ARTICLE IN PRESS
Journal of Mathematical Psychology 48 (2004) 239–246
On the (numerical) ranking associated with any finite binary relation
Michel Regenwetter* and Elena Rykhlevskaia
University of Illinois at Urbana-Champaign, Department of Psychology, 603 E. Daniel Street, Champaign, IL 61820, USA
Received 18 February 2003; revised 24 March 2004
Abstract
We generalize the concept of a ‘ranking associated with a linear order’ from linear orders to arbitrary finite binary relations. Using
the concept of differential of an object in a binary relation as theoretical primitive, we axiomatically introduce several measurement
scales, some of which include the generalized ranking as a special case. We provide a computational formula for this generalized
ranking, discuss its many elegant properties and offer some illustrating examples.
r 2004 Elsevier Inc. All rights reserved.
Keywords: Binary relation; Digraph; Order; Ranking; Rank position; Scale
1. Introduction
A binary relation defined on a set of objects specifies
how objects are compared (i.e., related) to each other in
a pairwise fashion. The ‘rank’ of an object with respect
to a linear order is a numerical score that describes the
position of this object relative to all other objects in the
linear order. The rank of an object c associated with a
linear order over n many objects is routinely defined to
be i if n i many objects appear after c in the linear
order, and, equivalently, if i 1 many objects appear
ahead of c in the linear order. (We formally state the
definition in the body of the paper.) A mapping that
assigns ‘ranks’ to all objects in a linear order is called a
‘ranking’. A ranking thus provides a quantitative,
numerical assignment to all objects in a way that
reflects the relative ‘positions’ of the objects in the given
linear order. Fig. 1 displays the numerical rank of each
object associated with the linear order fða; bÞ; ða; cÞ;
ða; dÞ; ðb; cÞ; ðb; dÞ; ðc; dÞg: Object a has rank 1; b has
rank 2; c has rank 3; and d has rank 4:
The present paper is dedicated to generating ‘ranks’
and an associated ‘ranking’ for any binary relation on a
specified finite set of n objects, in a fashion that includes
the standard ranking associated with a linear order as a
special case.
*Corresponding author.
E-mail addresses: [email protected] (M. Regenwetter),
[email protected] (E. Rykhlevskaia).
0022-2496/$ - see front matter r 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmp.2004.03.003
In many sports or board game tournaments, not every
pair of players or teams has an opportunity to compete
with each other directly. However, in a resulting
tournament rating list, the players or teams are
supposed to be ‘ranked’ according to their achievement
in a given competition. Fig. 2 shows a possible outcome
of such a hypothetical competition. The figure can be
read as follows: a has beaten b; who has beaten c; who,
in turn, has beaten a and d; etc. Only those pairs of
players (teams) have competed who are linked to each
other by an arrow in the directed graph. It is obvious
that the depicted binary relation is neither complete
(e.g., c and f have not competed against each other), nor
transitive (e.g., the fact that c has beaten d; and that d
has beaten e; does not necessarily imply that c has
beaten e: In fact, they did not compete). Furthermore,
the directed graph has a cyclic triple because a has
beaten b; b has beaten c; who, in turn, has beaten a:
Figs. 3 displays a weak order (left), a more general
semiorder (center) and a more general partial order
(right) over six objects (disregard the numbers in this
figure for the time being). Together with linear orders,
these are among the most commonly used binary
relations to describe judges’ preferences over a set of
objects (see, e.g., Roberts (1979) for standard definitions). In fact, any of the binary relations depicted in
Figs. 1–3 could represent individual voter preferences in
a political poll or ballots in an election. This assumes
that voters are permitted to provide preference information in a sufficiently flexible format, possibly even
ARTICLE IN PRESS
240
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
Fig. 1. Example of a linear order and its associated ranking.
Fig. 2. Example of a directed graph.
allowing for cyclic individual preferences. Many voting
methods that rely on rank order information (like the
famous Borda method), currently require linear order
input for lack of a general definition of rankings. The
present paper provides the foundations for generalizing
ranking based voting methods to ballots that provide
binary preference relations of any kind.
The problem of ranking objects relative to a binary
relation other than a linear order has been previously
tackled for some specific families of binary relations.
