Ranking by choosing 1

Ranking by choosing

1

Denis Bouyssou

CNRS – LAMSADE

2 September 2002

1_{I am grateful Bernard Monjardet, Herv´e Raynaud and Jean-Claude Vansnick} for helpful discussions. Special thanks go to Patrice Perny who introduced me to the subject and to Olivier Hudry for his help with Slater orders. The usual caveat applies.

2_{LAMSADE, Universit´e Paris Dauphine, Place du Mar´echal de Lattre de} Tas-signy, F-75775 Paris Cedex 16, France, tel: +33 1 44 05 48 98, fax: +33 1 44 05 40 91, e-mail: [email protected].

Abstract

Procedures designed to select alternatives on the basis of the results of pair-wise contests between alternatives have received much attention in literature. The particular case of tournaments has been studied in depth. More recently weak tournaments and valued generalizations thereof have been investigated. The purpose of this paper is to investigate to what extent these choice procedures may be meaningfully used to define ranking procedures via their repeated use, i.e. when the equivalence classes of the ranking are determined by successive applications of the choice procedure. This is ranking by choos-ing.

As could be expected, such ranking procedures raise monotonicity prob-lems. We analyze these problems and show that it is nevertheless possible to isolate a large class of well-behaved choice procedures for which failures of monotonicity are not overly serious. The hope of finding really attractive ranking by choosing procedures is however shown to be limited. Our results are illustrated on the case or tournaments.

Keywords: Ranking procedures, Choice procedures, Monotonicity, Strong Superset Property, Ranking by choosing, Tournaments.

1

Introduction

In many different contexts, it is necessary to make a choice between alterna-tives on the sole basis the results of several kinds of pairwise contests between these alternatives. Among the many possible examples, let us mention:

• Sports leagues (games usually involve two teams),

• Social choice theory, via the use of C1 or C2 Social Choice functions [31] in view of the well-known results in [50, 25],

• Multiple criteria decision making using “ordinal information” [5, 69] in view of the results in [17],

• Psychology with, e.g., the study of binary choice probabilities [47, 79]. This problem has received close attention in recent years most particularly when the result of the pairwise contests may be summarized by atournament (an excellent summary of this literature may be found in [45]) and much is known on the properties and interrelations of such choice procedures. This line of research has been recently extended to weak tournaments (ties are allowed) [61, 74] and valued generalizations of (weak) tournaments [7, 12, 30, 22, 29, 26, 33, 42, 46, 59, 60, 67] (intensity of preference or number of victories may be taken into account).

The related problem ofranking alternatives on the sole basis the results of several kinds of pairwise contests between these alternatives has compar-atively received much less attention in recent years (see, however, [72, 35]), although it generated classical studies [40, 41, 77] and is clearly in the spirit of Social Welfare Functions `a la Arrow [4]. This is a pity since most classical applications of choice procedures are also potential applications for ranking procedures. This is, e.g., clearly the case for sports since most leagues want to rank order teams at the end of season and not only to select the winner(s). This also the case in the many situations in which, although a choice between alternatives is to be made, alternatives may disappear (e.g. candidates for a position may withdraw), so that there is a necessity of building a waiting list.

The problem of devising sound ranking procedures for such situations can be studied without explicit reference to choice procedures [15, 18, 19, 20, 35, 72, 80]. This is in line with H. Moulin’s [56] advice to clearly distinguish the question of ranking alternatives from the one of selecting winners.

We shall be concerned in this paper with a quite different approach to ranking on the basis of pairwise contests. Several authors have suggested

[5, 69] that a ranking could well be devised by successive applications of a choice procedure. The most natural way to do so goes as follows:

• Apply the choice procedure to the whole set of alternatives. Define the first equivalence class of the ranking as the chosen elements in the whole set.

• Remove the chosen elements from the set of alternatives.

• Apply the choice procedure to the reduced set. Define the second equiv-alence class of the ranking as the chosen elements in the reduced set.

• Iterate the above procedure to define the following equivalence classes of the ranking until there are no more alternatives to rank.

This is ranking by choosing. An example may help clarify the process.

Example 1 (Ranking by choosing with Copeland)

Let X ={a, b, c, d, e, f, g}. Consider the tournament T onX defined by:

aT b, aT f bT c, bT d, bT e, bT f cT a, cT e, cT f, cT g dT a, dT c, dT e, dT f, dT g

eT a, eT f, eT g f T g, gT a, gT b.

Suppose that you want to use the Copeland choice procedure Cop(A, T) selecting the elements in A having a having maximal Copeland score in T

restricted to A as a basis for ranking alternatives.

Applying the above ranking by choosing algorithm successively leads to:

Cop(X, T) = {d}, Cop(X\ {d}, T) = {c}, Cop(X\ {d, c}, T) ={e}, Cop(X\ {d, c, e}, T) ={a, g}, Cop(X\ {a, d, c, e, g}, T) = {b}, Cop(X\ {a, b, c, d, e, g}, T) = {f}.

Hence we obtain the ranking (using obvious notations): d  c Âe  [a ∼

have obtained ranking alternatives using their Copeland scores in X, i.e.

d  [b ∼ c]  e  [a ∼ g]  f, although both rankings clearly coincide on

the first equivalence class. 3

Using ranking by choosing, we may associate a well-defined ranking proce-dure to every choice proceproce-dure. A natural question arises. If the choice procedure has “nice properties”, will it also be the case for the induced rank-ing procedure? This is the central question in this paper.

Most ranking procedures that are used in practice are not of this ranking by choosing type. Most often (take the example of most sports leagues) they are rather based on some kind ofscoring function that aggregates into a real number the results of the various pairwise contests, e.g. ranking alternatives according to their Copeland scores.

Although ranking procedures induced by choice procedures may seem complex when compared to those based on scoring functions, several authors have forcefully argued in favor of their reasonableness [5, 69] and many of them were proposed [5, 24, 49, 68]. They are, in general, easy to compute and rather easy to explain. They are—structurally—insensitive to a possible withdrawal of (all) best ranked alternatives [80]. Furthermore, if the answer to the preceding question were to be positive, there would be a clear interest in using well-behaved choice procedures as a basis for ranking procedures. The situation is however more complex.

