Scoring rules: A cooperative game-theoretic approach*

Emilio Calvo1, InÄaki GarcõÂa2, Esther GutieÂrrez2

1 Departamento de AnaÂlisis EconoÂmico, Universitat de Valencia, Avinguda delsTarongers s/n, EdifõÂcio Departamental Oriental, E-46022 Valencia, Spain(e-mail: emilio.calvo@uv.es)2 Departamento de EconomõÂa Aplicada, Facultad de Ciencias EconoÂmicas yEmpresariales, Universidad del PaõÂs Vasco-Euskal Herriko Unibertsitatea,Avenida Lehendakari Aguirre 83, E-48015 Bilbao, Spain

Received: 17 March 1997/Accepted: 12 May 1998

Abstract. In this work we de®ne the game of the alternatives for each preferencepro®le, and establish relations between scoring rules and cooperative solutionconcepts for that game, such as the family of semivalues and the family ofleast square values.

1 Introduction

Given a society facing a set of alternatives, a social welfare preordering (SWP)is a function which determines, on the basis of the preferences of the agents,the order of social preference in which the alternatives are ranked. The wellknown Arrow's (1963) impossibility result has focused most Social Choiceliterature into strategic considerations and stressed the need for a systematicanalysis of the strategic properties of mechanisms for collective decisionmaking.

Nevertheless, the importance of strategic considerations should not lead usto forget the cooperative point of view of the aggregation problem. From themoment that the agents have chosen what preferences to reveal, the SWP isthe way in which individuals agree on how to determine the social value of thealternatives (according to how they are ranked). Di¨erent SWPs yield di¨er-ent ways to agree, and they can be compared by means of the properties thatcharacterise each of them. This axiomatic approach is also a widespread

Soc Choice Welfare (1999) 16: 491±512

* Financial support by the Universidad del PaõÂs Vasco-EHU, grant UPV036.321-HA012/94, by the Gobierno Vasco, grant PI 95/101, and by the MEC, grant PB96-0247, is gratefully acknowledged.We are grateful for the constructive suggestions made by an anonymous referee, whichhelped us improve an earlier version of this paper.

methodology used in cooperative game-theoretical models, giving normativesupport to the di¨erent cooperative solution concepts proposed.

Here we go a step further: we show how to build a cooperative gameamong the alternatives in such a way that di¨erent SWPs can be rewritten asdi¨erent cooperative solutions applied to the same game.

The strategy we have used is as follows: given a set of alternatives, thecharacteristic function gives, for every subset of alternatives, its social worth

taking into account the preferences that agents have on the alternatives. Thisis done by merely counting the agents who are supporters1 of this subsetof alternatives. This is an ordinal de®nition and things such as intensity ofpreferences or cardinal utilities do not play any role. By applying an univaluedsolution concept to the game we ®nd the social value of each alternative. If werank the alternatives according to these values a social preordering is built up.Therefore, it is possible to de®ne SWPs by using cooperative solutions and,vice versa, we can ask for any particular SWP if it is possible associate a co-operative solution with it in this way2.

In this paper we restrict this analysis for a particular subfamily of SWPs,called scoring rules (SR). A scoring rule assigns a score of si to the ith mostpreferred alternative of each individual3 and gives to each alternative the totalscore obtained by adding up the scores which each alternative obtains amongall the agents. Social preference is obtained by ranking the alternatives ac-cording to these total scores. One example is the Borda (1781) rule, de®nedby making the di¨erences between two consecutive scores constant. It willbe shown that the cooperative solution associated to the Borda rule is theBanzhaf (1965) semivalue. Another well-known example is the plurality rule,which gives the score one to the most preferred alternative and zero to therest. We show that it has also a particular semivalue4. An interesting fact isthat the game of alternatives always has a nonempty core formed by a singlepayo¨ allocation. In this core allocation the value of each alternative isprecisely the score given by the plurality rule. The only e½cient semivalue isthe well-known Shapley (1953) value, which here yields a new scoring rulewhose properties deserve to be explored. In summary, our ®rst result is thefollowing:

Theorem A. Every semivalue applied to the game of alternatives induces a

scoring rule.

A query appears at this point: can any SR be represented by a semivalue?The answer is no. In Theorem 2 the necessary and su½cient conditions thatscores must satisfy for that are given. If we compare SRs that can be repre-

1 The formal de®nition will be given in Sect. 3.2 If we take the set of alternatives which maximizes its social value, we can de®ne asocial choice function instead of a SWP. A similar analysis could be made in thissetting.3 It is assumed here that the individual preferences are strict, i.e. ties between alter-natives are not allowed.4 The family of semivalues was introduced by Debey et al. (1981).

492 E. Calvo et al.

sented by semivalues with SRs that cannot, nothing especially pathological isexcluded. Hence, it would be of interest to ®nd other solution concepts relatedwith these remaining SRs.

Recently a new family of cooperative solutions has appeared in literature:the least square values [Ruiz et al. (1995)]. These help us to solve this problem.It is proved that there exists a close relationship between this family and thatof semivalues: the additive e½cient normalisation of any semivalue produces aleast square value. But it is a wider family in the sense that there exist leastsquare values which are not the additive normalisation of any semivalue. Wenot only prove that any least square value applied to the game of alternativesalso yields a SR, as semivalues do, but moreover -and surprisingly- the converseis also true here. This is the content of our main theorem:

Theorem B. There exists a 1-1 mapping between scoring rules and least squarevalues.

The signi®cance of Theorem B is the representativeness of the scoring rulesvia the least square values. This representativeness allows us to give an inter-pretation of the scores in probabilistic terms. Also, it allows us to extend thescoring rules to the domain in which individual preferences are preorderings.Our extension is di¨erent from the known in the literature as the average ex-tension. We also compare both extensions from the point of view of symmetry.

Let us brie¯y outline the contents. Section 2 gives some preliminaryde®nitions and notations. Section 3 de®nes the game of alternatives and showsthe relationship between scoring rules and semivalues. Section 4 introducesthe family of least square values and proves its isomorphism with the familyof scoring rules. Section 5 yields a probabilistic interpretation of scoring rulesbased on their relation with least square values. Section 6 shows how to extendthe scoring rules to the domain in which preferences of voters are preorderings.Section 7 considers the setting of a universe of alternatives, and characterisesthe family of ``consistent'' scoring rules, i.e., the family of scoring rules asso-ciated with the family of semivalues de®ned in an in®nite set of players. And,®nally, in Sect. 8 we give some concluding remarks.

