The document presents an axiomatic construction of an aggregation rule for combining individual three-graded rankings into a social ranking. It introduces four axioms: Pareto, Pairwise Compensation, Non-Compensatory Threshold, and Contraction. It proves that the only rule that satisfies these axioms is the "threshold rule", which aggregates rankings by comparing the number of "bad" and "average" evaluations across alternatives. The threshold rule defines a weak order over alternatives, and the social ranking is determined by the equivalence classes of this weak order.

1. Mathematical Social Sciences 53 (2007) 106 – 110
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A ‘threshold aggregation’ of three-graded rankings
Fuad Aleskerov a,b,⁎, Vyacheslav Yakuba b , Dmitriy Yuzbashev b
a
Department of Mathematics, State University “Higher School of Economics”,
Myasnitskaya Str., 20, Moscow 101000, Russia
b
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Str., 65, Moscow 117997, Russia
Received 1 September 2005; received in revised form 1 July 2006; accepted 1 September 2006
Available online 15 November 2006
Abstract
A problem of axiomatic construction of a social ranking is studied for the case when individual opinions
are given as three-graded rankings. It is shown that the only rule satisfying the axioms introduced is the
threshold rule.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Aggregation; Three-graded ranking; Threshold rule; Axiomatic construction
1. Introduction
We consider the following problem of construction social decision rule: a set of n agents is
given, each agent evaluates alternatives from a finite set A using trichotomous preferences (three-
graded rankings), i.e. using the scale ‘bad–average–good’,1 and the problem is how to construct a
social ranking over alternatives.
The problem of aggregation of such individual opinions has been considered in a few pub-
lications (Hillinger, 2004; Ju, 2005; Ôzyurt and Sanver, 2004).
Very often to use for such case the Borda rule is not correct since any summation of ranks rule has
a compensatory nature — for some alternative low evaluation by one agent can be compensated by
high evaluation of other. In other words, if one would like to take carefully into account the low
evaluation given by an agent, she may not use the Borda rule or its counterparts.
⁎Corresponding author. Department of Mathematics, State University “Higher School of Economics”, Myasnitskaya Str.,
20, Moscow 101000, Russia.
E-mail address: alesk@ipu.ru (F. Aleskerov).
1
Or, in terms of Hillinger (2004), ‘goods, bads, and nulls’.
0165-4896/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.mathsocsci.2006.09.002

2. F. Aleskerov et al. / Mathematical Social Sciences 53 (2007) 106–110 107
In Hillinger (2004) a method is studied in which the agents express their opinions via three-
graded rankings, and the aggregation rule is just the summation of ranks. Some axioms are
considered which a rule satisfies to.
Another characterization of aggregation rule for three-graded rankings is given in Ju (2005).
Several standard axioms are considered and it is shown that an aggregation rule is a version of
plurality procedure.
We give an axiomatic construction of the new aggregation rule called ‘threshold rule’. The
axioms used are Pareto one, Pair-wise compensation, Non-compensatory threshold and Contraction.
Pareto axiom means, that if for one alternative evaluations of all agents is not less than for another
alternative, and there is an agent whose evaluation of the first alternative is strictly higher than of the
second one, then in a social ranking the first alternative has a higher rank than the second one.
Pair-wise compensation axiom means that if all agents but two evaluate two alternatives
equally, and those two agents have ‘mutually inverse’ values, then the alternatives have the same
rank in the social ranking.
Non-compensatory threshold axiom reveals the very idea of threshold aggregation: if at least one
agent evaluates an alternative as ‘bad’, then in social ranking this alternative has lower rank that the
one evaluated as ‘average’ by all agents, despite of how many ‘good’ values has the first alternative.
Contraction axiom states that if for two alternatives the evaluations of some agent are equal
then she can be excluded from the consideration when social ranking is constructed, i.e.
everything will be decided by other agents' evaluations.
Then we prove the theorem that the threshold rule is the only rule satisfying those four axioms.
The threshold rule aggregates individual preferences as follows: first, if the number of bad
evaluations in one alternative is greater than that in the other one, then the first alternative has the
lower rank in the social ranking; if the number of ‘bads’ in both alternatives are equal, then the
comparison is made with respect to ‘average’ evaluations.