The potential drawbacks of these ranking by chosing procedures should be obvious: their very conception implies the existence of discontinuities together with a progressive impoverishment of information from one iteration to another. This is likely to create difficulties with most wanted normative properties as was forcefully shown by P. Perny [62]. The purpose of this paper is to explore the extent of these difficulties concentrating on monotonicity. An example will clarify how bad the situation can be.

Example 1 (continued)

Consider the tournament V identical to T except that aV d. It is clear that

Cop(X, V) = {b, c, d}.

We hadaÂbwithT. We now obtainb ÂawithV, while the position ofa

has clearly improved when going fromT toV. This is a serious monotonicity

problem. 3

The problem studied in this paper is reminiscent of the well-know mono-tonicity problems encountered in electoral procedures with “run-offs” (e.g. the French system of plurality with run-off, the Hare, Coombs and Nansson procedures see [57, 31]) that also involve discontinuities. It is well-known that they often have a disappointing behavior with respect to monotonicity

(see [78, 57, 31, 32, 73]). Although these difficulties are linked with our prob-lem, electoral procedures with run-offs are choice procedures and not ranking procedures. Hence the problem studied here has distinctive characteristics.

Although many ranking by choosing procedures have been suggested, their study has received limited attention so far. P. Perny [62] showed that most procedures of this type proposed in the literature violate monotonicity. He suggested to study the problem more in depth. Shortly after, we proposed [16] some results in that direction (since more powerful results appear difficult to obtain, this text is a revised and simplified version of [16]). More recently, the problem was tackled in [27, 38] in a Social Choice context.

Using a weak notion of monotonicity we show that, rather surprisingly, that there are non-trivial and rather well-behaved choice procedures leading to monotonic ranking by choosing procedures. The hope of finding really attractive ranking by choosing procedures is however shown to be limited.

The paper is organized as follows. The next section introduces our main definitions and elucidate our notation. Our results are collected in section 3. We apply our results to the classical case of tournaments in section 4. A final section discusses our findings.

2

The setting

Throughout the paper, Xwill denote afinite set with|X|=m ≥1 elements. Elements of X will be interpreted as alternatives that are to be compared on the basis the results of several kinds of pairwise contests. We denote by

P(X) the set of all nonempty subsets of X.

2.1

Pairwise contests between alternatives

Pairwise contests between alternatives arise in many different contexts. There-fore, it is not surprising that many different models have been proposed to summarize them. The most simple ones consist of binary relations (tour-naments [45, 56], weak tour(tour-naments [61], reflexive binary relations [80]). More sophisticated models use real-valued functions onX2_(weighted

Tourna-ments [26], comparison functions [29] or general valued relations [42, 33, 67]). Many of these models can be justified by results saying that some type of aggregation methods lead to all (or nearly all) instances of these models [17, 23, 25, 50].

Although our results can be extended to more general cases [16], we use throughout the paper the comparison function model presented in [29]. It is sufficiently flexible to include all complete binary relations (and, hence,

to deal with all C1 social choice functions) and all 0-weighted tournaments (and, hence, to deal with most C2 social choice functions, in fact with what [26] called C1.5 social choice functions). We refer to [29] for more possible interpretations.

A comparison function π onX is a skew-symmetric real-valued function on X2 _{(i.e. such that π(x, y) = −π(y, x), for all x, y ∈ X). The set of all}

comparison function onX is denotedG_{(X). We denote byπ}|A the restriction of π onA⊆X, i.e. the function π|A onAsuch that π|A(x, y) = π(x, y), for all x, y ∈A.

Example 2 (Weak Tournaments)

Aweak tournament V onXis a complete (xV yoryV y, for allx∈X) binary relation1 _{onX. A tournament is an antisymmetric (xV y andyV x ⇒x=y,}

for all x, y ∈X) weak tournament. We denote T_{(X) (resp.}WT_{(X)) the set}

of all tournaments (resp. weak tournaments) on X. A transitive tournament (resp. weak tournament) is a linear order (resp. weak order). We noteWO(X) the set of all weak orders on X.

The interest in weak tournaments is explained by McGarvey’s theorem [50] ensuring that any V ∈ WT(X) is the simple majority relation of some profile of linear orders.

Note that any comparison functionπ ∈G(X) induces a weak tournament

V ∈WT(X) letting xV y ⇔π(x, y)≥0.

Conversely, any weak tournament induces V ∈ WT_{(X) induces a}

com-parison function πV ∈G(X) is defined letting, for all x, y ∈X,

πV(x, y) =

  

 

1 if xV y and N ot[yV x],

0 if xV y and yV x,

−1 if yV x and N ot[xV y].

(1)

We sometimes abuse notations and write V instead of πV. 3

Example 3 (0-weighted tournaments)

A 0-weighted tournament on X is a clique in which each arc (x, y) has a skew symmetric integer valuation n(x, y). Using Debord’s theorem [25], any 0-weighted tournament with all n(x, y) having the same parity is the net preference matrix of some profile of linear order on X, i.e. n(x, y) is the number of linear orders in the profile for which x > y minus the number of

1_{We follow here the widely used terminology of [56, 61] although the term match} suggested by [53, 66] seems more satisfactory. Note that we work here, for commodity, with reflexive (weak) tournaments although most authors prefer the asymmetric version [45]. This has no consequences in what follows.

linear orders in the profile for which y > x. Clearly the set of comparison

functions includes all 0-weighted tournaments. 3

Letπ andπ0 _{be two comparison functions onX. We say thatπ}0 _improves x∈X (vis-`a-vis π) if for all y, z ∈X\ {x},

π0(y, z) = π(y, z) and π0(x, y)≥π(x, y).

We often denote πx↑ _{a comparison function improving x∈X via-`a-vis π.}

Let π ∈ G(X), A ⊆ X, x, y ∈ A. We say that x covers y in A if

π(x, y)> 0 and for all z ∈A\ {x, y}, π(x, z)≥ π(y, z). It is clear that the covering relation thus defined is asymmetric and transitive. Hence its has maximal elements. We denoteU C(A, π)⊆Athe set of maximal elements of the covering relation in A. This definition due to [29] extends to comparison functions a well-known concept due to [31, 51, 52].

