Mathematical Social Sciences 63 (2012) 193–196

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Mathematical Social Sciences

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A general scoring rule✩

Wulf Gaertner a,∗, Yongsheng Xu b

a University of Osnabrück, Germanyb Georgia State University, United States

a r t i c l e i n f o

Article history:Received 19 May 2011Received in revised form14 December 2011Accepted 14 January 2012Available online 24 January 2012

a b s t r a c t

This paper studies a ranking rule of the following type axiomatically: each voter places k candidates into ncategorieswith ranks from n to 1 attached to these categories, the candidate(s) with the highest aggregatescore is (are) the winner(s). We show that it is characterized by a monotonicity condition and a multi-stage cancellation property.

© 2012 Elsevier B.V. All rights reserved.

1. An introductory example

Imagine that you are one of the members of a committee thathas to decide among a certain number of research applications forfunding. Let us suppose that k proposals were submitted. Let usfurther assume that at the end of a longer discussion, the chairmanof your committee comes forward with the following procedure.He or she declares that there are n categories (from excellent to fail,let us say, with n − 2 categories in between), with ranks from n to1 attached to these categories. The chairperson asks all membersof your committee to allocate the k proposals to the n availablecategories. Furthermore, the chairperson announces that, as soonas eachmember has assigned the k applications to the n categories,he or she would count the rank numbers assigned to each proposaland then construct a ranking over the k proposals from the highestrank sum to the lowest, the proposal with the highest aggregatesum being the winner, though more than one proposal may beselected depending on the available budget.

In this paper, we study the aggregation rule illustratedabove axiomatically. In particular, we will show that thisrule is characterized by two simple properties, monotonicityand cancellation independence. Monotonicity requires that, ifeveryone ranks an alternative x no lower than another alternativey, then the group must rank x no lower than y, and if in addition,some individual ranks xhigher than y, then xmust be rankedhigherthan y by the group. The essence of cancellation independenceis that, the pair-wise comparison of alternatives depends only on

✩ We thank Wolfgang Leininger for raising a question that led to this paper. Forhelpful discussions and detailed comments, we are grateful to Nick Baigent, MarcusPivato, Don Saari and Harrie de Swart. We also wish to thank two referees and theeditor for various very helpful suggestions.∗ Corresponding author.

E-mail address:wulf.gaertner@uni-osnabrueck.de (W. Gaertner).

0165-4896/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.mathsocsci.2012.01.006

the rank differences. For example, consider two alternatives x andy, and two situations. In situation I, individual i places x three‘grades’ above y while individual j places y two grades above x,while in situation II, individual i places x one grade above y andindividual j assigns y and x the same rank, individual i’s placementof x in situation II is the same as individual i’s placement of x insituation I, and individual j’s placement of y in situation II is thesame as individual j’s placement of y in situation I, while all theother individuals’ grades of x and y in situation II remain the sameas their grades in situation I. Then, the group ranking of x and y inthe two situations must be the same.

We note that, if n were equal to k and all members ofthe committee had strict orderings over the proposals, thechairperson’s ranking procedure would be the well-known Bordamethod. But note that the chairperson did not require that everymember come up with a strict ordering. Nor did he or she requirethat all categories be filled by each and every committee member.So one of the members could, for example, decide to assign k

2proposals to the category ‘‘good’’ and k

2 proposals to the category‘‘acceptable’’, leaving thehighest rank, the lowest rank and all othercategories void. Then this is definitely not the canonical Bordarank-order method.