2 Preliminaries

Let V be the universe of agents (voters) and N � f1; 2; . . . ; ng a ®nite and®xed set of alternatives (candidates). We shall assume that each voter haspreferences on N represented by an ordering of N: this is a complete, transitiveand asymmetric relation on N. We denote by L�N� the set of orderings of N.

Let V HV be a ®nite set of agents, a preference pro®le on V is a functiono : V ! L�N � where o�u� are the preferences of agent u. We denote ou �o�u� and it can be represented by a permutation ou : N ! N where ou�i� � k

means that i is the kth ranked alternative for the agent u A V . Also, ou�i� �ou� j� means that alternative i is preferred to alternative j by agent u. The setof all preference pro®les on V is denoted by L�N�V .

Scoring rules 493

A social welfare preordering (SWP) is a map R : L�N�V ! P�N�, suchthat to each pro®le associates a (social) preordering of N (complete andtransitive), where P�N� is the set of preorderings of N. The di¨erence betweenthe individual and the collective preference is that in the latter social indi¨-erences (ties) are allowed. Thus, iR�o� j means that social welfare is not lowerat i than at j.

An example of such a function is a scoring rule. Let s � �s1; . . . ; sn� A Rn bea vector such that s1 V s2 V � � � V sn and s1 > sn; we call this a scoring vector.Scoring rule assigns a score of sk to the kth most preferred alternative of eachvoter and gives to each alternative the total score obtained by adding up thescores that it obtains among all the voters. Social preference is obtained byranking the alternatives according to these total scores. Formally, for allo A L�N�V and i A N, we de®ne the total score obtained by alternative i in the

pro®le o as so�i� �P

u AV

sou�i�. The scoring rule SR : L�N�V ! P�N� gives thefollowing preordering: iSR�o� j i¨ so�i�V so� j�, for all i; j A N and o AL�N�V . Note that the way in which the scoring rule assigns the scores doesnot depend on the name of the agents and on the name of the alternatives, i.e.,the scoring rule veri®es the two properties called anonymity among agents andneutrality among alternatives.

Notice that if s is a scoring vector and s 0 � �s 01; . . . ; s 0n� A Rn is such thats 0i � lsi � m with l > 0 and m A R, then s 0 is another scoring vector, but onewhich yields the same scoring rule than s. Because of this, scoring vectorsmust be considered as belonging to equivalence classes: s and s 0 are equivalentswhen they yield the same scoring rule for every set of voters V. Then we cantake a scoring vector arbitrarily as representative of the equivalence class itbelongs to.

3 Semivalues

A TU game with a ®nite set of players N is given by a characteristic functionv : 2N ! R with v�q� � 0. The set of all games in N is denoted by GN . Acooperative solution is a function f : G N ! RN .

In order to show the relationship between scoring rules and cooperativesolution concepts we start by constructing the following cooperative game.

De®nition 1. Let o be a pro®le on V, we de®ne the game of the alternatives for

the pro®le o as the game vo A G N given by:

i) vo�q� � 0ii) vo�S� � jfu A V : 6b j A N nS with ou� j� � ou�i� for some i A Sgj, for q0

S HN

iii) vo�N � � jV j.Notice that vo�S� is the number of voters whose most preferred alternatives

are those entire in S (the order in which they are preferred is not important),

494 E. Calvo et al.

i.e., vo�S� is the number of supporters of S. Hence, vo�S� is called the socialworth of S.

De®nition 2. Let f be a cooperative solution on G N . We de®ne the social welfare

preordering associated to f, Rf, in the following way:

iRf�o� j iff fi�vo�V fj�vo�; �i; j A N; o A L�N�V �

The aim of this paper is to establish the relationship between scoring rulesand the SWP associated with cooperative solutions. The following lemma is ofinterest, taking into account that scoring rules aggregate the scores of an al-ternative in a linear way.

Lemma 1. Let V and V 0 be two disjoint sets of voters, and o and o0 be pro®leson V and V 0 respectively, then it holds that vo�o 0 � vo � vo 0 , where o� o 0 is

the pro®le on V WV 0 given by:

�o� o 0��u� � o�u� if u A V

o 0�u� if u A V 0

�This lemma allows us to consider a pro®le o on V as the sum of the pro®les

o�u� for all u A V and, therefore, if f is a cooperative solution that satis®eslinearity, then fi�vo� �

Pu AV

fi�vo�u��, for all i A N and o A L�N�V . This is

why we are interested in linear cooperative solutions.A well-known family of linear solutions for TU games is the so-called

semivalues family. The following de®nition is the characterisation of thesesolutions given by Dubey et al. (1981).

De®nition 3. A semivalue is a function c : G N ! RN of the form

ci�v��Xnÿ1k�0

pk

XS JN nijSj�k

�v�S W i�ÿv�S��

0BB@1CCA� X

S JNnipjSj�v�S W i�ÿv�S��; �1�

for all i A N, and v A G N , where p � �p0; p1; . . . ; pnÿ1� A Rn is a vector such

that pk V 0 for all k � 0; . . . ; nÿ 1 andPnÿ1k�0

nÿ 1

k

� �pk � 1.

That is to say, P (de®ned by P�S � � ps, ES HN ) is a probability distributionover the coalitions not containing player i which assigns the same probabilityto coalitions of the same size. Thus, a semivalue assigns to each player his orher expected marginal contribution according to the probability distribution P.

The next two theorems show the relationship between semivalues andscoring rules.

Theorem 1. Let c be a semivalue associated with a vector p. Then Rc is a

scoring rule.

Scoring rules 495

Proof: We have to prove that there exists a scoring rule SR such that ci�vo�Vcj�vo� i¨ iSR�o� j, for all o A L�N�V and i; j A N. But, taking into accountthat semivalues satisfy linearity, this is equivalent to:X

u AV

ci�vo�u��VXu AV

cj�vo�u�� iffXu AV

sou�i�VXu AV

sou� j�;

for al i; j A N and o A L�N�V .Note that for all u A V , vo�u� is the permuted game5 of vo� , where

o� A L�N� is given by the identity permutation o��i�� i, i AN. So, as semi-values also satisfy anonymity, then we have ci�vou

��cou�i��vo� �, for all i A N.Therefore, we only have to prove that the vector s � �s1; . . . ; sn� A Rn given bysi � ci�vo� � is a scoring vector.

From (1), we have that c1�vo� � �Pnÿ1j�0

pj and ci�vo� � �Pnÿ1j�i

pj ÿPiÿ2j�1

pj, for

i � 2; . . . ; n. So, the system si � ci�vo� �, i � 1; . . . ; n may be expressed in amatrix way by s � Bp where:

B �

1 1 1 . . . 1

0 0 1 . . . 1

..

