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 \(x,y \in
X,(x,y) \in R \Rightarrow (y,x) \notin R\), and
- 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 \(N\), the set of all subsets or coalitions
of \(N\),
\[
S \in 2^N \quad \Leftrightarrow \quad S \subseteq N.
\]
\(\mathcal{T}(N)\) 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:
\[
\begin{array}{rrrr}
\succeq \ =\ \{(\{1,2\},\{1,2\}), & (\{1,2\},\{2\}), &
(\{1,2\},\emptyset), & (\{1,2\},\{1\}),\hphantom{\}}\\
& (\{2\}, \{2\}), & (\{2\}, \emptyset), &
\hphantom{1,}(\{2\}, \{1\}),\hphantom{\}}\\
& (\emptyset, \emptyset), & (\emptyset, \{2\}), &
(\emptyset, \{1\}),\hphantom{\}}\\
& & & (\{1\}, \{1\})\hphantom{,}\}
\end{array}
\]
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 \(\{\{1,2\}\} \succ \{\{2\}, \emptyset\} \succ
\{\{1\}\}\).
Note that the way the set \(\succeq\) is presented in the example is
somewhat deliberate to better visualize occurring symmetries and
asymmetries. This also lets us neatly represent a power relation in the
form of an incidence matrix.
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 <- PowerRelation(list(
list(c(1,2)),
list(2, c()),
list(1)
))
pr
## 12 > (2 ~ {}) > 1
## [1] "PowerRelation" "SingleCharElements"
Notice how coalitions such as \(\{1,2\}\) are written as 12
to
improve readability. Similarly, passing a string to the function
as.PowerRelation()
saves some typing on the user’s end by
interpreting each character of a coalition as a separate element. Note
that spaces in that function are ignored.
as.PowerRelation("12 > 2~{} > 1")
## 12 > (2 ~ {}) > 1
The compact notation is only done in PowerRelation
objects where every element is one digit or one character long. If this
is not the case, curly braces and commas are added where needed.
prLong <- PowerRelation(list(
list(c("Alice", "Bob")),
list("Bob", c()),
list("Alice")
))
prLong
## {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 how
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
## {1, 2} > ({2} ~ {}) > {1}
Internally a PowerRelation
is a list with four
attributes.
Attribute |
Description |
Value in pr |
elements |
Sorted vector of elements |
c(1,2) |
eqs |
List containing lists, each containing coalitions in the same equivalence class |
list(list(c(1,2)),
list(c(2), c()),
list(c(1))) |
coalitionLookup |
Function to determine a coalition's equivalence class index |
function(coalition) |
elementLookup |
Function to determine, which coalitions an element takes part in |
function(element) |
While coalitions are formally defined as sets, meaning the order
doesn’t matter and each element is unique, the package tries to stay
flexible. As such, coalitions will only be sorted during initialization,
but duplicate elements will not be removed.
prAtts <- PowerRelation(list(
list(c(2,2,1,1,2)),
list(c(2,1), c())
))
## Warning in createLookupTables(equivalenceClasses): Found 1 coalition that contain elements more than once.
## - 1, 2 in the coalition {1, 1, 2, 2, 2}
## 11222 > (12 ~ {})
## [1] 1 2
prAtts$coalitionLookup(c(1,2))
## [1] 2
prAtts$coalitionLookup(c(2,1))
## [1] 2
prAtts$coalitionLookup(c(2,1,2,1,2))
## [1] 1
## [[1]]
## [1] 1 1
##
## [[2]]
## [1] 1 1
##
## [[3]]
## [1] 1 1
##
## [[4]]
## [1] 2 1
Manipulating
PowerRelation
objects
It is strongly discouraged to directly manipulate
PowerRelation
objects, as its attributes are so tightly
coupled. This would require updates in multiple places. Instead, it is
advisable to simply create new PowerRelation
objects.