2. Grades, ranks and scores

In the Arrovian framework of social choice theory, each voter isassumed to provide a complete ranking of the alternatives at stake.Arrow’s social welfare function (Arrow, 1951, 1963) is a mappingfrom the set of all logically possible profiles of individual orderingsover a set of given alternatives into the set of all orderings overthese alternatives. In scoring rules like the Borda rule, voters alsoexpress a rank ordering of the alternatives or candidates on theirballots. This is different under plurality voting where a voter canassign one point to only one alternative, his or her most preferredcandidate, and zero points to all the other alternatives. In approval

194 W. Gaertner, Y. Xu / Mathematical Social Sciences 63 (2012) 193–196

voting (Brams and Fishburn, 1983), an agent can assign one orzero points to each given alternative independently and thereis no restriction on how many alternatives receive one point onthe voter’s ballot. So under both plurality and approval voting, aballot is not a (strict) ranking of the available options. Therefore,expressing such a rank ordering is by no means necessary in orderto make social choices (see also Zwicker, 2008).

Actually, apart from plurality and approval voting, there isa small literature which discusses exactly the following: votersattach scores to alternatives, and the way they do this canvary substantially among them. Balinski and Laraki (2007) haveproposed a social grading function that for each alternative focuseson its median grade and chooses the alternative with the bestmedian evaluation (see also Bassett and Persky, 1999). Moreprecisely, if, for any alternative, an odd number of grades issubmitted, the median grade is picked. For the case of an evennumber of grades, the lower of the two median grades is chosen.Felsenthal and Machover (2008) and Zahid and de Swart (2010)have shown that the Balinski–Laraki grading function can yieldcounter-intuitive results, not only in the case that the lower ofthe two medians is picked but also more generally.1 For example,a condition that is sometimes referred to as consistency, at othertimes as reinforcement (Young, 1974, 1975) meaning that if thereare two separate parts of an electorate and a certain candidatewins in both electorates, then this candidate must also win in theaggregate electorate, may not be satisfied by the Balinski–Larakigrading function. Zahid and de Swart take sides with a methodcalled range voting that focuses on the absolute rank sum (orthe average aggregate rank) of each candidate, but for reasons ofmanipulability, theywant to restrict the interval of possible gradesto a much narrower interval than was originally proposed by oneof the originators of range voting, Smith.2 Unfortunately, Smithdoes not come forward with a characterization of his aggregationscheme nor do Zahid and de Swart provide one for their modifiedproposal that they call Borda majority count.

Young (1974) has a result where scoring functions arecharacterized by anonymity, neutrality and consistency (orreinforcement) when voters have complete rankings of theunderlying alternatives. Myerson (1995) drops the assumptionthat ballots consist of rank orderings. This author restricts the setof possible votes of an agent to a nonempty finite set. In such acontext, a voting rule is intended to provide a nonempty subsetof candidates as the chosen set and Myerson shows that if thisvoting rule satisfies neutrality, reinforcement (consistency) and acondition called overwhelming majority, it is representable as ascoring rule. Overwhelmingmajoritymeans that if the set of voterscan be partitioned into n + 1 subsets, n of which submit the samevote distribution yielding a particular result while the remainingsubset has a different vote distribution, then, when n becomes verylarge, the latter subset becomes a very small part of the overallelectorate and therefore should not overturn the decision of then subsets with the same vote distribution.3

Let us return to our introductory example. It seems to be arather straightforward and transparent voting method, apparentlywidely applied in committees and other smaller bodies. Theinteresting point about the described procedure is that since eachand every committee member is totally free in the way how toassign candidates to categories, this is tantamount to saying that

1 Another critical analysis of choosing the best median evaluation and acomparison of this principle with majoritarian ideas can be found in Laslier (2011).2 Smith entertains a lively web site under rangevoting.org.3 It should be noted that both Young (1974) and Myerson (1995) work in a

framework with a variable electorate, while in our framework, the electorate isfixed. See our next section.

each and every voter chooses his or her own personal assignmentof scores so that the profile of rank assignments by all voters isbased on a list of individual scoring functions. This feature rendersthe present procedure quite different from the Borda method orsome of the other known scoring functions.4

In recent years there have been a couple of field experimentsconducted along the French presidential elections in 2002 and2007. Some of these experiments were based on Balinski andLaraki’s grading method, others examined voters’ opinions onthe basis of approval voting and evaluation voting with a simple(2, 1, 0) scale. All these voting schemes are in contrast to theactually applied two-round plurality vote used in France. Notsurprisingly, these schemes ‘‘produced’’ results which differedquite substantially from the official French election results. Sinceour contribution has a different focus, we abstain from providingdetailed findings but refer, for a fairly recent survey, to Baujard andIgersheim (2010).