For example, for weak orders, a straightforward
solution is to allow rational valued ranks for ties, as
depicted in the left hand side of Fig. 3. Ranking methods
leading to numerical scores have previously been
axiomatized for tournaments (Henriet, 1985; Rubinstein, 1980). Furthermore, a host of work has been done
on finding either a linear order, or a (strict) weak order
that is, in some appropriate sense, ‘closest’ to a binary
relation of some specified type. For instance, it is well
known that there is a compatible weak order associated
with every semiorder (Luce, 1956; Scott & Suppes,
1958). There are multiple papers by the European school
of multi-criteria decision-aiding on so-called ‘outranking’ methods for valued binary relations (Bouyssou,
1992; Bouyssou & Vincke, 1997; Brans & Vincke, 1985;
Brans, Vincke, & Mareschal, 1986; Vincke, 1999) and
for irreflexive binary relations (Vincke, 1992a). There
are also methods for ranking the elements of a partial
order so as to minimize the difference in ranks of
incomparable elements (Tannenbaum, Trenk, & Fishburn, 2001). A related line of work is the study of the
smallest weak order(s) containing a given partial order
(Doble, Doignon, Falmagne, & Fishburn, 2001). There
is a related literature in statistics on finding a ‘consensus
ranking’ based on notions of distance between rankings
or partial rankings (see, e.g., Critchlow, 1980; Critchlow, Fligner, & Verducci, 1991; Kendall, 1962).
In contrast to much of the work just cited, which
either generates only a weak order, puts constraints on
the allowable binary relations, or aggregates multiple
relations, we are interested in assigning a numerical score
to each object on the basis of any single binary (‘crisp’)
relation. This score should, in particular, boil down to
the standard ranking in the case of linear orders.
The rest of the paper is organized as follows: we first
introduce the basic concepts, notation and definitions.
In Section 3 we state a collection of axioms motivated by
various nice properties of the standard ranking for
linear orders. We also study various classes of numerical
scoring functions that satisfy various combinations of
these axioms, and we lay out the measurement scales
associated with these collections of functions. In Section
4 we axiomatize and derive a formula for ranking
objects with respect to any finite binary relation (defined
over the same objects). Section 5 is devoted to the
discussion of some properties of the generalized concept
of ranking.
2. Notation and definitions
In all theoretical developments throughout the paper,
we consider a fixed finite base set C; which we always
assume to contain n objects. Some illustrations specify
particular choices for C and, thus, for n:1 A binary
relation B on C is a collection of ordered pairs of
elements of C; i.e., BDC C: For any pair ða; bÞAB we
also routinely write aBb: As mentioned before, we do
not put any constraints on the nature and properties of
the arbitrary binary relations B: When we use a binary
relation B to describe preferences, it is standard to read
ða; bÞAB as ‘‘a is preferred to (better than) b’’.
In our graphic representation of general binary
relations, as in Fig. 2, we draw an arrow from x to y
to denote a directed edge ðx; yÞ in the binary relation.
For partial orders, as in Fig. 3, we draw Hasse diagrams,
where we replace arrows by lines and omit any lines
implied by transitivity. One can get from a to b by
following a downward path in the graph whenever ða; bÞ
belongs to the binary relation.2
We need further concepts and notation. Because
linear orders are of special interest in this paper, we use
p to denote a linear order on C and P for the collection
of all linear orders on C: For any binary relation B on C
1
Although we do not always technically require it, the most
interesting case is when n41:
2
When neither ða; bÞ nor ðb; aÞ belong to the relation, or both belong
to the relation, we draw one object higher than the other when the
latter is given a lower generalized rank—see later.
ARTICLE IN PRESS
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
241
Fig. 3. Examples of a weak order, a semiorder, a partial order and their associated generalized rankings.
and for cAC; let Bc ¼ B-ðC fcgÞ ¼ fða; cÞjða; cÞABg
and cB ¼ B-ðfcg CÞ ¼ fðc; bÞjðc; bÞABg: The in-degree IB ðcÞ of any element c with respect to a binary
relation B is equal to jBcj: The out-degree OB ðcÞ of c
with respect to B is jcBj (Harary, 1969).
We now introduce our central theoretical primitive.
Definition 1. The differential DB ðcÞ of any element
cAC with respect to a binary relation BDC C is
given by
DB ðcÞ ¼ IB ðcÞ OB ðcÞ ¼ jfaACjða; cÞABgj
jfbACjðc; bÞABgj;
ð1Þ
that is, the differential is the difference between the indegree and the out-degree. What we call the differential,
has been labelled the score elsewhere (Vincke, 1992a).