We say that x sign-covers y in A for π if it covers y for the comparison function πsign defined by

πsign(x, y) =   

 

1 if π(x, y)>0,

0 if π(x, y) = 0,

−1 if π(x, y)<0,

for allx, y ∈X, i.e. the comparison function induced by the weak tournament induced by π. It is clear that the sign covering relation is asymmetric and transitive and, therefore, has maximal elements. We denote SU C(A, π)⊆A

the set of maximal elements of the sign covering relation in A. It is easy to see that SU C(A, π)⊆U C(A, π).

ACondorcet winner inA∈P(X) for a comparison functionπ ∈G(X) is an alternative x that defeats all other alternatives inA in pairwise contests, i.e. such that

π(x, y)> π(y, x), for all y∈A\ {x}.

It is clear that the set of Condorcet Winners Cond(A, π) is either empty or is a singleton.

Remark 1

When there is a Condorcet winner, it is clear that Cond(X, π) = SU C(A, π) and, hence, Cond(X, π) ⊆ U C(A, π). The uncovered set U C(A, π) may

2.2

Ranking procedures

A ranking procedure (for comparison functions onX)%associates with each comparison functionπ onX a weak order %(π)∈WO_{(X), i.e. is a function}

fromG(X) into WO(X). The asymmetric (resp. symmetric) part of%(π) is denoted Â(π) (resp.∼(π)).

Example 4 (Ranking procedures induced by a scoring function)

Many ranking procedures are based on scoring functions on X. A scoring function associates with each π ∈ G(X) and each x ∈ A a real number ScoreF(x, π) =F(π(x, y)y∈X\{x}), whereF is a real-valued function onR|X|−1

being symmetric in its arguments and nondecreasing in all its arguments. The ranking procedure %F associated to scoreF is defined by

x%F (π)y ⇔ScoreF(x, X, π)≥ScoreF(y, X, π) (2)

for all x, y ∈X and all π∈G(X).

Two scoring functions that are often used are:

• the Copeland score in which F =P

,

• the Kramer score in which F = min.

Note that using the Copeland score on a 0-weighted tournament correspond-ing to a net preference matrix of a profile of linear orders amounts to rankcorrespond-ing alternatives according to their Borda score [81].

Observe that in a ranking procedure base on a scoring function, the func-tion F used to compute ScoreF(x, π) is independent of x and symmetric in

its arguments. Therefore, such ranking procedures do not depend on a par-ticular labelling of the alternatives. Furthermore, since F is nondecreasing in all its arguments, the ranking will respond in the expected direction to an

improvement of xin π. This is formalized below. 3

Let Σ(X) be the set of all one-to-one functions on X (i.e. permutations). Given a comparison function π and a permutation σ ∈ Σ(X), we define πσ

as the comparison function defined letting, for all x, y ∈X πσ_{(σ(x), σ(y)) =}

π(x, y).

Definition 1 (Neutral ranking procedures)

A ranking procedure % on X is said to be neutralif, for all for all π∈G(X) and all σ ∈Σ(X),

Observe that with a neutral ranking procedure, if the comparison function is totally indecisive, i.e. if π(x, y) = π(y, x) = 0, for all x, y ∈ X, then this indecisivity is reflected in the weak order%(π), i.e.x%(π)y, for allx, y ∈X.

Definition 2 (Monotonic ranking procedure)

A ranking procedure % on X is said to be

• monotonic, if

x%(π)y ⇒x%(π0)y and (3)

xÂ(π)y ⇒xÂ(π0)y, (4)

• weakly monotonic if (3) holds,

• very weakly monotonic if

xÂ(π)y⇒x%(π0)y, (5)

for all x, y ∈X and all π, π0 _{∈G(X) such that π}0 _{improves x vis-`a-vis π.}

It is clear that monotonicity implies weak monotonicity which in turn im-plies very weak monotonicity. As already observed, it is easy to build a monotonic ranking procedure using a scoring function. This will clearly be more difficult with ranking by choosing procedures in view of example 1. In a weakly monotonic ranking procedure, “efforts do not hurt”, since the posi-tion of the improved alternative cannot deteriorate: it may only happen that beaten alternatives now tie with the improved one. Very weak monotonicity only forbids strict reversals in % after an improvement. Although this is a very weak condition, example 1 show that it can be violated with seemingly reasonable ranking by choosing procedures.

Remark 2

Monotonicity can be strengthened to strict monotonicity adding the require-ment that that if x ∼ (π)y and x is strictly improved in πx↑ _{(i.e. π}x↑ _{6= π)}

then xÂ(πx↑_)y.

This is a very strong condition, although it proves useful to characterize ranking procedures based on scoring functions F that are increasing in all arguments [15, 35, 72]. [27] proves, in a social choice context, that there is no strictly monotonic ranking by choosing procedure, as soon as one is willing to consider only “reasonable” choice procedures. The use of strict monotonicity

Consider a transitive weak tournament W ∈ WT_{(X) and its associated}

comparison functionπW as defined by (1). SinceW is a weak order, it seems

obvious to require that any reasonable ranking procedure should not alter this ranking.

Definition 3 (Faithful ranking procedure)

A ranking procedure % on X is said to be faithful if, for all transitive weak tournaments W ∈WT_{(X) and all x, y} ∈X,

x%(πW)y⇔xW y. (6)

A ranking procedure is said to be faithful for linear orders if the above con-dition holds for antisymmetric weak orders, i.e. linear orders.

Many other conditions can obviously be defined for ranking procedures (for an overview see [20, 35, 72, 80]). They will not be useful here.

This analysis of ranking by choosing procedures clearly calls now for a closer look at choice procedures.

2.3

Choice procedures

A choice procedure (for comparison functions on X) S associates with each comparison function π ∈ G(X) and each nonempty subset A ∈ P(X) a nonempty set of chosen2 _{alternatives included inA. More formally, achoice}

procedure S onX is a function from P(X)×G(X) into P(X) such that, for all A∈P(X) and all π ∈G(X),S(A, π)⊆A.

We say thatS0 _{refines Sif, for allA∈P(X) and allπ∈G(X),S}0_{(A, π)⊆}

S(A, π).