We shall provide a very simple and rather straightforwardcharacterization of scoring rules that largely takes advantage of theobservation that for any pair of alternatives or candidates, rank orgrade differences of opposite sign can be appropriately reduced.This fact could also have been employed in a characterization ofthe Borda method or even in a characterization of the utilitarianrule (see e.g. Maskin, 1978) though the underlying bits of (ranking)information are quite different in either case and the latter rulewould require that some form of continuity be introduced.

3. Basic notation, definitions and axioms

Let X be the universal set of alternatives containing a finitenumber of elements, and let N be the set of n individuals withn > 1. Let E = {1, . . . , E} with E > 1 be a set of given positiveintegers from 1 to E. Note that instead of set E of positive integers,we could have proposed a scoring interval between 1 and 10, letus say, or could have proposed the zero–one interval of the reals.This would not affect the theoretical analysis that follows. Also,the choice of E is independent of the number of alternatives to beevaluated.

Each individual i ∈ N chooses a scoring function si : X → Esuch that, for all x ∈ X , si(x) indicates the score that i assignsto x. Let Si be the set of all possible scoring functions of individuali. The assignment of ranks according to si(·) clearly is determinedby some underlying preference ordering of agent i. If this orderingwere purely ordinal, a multitude of rank assignments wouldcorrespond to the same preference ordering.

However, given set E our rank assignment rule allows the agentto express preference intensities. So if i prefers candidate a tocandidate b, this can be expressed by assigning a and b to adjacentranks or categories or by putting a several categories above b. Theagent is totally free in deciding how many categories or ranksfinally are between the two alternatives, given that there areenough categories to be potentially filledwith a and b, respectively.Only the case of indifference forces the agent to assign equivalentobjects to the same rank position.

So different persons can express a strict preference for a over b,let us say, in many different ways, given that set E is rich enough.This, of course, opens the way to manipulation or dishonestvoting. Overstating a given strict preference for a over b by not

4 To a certain degree, so-called relative utilitarianism uses individual scoringfunctions where each individual chooses a number between 0 and 1 with 0indicating the worst alternative and 1 representing the best alternative. Theaxiomatic derivations of relative utilitarianism found in the literature (see, forexample, Dhillon and Mertens (1999) and Segal (2000)), however, use the Arrovianframework in which individuals’ preferences over lotteries are representable bynormalized von Neumann–Morgenstern utilities between 0 and 1.

W. Gaertner, Y. Xu / Mathematical Social Sciences 63 (2012) 193–196 195

allocating the two options to adjacent categories but by leavingmany categories blank between a and b will hurt b and the richerset E, the more possibilities an agent has to overstate his or herpreference. So restricting the cardinality of set E, as suggestedby Zahid and de Swart in their own set-up, is a way to limitthe possibility of manipulation without, however, being able toeliminate strategic voting altogether.

Henceforth, we shall leave aside the issue of strategicallyranking candidates or alternatives and assume that voters acthonestly.5

Let ℘ be the set of all orderings over X .A profile s = (s1, . . . , sn) is a list of scoring functions, one for

each individual. An aggregation rule f is defined as a mapping:S1 × · · · × Sn → ℘. Let S = S1 × · · · × Sn. f is said to be anE-based scoring rule, to be denoted by fE , iff, for any s ∈ S, and anyx, y ∈ X , x % y ⇔

i∈N si(x) ≥

i∈N si(y), where %= f (s). The

asymmetric and symmetric parts of % will be denoted by ≻ and ∼,respectively.

For any s, s′ ∈ S, any i, j ∈ N and any x, y ∈ X , we say that sand s′ are (i, j)-variant with respect to (x, y) if sk(x) = s′k(x) andsk(y) = s′k(y) for all k ∈ N − {i, j}.