We do not use the word ‘score’ because we consider
multiple kinds of scores. The analogous concept for
valued binary relations has been axiomatized and
labelled the net flow (score) (Bouyssou, 1992; Bouyssou
& Vincke, 1997). By definition the differential DB takes
values in D ¼ f1 n; 2 n; y; n 1g: Since C is a fixed
set of n objects, D is also fixed throughout the paper.
Our general definition of rank crucially depends on
this concept of differential. The general idea is to define
the rank of an object as a function which is increasing in
the in-degree and decreasing in the out-degree, with the
additional proviso that each ordered pair in B should be
given equal weight (importance). The differential itself
clearly satisfies these requirements. Therefore, we study
transformations f 3DB of the differential. We now state
the standard formal definition of the ranks and ranking
associated with a linear order.
The rest of the paper moves from linear orders to general
binary relations and is dedicated to representations
fB ðcÞ ¼ ð f 3DB ÞðcÞ ¼ f ðDB ðcÞÞ;
ð3Þ
for various functions f and for arbitrary binary relations
BDC C: The two most important special cases are (1)
the case of f ðxÞ ¼ nþ1þx
2 ; which, in the case that B is a
linear order, is the traditional concept of ranks associated
with linear orders given in Eq. (2) above, and (2) the case
where f is the identity mapping.
First, we state and study a series of axioms which
define various families of such functions. Our choice of
axioms is motivated by properties of Rankp that we may
wish to preserve for any general concept of ranking, but
also by considerations of scale families of measurement.
More specifically, we rely on the simple facts that Rankp
is a one-to-one transformation of DB ; it is strictly
monotonically increasing in DB and it preserves ratios of
differences between values of DB : Beyond these three
properties, we also consider additional potential constraints on potential general scoring functions, say, on
the average, maximal or minimal score that should be
assigned to the elements of C for any fixed binary
relation on C:
3. An axiomatic derivation of numerical scoring functions
for arbitrary binary relations
A numerical scoring function associated with a binary
relation B is a function fB of the form
fB : C- R
Definition 2. Let cAC; and let pDC C be a linear
order over C: The rank of c with respect to p is denoted
by Rankp ðcÞ and routinely defined as follows:
Rankp ðcÞ ¼ i 3 Ip ðcÞ ¼ i 1 3 Op ðcÞ ¼ n i:
Using the concept of a differential, and for purposes
of our later generalizations, we can write instead that
Rankp ðcÞ ¼
n þ 1 þ Dp ðcÞ
:
2
ð2Þ
c/ fB ðcÞ:
ð4Þ
We only consider scoring functions that can be written
as a transformation f of DB ; i.e., fB ¼ f 3DB ; 8B: In
particular, we require the transformation f : D-R to be
the same for all B: Note that DB : C-D and, in
particular, that DB ðCÞ ¼ ,cAC fDB ðcÞgDD; 8B:
We consider a series of axioms and show how, by
consecutively adding one axiom at a time, we increasingly constrain the possible form of such a numerical
ARTICLE IN PRESS
242
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
scoring function fB : Our choice of axioms allows us to
move stepwise from the most general and least
constrained numerical functions to more restricted ones.
We derive well specified families of scoring functions
that correspond to traditional scale families (restricted
to a finite domain). (For a standard introduction and
overview of scale families, see Roberts, 1979).
Axiom 1 (Discriminability). The scoring function fB
discriminates between elements with respect to their
differentials. Formally, 8a; bAC; 8BDC C;
DB ðaÞaDB ðbÞ ) fB ðaÞafB ðbÞ:
ð5Þ
Axiom 2 (Monotonicity). The scoring function fB is
strictly monotonically increasing in the differential.