Example 5 (Choice procedures induced by scoring functions)

Like with ranking procedures, many choice procedures are based on scoring functions. A scoring function for choice procedures associates with each

π ∈ G_{(X) each A} ∈ P_{(X) and each x}∈ A a real number ScoreF(x, A, π) =

F|A|(π(x, y)y∈A\{x}) where eachF|A| is a real-valued function onR|A|−1 being

symmetric in its arguments and nondecreasing in all its arguments. We simply have, for all A∈P_{(X) and all x}∈A,

x∈ SF(A, π)⇔ScoreF(x, A, π)≥ScoreF(y, A, π), for all y∈A. (7)

Such choice procedures are clearly independent of the labelling of alternative and have obvious monotonicity properties. Furthermore, the chosen elements

in A only depends on the restriction π|A of π toA 3

Definition 4 (Properties of a choice procedure)

A choice procedure S on X is said to be:

• neutral if

x∈ S(A, π)⇔σ(x)∈ S(A, πσ_), • local if

[π|A =π0|A]⇒ S(A, π) =S(A, π0).

• Condorcet if

Cond(A, π)6=∅ ⇒ S(A, π) = Cond(A, π),

• monotonic if

x∈ S(A, π)⇒x∈ S(A, πx↑_),

• properly monotonic if it is monotonic and

y /∈ S(A, π)⇒y /∈ S(A, πx↑),

for all π, π0 _{∈ G(X), all A ∈ P(X), all σ ∈ Σ(X), all x, y ∈ X and all} πx↑ _{∈G(X) improving x vis-`a-vis π.}

We refer to [29, 26, 35, 45, 56, 61] for a thorough overview of the variety and the properties of neutral, local,Condorcet and monotonic choice procedures. An example of such procedures is SU C(A, π) [29].

Remark 3

Note that with the question of ranking by choosing procedures in mind, only local choice procedures raise problems. Using a non local choice procedure, e.g. the one one selection in all A∈P_{(X) alternatives of maximal Copeland}

score in X, instead of A, will clearly lead to a monotonic pseudo ranking by

choosing procedure. •

Choice procedures may be viewed as associating a choice function on

X to every comparison function π defined on X. Hence, when π is kept fixed, classical properties of choice functions may be transferred to choice procedures. We recall some of them below, referring the reader to [1, 2, 48, 55, 76] for a detailed study of these conditions and their relations to the classical one guaranteeing that a choice functions can be rationalized, i.e. that there is a complete binary relation on X such that chosen elements in any subset are the greatest elements of this binary relation restricted to that subset.

Definition 5 (Choice functions properties of choice procedures)

A choice procedure S on X is said to satisfy:

• Strong Superset Property (SSP) if [S(A, π)⊆B ⊆A]⇒ S(B, π) = S(A, π),

• Aizerman if

[S(A, π)⊆B ⊆A]⇒ S(B, π)⊆ S(A, π),

• Idempotency if

S(S(A, π), π) = S(A, π),

• β+ _if

[A⊆B and A∩ S(B, π)=6 ∅]⇒ S(A, π)⊆ S(B, π), for all π ∈G(X) and all A, B ∈P(X).

Remark 4

We follow here the terminology of [55] that has gained wide acceptance. Let us however observe that the nameAizerman, is especially unfortunate since, in fact, M.A. Aizerman and his collaborators apparently never used this condition in their classical works on choice functions (on the contrary, they made central use of SSP under the nameOutcast [3, 1, 2]). We follows [76] for β+_.

Let us observe thatSSP clearly impliesAizerman andIdempotency. The reverse implication is also true [2, 29, 55]. On the other hand, SSP and β+

are independent conditions [1, 2, 48, 76]. Clearly, none of these conditions is sufficient to imply that the choice function can be rationalized (for such

conditions see [2, 55, 76]). •

Remark 5 (Refining choices)

LetS be a choice procedure onXand defineS1 _{=S. For all positive integers} k ∈N∗, we define Sk and S∞ letting, for allA∈P(X) and all π ∈G(X),

Sk

(A, π) =S(Sk−1_{(A, π), π) and}

S∞(A, π) = \

k∈N∗

Sk(A, π).

It is clear that Sk _{and S}∞ _{are choice procedures. They are obtained by}

successive refinements of S. It is well-known that when S is monotonic but not idempotent, it may happen that S∞ _{is not monotonic. This the case}

An apparently open question is to find necessary and sufficient conditions on S so that this is the case. This problem is clearly related to the already-mentioned monotonicity problems encountered in electoral procedures with

runoffs. We do not study it here. •

2.4

Ranking procedures induced by choice procedures

Ranking by choosing procedures build a weak order by successive application of a choice procedure, its first equivalence class consisting of the elements chosen in X, the second equivalence class of the elements chosen after the elements chosen at the first step are removed from X and so on.

In order to be a little more formal, we need more notation. LetW be a weak order on a setY. We denote byClk(Y, W) (wherek∈N∗) the elements

in the k-th equivalence class of W, i.e.

Cl1(Y, W) ={x∈Y :xW y,∀y∈Y}

Clk(Y, W) = {x∈Zk−1 =Y \[

k−1 [

`=1

Cl`(Y, W)] :xW y,∀y ∈Zk−1}.

Note that Cl1(Y, W) is always nonempty and that a weak order is clearly

uniquely defined by its ordered set of equivalence classes.

Similarly, we denoteRk(X,S, π), the unchosen elements inX withπafter

k ∈N applications ofS, i.e.

R0(X,S, π) =X,

Rk(X,S, π) =Rk−1(X,S, π)\ S(Rk−1(X,S, π), π),

with the understanding that S(∅, π) =∅. Note that R0(X,S, π) is nonempty

by construction.

Definition 6 (Ranking procedure induced by a choice procedure)

Let S be a choice procedure on X. The ranking procedure %S induced by S

is the ranking procedure such that, for all π ∈G(X) and all k∈N∗,

Clk(X,%S) =S(Rk−1(X,S, π), π).

Some properties of S are easily transferred to %S.

Lemma 1 (Transferring properties from choice to ranking)

• If S is a neutral then %S is neutral,

• If S is based on a scoring function with all F|A| being increasing in all

arguments then %S is faithful.