We now introduce two properties to be imposed on anaggregation rule f .Monotonicity. For all s ∈ S and all x, y ∈ X , if si(x) ≥ si(y) for alli ∈ N then x % y and if si(x) ≥ si(y) for all i ∈ N and sj(x) > sj(y)for some j ∈ N , then x ≻ y.

Monotonicity is fairly straightforward. It requires that, inranking two alternatives x and y, if the score assigned to x by eachindividual i in N is at least as great as the score assigned to y by thesame individual i, then xmust be ranked at least as high as y by thegroup, and, if in addition, some individual assigns a higher score tox than to y, then x must be ranked higher than y by the group.Cancellation independence (CI). For all s, s′ ∈ S, all x, y ∈ X andall i, j ∈ N , if s and s′ are (i, j)–variant with respect to (x, y),si(x) − si(y) = a, sj(y) − sj(x) = b, s′i(x) = si(x), s′j(y) = sj(y),s′i(y) = si(y)+γ and s′j(x) = sj(x)+γ where γ = min(a, b) whena ≥ 0 and b ≥ 0 and γ = max(a, b) when a < 0 and b < 0, thenx % y ↔ x%′ y, where %= f (s) and %′

= f (s′).Cancellation independence makes use of the fact, as stated

earlier, that for any pair of alternatives, rank differences of oppositesign can be reduced without changing the aggregate outcome ofthe grading procedure. So in condition CI, vectors s and s′ definescoring profiles that are aggregate-rank equivalent with respect toany x, y ∈ X . We call s′ an s-reduced scoring profile. Condition CItherefore requires that f (s) and f (s′) order x and y in exactly thesame way. Note that if a = b, we obtain s′h(x) = s′h(y) for h ∈

{i, j}. Note also that the conditions of anonymity and neutrality areeffectively built into the cancellation procedure described above.Note furthermore that in relation to our introductory example, aand bwould be integer-valued. However, as stated before, this doesnot necessarily have to be the case.

Condition CI makes an implicit assumption about ‘‘inter-personal comparisons of scores’’ which is crucial in utilitarianism:here CI requires that if, when comparing two alternatives x and ythat are ranked opposite by two voters, both voters raise the scoresfor their respective lower ranked alternative by the same numberof scores, then the group ranking over the two alternatives shouldnot change. Also condition CI applies trivially to individuals whoare indifferent between any two alternatives. These individualsdo not matter and can, therefore, be disregarded (separability ofindifferent voters).

5 Núñez and Laslier (2011) study the concept of strategic equivalence of votingrules and show that approval voting and evaluative voting are strategicallyequivalent. The same holds for plurality voting and cumulative voting.

4. A characterization

In this section, we present a simple characterization of theE-based aggregation rule.

Theorem. f = fE if and only if f satisfies monotonicity and cancel-lation independence.

Proof. It can be checked that fE satisfies monotonicity andcancellation independence. In what follows, we shall show that if fsatisfiesmonotonicity and cancellation independence, then itmustbe fE .

Let f satisfy monotonicity and cancellation independence. Lets ∈ S and x, y ∈ X , and %= f (s). We shall show that (i) if

i∈N

si(x) =

i∈N si(y), then x ∼ y, and (ii) if

i∈N si(x) >

i∈N si(y),then x ≻ y. Note that, for any s ∈ S,

i∈N si(x) is a number, and

the binary relation ≥ over the set of all real numbers is reflexive,transitive and complete. Once (i) and (ii) are shown to hold, thefollowing follows easily from the preceding remark: for all s ∈ Sand x, y ∈ X , (iii) x ∼ y ⇒

i∈N si(x) =

i∈N si(y), and (iv)

x ≻ y ⇒

i∈N si(x) >

i∈N si(y).Consider the case,

i∈N si(x) =

i∈N si(y), first. If si(x) = si(y)

for all i ∈ N , then, by monotonicity, we have x % y and y % ximplying that x ∼ y. Suppose it is not the case that [si(x) = si(y)for all i ∈ N]. Then, there must exist two individuals j, k ∈ N suchthat sj(x) > sj(y) and sk(y) > sk(x).