Formally, 8a; bAC; 8BDC C;
DB ðaÞoDB ðbÞ ) fB ðaÞofB ðbÞ:
ð6Þ
Axiom 3 (Preservation of ratios of differences). The
scoring function fB preserves all ratios of differences with
respect to the differential. Formally, 8a; b; c; dAC;
8BDC C;
fB ðaÞ fB ðbÞ DB ðaÞ DB ðbÞ
¼
;
fB ðcÞ fB ðdÞ DB ðcÞ DB ðdÞ
ð7Þ
1. Axiom 1 holds iff
fB ¼ f 3DB ;
Axiom 4 (Averaging). For fixed B; the (arithmetic)
average score under fB ; over all elements of C; is equal
to zero. Formally, 8BDC C;
P
cAC fB ðcÞ
¼ 0:
ð8Þ
n
Axiom 5 (Preservation of differences). The scoring
function fB preserves all differences in the values of the
differential. Formally, 8a; bAC; 8BDC C;
ð9Þ
Observation 1. Preservation of differences (Axiom 5)
implies preservation of ratios of differences (Axiom 3).
Proof. This follows trivially by substituting (9) into
(7). &
The reason we state two distinct axioms, of which one
implies the other, is that they translate into properties of
the possible numerical scoring functions that define
more and more specific (i.e., constrained) classes of
functions fB and associated measurement scales.
Theorem 1. Let us consider all possible numerical scoring
functions (assuming n41) that are functions of the
ð10Þ
where f is any one-to-one mapping on D: In other
words, the representation fB is a nominal scale, because
the collection of all admissible transformations of DB is
the class of all one-to-one mappings on D:
2. Axioms 1 and 2 simultaneously hold iff
fB ¼ h3DB ;
ð11Þ
where h is any strictly monotonically increasing
function on D: In other words, fB is an ordinal scale,
because the collection of all admissible transformations
of DB is the class of all strictly increasing functions
on D:
3. Axioms 1–3 simultaneously hold iff
fB ¼ aDB þ b;
ð12Þ
with any choice of a; bAR s.t. a40: Thus, fB is an
interval scale, because the collection of all admissible
transformations of DB is the class of all positive linear
functions on D:
4. Axioms 1–4 simultaneously hold iff
fB ¼ aDB ;
whenever both ratios are well defined.
fB ðaÞ fB ðbÞ ¼ DB ðaÞ DB ðbÞ:
differential, DB for any binary relation BDC C: The
following properties hold:
ð13Þ
where 0oaAR: In other words, fB is a ratio scale,
because the collection of all admissible transformations
of DB is the class of all multiplications by a positive
constant, on D:
5. Axioms 1–3 and 5 simultaneously hold iff
fB ¼ DB þ b;
ð14Þ
for any bAR: In other words, fB is a difference scale,
because the collection of all admissible transformations
of DB is the class of all transpositions by a real number,
on D:
6. Axioms 1–5 simultaneously hold iff
fB ¼ DB :
ð15Þ
In other words, fB is an absolute scale, because the only
admissible transformation of DB is the identity
mapping on D:
Proof. The proof of Part 1 follows directly from the
definition of a one-to-one mapping. Similarly, to prove
Part 2, we point out that, by definition, fB is a strictly
monotonically increasing function in D (i.e., a strictly
monotonically increasing transformation of DB ). To
prove Part 3, we first point out that the ratios in (7) are
well defined whenever cad; because of the ‘Discriminability’ Axiom (Eq. (5)). We now rewrite the statement
ARTICLE IN PRESS
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
243
of the ‘Preservation of ratios of differences’ Axiom
(Eq. (7)) in the following way:
4. An axiomatic generalization of ‘Ranks’ for arbitrary
binary relations
f ðxÞ f ðyÞ f ðzÞ f ðwÞ
¼
;
xy
zw
8x; y; z; wAD; xay; zaw;
To define the ‘rank’ for an object with respect to an
arbitrary binary relation, we introduce three new axioms
that are noteworthy properties of Rankp :
ð16Þ
where f is the (unknown) transformation in fB ¼ f 3DB :
It follows that
f ðxÞ f ðyÞ
¼ constant; 8x; yAD; xay:
xy
ð17Þ
We denote the ‘constant’ by a; and we note that a40 iff
f ðxÞ is strictly increasing (as required by Axioms 1 and
2). Assuming for a moment that y ¼ 0; and denoting
b ¼ f ðyÞ; we obtain, 8xAD;
f ðxÞ f ð0Þ
¼ a 3 f ðxÞ ¼ ax þ b:
x0
ð18Þ
Parts 4–6 of the theorem hold trivially by substituting
Axiom 4 (Eq. (8)) and/or Axiom 5 (Eq. (9)) into the
linear function in the right-hand side of Eq. (12). &
We have shown how various families of scoring
functions fB ðcÞ ¼ f 3DB ðcÞ follow from different assumptions restricting the nature of such functions. Clearly,
the numerical scoring function DB ; when viewed as an
absolute scale, is a very natural and intuitive way to
represent each object’s ‘relative position’ in an arbitrary
binary relation. Indeed, by definition, negative or
positive values of DB are assigned to an object c
depending on whether its out-degree OB ðcÞ or its indegree IB ðcÞ is more prevalent. DB ðcÞ ¼ 0 represents the
score of an object c for which the in-degree and the outdegree values are ‘at balance’, i.e., their values are equal.