. �ÿ1� . .. ��1� ..

.

0 ÿ1 ÿ1 . ..

1

0 ÿ1 ÿ1 . . . 0

0BBBBBBB@

1CCCCCCCA:

Therefore, we have that s1 V s2 V � � � V sn because si ÿsi�1� piÿ1� pi V 0for all i � 1; . . . ; nÿ 1, and, since there exists j such that pj > 0, then s1 > sn,i.e., s is a scoring vector. r

Observe that di¨erent semivalues may yield the same scoring rule.Now a question arises: is the converse true? The next theorem shows that

only a subfamily of scoring rules can be related with semivalues.

Theorem 2. A scoring rule given by a scoring vector s is the SWP associated

with a semivalue (not necessarily unique) if and only if s satis®es:Xi�j

k�i

�ÿ1�k�isk V

Xi�jÿ1

k�i�1�ÿ1�k�i�1

sk;

for 1U i U nÿ 1; 3U j U nÿ i and j odd.

Proof: As s can be any element of the equivalence class that represents the

scoring rule, consider s as verifyingPnÿ1i�1

nÿ2iÿ1

� ��si ÿ si�1� � 1. If s' is any other

scoring vector of the same equivalence class satisfying the same condition, it isof the form s 0i � si � m, i � 1; . . . ; n, with m A R.

5 For all permutation p and v A G N , p�v� is the permuted game of v, de®ned by:p�v��S� � v�p�S��, S JN.

496 E. Calvo et al.

The aim of Theorem 2 is precisely to solve the converse of theorem 1, thatis, given s, to ®nd p such that Bp � s and p satisfying pk V 0 for all k � 0; . . . ;

nÿ 1 andPnÿ1k�0

nÿ 1

k

� �pk � 1.

In order to solve the system, we distinguish between n being odd and even.If n is odd then jBj � 1, so the system has a unique solution: p � Bÿ1s, where

Bÿ1 �

1 ÿ1 1 ÿ1 . . . 1

0 0 ÿ1 1 . . . ÿ10 1 0 ÿ1 . . . 1

..

. ... ..

. . .. ..

.

..

. ... ..

. . .. ÿ1

0 1 ÿ1 1 . . . 0

0BBBBBBBBBB@

1CCCCCCCCCCA:

That is to say:

p0 �Xn

j�1�ÿ1� j�1sj

pk �Xk

j�2�ÿ1� j�k

sj �Xn

j�k�2�ÿ1� j�1�k

sj; k � 1; . . . ; nÿ 1

8>>>>><>>>>>:�2�

Now we have to check if pk V 0 for all k � 0; . . . ; nÿ 1 andPnÿ1k�0�

nÿ 1

k

� �pk � 1. From (2), we have:

Xnÿ1k�0

�nÿ 1

k

�pk

�Xn

j�2sj

�Xjÿ2k�0

�nÿ1

k

��ÿ1� j�1�k �

Xnÿ1k�j

�nÿ1

k

��ÿ1� j�k

���

nÿ10

�s1:

And, using the following formula:

nÿ 1

k � 1

� �ÿ nÿ 1

k

� �� nÿ 2

k � 1

� �ÿ nÿ 2

k ÿ 1

� �; �k � 1; . . . ; nÿ 3�; �3�

we obtain:Xnÿ1k�0

nÿ 1

k

!pk �

Xnÿ1j�2

sj

nÿ 2

j ÿ 1

!ÿ nÿ 2

j ÿ 2

!" #

ÿ sn �nÿ 1

0

!s1 �

Xnÿ1i�1

nÿ 2

i ÿ 1

!�si ÿ si�1� � 1:

In general, we do not have that pk V 0 for all k � 0; . . . ; nÿ 1. In order to

Scoring rules 497

have it, we must see if there exists m A R such that p � Bÿ1s 0V 0, with s 0i �si � m, i � 1; . . . ; n. Replacing s by s 0 in (2), we have:

p0 �Xn

j�1�ÿ1� j�1

sj � m

pk �Xk

j�2�ÿ1� j�ksj �

Xn

j�k�2�ÿ1� j�1�ksj � �ÿ1�km; k � 1; . . . ; nÿ 1

8>>>>><>>>>>:Therefore, pV 0 happens if and only if:

Max

�Xn

j�1�ÿ1� jsj;

Xi

j�2�ÿ1� j�1sj �

Xn

j�i�2�ÿ1� jsj

�for i even; i � 2; . . . ; nÿ 3�;Xnÿ1j�2�ÿ1� j�1sj

�U m

UMin

�Xn

j�3�ÿ1� jsj;

Xi

j�2�ÿ1� j�1sj �

Xn

j�i�2�ÿ1� jsj

�for i odd; i � 3; . . . ; nÿ 2��:

It can be easily checked that this condition is equivalent to the one given inthe theorem. Moreover, for each m satisfying this condition, we have a di¨erentp and, therefore, a di¨erent semivalue associated with the same scoring rule.

Now let us study the system for n even. The system may not be solvablebecause the rank of the matrix B is nÿ 1. It is solvable if and only if

s1 1 1 . . . 1

s2 0 1 . . . 1

..

. �ÿ1� . .. ��1� ..

.

snÿ1 ÿ1 ÿ1 . ..

1

sn ÿ1 ÿ1 . . . 0

��������������

��������������� 0:

It is easy to see that this condition is equivalent to sn�Pnÿ1i�2�ÿ1� i�1si, so consider s

the only element in the equivalence class satisfying also the last condition.Then the solution of the system is of the form f�p0; p1; . . . ; pnÿ1� : p0 A Rgwhere:

pk � ÿsk�1 � 2Xk

j�2�ÿ1� j�k

sj � �ÿ1�k�p0 ÿ s1�; for k � 1; . . . ; nÿ 1:

498 E. Calvo et al.

Therefore:

Xnÿ1k�0

nÿ 1

k

!pk �

nÿ 1

0

!p0 ÿ

Xnÿ1k�1

nÿ 1

k

!sk�1 � �p0 ÿ s1�

�Xnÿ1k�1

nÿ 1

k

!�ÿ1�k � 2

Xnÿ1k�2

nÿ 1

k

! Xk

j�2�ÿ1� j�ksj

" #

� p0 � �p0 ÿ s1�Xnÿ1k�1

nÿ 1

k

!�ÿ1�k

�Xnÿ1j�2

sj ÿnÿ 1

j ÿ 1

!� 2

Xnÿ1k�j

�ÿ1� j�knÿ 1

k

!" #ÿ sn:

And, using formula (3), we obtain:

Xnÿ1k�0

nÿ 1

k

!pk � p0 ÿ �p0 ÿ s1� �

Xnÿ1j�2

sj ÿnÿ 1

j ÿ 1

!� 2

nÿ 2

j ÿ 1

!" #ÿ sn

� s1 �Xnÿ1j�2

sj

nÿ 2

j ÿ 1

!ÿ nÿ 2

j ÿ 2

!" #ÿ sn

�Xnÿ1i�1

nÿ 2

i ÿ 1

!�si ÿ si�1� � 1:

Finally, pk V 0 for all k � 0; . . . ; nÿ 1, if and only if:

Max 0; s1 � si�1 � 2Xi

j�2�ÿ1� j�1

sj �for i even; i � 2; . . . ; nÿ 2�( )

U p0

UMin s1 ÿ si�1 � 2Xi

j�2�ÿ1� j�1sj �for i odd; i � 1; . . . ; nÿ 1�

( ):

Which is equivalent to the condition of the theorem. Moreover, for each p0

satisfying the last condition, we have a di¨erent semivalue associated with thesame scoring rule. r

Examples

Plurality rule. This is a scoring rule de®ned by taking as scoring vector: s1 � 1and si � 0 for all i0 1. It is easily checked that this scoring vector satis®es thecondition of theorem 2, and then its associated semivalue is the one given byp0 � 1 and pi � 0 for all i 0 0, so ci�v� � v�i� for all i A N.

Scoring rules 499

An interesting fact is that the core6 of the game of alternatives is alwaysnonempty and it is formed by a unique payo¨ allocation. In this core allocation,the value of each alternative is exactly the score given by the plurality rule.This is the content of next proposition.

Proposition 1. For any o A L�N�V it holds that C�vo� � f�vo�1�; . . . ; vo�n��g.Proof: First note that

Pi AN

vo�i� � jV j � vo�N�, then �vo�1�; . . . ; vo�n�� is theonly imputation7 of the game vo. Moreover, for every coalition S,q0SJN,we have that vo�S�U jV j ÿ

Pj AN nS

vo� j��Pi AS

vo�i�, hence C�vo� � f�vo�1�; . . . ;vo�n��g. r

Borda score. The Borda (1781) rule is de®ned by a scoring vector satisfyingsi ÿ si�1 � k, for all i � 1; . . . ; nÿ 1. In the proof of Theorem 1 we found thatthe score associated with each semivalue must satis®es sk ÿ sk�1 � pkÿ1 � pk

for all k�1; . . . ; nÿ1. The Banzhaf (1965) semivalue computes the expectationof the marginal contributions of player i, v�S W i� ÿ v�S�, assuming that all

coalitions S are equally likely to be formed, i.e., pk �1

2nÿ1, for k � 0; . . . ;

nÿ 1. Then clearly, the SR associated with the Banzhaf semivalue is theBorda rule.

The best known semivalue is the Shapley (1953) value. It computes theexpected marginal contributions, assuming that ®rst, all coalitions of the samesize are equally likely, and second, all sizes are equally likely, i.e., pk �1

n

nÿ1k

� �ÿ1, for k�0; . . . ; nÿ1. Hence, it means that coalitions of interme-

diate size are less likely to happen than coalitions placed at the extremesof the rank-size. This semivalue de®nes a new SR whose scores are s1 �1

n

Pnÿ1j�0

nÿ 1

j

� �ÿ1and sk � 1

n

Pnÿ1j�k

nÿ1j

� �ÿ1ÿPkÿ2

j�1

nÿ1j

� �ÿ1" #, for k � 2; . . . ; n.

We call this SR the Shapley rule. As example, the scores for 3 alternativesare s1 � 5=6, s2 � 1=3 and s3 � ÿ1=6, (note that it is the same as the Bordarule for n � 3); and for 4 alternatives are s1 � 2=3, s2 � 1=3, s3 � 1=6 ands4 � ÿ1=6.

By using again the relation sk ÿ sk�1 � pkÿ1 � pk, the Shapley and Bordarules can be compared. Consider two di¨erent pro®les o;o 0 A L�N�V with oand o 0 di¨ering only for one voter u A V , in such a way that for the alternative

6 The core C�v� of a TU-game v (Gillies 1959) is the set of e½cient allocations thatcannot be improved upon by any coalition, i.e.

C�v� � y A RN :Xi AN

yi � v�N� andXi AS

yi V v�S�; S JN

( ):

7 A payo¨ vector which is e½cient and individually rational.

500 E. Calvo et al.

i A N, ou�i� � k and o 0u�i� � k � 1. The variation in the total score obtainedby alternative i in both pro®les is so�i�ÿso 0�i��skÿsk�1. This can be inter-preted as the marginal e¨ect in the total score due to the one-jump variation inthe position of this alternative within the ranking of one individual. For theBorda rule, this marginal impact is the same, regardless of the place in theranking at which this jump is made. However, for the Shapley rule, the impactvaries according to the place at which this jump happens: it is less important ifthe jump happens in the middle part of the ranking and it is more important ifit happens at the top or bottom of the ranking.

4 Least square values

Other interesting family of linear solutions is the family of least square values,which is obtained by minimizing the weighted variance of the coalitionalexcesses on the preimputation set.

De®nition 4 (Ruiz et al. 1995). Let m � �m1; . . . ;mnÿ1� A Rnÿ1 be a vector such

that mi V 0 for all i � 1; . . . ; nÿ 1 and m0 0. The least square value asso-

ciated with the vector m �LSm ÿ value� is the function LS m : G N ! RN where,

for all v A G N , LS m�v� is the solution of the problem:

minx

XS JNS 0N

�e�S; x��2 mjSj

Xn

i�1xi � v�N�

8>>>><>>>>:where e�S; x� � v�S� ÿ x�S�, and x�S� � P

i AS

xi.

Its explicit formula is given by:

LS mi �v� �

v�N�n� 1

annb m

i �v� ÿXn

j�1b m

j �v�" #

;

where b mi �v� �

PS JN;S 0N

i A S

mjSjv�S� for i � 1; . . . ; n and a � Pnÿ1s�1

nÿ 2

sÿ 1

� �ms.