• If S is a local, neutral, Aizerman and refines U C then %S is faithful.

Proof

The first three assertions are obvious from the definitions. Let us prove the last one. Suppose that W is a weak order. It is clear that U C(X, W) = Cl1(X, W). Since S refines U C we must have S(X, W) ⊆ Cl1(X, W). We

haveS(X, W)⊆Cl1(X, W)⊆X. Hence, sinceSisAizerman,S(Cl1(X, W), W)⊆

S(X, W).

SinceSis local and neutral, we know thatS(Cl1(X, W), W) = Cl1(X, W).

Hence, S(X, W) = Cl1(X, W). The conclusion follows from a repetition of

this argument. 2

Unfortunately, as shown in example 1 above, monotonicity is not transferred as easily from choice procedures to ranking procedures. Since monotonicity seems to be a vital condition for the reasonableness of a ranking procedure, we investigate in this paper which choice procedures S have an associated ranking procedure %S that is monotonic or weakly monotonic.

Remark 6

It should be observed that given a scoring function ScoreF %F and%SF may

have quite different properties. Considering for instance the Kramer score Scoremin and its extension to choice procedures, it is easy to see that%min is

not faithful (since all alternatives not belonging to the first equivalence of a weak order are tied with %min). On the contrary, it is clear that that %S_min

is indeed faithful. •

3

Results

3.1

Weak monotonicity

Our aim is to find conditions on choice procedures that would guarantee that the ranking procedures they induce is weakly monotonic. As already shown by example 1, there are choice procedure S that are neutral, local, prop-erly monotone and Condorcet while %S is not even very weakly monotonic.

Guaranteeing that %S is weakly monotonic is therefore not as trivial a task

as it might appear at first sight.

Our central result in this section says that any local and monotonic choice procedure satisfyingSSP generates a ranking procedure that is weakly mono-tonic.

Proposition 1 (SSP and weak monotonicity)

If S is local, monotonic and satisfies SSP then %S is weakly monotonic.

Proof

Suppose that S is local, monotonic and satisfies SSP and that %S is not

weakly monotonic. By definition this implies that for some π ∈ G(X) and some πx↑ _{improving x∈X vis-`a-visπ, we have x%}

S (π)y and y ÂS (πx↑)x.

SinceS is monotonic, it is impossible that x∈Cl1(X,%S (π)) =S(X, π)

since this would imply x ∈ Cl1(X,%S (πx↑)) = S(X, πx↑), violating y ÂS

(πx↑_{)x. By construction, we know that x /∈Cl}

1(X,%S (πx↑)) =S(X, πx↑).

Suppose that x∈Clr(X,%S (π)) for some r >1 and let Z =X\ {x}.

We haveπ|Z =πx↑_{|Z. Since S is local, this impliesS(Z, π) =S(Z, π}x↑_).

Since x /∈ S(X, π), we have S(X, π) ⊆ Z ⊆ X and SSP implies S(X, π) =

S(Z, π).

Similarly, we know that x /∈ S(X, πx↑_{) so that S(X, π}x↑_{) ⊆ Z ⊆ X and} SSP impliesS(X, πx↑_{) =S(Z, π}x↑_).

Because S(Z, π) = S(Z, πx↑_{) we now know that S(X, π}x↑_{) = S(X, π).}

Hence, Cl1(X,%S (π)) = Cl1(X,%S (πx↑)). Note, in particular that y /∈

Cl1(X,%(πx↑)).

Ifx ∈Cl2(X,% (πx↑)), it is impossible that yÂ(πx↑)x. Hence, we must

haver >2. BecauseS is local, it is easy to see that the above reasoning onX

can now be applied to R1(X,S, π) = R1(X,S, πx↑). We therefore conclude:

Cl2(X,%S (π)) =Cl2(X,%S (πx↑)) andy /∈Cl2(X,%(πx↑)).

Iterating the above reasoning shows that the violation of weak mono-tonicity of %S can only happen if Clk(X,%S (π)) = Clk(X,%S (πx↑)),

for k = 1,2, . . . , r − 1. Let W = X \S

k=1,2,...,r−1Clk(X,%S (π)) = X \ S

k=1,2,...,r−1Clk(X,%S (πx↑)). We know that x, y ∈ W. By hypothesis, x ∈ Clr(X,%S (π)) so that x ∈ S(W, π). Since S is monotonic, we must

have x∈ S(W, πx↑_{). Therefore it is impossible thatyÂ(π}x↑_{)x. 2}

Let us note that in the literature on tournaments it is possible to find very well-behaved choice procedures that are neutral, local, monotonic while sat-isfyingSSP [45] (e.g.M CS,BP, as defined below). For general comparison functions, [29] also present several such procedures. Proposition 1, there-fore shows that there are many well-behaved weakly monotonic ranking pro-cedures induced by choice propro-cedures. Let us give an example of such a procedure.

Example 6 (Sign Essential set)

The bipartisan setBP defined for tournaments [43] has recently been general-ized to comparison functions [29, 26]. Observe that any comparison function

π induces a symmetric two-person zero-sum game on the set of “players” X. The same is clearly true for πsign.

It is well-known that all such games have Nash equilibria in mixed strate-gies [58]. The sign Essential Set SES consists in all “strategies” that are played with strictly positive probability in one of these equilibria in the two-person zero-sum game induced by πsign.

[29] show thatSESdefines a choice procedure that is monotonic,Condorcet and satisfies SSP, on top of being clearly local and neutral. It is not dif-ficult to see that it refines U C (as well as several other reasonable choice procedures).

Hence, %SES is a neutral, faithful and weakly monotonic ranking

proce-dure which appears reasonable. 3

Remark 7 (Aizerman _{cannot be substituted to SSP)}

The above proposition does not hold ifAizerman is substituted toSSP. It is well-known that SU C is monotonic and satisfiesAizerman but violatesSSP

(see [45]). The following example shows that %SU C is not even very weakly

monotonic.