In other words, such an object c can be thought of as
having a relative position in the ‘middle’ of the binary
relation. However, as we saw in the Introduction, the
concept of Rankp is the more standard concept used for
linear orders and it has it’s own intuitive appeal, such as
the fact, that for a linear order defined on n objects, the
minimal value of Rankp is 1, and the maximal value is n:
We now spell out the exact relationship between
Rankp and the functions fB axiomatized in this section.
Observation. Let n41: The standard ‘rank’ of an object
c; Rankp ðcÞ; associated with a linear order p on C; can be
written as a transformation fp ¼ f 3Dp iff fp ¼ aDp þ b
with a ¼ 12; and b ¼ nþ1
2 ; 8pAP: Furthermore, Rankp as a
transformation of Dp satisfies Axioms 1–3 and violates
Axioms 4 and 5.
We omit the simple proof for reasons of brevity. We
now proceed to define a ‘ranking’ associated with any
binary relation by considering functions of DB that are
intuitively similar to the standard Rankp ; and which
coincide with Rankp whenever B ¼ pAP:
Axiom 4 (Averaging). For fixed B; the scoring function
fB assigns an average value of nþ1
2 : Formally, 8BDC C;
P
nþ1
cAC fB ðcÞ
¼
:
ð19Þ
n
2
Axiom 5 (Minimum). The smallest possible score of an
object in an arbitrary binary relation is equal to 1.
Formally,
min
cAC
ð fB ðcÞÞ ¼ 1:
ð20Þ
BDC C
Axiom 6 (Maximum). The largest possible score of an
object in an arbitrary binary relation is equal to n.
Formally,
max ð fB ðcÞÞ ¼ n:
cAC
BDC C
ð21Þ
Note that we do not require fB to achieve either the
‘minimum’ or the ‘maximum’ for every B:
Theorem 2. Let cAC and let BDC C be any binary
relation defined on C; and fB be any numerical scoring
function that can be written as a transformation fB ¼
f 3DB of the differential. Axioms 1–3 and 4 and 5
simultaneously hold iff Axioms 1 3; 4 and 6
simultaneously hold iff fB is defined as follows: For 8cAC;
n þ 1 þ DB ðcÞ
;
ð22Þ
fB ðcÞ ¼
2
where DB ðcÞ is the differential of the element cAC with
regard to the binary relation BDC C; as defined in (1).
We omit the simple proof for reasons of brevity. It
follows from the axioms in conjunction with Theorem 1.
Definition 3. Let BDC C be any binary relation on C:
We define the general concept of rank of cAC with
respect to B; denoted by RankB ðcÞ; as
n þ 1 þ DB ðcÞ
:
ð23Þ
RankB ðcÞ ¼
2
The mapping RankB defined on C is called the (generalized) ranking associated with the binary relation B:
Fig. 3 illustrates how RankB assigns ranks to objects
in any given set endowed with different binary relations
B: The numerical values of RankB for the weak order,
ARTICLE IN PRESS
244
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
1. The average rank assigned to a fixed object over all
binary relations in B equals nþ1
2 : Formally, 8cAC;
P
nþ1
BAB ðRankB ðcÞÞ
¼
:
ð24Þ
jBj
2
Fig. 4. Example of a directed graph and its associated generalized
ranking.
the semiorder and the partial order in Fig. 3 are reported
in parentheses next to the name of each object. For the
tournament example illustrated earlier by Fig. 2, we
have generated a new directed graph (given in Fig. 4).
That figure shows the generalized ranking associated
with the binary relation. We have also rearranged
all objects in a fashion that reflects the strict weak
order induced by the associated ranking. In fact, the
graphic displays in Figs. 3 and 4 are both organized in
such a way that objects with smaller ranks are displayed
higher.
