(\succeq)\). In other terms, the score of every element is the maximum CP-Majority comparison score against all other elements.
This is slightly different from the definition found in [10], where the CP-Majority comparison \(d_{ii}(\succeq)\) is also considered. Since by definition \(d_{ii}(\succeq) = 0\), the Kramer-Simpson scores in those cases will never be negative, possibly discarding valuable information.
To still account for the original definition in [10], the functions
kramerSimpsonScoresandkramerSimpsonRankingoffer acompIvsIparameter that can be set toTRUEif one wishes for \(d_{ii}(\succeq)\) to be included in the comparisons.3.3.4 Lexicographical Excellence Solution
The idea behind the lexicographical excellence solution (Lexcel) is to reward elements appearing more frequently in higher ranked equivalence classes.
For a given power relation \(\succeq\) and its quotient order \(\sum_1 \succ \dots \succ \sum_m\), we denote by \(i_k\) the number of coalitions in \(\sum_k\) containing \(i\):
\[\begin{equation} i_k = |\{S \in \textstyle \sum_k: i \in S\}| \end{equation}\]
for \(k \in \{1, \dots, m\}\). Now, let \(\text{Score}_\text{Lex}(i)\) be the \(m\)-dimensional vector \(\text{Score}_\text{Lex}(i) = (i_1, \dots, i_m)\) associated to \(\succeq\). Consider the lexicographic order \(\geq_\textrm{Lex}\) among vectors \(\mathbf{i}\) and \(\mathbf{j}\): \(\mathbf{i} \geq_\textrm{Lex} \mathbf{j}\) if either \(\mathbf{i} = \mathbf{j}\) or there exists \(t : i_r = j_r, r \in \{1,\dots,t-1\}\), and \(i_t > j_t\).
(Lexicographic-Excellence relation [8]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\) with its corresponding quotient order \(\sum_1 \succ \dots \succ \sum_m\). The Lexicographic-Excellence relation is the binary relation \(R^\succeq_\textrm{Lex} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{Lex}j \Leftrightarrow \text{Score}_\text{Lex}(i) \geq_{\textrm{Lex}} \text{Score}_\text{Lex}(j) \end{equation}\]
For some generalizations of the Lexcel solution see also [9].
Lexcel score vectors are very similar to the cumulative score vectors (see section on Cumulative Dominance) in that the number of times an element appears in a given equivalence class is of interest. In fact, applying the base function
cumsumon an element’s lexcel score gives us its cumulative score.3.3.5 Dual Lexicographical Excellence Solution
Similar to the Lexcel ranking, the Dual Lexcel also uses the Lexcel score vectors from definition 9 to establish a ranking. However, instead of rewarding higher frequencies in high ranking coalitions, it punishes higher frequencies in lower ranking coalitions, or, it punishes mediocrity[14].
Take the values \(i_k\) for \(k \in \{1, \dots, m\}\) and the Lexcel score vector \(\text{Score}_\text{Lex}(i)\) from section Lexicographical Excellence Solution. Consider the dual lexicographical order \(\geq_\textrm{DualLex}\) among vectors \(\mathbf{i}\) and \(\mathbf{j}\): \(\mathbf{i} \geq_\textrm{DualLex} \mathbf{j}\) if either \(\mathbf{i} = \mathbf{j}\) or there exists \(t: i_t < j_t\) and \(i_r = j_r, r\in\{t+1, \dots, m\}\).
(Dual Lexicographical-Excellence relation [14]) Let \(\succeq \in \mathcal{T}(\mathcal{P})\). The Dual Lexicographic-Excellence relation is the binary relation \(R^\succeq_\textrm{DualLex} \subseteq N \times N\) such that for all \(i,j \in N\):
\[\begin{equation} iR^\succeq_\textrm{DualLex}j \Leftrightarrow \text{Score}_\text{Lex}(i) \geq_\textrm{DualLex} \text{Score}_\text{Lex}(j) \end{equation}\]
The S3 class
LexcelScoresdoes not account for Dual Lexcel comparisons. Instead-rev(x)is called on a Lexcel score vectorxsuch that the resulting comparisons produces a Dual Lexcel ranking solution.