To permutate the order of equivalence classes, it is possible to take
the equivalence classes in $eqs
and use a vector of indexes
to move them around.
pr <- as.PowerRelation("12 > (1 ~ {}) > 2")
PowerRelation(pr$eqs[c(2, 3, 1)])
## (1 ~ {}) > 2 > 12
PowerRelation(rev(pr$eqs))
## 2 > (1 ~ {}) > 12
For permutating individual coalitions, using
as.PowerRelation.list()
may be more convenient since it
doesn’t require nested list indexing.
coalitions <- unlist(pr$eqs, recursive = FALSE)
compares <- c(">", "~", ">")
as.PowerRelation(coalitions[c(2,1,3,4)], comparators = compares)
## 1 > (12 ~ {}) > 2
# notice that the length of comparators does not need to match
# length(coalitions)-1
as.PowerRelation(rev(coalitions), comparators = c("~", ">"))
## (2 ~ {}) > (1 ~ 12)
# not setting the comparators parameter turns it into a linear order
as.PowerRelation(coalitions)
## 12 > 1 > {} > 2
appendMissingCoalitions()
Let \(\succeq \in
\mathcal{T}(\mathcal{P})\). We may have not included all possible
coalitions, such that \(\mathcal{P} \subset
2^N, \mathcal{P} \neq 2^N\).
appendMissingCoalitions()
appends all the missing
coalitions \(2^N - \mathcal{P}\) as a
single equivalence class to the end of the power relation.
pr <- PowerRelation(list(
list(c("AT", "DE"), "FR"),
list("DE"),
list(c("AT", "FR"), "AT")
))
pr
## ({AT, DE} ~ {FR}) > {DE} > ({AT, FR} ~ {AT})
# since we have 3 elements, the super set 2^N should include 8 coalitions
appendMissingCoalitions(pr)
## ({AT, DE} ~ {FR}) > {DE} > ({AT, FR} ~ {AT}) > ({AT, DE, FR} ~ {DE, FR} ~ {})
makePowerRelationMonotonic()
A power relation \(\succeq \in
\mathcal{T}(\mathcal{P})\) is monotonic if
\[
S \succeq T \quad \Rightarrow \quad T \subset S
\]
for all \(S, T \subseteq N\). In
other terms, given a monotonic power relation, for any coalition, all
its subsets cannot be ranked higher.
makePowerRelationMonotonic()
turns a potentially
non-monotonic power relation into a monotonic one by moving and
(optionally) adding all missing coalitions in \(2^N - \mathcal{P}\) to the corresponding
equivalence classes.
pr <- as.PowerRelation("a > b > c ~ ac > abc")
makePowerRelationMonotonic(pr)
## (abc ~ ab ~ ac ~ a) > (bc ~ b) > c
makePowerRelationMonotonic(pr, addMissingCoalitions = FALSE)
## (abc ~ ac ~ a) > b > c
# notice how an empty coalition in some equivalence class
# causes all remaining coalitions to be moved there
makePowerRelationMonotonic(as.PowerRelation("ab > c > {} > abc > a > b"))
## (abc ~ ab) > (ac ~ bc ~ c) > (a ~ b ~ {})
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
PowerRelation
or as.PowerRelation
, but also
formats the function call in such a way that makes it easy to rearrange
the ordering of the coalitions. RStudio offers shortcuts such as 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"),
result = "print"
)
## as.PowerRelation("
## abc
## > ab
## > ac
## > bc
## > a
## > b
## > c
## > {}
## ")
Using Alt+Up or Alt+Down to move one or more
lines of code
By default, createPowerset()
returns the power set in
the form of a list. This list can be passed directly to
as.PowerRelation()
to create a linear order.
ps <- createPowerset(1:2, includeEmptySet = FALSE)
ps
## [[1]]
## [1] 1 2
##
## [[2]]
## [1] 1
##
## [[3]]
## [1] 2
## 12 > 1 > 2
# equivalent
PowerRelation(list(ps))
## (12 ~ 1 ~ 2)
as.PowerRelation(createPowerset(letters[1:4]))
## abcd > abc > abd > acd > bcd > ab > ac > ad > bc > bd > cd > a > b > c > d > {}
Generating
PowerRelation
objects
Given a list of coalitions, it is possible to loop through all
possible permutations of power relations using
powerRelationGenerator()
. Calling gen()
in the
example below always produces a unique PowerRelation
object. If all permutations have been exhausted, NULL
is
returned.