Overall, starting from any finite binary relation, we
obtain the associated ranking of the objects, which, in
turn, induces a strict weak order on the same objects.
For example, player (team) f ; who has beaten two other
players (teams), has the rank RankB ð f Þ ¼ 2:5: The fact
that f has the smallest rank is consistent with the
conclusion that f could be viewed as the ‘winner’ of the
tournament (relative to all others who participated, and
ignoring any information other than that given by the
binary relation).
While Axioms 1–3 and 4 –6 already give RankB
intuitive appeal by making it such a natural generalization of Rankp ; we now show that RankB has
additional simple and elegant properties. We also
discuss how these properties may change depending on
whether or not B is a linear order.
5. Some striking properties of RankB for arbitrary binary
relations
Given a binary relation BDC C; we write B1 for
the inverse of B; defined as B1 :¼ fðb; aÞjða; bÞABg:
Recall also that binary relations are often used to
mathematically capture preferences, where ða; bÞAB
usually denotes that a is preferred to b in preference
state B:
Observation 3. Let BD2CC be a collection of binary
relations which is closed under inversion, i.e., 8BDC C;
we have BAB3B1 AB:
Notice that this holds for any BD2CC that is closed
under inversion. In particular, when B ¼ 2CC ; we obtain
that the average value of RankB over all binary relations
on C is equal to nþ1
2 : Alternatively, B could be the
collection of all (strict) weak orders, the collection of all
semiorders, the collection of all (strict) partial orders, etc.
In other words, the average rank of an object over all weak
orders (or over all semiorders, or over all interval orders,
or over all partial orders, or over all partial orders of a
given dimension) is always equal to nþ1
2 : Notice also that
this average rank is identical to the average rank
computed over all objects for a fixed binary relation, as
specified by Axiom 4 :
2. RankB has the following ‘symmetry’ property:
8cAC; 8BDC C;
RankB1 ðcÞ ¼ n þ 1 RankB ðcÞ:
ð25Þ
Moreover, it follows from (25), that for any binary
relation that equals its inverse relation, i.e., for any
symmetric binary relation (Roberts, 1979), every object in
the base set will get the average rank value: 8BDC C;
nþ1
1
;
B ¼ B ) RankB ðcÞ ¼ RankB1 ðcÞ ¼
2
8cAC:
ð26Þ
In particular, all objects in the complete indifference
relation | are given average rank.
3. RankB has the ‘minimality’ property: an object c has
rank 1 (i.e., the smallest possible value) iff c is in relation
B with all other objects x in C and no other objects xAC
are in the relation B with c. Formally, RankB ðcÞ ¼
13½8xAC fcg; ðc; xÞAB and ðx; cÞeB: In other
words, 8cAC; 8BDC C;
RankB ðcÞ ¼ 13 ½ðOB ðcÞ ¼ n 1Þ4ðIB ðcÞ ¼ 0Þ
3 ½DB ðcÞ ¼ 1 n:
Thus, in the context of preference relations, an object is
ranked first iff it is strictly preferred to all other objects.
4. RankB has the ‘maximality’ property: object c has
rank n (i.e., the largest possible value) iff all other objects
x in C are in relation B with c; and there are no other
objects xAC that c is in relation with. Formally,
RankB ðcÞ ¼ n3½8xAC fcg; ðx; cÞAB and ðc; xÞeB:
In other words, 8cAC; 8BDC C;
RankB ðcÞ ¼ n3 ½ðIB ðcÞ ¼ n 1Þ4ðOB ðcÞ ¼ 0Þ
3 ½DB ðcÞ ¼ n 1:
Thus, in the context of preference relations, an object
receives rank n among n objects iff all (other) objects are
strictly preferred to it.
ARTICLE IN PRESS
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
245
5. The generalized rank, RankB ; of object c is equal to
iff the number of other objects x in C which are in
relation with c is equal to the number of other objects x
which c is in relation with. Formally, 8cAC; 8BDC C;
nþ1
2
RankB ðcÞ ¼
nþ1
3 IB ðcÞ ¼ OB ðcÞ:
2
In particular, if there is an object c which is not related to
any other object x in C; and to which no other object is
related, then c has average generalized rank. (For
example, object f in the right hand side of Fig. 3 is
incomparable to all other objects and obtains the ‘average’
score of 3:5). Similarly, objects in a perfect cycle (say,
aBbBcBdBa on C ¼ fa; b; c; dg) are given average rank.