Example 7 (%U C is not very weakly monotonic)

Let X ={a, b, c, d, e, f, g}. Consider the tournament T onX defined by:

aT b, aT d, aT e, aT f, aT g bT c, bT d, bT e, bT f, bT g

cT a, cT e, cT f, cT g dT c, dT e,

eT f f T d, f T g gT d, gT e.

It is easy to check, using the comparison function defined by (1), that:

U C(X, T) = {a, b, c}, U C(X \ {a, b, c}, T) = {e, f, g}. Hence, we have

f ÂU C (T)d.

Consider now the tournamentV identical toT except thateV b. We have:

U C(X, V) = {a, b, c, d}, so that d ÂU C (V)e. This shows that %U C is not

very weakly monotonic. 3

•

It is clearly tempting to look for a result similar to proposition 1 involving the monotonicity of%S. This problem is far more difficult than with weak

mono-tonicity and we only have negative results on that point. Proposition 3 below implies that proposition 1 is no longer true if monotonicity is substituted to

weak monotonicity. •

Remark 9 (SSP is not necessary)

For local and monotone choice procedures π, SSP is a sufficient condition for %S to be weakly monotonic. It is not necessary however, even when

attention is restricted to the, well-structured, case of tournaments. Let us consider that case and show that there are, on some setsX, choice procedures violatingSSP while being weakly monotonic. We abuse notations and write

T instead ofπT.

Suppose that |X| = 5. The following example shows that SU C may violate SSP.

Example 8 (SU C violates SSP when |X| = 5)

Let X ={a, b, c, d, e}. Consider the tournament T onX defined by:

aT b, aT d, bT c, bT e, cT a, cT d, cT e,

dT b, dT e, eT a.

We haveSU C(X, T) ={a, b, c, d}(eis covered byc) andSU C({a, b, c, d}, T) =

{a, b, c}(dis covered bya). This violatesSSP sinceSU C(X, T)⊆ {a, b, c, d} ⊆

X but SU C({a, b, c, d}, T) ={a, b, c} 6=SU C(X, T) ={a, b, c, d}. 3

Let us now show that, when |X| ≤ 5, %SU C is weakly monotonic. The

proof uses the following well-knows facts on uncovered elements in a tourna-ment.

Lemma 2 ([56])

1. x ∈ SU C(A, T) iff for all y ∈ A\ {x}, either xT y or [xT z and zT y], for some z ∈A (2-step principle).

2. SU C(A, T) = {x} iff xT y for all y ∈A\ {x}.

3. If |SU C(A, T)| 6= 1 then |SU C(A, T)| ≥3 and Cond(A, T|A) = ∅.

Lemma 3

Proof

If |X| = 3, the proof easily follows from lemma 2 and the monotonicity of

SU C. If |X|= 4, three cases arise by lemma 2.

• If |Cl1(X,%SU C (T))| = 1. Let {a} = Cl1(X,%SU C) = SU C(X, T).

Since a is a Condorcet winner in X, it is impossible to improve a. If any b 6= a is improved wrt to a, it becomes uncovered and weak monotonicity of%SU C cannot possibly be violated. Ifb6=ais improved

wrt to an alternative different from a, then a remains the Condorcet winner and it is clear that weak monotonicity of %SU C cannot possibly

be violated.

• If|Cl1(X,%SU C (T))|= 3 and therefore|Cl2(X,%SU C (T))|= 1. Weak

monotonicity of %SU C can only violated if an element in Cl1(X,%SU C

(T)) = SU C(X, T) is improved. Since SU C is monotonic, this im-proved element will remain uncovered in X. Thus, weak monotonicity cannot possibly be violated.

• If |Cl1(X,%SU C (T))| = 4, weak monotonicity of %SU C follows from

the monotonicity of SU C. 2

Lemma 4

If |X|= 5, %SU C is weakly monotonic.

Proof

Four cases arise by lemma 2.

• If |Cl1(X,%SU C (T))| = 1. Let {a} = Cl1(X,%SU C) = SU C(X, T).

Sincea is a Condorcet winner inX, it is impossible to improvea. If an alternative not inCl1(X,%SU C (T)) is improved vis-`a-visa, it becomes

uncovered, because of part 1 of lemma 2. Thus weak monotonicity can-not be violated. If an alternative can-not inCl1(X,%SU C (T)) is improved

vis-`a-vis another alternative not in Cl1(X,%SU C (T)), it is clear that

after the improvement a remains a Condorcet winner and thus chosen alone inX. In view of lemma 3, weak monotonicity cannot possibly be violated.

• If |Cl1(X,%SU C (T))| = 3 and, therefore |Cl2(X,%SU C (T))| = 1 and |Cl3(X,%SU C (T))| = 1. Let X = {a, b, c, d, e} and suppose wlog

that Cl1(X,%SU C (T)) =SU C(X, T) = {a, b, c}, Cl2(X,%SU C (T)) =

SU C({d, e}, T) = {d} and Cl3(X,%SU C (T)) = {e}. We must then

It is impossible to improveeand to violate weak monotonicity. In view of part 1 of lemma 2, observe thatdcan beat at most one alternative in

{a, b, c} because we know that d /∈SU C(X, T). Ifd beats exactly one alternative in {a, b, c} any improvement of d will make it uncovered. Hence, weak monotonicity cannot be violated. Suppose therefore that

d does not beat any alternative in{a, b, c}. Becausee /∈SU C(X, T),e

can beat at most one alternative in {a, b, c}.

Suppose first that e does not beat any alternative in {a, b, c}. In any

T0 _{improvingd, it is clear that %}

SU C (T0) =%SU C (T) and no violation

of weak monotonicity can occur.

Suppose then thatebeats one alternative in{a, b, c}and suppose wlog that eT a. If T0 _{improves d vis-`a-visa, we still have %}

SU C (T0) =%SU C

(T). If T0 _{improves d vis-`a-vis b or c then Cl}

1(X,%SU C (T0)) = {a, b, c, d} so that no violation of weak monotonicity can occur.

• If |Cl1(X,%SU C (T))| = 4. We have |Cl2(X,%SU C (T))| = 1. Weak

monotonicity of %SU C can only violated if an element in Cl1(X,%SU C

(T)) = SU C(X, T) is improved. Since SU C is monotonic, this im-proved element will remain uncovered in X. Thus, weak monotonicity cannot possibly be violated.