coalitions <- list(c(1,2), 1, 2)
gen <- powerRelationGenerator(coalitions)
while(!is.null(pr <- gen())) {
print(pr)
}
## (12 ~ 1 ~ 2)
## (12 ~ 1) > 2
## (12 ~ 2) > 1
## (1 ~ 2) > 12
## 12 > (1 ~ 2)
## 1 > (12 ~ 2)
## 2 > (12 ~ 1)
## 12 > 1 > 2
## 12 > 2 > 1
## 1 > 12 > 2
## 2 > 12 > 1
## 1 > 2 > 12
## 2 > 1 > 12
Permutations over power relations can be split into two parts:
- generating partitions, or, generating differently sized equivalence
classes, and
- moving coalitions between these partitions.
In the code example above, we started with a single partition of size
three, wherein all coalitions are considered equally preferable. By the
end, we have reached the maximum number of partitions, where each
coalition is put inside an equivalence class of size 1.
The partition generation can be reversed, such that we first receive
linear power relations.
gen <- powerRelationGenerator(coalitions, startWithLinearOrder = TRUE)
while(!is.null(pr <- gen())) {
print(pr)
}
## 12 > 1 > 2
## 12 > 2 > 1
## 1 > 12 > 2
## 2 > 12 > 1
## 1 > 2 > 12
## 2 > 1 > 12
## 12 > (1 ~ 2)
## 1 > (12 ~ 2)
## 2 > (12 ~ 1)
## (12 ~ 1) > 2
## (12 ~ 2) > 1
## (1 ~ 2) > 12
## (12 ~ 1 ~ 2)
Notice that the “moving coalitions” part was not reversed, only the
order the partitions come in.
Similarly, we are also able to skip the current partition.
gen <- powerRelationGenerator(coalitions)
# partition 3
gen <- generateNextPartition(gen)
# partition 2+1
gen <- generateNextPartition(gen)
# partition 1+2
gen()
## 12 > (1 ~ 2)
Note: the number of possible power relations grows tremendously fast
as the number of coalitions rises. To get to that number, first consider
how many ways \(n\) coalitions can be
split into \(k\) partitions, also known
as the Stirling number of second kind,
\[
S(n,k) = \frac{1}{k!}\ \sum_{j=0}^k (-1)^j \binom{k}{j}(k-j)^n.
\]
The number of all possible partitions given \(n\) coalitions is known as the Bell number
(see also numbers::bell()
),
\[
B_n = \sum_{j=0}^k S(n,k).
\]
Given a set of coalitions \(\mathcal{P} \in
2^N\), the number of total preorders in \(\mathcal{T}(\mathcal{P})\) is
\[
|\mathcal{T}(\mathcal{P})| = \sum_{k=0}^{|\mathcal{P}|} k!\ *\
S(|\mathcal{P}|, k)
\]
1 |
1 |
1 |
2 |
2 |
3 |
3 |
5 |
13 |
4 |
15 |
75 |
5 |
52 |
541 |
6 |
203 |
4.683 |
7 |
877 |
47.293 |
8 |
4.140 |
545.835 |
9 |
21.147 |
7.087.261 |
10 |
115.975 |
102.247.563 |
11 |
678.570 |
1.622.632.573 |
12 |
4.213.597 |
28.091.567.595 |
13 |
27.644.437 |
526.858.348.381 |
14 |
190.899.322 |
10.641.342.970.441 |
(\(2^4-1\)) 15 |
1.382.958.545 |
230.283.190.977.959 |
16 |
10.480.142.147 |
5.315.654.681.940.580 |
SocialRanking
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 literature, \(i I^\succeq j\) and
\(i P^\succeq j\) are often used to
denote the symmetric and asymmetric part, respectively. \(i I^\succeq j\) therefore means that \(i R^\succeq j\) and \(j R^\succeq i\), whereas \(i P^\succeq j\) implies that \(i R^\succeq j\) but not \(j R^\succeq j\).
In section 3.1 we show how a general
SocialRanking
object can be constructed using the
doRanking
function. In the following sections, we will
introduce the notion of dominance[4],
cumulative dominance[12] 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[13] (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 lexcelRanking
a > c > b
for dualLexcelRanking
a > b ~ c
for copelandRanking
and
kramerSimpsonRanking
a ~ c > b
for
ordinalBanzhafRanking
Creating
SocialRanking
objects
A SocialRanking
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
SocialRanking
objects. Given a sortable vector or list of
scores it determines the power relation between all players, where the
names of the elements are determined from the names()
attribute of scores
. Hence, a PowerRelation
object is not necessary to create a SocialRanking
object.