We omit the simple proof for brevity. Notice that the
converse of (26) does not hold. For instance, in the case
of a perfect cycle B we have BaB1 :
Sometimes it is interesting to consider a binary
relation R which is the result of combining two other
binary relations. For instance, let B ¼ fðe; f Þ; ðe; gÞg
on C ¼ fe; f ; gg and B0 ¼ fða; bÞ; ða; cÞ; ða; dÞ; ðb; cÞ;
ðb; dÞ; ðc; dÞg on C0 ¼ fa; b; c; dg; be preference relations
as depicted in Fig. 5. Here n ¼ 3 and n0 ¼ 4: If a
judge has no reason to prefer any object in C to any
object in C0 ; or vice versa, a suitable preference
relation over all objects fa; b; c; d; e; f ; gg might be
R ¼ B,B0 : When all objects in C are preferable to all
objects in C0 ; the natural overall preference relation is
R ¼ ðB,B0 Þ,ðC C0 Þ:
We thus proceed to considering some properties of
RankR associated with such a new relation R in the two
cases where 1) R ¼ B,B0 ; or 2) R ¼ ðB,B0 Þ,ðC C0 Þ:
Observation 4. Suppose, C-C0 ¼ |; jC0 j ¼ n0 ; BDC C; and B0 DC0 C0 : Then the following properties hold:
1. If R ¼ B,B0 then, 8cAC,C0 ;
8
n0
>
< RankB ðcÞ þ
if cAC;
2
ð27Þ
RankR ðcÞ ¼
>
: RankB0 ðcÞ þ n if cAC0 :
2
In particular, if C0 ¼ fcg; then n0 ¼ 1 and
RankR ðcÞ ¼ 1 þ
n n þ n0 þ 1
¼
;
2
2
ð28Þ
Fig. 5. Two binary relations B (left hand side) and B0 (right hand side)
and their respective (separate) associated generalized ranks.
Fig. 6. Example of various generalized rankings associated with
preference relations obtained by combining two preference relations
B; B0 in different ways.
i.e., we have the situation discussed in Statement 5 of
Observation 3.
2. As a consequence, if R ¼ B,B0 then the possible
values of RankR for objects in C and C0 are bounded from
above and below as follows:
n0
n0
1 þ p minðRankR ðcÞÞp maxðRankR ðcÞÞpn þ ;
cAC
2 cAC
2
n
n
0
1 þ p min0 ðRankR ðcÞÞr max0 ðRankR ðcÞÞrn þ :
cAC
2 cAC
2
3. If R ¼ B,B0 ,½C C0 then, 8cAC0 ,C;
RankB ðcÞ
if cAC;
RankR ðcÞ ¼
RankB0 ðcÞ þ n if cAC0 :
ð29Þ
4. As a consequence, if R ¼ B,B0 ,½C C0 then the
possible values of RankR for objects in C and C0 are
bounded from above and below as follows:
1p minðRankR ðcÞÞp maxðRankR ðcÞÞpn;
cAC
cAC
n þ 1p min0 ðRankR ðcÞÞp max0 ðRankR ðcÞÞpn þ n0 :
cAC
cAC
Proof. These statements follow directly from the definition of RankB : &
Fig. 6 shows the generalized ranking of all objects in
fa; b; c; d; e; f g with respect to the combined relation R
in the two cases where 1) R ¼ B,B0 (left hand side), or
2) R ¼ B,B0 ,½C C0 (right hand side), where B; B0
are the same as in Fig. 5.
6. Conclusions
We have studied some ways of deriving a numerical
rank for each object in a finite set C from an arbitrary
binary relation defined on C:
First, we have introduced the differential DB as a basic
measure of an object’s position in a relation. Next, we
have studied classes of numerical scoring functions
which can be written as transformations of the
differential. We have shown how various families of
such scoring functions satisfy different sets of axioms
ARTICLE IN PRESS
246
M. Regenwetter, E. Rykhlevskaia / Journal of Mathematical Psychology 48 (2004) 239–246
constraining the nature of such a transformation.
Finally, we have axiomatized and derived RankB as a
concept of a ‘numerical ranking associated with an
arbitrary binary relation’. We have considered some
interesting properties of RankB ; and illustrated this
concept on some examples.