Note that constraintPni�1

xi � v�N� only induces a speci®c normalisation of

the scoring vector:Pni�1

si � 1. ConstraintPni�1

xi � v�N� has a particular sense in

resource allocation, but actually each least square value induces its corre-

sponding rank in the payo¨s of the players. If we putPni�1

xi � k instead ofPni�1

xi � v�N�, the relationship between the new payo¨s Lki �v�, and the former

Scoring rules 501

LS mi �v�, is:

Lki �v� � LS m

i �v� �k ÿ v�N�

n; �i A N; v A G N�:

Ruiz et al. (1995) proved that there exists a close relationship between thisfamily and that of semivalues: the additive e½cient normalisation of anysemivalue is a least square value, that is, given a semivalue c there exists m,given by mi � piÿ1 � pi, i � 1; . . . ; nÿ 1, such that:

LS mi �v� � ci�v� �

1

nv�N� ÿ

Xj AN

cj�v� !

; �i A N; v A G N�:

But the converse is not true: there exist LS m ± values which are not theadditive normalisations of any semivalue. The two families have only one el-ement in common: the Shapley value.

Theorem 3. There exists a bijection between scoring rules and the SWP's asso-

ciated with the family of least square values.

Proof: As the LS m ± values also satisfy linearity and anonymity, we only haveto prove it for a set of only one voter and for the pro®le o� given by theidentity permutation, that is to say, we only have to see that, given a scoringrule with scoring vector s, there exists a unique LS m ± value such thatLS m

i �vo� � � si for i � 1; . . . ; n, and converse.As s can be any element of the equivalence class which represents

the scoring rule, take as s the only representative of the class verifyingPnÿ1i�1

nÿ 2

i ÿ 1

� ��si ÿ si�1� � 1 and

Pni�1

si � 1.

For the game vo� , it holds that bmi �vo� � �

Pnÿ1k�i

mk for all i � 1; . . . ; nÿ 1,

b mn �vo� � � 0 and vo� �N� � 1. We can assume that a � Pnÿ1

s�1

nÿ 2

sÿ 1

� �ms � 1.

So, the system LS mi �vo� � � si, i � 1; . . . ; n, is a system of n linear equations

which can be expressed in a matrix way by Cm � s' where:

C �

nÿ 1 nÿ 2 � � � 2 1

ÿ1 nÿ 2 � � � 2 1

ÿ1 ÿ2 . ..

2 1

..

. ... . .

. ...

ÿ1 ÿ2 � � � ÿ�nÿ 2� 1

ÿ1 ÿ2 � � � ÿ�nÿ 2� ÿ�nÿ 1�

266666666664

377777777775and s 0i � nsi ÿ 1, i � 1; . . . ; n.

It is easy to prove that the rank of C is nÿ 1, and that the system is

solvable if and only ifPni�1

si � 1. Hence, the system has a unique solution,

502 E. Calvo et al.

given by mi � si ÿ si�1 for i � 1; . . . ; nÿ 1. It is straightforward that mi V 0for all i � 1; . . . ; nÿ 1 and there exists j with mj > 0.

In order to prove that converse, we only have to see that the vector s givenby si � LS m

i �vo� � for i � 1; . . . ; n is a scoring vector. Indeed, we have thats1 V s2 V . . . V sn and s1 > sn because si ÿ si�1 � mi=aV 0 for all i � 1; . . . ;nÿ 1. r

Remark 1. In the proof of the previous theorem, we have seen that the rela-tionship between scoring rules and LS m ± values is given by mi � si ÿ si�1 fori � 1; . . . ; nÿ 1. Therefore, there also exists a bijection between the scoringrules which have scoring vector satisfying s1 > s2 > � � � > sn and the LS m ±values associated to m > 0. These scoring rules satisfy Pareto optimality: ifalternative i is unanimously preferred to alternative j, then j should not besocially preferred to i.

Remark 2. Given a scoring rule which satis®es the condition of Theorem 2, therelation between it and the associated semivalue and least square value is mi �siÿsi�1� piÿ1� pi, i � 1; . . . ; nÿ1, therefore, the least square value associatedwith the scoring rule is the additive normalisation of the semivalue. For ex-ample, the least square value associated with the plurality rule is the center of

the imputation set (CIS):

CISi�v� � v�i� � 1

nv�N� ÿ

Xj AN

v� j� !

; �i A N�:

(Although for the game vo, CISi�vo� � vo�i�, i A N, because vo�N� � jV j �Pj AN

vo� j�). And the one associated with the Borda rule is given by m ��1; . . . ; 1� and is called the least square premucleolus. This is the normalisationof the Banzhaf semivalue, and was studied and characterised in Ruiz et al.(1996).

Another example is the Shapley value, because it is an e½cient semi-

value. The Shapley value, written as a least square value, is given by

mi � 1

nÿ 1

nÿ 2

i ÿ 1

� �ÿ1; i � 1; . . . ; nÿ 1.

5 Probabilistic interpretation

Taking into account Theorem 3 we have that, given a scoring rule SR, thereexists a unique LS m ± value such that:

iSR�o� j iff LS mi �vo�VLS m

j �vo�; �i; j A N; o A L�N�V �;or equivalently:

Scoring rules 503

iSR�o� j iffX

S JN;S 0Ni A S

mjSjvo�S�VX

S JN;S 0Nj A S

mjSjvo�S�

�i; j A N; o A L�N�V �: �4�That is to say, the scoring rule ranks the alternatives depending on the numbers

b mi �vo� �

PS JN;S 0N

i A S

mjSjvo�S� for i � 1; . . . ; n. This property characterise the

set of all scoring rules, i.e., any social welfare preordering R, for which thereexists a vector m A Rnÿ1, with mi V 0 for all i � 1; . . . ; nÿ 1 and m0 0, suchthat:

iR�o� j iff b mi �vo�V b m

j �vo�; �i; j A N; o A L�N�V �;is a scoring rule. Moreover, the relation between the scoring vector s and thevector m is given by mi � si ÿ si�1 for i � 1; . . . ; nÿ 1 (assuming a � 1).

Let us see now what is the meaning of the number b mi �vo� and of the re-

lation (4) in the particular cases of the plurality rule, Borda rule, antipluralityrule and Shapley rule.

Plurality rule. The vector m associated is given by m1 � 1 and mi � 0 for alli 0 1. Therefore, we have that:

b mi �vo� � vo�i� � jfu A V=ou�i� � ou� j�; for all j 0 igj:

So, relation (4) means that the plurality rule ranks an alternative depending onthe number of voters who elect it as their favourite.