# we define some arbitrary score vector where "a" scores highest.
# "b" and "c" both score 1, thus they are indifferent.
scores <- c(a = 100, b = 1, c = 1)
doRanking(scores)
## a > b ~ c
# we can also tell doRanking to punish higher scores
doRanking(scores, decreasing = FALSE)
## b ~ c > a
When working with types that cannot be sorted (i.e.,
lists
), a function can be passed to the
compare
parameter that allows comparisons between arbitrary
elements. This function must take two parameters (i.e., a
and b
) and return a numeric value based on the
comparison:
compare(a,b) > 0
: a
scores higher than
b
,
compare(a,b) < 0
: a
scores lower than
b
,
compare(a,b) == 0
: a
and b
are equivalent.
scores <- list(a = c(3, 3, 3), b = c(2, 3, 2), c = c(7, 0, 2))
doRanking(scores, compare = function(a, b) sum(a) - sum(b))
## a ~ c > b
# a and c are considered to be indifferent, because their sums are the same
doRanking(scores, compare = function(a,b) sum(a) - sum(b), decreasing = FALSE)
## b > a ~ c
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 <- as.PowerRelation("3 > 1 > 2 > 12 > 13 > 23")
# 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 <- as.PowerRelation("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 [12] 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 <- as.PowerRelation("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 <- as.PowerRelation("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 \(S \notin
\mathcal{P}\) or \(S \cup \{i\} \notin
\mathcal{P}\), \(m_i^S(\succeq) =
0\).
The function ordinalBanzhafScores()
returns three
numbers for each element,
- the number of coalitions \(S\)
where a player’s contribution has a positive impact,
- the number of coalitions \(S\)
where a player’s contribution has a negative impact, and
- the number of coalitions \(S\) for
which no information can be gathered, because \(S \notin \mathcal{P}\) or \(S \cup \{i\} \notin \mathcal{P}\).
The sum of the first two numbers determines the score of a player.
Players with higher scores rank higher.
pr <- as.PowerRelation(list(c(1,2), c(1), c(2)))
pr
## 12 > 1 > 2
# both players 1 and 2 have an Ordinal Banzhaf Score of 1
# therefore they are indifferent to one another
# note that the empty set is missing, as such we cannot compare {}u{i} with {}
ordinalBanzhafScores(pr)
## $`1`
## [1] 1 0 1
##
## $`2`
## [1] 1 0 1
##
## attr(,"class")
## [1] "OrdinalBanzhafScores"
ordinalBanzhafRanking(pr)
## 1 ~ 2
pr <- as.PowerRelation("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[c(1,2)]))
## 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[14]. 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 <- as.PowerRelation("(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[15, 16], 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 <- as.PowerRelation("(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 <- as.PowerRelation("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 <- as.PowerRelation("12 > (123 ~ 23 ~ 3) > (1 ~ 2) > 13")
# 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)
## list()
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[13].
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 [13]) 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 <- as.PowerRelation("12 > (123 ~ 23 ~ 3) > (1 ~ 2) > 13")
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 <- as.PowerRelation("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 <- as.PowerRelation("12 > (1 ~ 2)")
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.
as.PowerRelation("12 > 2 > (1 ~ 2) > 12")
## Warning in createLookupTables(equivalenceClasses): Found 2 duplicate coalitions, listed below. This violates transitivity and can cause issues with certain ranking solutions. You may want to take a look at socialranking::transitiveClosure().
## - {2}
## - {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(as.PowerRelation(list(1, 2, 1)))
pr
## 1 > 2 > 1
## (1 ~ 2)
# two cycles, (1>3>1) and (2>23>2)
pr <- suppressWarnings(
as.PowerRelation("1 > 3 > 1 > 2 > 23 > 2")
)
transitiveClosure(pr)
## (1 ~ 3) > (2 ~ 23)
# overlapping cycles
pr <- suppressWarnings(
as.PowerRelation("c > ac > b > ac > (a ~ b) > abc")
)
transitiveClosure(pr)
## 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
solutions. In: Bessiere C, editor. Proceedings of the twenty-ninth
international joint conference on artificial intelligence,
IJCAI-20. International Joint Conferences on Artificial
Intelligence Organization; 2020. p. 17–23. doi:
10.24963/ijcai.2020/3.