• If |Cl1(X,%SU C (T))| = 5, weak monotonicity of %SU C follows from

the monotonicity of SU C. 2

•

Remark 10

It easily follows from [56] that there is no local and Condorcet choice proce-dure satisfying Chernoff, i.e., for all π∈G(X) and all A, B ∈P(X),

[A⊆B]⇒ S(B, π)∩A⊆ S(A, π).

Hence, since Chernoff is a necessary condition for S to be rationalized (and given the correspondence between C1 and (most) C2 social choice func-tions [31, 26] and choice procedures for comparison funcfunc-tions), this tends to severely limit the scope of the Theorem 1 in [38]. •

Remark 11

As conjectured by Patrice Perny [63], β+_{is an alternative sufficient condition}

on S guaranteeing the weak monotonicity of %S. Proposition 2 (β+ and weak monotonicity)

If S is monotonic and satisfies β+ _{then %}

Proof

We use the following claim.

Claim ([63])

If S is monotonic and satisfies β+ _{then %}

S is very weakly monotonic.

Proof

In violation of the claim suppose that a ÂS (π)b and b ÂS (πa↑)a. Since b ÂS (πa↑)a, we have, for some A⊆X such that a, b∈A, b∈ S(A, πa↑) and a /∈ S(A, πa↑_).

Using β+_{, we know that {b} = S({a, b}, π}a↑_{). Since S is monotonic,}

this implies {b} = S({a, b}, π). Using β+_{, this implies that for all B ⊆ X,} a, b∈B ⇒b∈ S(B, π). Therefore, it cannot be true that aÂS (π)b. 2

In view of the above claim, it remains to prove thata∼S (π)b⇒a%S (πa↑)b.

LetW be a weak order on X and x ∈X. Define X≥_{(x, W) = {y ∈ X :} xW y}, X>_{(x, W) = {y ∈ X : xα[W]y}, X}=_{(x, W) = {y ∈ X : xι[W]y},}

where α[W] (resp. ι[W]) denotes the asymmetric (resp. symmetric) part of

W.

Take any c∈ X=_(a,%

S (π)) such that c%S (πa↑)d, for all d∈X=(a,%S

(π)). Suppose that cÂS (πa↑)a.

By construction, we haveX=_(a,%

S (π))⊆X≥(c,%S (πa↑)) andX≥(a,%S

(πa↑_))⊆X≥_(c,%

S (πa↑)).

Using the above claim, we know thatX>_(a,%

S (π))⊆X≥(a,%S (πa↑)).

Therefore, X≥_(a,%

S (π))⊆ X≥(c,%S (πa↑)). By construction, we know

thata ∈ S(X≥_(a,%

S (π)), π). SinceSis monotonic, we havea∈ S(X≥(a,%S

(π)), πa↑_{). Using β}+_{, we must have a ∈ X}≥_(c,%

S (πa↑)) contradicting

cÂS (πa↑)a. 2

It is well-known [56, 76] that β+ _{is a very strong condition. For instance, in}

the case of tournaments, any choice procedureS satisfyingβ+_andCondorcet

must include the top cycle T C, i.e. the choice procedure selecting in A the maximal elements of the asymmetric part of the transitive closure on A of

T. Clearly, choice procedures are highly undiscriminating.

Therefore, althoughβ+_{andSSP are independent conditions, we feel that}

the above proposition is far more satisfactory than proposition 1. Since there are local, monotonic choice procedures satisfying SSP but violating β+ _(e.g. SES), it is clear that, hopefully, β+ _{is not a necessary condition for weak}

monotonicity. •

It is not difficult to observe that the proof of propositions 1 and 2 makes no use of the skew-symmetry property of comparison functions (when mono-tonicity is properly redefined). They can be easily generalized to cover more general cases [16], e.g. genreral valued (or fuzzy) binary relations [7, 14, 19].

We do not explore this point here. •

3.2

Monotonicity

Let us consider the case of tournaments [45, 56]. There are neutral, mono-tonic and Condorcet choice procedures S such that %S is monotonic. This

is clearly the case for T C which satisfies both SSP and β+_{. We already}

observed that T C is a very undiscriminating choice procedure for tourna-ments. It would therefore be of interest to find more discriminating choice procedures S so that %S is monotonic. As show below, this proves difficult

however.

Proposition 3 (Covering compatibility and Aizerman₎

LetS be a local, neutral and monotonic choice procedure satisfying Aizerman. If S refines U C then %S is not monotonic.

Proof

The claim will be proved if we can show that, for all neutral and monotonic choice procedures refiningU C and satisfyingAizerman, there is a comparison function π such that a∈ S(X, π) b /∈ S(X, π) and b ∈ S(X, πa↑_{), i.e. that S}

is not properly monotonic. The following example suffices.

Example 9

Let X ={a, b, c, d, e}. Consider the tournament T onX defined by:

aT d, aT e, bT a, cT a, cT b, dT b, dT c, dT e

eT b, eT c.

We have U C(X, T) = {a, c, d} and aT d, dT c and cT a. Therefore, since S

refines U C, we have S(X, T) ⊆ {a, c, d} ⊆ X. Since S satisfies Aizerman,

S({a, c, d}, T)⊆ S(X, T).

Because S is local and neutral, we know that S({a, c, d}, T) = {a, c, d}. Hence we must have S(X, T) ={a, c, d}.

Consider now the tournament V identical to T except that aV c. Using the same reasoning as above, it is easy to check thatS(X, V) = U C(X, V) =

2

Remark 13

P. Perny [64, 65] has proposed a different impossibility result using a “positive discrimination” axiom on choice procedures that says that, starting with any comparison function, it is always possible to obtain any alternative as the unique choice provided this alternative is “sufficiently” improved. This negative result only deals with weak monotonicity of %S however. • Remark 14

The above result is fairly negative as long asAizermanis considered as a vital property. If this not the case, new possibilities are easily seen to emerge. Consider for instance, the well known T C∗ _{choice procedure [74] for weak}

tournaments selecting in any subset, the maximal elements of the asymmetric part of the transitive closure (on that subset) of the asymmetric part of the weak tournament. It is well-known that T C∗ _{violates both Aizerman and} β+_{. For the sake of completeness, we give an example below.}

Example 10 (T C∗ _violates Aizerman _{and β}+

)

Let X ={a, b, c}. Consider the weak tournament V on X defined by:

aV b, aV c, bV c, cV a.