Borda rule (B). The vector m associated satis®es mi � mj for all i; j � 1; . . . ;nÿ 1 and therefore, in relation (4) we can assume that mi � 1 for all i �1; . . . ; nÿ 1. Hence, we have that:

b mi �vo� �

XS JN;S 0N

i A S

vo�S� �Xk 0 i

pik�o�;

where pik�o� is the number of voters in o preferring i to k. Then, relation (4) isequivalent to:

iB�o� j ,Xk 0 i

pik�o� ÿXk 0 j

pjk�o�V 0; �i; j A N; o A L�N�V �:

Since,Pk 0 i

pik�o� ÿP

k 0 j

pjk�o��1

2

Pk 0 i

�pik�o� ÿ pki�o�� ÿP

k 0 j

�pjk�o� ÿ pkj�o��" #

,

the cancellation property8 of Young (1974) follows from (4). Moreover, wecan say not only when all the alternatives tie, but also when two any alter-

8 Cancellation property: if pij�o� � pji�o� for all i 0 j then a tie between all alternativesis yield.

504 E. Calvo et al.

natives tie: a tie between alternatives i; j is yield if and only ifP

k 0 i

pik�o� �Pk 0 j

pjk�o�.

Antiplurality rule. Its scoring vector is si � 1, for all i 0 n and sn � 0. Then,mi � 0 for all i0 nÿ 1 and mnÿ1 � 1. Therefore, we have that:

b mi �vo� �

Xj 0 i

vo�N n j� � jV j ÿ jfu A V=ou�i� � ou� j� for all j 0 igj;

which means that the smaller the number of voters who rank an alternativethe last, the better the antiplurality rule ranks it.

Shapley rule (S). The vector m associated is given by mi � 1

nÿ 1

nÿ 2

i ÿ 1

� �ÿ1,

i � 1; . . . ; nÿ 1. De®ning b mij �vo� �

PSJN;S 0N

i AS; j BS

mjSjvo�S� �P

SJN;S 0Ni AS; j BS

1

nÿ 1�

nÿ2jSjÿ1

� �ÿ1vo�S�, relation (4) is equivalent to:

iS�o� j iff bmij �vo�V bm

ji �vo�; �i; j A N;o A L�N�V �:

It is possible to yield a probabilistic interpretation of the number b mij �vo�.

For that, recall thatP

S JN;S 0Ni A S; j B S

mjSj � 1 for any least square value. Hence, b mij �

�vo� is the expected number of supporters that alternative i obtains joining aset of alternatives in Nnfi; jg, being mjSj the probability of forming a coalitionS. For the Shapley rule it is assumed that all coalitions of the same size havethe same probability of joining alternative, i, and second, all sizes are equallyprobable.

We can also give an analogous interpretation in the previous cases, butchaning the corresponding probability distribution: for the Borda rule, all thecoalitions joining i are equally probable, for the plurality rule, only coalitionfig can be formed and, in the antiplurality rule, alternative i only can joinN nfi; jg.

6 Scoring rules on P�N�V

In the following de®nition, we show a way to extend the scoring rules toP�N�V by generalising relation (4).

De®nition 5. Given SR a scoring rule with scoring vector s, we de®ne its exten-

sion to P�N�V in the following way:

Scoring rules 505

iSR�o� j iffX

SJN;S 0Ni AS

mjSjvo�S�VX

SJN;S 0Nj AS

mjSjvo�S�;

�i; j A N;o A P�N�V �;where mi � si ÿ si�1 for i � 1; . . . ; nÿ 1.

Or equivalently, in terms of scores, the total score obtained by alternative i inthe pro®le o is so�i� � b m

i �vo� �P

u AV

bmi �vou

�.

Example. Suppose that the preferences of a voter u A V are the natural orderin N except for a tie between iÿ1 and i, that is, oa�1�>oa�2� � � �>oa�iÿ1�@oa�i� > oa�i � 1� > � � � > oa�n�9. Then:

bmj �vou

� �Xnÿ1k�j

mk �miÿ1 �Xnÿ1k�j

�sk ÿ sk�1� � siÿ1 ÿ si

� sj ÿ sn � siÿ1 ÿ si; for 1U j U i ÿ 2

bmiÿ1�vou

� �Xnÿ1

k�iÿ1mk �

Xnÿ1k�iÿ1

�sk ÿ sk�1� � siÿ1 ÿ sn

bmi �vou

� �Xnÿ1

k�iÿ1mk � bm

iÿ1�vou�

bmj �vou

� �Xnÿ1k�j

mk �Xnÿ1k�j

�sk ÿ sk�1� � sj ÿ sn; for i � 1U j U nÿ 1

bmn �vou

� � 0

Or, equivalently, adding to all of them the quantity m � sn, the scores that thescoring rule assigns are:

sou� j� � sj � �siÿ1 ÿ si�; for j � 1; . . . ; i ÿ 2

sou�i ÿ 1� � sou

�i� � siÿ1

sou� j� � sj; for j � i � 1; . . . ; n

That is, comparing with the situation in which there is not a tie, for all thealternatives 1U j U i ÿ 2 and i, their scores are increased in the quantityh � siÿ1 ÿ si, and the rest of the alternatives keep their scores.

This extension is di¨erent form the average extension, which, in the previousexample, keeps the scores of all alternatives except for i ÿ 1 and i, which givesthem the average between their scores siÿ1 and si.

A property that least square values satisfy is:

Symmetry: For all i; j A N such that v�S W fig� � v�S W f jg�, ES HNnfi; jg,then fi�v� � fj�v�.

9 ou�i�@ou� j� means that alternative i is indi¨erent to alternative j for agent u.

506 E. Calvo et al.

In this case, players i; j are said substitutes in the game v, for this reasonthis property is also called equal treatment for substitute players. In our con-text this property can be rewritten as:

Equal treatment: For any o A P�N�V and any i; j A N, if vo�S W fig� �vo�S W f jg�, for all S HNnfi; jg, then i and j tie in R�o�.

The interpretation of this property is the following: if whenever two alter-natives i; j join other alternatives in N nfi; jg, they obtain the same number ofsupporters, then, i and j must be socially indi¨erent.