Without loss of generality, let us assume that sj(x) − sj(y) =

ao ≥ 0 and sk(y) − sk(x) = bo ≥ 0. Consider s1 ∈ S such thats and s1 are (j, k)-variant. Furthermore, let s1j (y) = sj(y) + γ1 ands1k(x) = sk(x)+γ1, where γ1 = min(ao, bo). From the construction,it is clear that

i∈N s1i (x) =

i∈N s1i (y), either s1j (x) = s1j (y) or

s1k(x) = s1k(y). By cancellation independence, x%1 y ↔ x % y,where %1

= f (s1). If already at this stage, s1i (x) = s1i (y) for alli ∈ N , then by monotonicity, x∼

1 y implying x ∼ y. If it is notthe case that [s1i (x) = s1i (y) for all i ∈ N], then, by repeatingthe above argument p more steps if necessary, there must existsp ∈ S such that, sp and sp−1 are (jp, kp)-variant for some pair ofvoters jp, kp ∈ N , spjp(y) = sp−1

jp (y) + γp and spkp(x) = sp−1kp (x) + γp

where γp = min(ap−1, bp−1), and [spi (x) = spi (y) for all i ∈ N]. Bycancellation independence, x%p y ↔ x%p−1 y. Note that from thepreceding steps, we have x%p−1 y ↔ x%p−2 y, . . ., x%1 y ↔ x % y.By monotonicity and from [spi (x) = spi (y) for all i ∈ N], we getx∼

p y, and, therefore, x ∼ y.To complete the proof, we consider case (ii):

i∈N si(x) >

i∈N si(y). By employing an analogous proof strategy and giventhat

i∈N si(x) >

i∈N si(y), from cancellation independence,

there must exist s′ ∈ S such that [s′i(x) ≥ s′i(y) for all i ∈ N ands′i(x) > s′i(y) for some i ∈ N] and x%′ y ↔ x % y. By monotonicity,x≻

′ y. Therefore, x ≻ y. �

5. Concluding remarks

The basic idea that underlies the proof of our theorem, is tomodify and simplify an originally given ranking profile over severalconsecutive steps in such a way that the relation between theaggregate rank sums of any two alternatives x and y, let us say,is left unchanged6 (while the aggregate rank sums for the twoalternatives themselves continually change from step to step). Thisis done up to a point where the monotonicity condition as definedcan be ‘‘successfully’’ applied. It is easy to see that our scoring rule

6 This is somewhat reminiscent of the reduction procedures that were proposedby Gaertner and Heinecke (1978) and, independently, by Slutsky (1977) in relationto the simple majority rule.

196 W. Gaertner, Y. Xu / Mathematical Social Sciences 63 (2012) 193–196

satisfies Young’s reinforcement or consistency condition as well asMyerson’s overwhelmingmajority property suitably formulated inour framework with a fixed electorate.

There have been several characterizations of the linear Bordarule in the literature. Young (1974) used the properties of neu-trality, consistency, faithfulness and the cancellation property inorder to characterize this rule. Nitzan and Rubinstein (1981) usedneutrality, consistency, a monotonicity property and cancellationfor their characterization. The latter axiom says that if, wheneverthere is a profile such that for all pairs of alternatives (ai, aj), thenumber of voters preferring ai to aj equals the number who pre-fer aj to ai, then all alternatives are ranked equally. This property isnot in general satisfied by our rule except for the special case thatall individual scoring functions are identical. Consider the case thatk−2 voters are indifferent between ai, aj and ak, person i prefers aito aj to ak, person j has the opposite ranking, person i’s ranking overthe three alternatives is spread over three consecutive categoriesor ranks while j’s ranking over the three alternatives is spread overfour ranks. Then our rule does not yield complete social indiffer-ence among the three alternatives.