We have T C∗_{(X, V) = {a} and T C}∗_{({a, c}, V) = {a, c}. This violates both}

Aizerman and β+_{. 3}

[80] proves that %T C∗ is monotonic (see also [38]). One should however

notice that %T C∗ is a very particular ranking by choosing procedure since

the transitive closure operation has a clearly global character, in spite of the progressive restriction on the set of alternatives. This type of ranking by

choosing procedures are studied in detail in [38]. •

4

Application: the case of tournaments

In this section we apply the above results and observations to the case of tournaments, i.e. we only consider choice procedures defined for comparisons functions derived for tournaments. This case of particular interest because

such choice procedures have been analyzed in depth and, in spite of the re-stricteveness of the antisymmetry hypothesis, the underlying choice problem is encountered in many different and important settings.

[45] studies in detail seven3 _{different choice procedures. We briefly}

present them below referring the reader to [44, 45, 56] for precise definitions and results.

Top Cycle T C selecting in A the element of the first equivalence class of the weak order being the transitive closure of T onA,

Copeland Cop selecting in A the alternatives with the highest Copeland score in the tournament restricted to A,

Slater SL selecting in A all alternatives having the first rank in a linear order on A at minimal distance of the restriction of T onA,

Uncovered Set U C selecting all the uncovered alternatives in A [31, 51],

Banks B selecting all alternatives in A starting a maximal transitive path of T onA [6],

Minimal Covering Set M CS selecting all alternatives in the unique cov-ering set included in A of minimal cardinality [28],

Bipartisan Set BP selecting in A all alternatives in the support of the unique Nash equilibrium of the symmetric two-person zero-sum game on A induced byT [43].

Proposition 4 (Ranking by choosing procedures for Tournaments)

1. %T C is monotonic,

2. %M CS and %BP are weakly monotonic but not monotonic,

3. %U C, %B, %COP and %SL are not very weakly monotonic.

Proof

Part 1 is left to reader as an, easy, exercise. The weak monotonicity of

%M CS and %BP results from proposition 1, since it is well-known that both

procedures are neutral, local, monotonic and satisfySSP. The fact that they are not monotonic follows from proposition 3 since they both refine U C.

Part 3. We respectively showed in examples 1 and 7 that %Cop and

%U C are not very weakly monotonic. It is easy to see that example 7 also

3_{Since the Tournament Equilibrium Set introduced in [75] may not be a monotonic} choice procedure, we do not envisage it here

shows that %B is not very weakly monotonic (we have %U C=%B for both

tournaments). It remains to show that %SL is not very weakly monotonic.

We do not give details since they are rather cumbersome.

Example 11 (%SL is not very weakly monotonic)

LetX ={a, b, c, d, e, f, g, h, i}. Consider the tournamentT onX defined by:

aT b, aT e, aT g, aT h, aT i, bT c, bT e, bT f, bT g, bT i,

cT a, cT d, cT e, cT f dT a, dT b, dT e, dT i

eT f, eT h,

f T a, f T d, f T h, f T i, gT c, gT d, gT e, gT f, gT h, hT b, hT c, hT d, hT i

iT c, iT e, iT g.

Linear orders at minimal distance ofT are at distance d= 10. There are exactly 40 such orders and we have SL(X, T) = {a, b, d, f, g, h}. It is clear that the restriction of T to {c, e, i} is the linear order iT c, cT e, iT e. Hence, we have iÂSL(T)c.

Consider now the tournament V identical to T except that iV a. Again skipping details, linear orders at minimal distance ofT are at distanced = 10. There are exactly 10 such orders. We have SL(X, V) = {b, g, h}. Similarly, we obtainSL(X\ {b, g, h}, V) = {c}. ThereforecÂSL(V)i. This shows that

%SL is not very weakly monotonic. 3

2

5

Discussion

Using a ranking by choosing procedure raises serious monotonicity prob-lems. Rather surprisingly, however, it possible to isolate a class of quite well-behaved choice procedures that lead to weakly monotonic ranking by choosing procedures. If weak monotonicity is considered as an attractive property, these ranking procedures may well be good candidates to compete with other ranking procedures. If monotonicity is considered of vital impor-tance, then the situation is more critical since there are no local, neutral, monotonic and Aizerman choice procedure that is “sufficiently” discrimina-tory being included in U C and inducing a monotonic ranking procedure.

It would clearly be interesting to look for necessary and sufficient condi-tions on S for %S to be (weakly) monotonic. In view of remark 9, this task

is likely to be complex since the repeated use of S in order to build %S only

uses the result of the application of S on a relatively small number of sub-sets. Another intriguing problem would be to look for connections between the problem studied here and the one of finding necessary and sufficient con-ditions guaranteeing that S∞ _{is monotonic. More research in this direction}

is clearly needed.

The difficulties encountered with ranking procedures induced by choice procedures may also be considered as an incentive to study ranking proce-dures for their own sake, i.e. independently of any choice procedure. Research in that direction has already started [15, 18, 19, 20, 35, 33, 34, 39, 72, 80] mainly considering ranking procedures based on scoring functions. This is at variance with H. Moulin’s advice [56] to focus research on ranking procedure based on the approximation of a tournament (or a comparison function) by linear orders (or weak orders). This idea dates back to [40, 41, 77]. Although it raises fascinating deep combinatorial questions and difficult algorithmic problems [9, 10, 11, 13, 21, 36, 37, 54], this line of research raises other difficulties. As argued in [62, 70],

• the choice of the distance function should be analyzed with care as soon as one leaves the, easy, case of a distance between tournament and linear orders [8, 71],

• the likely occurrence of multiple optimal solutions to the optimization problem underlying the approximation is not easily dealt with,

• the normative properties of such procedures are not easy to analyze. Hence, studying simpler procedures, e.g. the ones based on scoring functions maybe a good starting point. In many common situations, ranking and not choosing is the central question and there is a real need for a thorough study of ranking procedures.

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