It is straightforward that any SR extended to P�N�V in the way ofDe®nition 5 preserves this property, due to the fact that least square valuessatisfy symmetry. Now, we are going to show that the average extension of

scoring rules do not verify equal treatment on P�N�V . For that, consider as

counterexample, the following pro®le o A P�N�V with N � f1; 2; 3g and V �fu; u 0g:

ou ou 0

1; 2 3

3 1; 2

that is, ou�1�@ou�2� > ou�3� and ou 0 �3� > ou 0 �1�@ou 0 �2�, and the scoringrule SR given by s1 � 10, s2 � 7 and s3 � 1. It is easily checked that:

vo�1� � vo�2� � vo�3� � 1

vo�12� � vo�13� � vo�23� � 1

So, candidates 1, 2 and 3 are substitutes and, therefore, they must tie in SR�o�.Nevertheless, with the average extension, we have that:

sou�1� � sou

�2� � 10�72� 8:5; sou

�3� � 1

sou 0 �3� � 10; sou 0 �1� � sou 0 �2� � 7�12� 4

)

) so�1� � so�2� � 8:5� 4 � 12:5; so�3� � 10� 1 � 11

That is, 1 and 2 tie in ®rst position and 3 is ranked second.

Remark 3. Approval voting is a system in which a voter can vote for any

number of candidates. The de®nition of scoring rule here excludes approvalvoting, but, in some way, we can consider it as the extension of the pluralityrule to P�N�V . That is to say, suppose that the set of alternatives is fa; b; c; dgand a voter vote for fb; cg. This is equivalent to applying plurality rule assumingthat the preferences of this voter are, for example, b@ c � a � d, because thusb and c obtain one point and a and d no point.

7 Variable number of alternatives

Up to now, we have considered the number of alternatives as ®xed, but we canstudy the problem when the number of alternatives changes.

Scoring rules 507

Suppose that we give a voting method to establish an order amongalternatives. It is reasonable that this method ought to be well speci®ed forany number of alternatives presented.

Let U be an in®nite set which represents the universe of alternatives. Let ussee how to extend the de®nitions of Sects. 2 and 3 to this context.

Let L�U� � QAHUA finite

L�A� and P�U� � QAHUA finite

P�A�. A preference pro®le on V

is a function o : V ! L�U� where for all u A V , o�u� � foA�u�gAHUA finite

satis®es

the following coherent condition: for each pair of ®nite sets A;BHU suchthat jAXBjV 2, and for all ak; aj A AXB, it is satis®ed that ak is preferred toaj in oA�u� if and only if ak is preferred to aj in oB�u�. L�U ;V� denotes the setof all preference pro®les on V. Hence, we have that L�U ;V�HL�U�V �QAHUA finite

L�A�V .

A social welfare preordering is the restriction to L�U ;V� of a map R :L�U�V ! P�U� such that R � Q

AHUA finite

RA, where RA : L�A�V ! P�A�.

A scoring vector on U is a family fsngn AN such that, for all n A N, it issatis®ed that sn A Rn, sn

1 V � � � V snn , and sn

1 > snn . The associated scoring rule

is de®ned as SR � QAHUA finite

SRA where for each ®nite set AHU , SRA is the

scoring rule given by the vector sjAj A RjAj.A game on U is given by a characteristic function v : 2U ! R with v�q� �

0. A set AHU is a support of v if, for each S HU , v�S� � v�S XA�. A ®nitegame is a game that has a ®nite support. We denote by G the set of all ®nitegames, and by G A the subset of G consisting of games with support A. A co-operative solution on G is a function f : G ! 6

n AN

Rn such that f�v� A RjAj

whenever v A G A, for all ®nite set AHU . The SWP associated to f is de®nedas follows: for each o A L�U ;V� and for each ®nite set AHU ,

i�Rf�A�oA� j iff fi�voA�V fj�voA

�; for all i; j A A

The de®nition of a semivalue on G is also due to Dubey et al. (1981), whogave the following characterisation. Let fpngn AN be a family of vectors such

that, for all n A N, pn A Rn, pn V 0,Pnÿ1i�0

nÿ 1

i

� �pn

i � 1 and pni � pn�1

i �pn�1

i�1 10, for all i � 0; . . . ; nÿ 1. The cooperative solution de®ned, for each®nite set AHU with jAj � n, by:

ci�v� �X

S JA

pnjSj�v�S W i� ÿ v�S��;

is a semivalue on G. And, moreover, every semivalue on G is of this form.

10 For each family fpngn AN in such a way, there exists a probability measure x on

�0; 1� such that pni �

� 10 t i�1ÿ t�nÿiÿ1

dx�t�, for all i A N and n A N, and conversely.

508 E. Calvo et al.

Taking into account Theorem 1, it is straightforward that if c is asemivalue on G, Rc is a scoring rule on U. The next theorem establishes theconditions for the converse.

Theorem 4. Let fsngn AN be a scoring vector on U satisfying for each n A N:

i)Pnÿ1i�1

nÿ 2

i ÿ 1

� ��sn

i ÿ sni�1� � 1,

ii) snn � sn�1

n ÿ sn�12

iii) sn�1i ÿ sn�1

i�2 � sni ÿ sn

i�1, for all i � 1; . . . ; nÿ 1

Then, there exists a semivalue on G such that the SWP associated to it is the

scoring rule given by fsngn AN.

Proof: For each n A N, let A � fa1; . . . ; ang any subset of U with jAj � n. Weidentify the element ai with i, i � 1; . . . ; n, and let o� A L�A� the preferencesrepresented by the identity permutation. Because of the linearity and anonymityof semivalues, we only have to prove that there exists a semivalue c such that,for each n A N, ci�vo� � � sn

i , for i � 1; . . . ; n.

Consider, for each n A N, the system ci�vo� � � sni , for i � 1; . . . ; n. We are

going to use the proof of Theorem 2:

a) If n is odd, the system has a unique solution pn A Rn, which satis®esPnÿ1i�0

nÿ 1

i

� �pn

i � 1 because of condition i). Its explicit form is:

pn0 �

Xn

j�1�ÿ1� j�1

snj ;

pni �

Xi

j�2�ÿ1� j�i

snj �

Xn

j�i�2�ÿ1� j�i�1

snj ; for i V 1:

Using condition iii), we obtain:

pni � sn�1

i�1 ÿ sn�1i�2 � �ÿ1� i�sn

n � sn�12 ÿ sn�1

n �; for i � 0; . . . ; nÿ 1:

And condition ii) implies that pni � sn�1

i�1 ÿ sn�1i�2 , for i � 0; . . . ; nÿ 1.

b) If n is even, the system is solvable if and only if snn �

Pnÿ1i�2�ÿ1� i�1sn

i . This

condition is equivalent to condition ii), using iii):

snn �

Xnÿ1i�2�ÿ1� i�1sn

i �Xnÿ2i�2

i even

�ÿsn�1i � sn�1

i�2 � � ÿsn�12 � sn�1

n :

Then, the solution of the system is a 1-dimensional a½ne subspace (depending

on pn0 ). Moreover, each solution vector pn satis®es

Pnÿ1i�0

nÿ 1

i

� �pn

i � 1. Take

Scoring rules 509

the solution pn such that pn0 � sn�1

1 ÿ sn�12 . Then:

pni � ÿsn

i�1 � �ÿ1� i 2Xi

j�2�ÿ1� jsn

j ÿ sn1 � sn�1

1 ÿ sn�12

!; for iV 1:

And, using condition iii), we obtain pni � sn�1

i�1 ÿ sn�2i�2 , for i V 1.