2. Formal statement of the problem and its solution
Let N = {1, …, n}, n N 1 be a set of agents, each agent i evaluates an alternative as xi, xi ∈ {1, 2, 3}.
The set of alternatives A is the set of all 3n vectors of the form (x1, …, xn).
The problem is to construct a function φ : A → R.
A function φ satisfies several axioms.
Axiom 1. (Pareto)
∀x, y ∈ A ∀i ∈ {1, …, n}: x ≥ y and ∃i0: xi0 N yi0 ⇒ φ(x) N φ(y).
Axiom 2. (Pair-wise compensation)
∀x, y ∈ A ∃i, j ∈ {1, …, n}: xi = yj, xj = yi and ∀k ≠ i, j xk = yk ⇒ φ(x) = φ(y).
Note that, in the present context, where all alternatives are admissible, Pairwise Compensation
is equivalent to the usual Anonymity requirement, i.e., φ(x) = φ(y) if y is obtained from x by a
permutation of the indices.
Axiom 3. (Non-compensatory threshold)
∀x ∈ A: ∃i0 ∈ (1,…, n): xi0 = 1φ(2, …, 2) N φ(x).

3. 108 F. Aleskerov et al. / Mathematical Social Sciences 53 (2007) 106–110
Axiom 4. (Contraction)
∀x, y ∈ A ∃i: xi = yi ⇒ φ(x) N φ(y) ⇔ φ(x1, …, xi−1, xi+1, …, xn) N φ(y1, …, yi−1, yi+1, …, yn).
Let us illustrate the Axiom 4 via an example. Suppose n = 2, and without loss of generality put
x1 = y1 for alternatives (x1, x2) and (y1, y2). Then Axiom 4 states that φ(x) N φ(y) ⇔ x2 N y2.
Denote as Wtresh ⊆ A × A a binary relation, which is constructed on A as follows:
Wtresh ¼ fðx; yÞjv1 ðxÞbv1 ðyÞ or v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞbv2 ðyÞg;
where v1(x) — is the number of ‘ones’ in the vector x, and v2(x) — is the number of ‘twos’.
The rule of construction of Wtresh we will call a threshold rule. Thus, Wtresh is a set of pairs of
alternatives satisfying two conditions: either in the first alternative the number of ‘bads’ in the
agents' evaluation is less than in the second one, or if the number of ‘bads’ is equal, then the
number of ‘nulls’ in the first alternative is less than in the second one.
One can prove that the relation Wtresh is a weak order on R, i.e., an irreflexive, transitive and
negatively transitive ((x, y) ∉ Wtresh, (y, z) ∉ Wtresh ⇒ (x, z) ∉ Wtresh) binary relation.
Every weak order P is defined by its set of equivalency classes, so xPy ⇔x ∈IiP, y ∈II иi N l, where
P
s
IsP ¼ fxaAj IayaX : yPxgIs−1 ¼ fxaAqIs j IayaX : yPxg N I1 ¼ Aq [ Ij
P P
j¼2
Consider aggregation procedure φ on the set A of vectors of dimension n.
Theorem. A procedure φ : A → R, satisfying Axioms 1–4 is defined by the system of equivalency
classes of a weak order generated by the threshold rule, so that uðxÞNuðyÞ ( xaIk tresh ;
) W
yaIjWtresh
8kNj.
We prove two lemmas before proving the theorem.
Note first, that the proof of the theorem and the lemmas based on the following procedure of
reordering the values of the vector of evaluations for an alternative. Due to the Axiom of pair-wise
compensation (Axiom 2) the values of the vector for an alternative can be reordered form left to
right where ‘ones’ are placed first, then ‘twos’, and, finally, ‘threes’, if any. Such reordering is
equivalent w.r.t. φ due to Axiom 2.
Lemma 1. If v1(x) = v1(y) and v2(x) = v2(y), then φ(x) = φ(y).
Proof. From the conditions v1(x) = v1(y) and v2(x) = v2(y) it implies that both x and y consist of
equal quantity of 1s, 2s, and 3s. Thus, the vectors can be transformed to each other by the
reordering procedure due to Axiom 2.