This approach raises a variety of open questions, such
as comparing RankB with alternative ‘ranking’ methods
that have been developed for various particular families
of relations and derived from alternative theoretical
primitives. In particular, it may be interesting to
compare the ‘optimality’ of RankB with respect to
benchmarks that have been proposed in the literature
(Bouyssou, 1992; Bouyssou & Vincke, 1997; Brans et al.,
1986; Henriet, 1985; Tannenbaum et al., 2001; Vincke,
1992a, b, 1999).
As mentioned in the Introduction, the generalized
ranking has useful and important applications in social
choice theory. In a follow-up paper, we investigate social
choice scoring functions, such as the famous Borda score
and plurality rule. Social choice scoring rules are
traditionally defined only for linear order preference
profiles, although there are some noteworthy exceptions
for the Borda score, such as Marchant (2000) and
Young (1974). Using RankB we can expand social choice
scoring rules to situations where individual voter ballots
provide finite binary relations of any kind. This is
extremely important from a practical, policy, and data
analysis point of view, because, in practice, virtually no
voting or polling methods actually collect complete
linear preference orders (over all candidates) from all, or
even any, voters.
Acknowledgments
We thank the National Science Foundation for
partially funding this research through NSF Grant
SBR 97-30076 to Regenwetter and the University of
Illinois Research Board for funding our collaboration.
We are indebted to John Boyd, Bill Batchelder, Peter
Fishburn, Tony Marley, Sasa Pekeč and Don Saari for
helpful pointers and comments on various aspects of
this work. We are also grateful to the action editor and
two referees for their helpful feedback.
References
Bouyssou, D. (1992). Ranking methods based on valued preference
relations: A characterization of the net flow method. European
Journal of Operational Research, 60, 61–67.
Bouyssou, D., & Vincke, P. (1997). Ranking alternatives on the basis
of preference relations: A progress report with special emphasis on
outranking relations. Journal of Multi-Criteria Decision Analysis,
6, 77–85.
Brans, J. P., & Vincke, P. (1985). A preference ranking organization
method. Management Science, 31, 647–656.
Brans, J. P., Vincke, P., & Mareschal, B. (1986). How to select and
how to rank projects: The PROMETHEE method. European
Journal of Operational Research, 24, 228–238.
Critchlow, D. (1980). Metric methods for analyzing partially ranked
data. New York: Springer.
Critchlow, D. E., Fligner, M. A., & Verducci, J. S. (1991). Probability
models on rankings. Journal of Mathematical Psychology, 35,
294–318.
Doble, C. W., Doignon, J.-P., Falmagne, J.-C., & Fishburn, P. C.
(2001). Almost connected orders. Order, 18, 295–311.
Harary, F. (1969). Graph theory. Reading, MA: Addison-Wesley.
Henriet, D. (1985). The Copeland choice function. Social Choice and
Welfare, 2, 49–63.
Kendall, M. (1962). Rank correlation methods (3rd ed.). New York:
Hafner.
Luce, R. D. (1956). Semiorders and a theory of utility discrimination.
Econometrica, 26, 178–191.
Marchant, T. (2000). Does the Borda rule provide more than a
ranking? Social Choice and Welfare, 17, 381–391.
Roberts, F. S. (1979). Measurement theory. London: Addison-Wesley.
Rubinstein, A. (1980). Ranking the participants in a tournament.
SIAM Journal of Applied Mathematics, 38, 108–111.
Scott, D., & Suppes, P. (1958). Foundational aspects of theories of
measurement. Journal of Symbolic Logic, 23, 113–128.
Tannenbaum, P. J., Trenk, A. N., & Fishburn, P. C. (2001). Linear
discrepancy and weak discrepancy of partially ordered sets. Order,
18, 201–225.
Vincke, P. (1992a). Exploitation of a crisp relation in a ranking
problem. Theory and Decision, 32, 221–240.
Vincke, P. (1992b). Multi-criteria decision-aid. Chichester: Wiley.
Vincke, P. (1999). Robust and neutral methods for aggregating
preferences into an outranking relation. European Journal of
Operational Research, 112, 405–412.
Young, H. P. (1974). An axiomatization of Borda’s rule. The Journal
of Economic Theory, 9, 43–52.