So, we have that, for each n A N, the solution of the system is pni � sn�1

i�1 ÿsn�1

i�2 , for i � 0; . . . ; nÿ 1. Clearly, pni V 011 and, from iii), it is also satis®ed

that pni � pn�1

i � pn�1i�1 , for all i � 0; . . . ; nÿ 1. Hence, the family fpngn AN

de®nes a semivalue on G such that, for each n A N, ci�vo� � sni , for i �

1; . . . ; n. r

Conditions i) and ii) of Theorem 4 are merely normalisation conditions inorder to determine the representative scoring vector of the equivalence class.Condition iii) is the restriction which guarantees the representation of thescoring rule by a semivalue on G, but its expression would be di¨erent andmore complicated if conditions i) and ii) were not speci®ed.

A scoring rule on L�U ;V� which satis®es conditions of Theorem 4 induces,for each n A N, a scoring rule on L�N�V which corresponds to a semivalue onG N . But the converse is not true: there exist scoring rules that, for each n A N,are associated to a semivalue on G N but, as SWP on L�U ;V�, are not relatedwith any semivalue on G, that is to say, the family fpngn AN do not satis®espn

i � pn�1i � pn�1

i�1 for all i � 0; . . . ; nÿ 1 and all n A N.As examples of scoring vectors satisfying conditions of Theorem 4 we

have:

1. Plurality rule: sn1 � 1 and sn

i � 0, for all i � 2; . . . ; n and n A N.

2. Borda rule: sn1 �

n

2nÿ1 and sni �

nÿ 2i � 2

2nÿ1 , for all i � 2; . . . ; n and n A N.

3. Shapley rule: sn1 �

1

n

Pnÿ1j�0

nÿ 1

j

� �ÿ1and sn

i �1

n

� Pnÿ1j�i

nÿ 1

j

� �ÿ1ÿPiÿ2

j�1�

nÿ 1

j

� �ÿ1�, for all i � 2; . . . ; n and n A N.

On the other hand, given fsngn AN a scoring vector on U, we have seen inSect. 4 that, for each n A N, the scoring rule given by sn is associated to aunique least square value de®ned by mn

i � sni ÿ sn

i�1, i � 1; . . . ; nÿ 1. So, thescoring vector fsngn AN gives us a family fmngn AN that de®nes a least squarevalue for any number of alternatives. Furthermore, if fsngn AN satis®esconditions i) and iii) of Theorem 4, then fmngn AN veri®es the followingconditions:

11 In this case, we do not need the condition of Theorem 2 to guarantee the non-negativity. Conditions ii) and iii) of Theorem 4 imply that pn

i � sn�1i�1 ÿ sn�1

i�2 , fori � 0; . . . ; nÿ 1 and n A N, is a solution of the system and, therefore, a non-negativesolution.

510 E. Calvo et al.

a) mni � mn�1

i �mn�1i�1 , for all i � 1; . . . ; nÿ 1 and n A N

b)Pnÿ1i�1

nÿ 2

i ÿ 1

� �mn

i � 1, for all n A N

Ruiz et al. (1995) called it a consistent least square value, and proved that anyconsistent LS-value is the additive e½cient normalisation of some semivalueon G, and conversely, the normalisation of a semivalue on G is a consistentLS-value. Therefore, a scoring rule that veri®es conditions of Theorem 4 iscalled a consistent scoring rule.

8 Concluding remarks

(1) Using an axiomatic approach, it could be of interest to complement theanalysis done here. In particular, which set of axioms yields the charac-terisation of the Shapley rule? Dubey et al. (1981) have an axiomatizationof the family of semivalues on G, Ruiz et al. (1995) also have an axioma-tization of the family of consistent least square values on G. An analogousstudy should be done for the family of consistent scoring rules de®ned onL�U ;V�.

(2) A lot of work has been done in the framework of voting12. Possibly, byusing the game vo as a bridge, some of these sophisticated voting rulescould introduce interesting new solution concepts into the cooperativeGame Theory setting.

References

1. Arrow KJ (1963) Social Choice and individual values, 2nd. edn. John Wiley andSons, New York

2. Banzhaf JF (1965) Weighted voting doesn't work: a mathematical analysis. RutgersLaw Rev. 19: 317±343

3. Borda JC (1781) MeÂmoire sur les eÂlections au scrutin. Histoire de l'AcademieRoyale des Sciences, Paris

4. Dubey P, Neyman A, Weber RJ (1981) Value Theory without e½ciency. MathOperat Res 6: 122±128

5. Gillies DB (1959) Solutions to General Non-Zero-Sum Games. In: Tucker AW,Luce RD (eds) Contributions to the Theory of Games, Volume IV (Annals ofMathematical Studies, 40) Princeton University Press, Princeton, pp 47±85

6. Hart S, Mas-Colell A (1996) Bargaining and Value. Econometrica 64: 357±3807. Moulin H (1994) Social Choice. In: Aumann RJ, Hart S (eds) Handbook of Game

Theory, Vol 2. Elsevier Science, Amsterdam, pp 1091±11258. Nash JF (1951) Non-Cooperative Games. Ann Math 54: 286±2959. Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica

50: 97±109

12 A good survey of this literature can be found in Moulin (1994).

Scoring rules 511

10. Ruiz L, Valenciano F, Zarzuelo JM (1995) The family of least square values forTU games. Working paper Economia Aplicada No. 5/1995, Universidad del PaõÂsVasco/EHU, Bilbao. Games Econ Beh (forthcoming)

11. Ruiz L, Valenciano F, Zarzuelo JM (1996) The Least Square Prenucleolus and theLeast Square Nucleolus. Two Values for TU Games based on the Excess Vector.Int J Game Theory 25: 113±134

12. Shapley LS (1953) A value for n-Person Games. Ann Math Study 28: 307±31713. Young HP (1974) An Axiomatization of Borda's rule. J Econ Theory 9: 43±52

512 E. Calvo et al.