11. Meyer D, Hornik K. Relations: Data structures and algorithms for
relations. 2022.
https://CRAN.R-project.org/package=relations.
12. Moretti S. An axiomatic approach to social ranking under coalitional
power relations. Homo Oeconomicus. 2015;32:183–208.
13. Serramia M, López-Sánchez M, Moretti S, Rodríguez-Aguilar JA. On the
dominant set selection problem and its application to value alignment.
Autonomous Agents and Multi-Agent Systems. 2021;35:1–38.
14. Copeland AH. A reasonable social welfare function. mimeo, 1951.
University of Michigan; 1951.
15. Simpson PB. On defining areas of voter choice: Professor tullock on
stable voting. The Quarterly Journal of Economics. 1969;83:478–90.
16. Kramer GH. A dynamical model of political equilibrium. Journal of
Economic Theory. 1975;16:310–34.
3
SocialRanking
ObjectsThe 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 literature, \(i I^\succeq j\) and \(i P^\succeq j\) are often used to denote the symmetric and asymmetric part, respectively. \(i I^\succeq j\) therefore means that \(i R^\succeq j\) and \(j R^\succeq i\), whereas \(i P^\succeq j\) implies that \(i R^\succeq j\) but not \(j R^\succeq j\).
In section 3.1 we show how a general
SocialRanking
object can be constructed using thedoRanking
function. In the following sections, we will introduce the notion of dominance[4], cumulative dominance[12] 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[13] (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
forlexcelRanking
a > c > b
fordualLexcelRanking
a > b ~ c
forcopelandRanking
andkramerSimpsonRanking
a ~ c > b
forordinalBanzhafRanking
3.1 Creating
SocialRanking
objectsA
SocialRanking
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 theequivalenceClasses
attribute inPowerRelation
objects.The function
doRanking
offers a generic way of creatingSocialRanking
objects. Given a sortable vector or list of scores it determines the power relation between all players, where the names of the elements are determined from thenames()
attribute ofscores
. Hence, aPowerRelation
object is not necessary to create aSocialRanking
object.When working with types that cannot be sorted (i.e.,
lists
), a function can be passed to thecompare
parameter that allows comparisons between arbitrary elements. This function must take two parameters (i.e.,a
andb
) and return a numeric value based on the comparison:compare(a,b) > 0
:a
scores higher thanb
,compare(a,b) < 0
:a
scores lower thanb
,compare(a,b) == 0
:a
andb
are equivalent.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 valueTRUE
ife1
dominatese2
, elseFALSE
. Note thate1
not dominatinge2
does not indicate thate2
dominatese1
, nor does it imply thate1
is 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
includeEmptySet
can be set toFALSE
such 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 [12] 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
objectpr
and two elementse1
ande2
,cumulativeScores(pr)
returns the vectors described in definition 2 for each element,cumulativelyDominates(pr, e1, e2)
returnsTRUE
orFALSE
based 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 betweene1
ande2
.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 \(S\cup\{i\}\succ S\cup\{j\}\).Coincidentally,
cpMajorityComparisonScore
withstrictly = TRUE
can be used to determine ife1
(strictly) dominatese2
.cpMajorityComparisonScore
should 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 \(S \notin \mathcal{P}\) or \(S \cup \{i\} \notin \mathcal{P}\), \(m_i^S(\succeq) = 0\).
The function
ordinalBanzhafScores()
returns three numbers for each element,The sum of the first two numbers determines the score of a player. Players with higher scores rank higher.
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[14]. 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[15, 16], 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
kramerSimpsonScores
andkramerSimpsonRanking
offer acompIvsI
parameter that can be set toTRUE
if 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
cumsum
on 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[13].
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 [13]) 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 vectorx
such that the resulting comparisons produces a Dual Lexcel ranking solution.