Gärdenfors (1973) proposed a different set of axioms. Heshowed that for at least three alternatives and at least threevoters, neutrality, strong monotonicity and stability characterizethe Borda rule. The stability condition which is a bit lengthy todefine prevents radical changes in the social rankings. According toGärdenfors, two preference profiles a and b are almost equal withrespect to alternative x iff there is at most one person i and at mostone alternative z such that i’s ranking in a is the same as in b on theset A−{z} and either i prefers z to x in a and is indifferent betweenthem in profile b or is indifferent between them in a and prefers xto z in b. Stability according to Gärdenfors is satisfied by a votingfunction or scoring rule iff it is not possible that y is preferred to x inprofile a and x preferred to y in b for any y = z, where z is as givenabove, whenever profiles a and b are almost equal with respect tox. It is very easy to construct two ranking profiles such that stabilityis violated under our general ranking rule. One just has to put y and

x several ranks apart in profile a with y preferred to x and let therank difference between y and x shrink appropriately in profile b.

References

Arrow, K., 1951,1963. Social Choice and Individual Values. JohnWiley and Sons, NewYork.

Balinski, M., Laraki, R., 2007. A theory of measuring, electing, and ranking.Proceedings of the National Academy of Sciences (PNAS) 104, 8720–8725.

Bassett Jr., G.W., Persky, J., 1999. Robust voting. Public Choice 99, 299–310.Baujard, A., Igersheim, H., 2010. Framed field experiments on approval voting:

lessons from the 2002 and 2007 French presidential elections. In: Laslier, J.-F.,Sanver, R. (Eds.), Handbook on Approval Voting. Springer-Verlag, Heidelberg,Berlin, pp. 357–395.

Brams, S.J., Fishburn, P.C., 1983. Approval Voting. Birkhäuser, Boston.Dhillon, A., Mertens, J.-F., 1999. Relative utilitarianism. Econometrica 67, 471–498.Felsenthal, D.S., Machover, M., 2008. The majority judgement voting procedure: a

critical evaluation. Homo oeconomicus 25, 319–334.Gaertner, W., Heinecke, A., 1978. Cyclically mixed preferences: a necessary

and sufficient condition for transitivity of the social preference relation.In: Gottinger, H.W., Leinfellner, W. (Eds.), Decision Theory and Social Ethics,Issues in Social Choice. Reidel, Dordrecht, pp. 169–185.

Gärdenfors, P., 1973. Positionalist voting functions. Theory and Decision 4, 1–24.Laslier, J.-F., 2011. On choosing the alternative with the best median evaluation.

Public Choice, (in press).Maskin, E., 1978. A theorem of utilitarianism. The Review of Economic Studies 45,

93–96.Myerson, R.B., 1995. Axiomatic derivation of scoring rules without the ordering

assumption. Social Choice and Welfare 12, 59–74.Nitzan, S., Rubinstein, A., 1981. A further characterization of Borda rankingmethod.

Public Choice 36, 153–158.Núñez, M., Laslier, J.-F., 2011. Overstating: a tale of two cities. Working Paper, CNRS

and Ecole Polytechnique, Paris.Segal, U., 2000. Let’s agree that all dictatorships are equally bad. Journal of Political

Economy 108, 569–589.Slutsky, St., 1977. A characterization of societies with consistent majority decision.

Review of Economic Studies 44, 211–225.Smith, W.D., Range voting, http://rangevoting.org/.Young, P., 1974. An axiomatization of Borda’s rule. Journal of Economic Theory 9,

43–52.Young, P., 1975. Social choice scoring functions. SIAM Journal of Applied

Mathematics 28, 824–838.Zahid, M.A., de Swart, H., 2010. The Borda majority count. Manuscript, Tilburg

University, Dept. of Philosophy.Zwicker, W.S., 2008. Consistency without neutrality in voting rules: when is a vote

an average? Mathematical and Computer Modelling 48, 1357–1373.