Lemma 2. If φ(x) = φ(y), then v1(x) = v1(y).
Proof. Suppose to the contrary, that v1(x) b v1(y). Consider the following reorderings: x Vfor x, and
y Vfor y such that: xV= (1, …, 1|*, …, *), where the vertical line divides elements on positions v1(x)
and v1(x) + 1, and ‘*’ marks for 2s or 3s; y V= (1, …, 1|1, …, 1|*, …, *), where the left vertical line
divides elements on positions v1(x) and v1(x) + 1, right vertical line divides elements on positions
v1(y) and v1(y) + 1 and ‘*’ marks for 2s or 3s also.
Consider the vector z = (1, …, 1|2, …, 2), in which the vertical line divides elements on positions
v1(x) and v1(x) + 1. Then φ(x V) ≥ φ(z) due to Axiom 1 (Pareto). It also can be seen that φ(z) N φ( yV)
by applying Axiom 3 (Non-compensatory threshold) after truncating first v1(x) elements in both z
and yVdue to Axiom 4. (Contraction). Combining all, it follows: φ(x) = φ(x V) ≥ φ(z)Nφ(y V) = φ(y)
which leads to contradiction.

4. F. Aleskerov et al. / Mathematical Social Sciences 53 (2007) 106–110 109
The case for v1(x) N v1(y) has symmetrical proof.
Proof of the Theorem. We need to prove the following:
uðxÞNuðyÞ ( v1 ðxÞbv1 ðyÞ or ½v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞbv2 ðyÞŠ
)
We prove the sufficiency by the rule of contraries.
Let ∃x, y : v1(x) b v1(y) or [v1(x) = v1(y)and v2(x) b v2(y)].
Consider two cases:
a) v1(x) b v1(y). As it was proved in Lemma 2: φ(x) N φ(y).
b) v1(x) = v1(y) and v2(x) b v2(y)
Consider the following reorderings: x Vfor x, and y Vfor y such that:
xV¼ ð1; N ; 1j2; N ; 2j3; N ; 3j3; N ; 3Þ
yV¼ ð1; N ; 1j2; N ; 2j2; N ; 2j3; N ; 3Þ;
where in both vectors the leftmost vertical line divides the v1(x) and v1(x) + 1 positions (note, v1
(x) = v1(y)), middle vertical line divides v1(x) + v2(x) and v1(x) + v2(x) + 1 positions, and rightmost
vertical line divides v1(y) + v2( y) and v1( y) + v2(y) + 1 (note, v2(x) b v2(y)). Note also, that there
might not be 1s in both x, and y, and there might not be 2s in x.
It implies from Axioms 1 and 2 that φ(x) = φ(x V N φ( yV = φ(y) which leads to contradiction.
) )
To prove the necessity, we need to prove the following:
uðxÞNuðyÞZ v1 ðxÞbv1 ðyÞ or ½v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞbv2 ðyÞŠ
To prove it by contradiction, consider the negation of the right hand side of the implication.
Iðv1 ðxÞbv1 ðyÞ or v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞbv2 ðyÞÞ
¼ v1 ðxÞzv1 ðyÞ and ðv1 ðxÞpv1 ðyÞ or v2 ðxÞzv2 ðyÞÞ
¼ v1 ðxÞzv1 ðyÞ and v1 ðxÞpv1 ðyÞ or v1 ðxÞzv1 ðyÞ and v2 ðxÞzv2 ðyÞ
¼ v1 ðxÞNv1 ðyÞ or v1 ðxÞNv1 ðyÞ and v2 ðxÞzv2 ðyÞ or v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞzv2 ðyÞ
¼ ðv1 ðxÞNv1 ðyÞ or v1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞNv2 ðyÞÞ or
ðv1 ðxÞ ¼ v1 ðyÞ and v2 ðxÞ ¼ v2 ðyÞÞ
Since v1(x) =v1(y) and v2(x) =v2(y) ⇒ φ(x) =φ(y) by Lemma 1, then by contradiction, we have
¬φ(x) =φ(y) ⇒ ¬(v1(x) =v1(y) and v2(x) =v2(y)), or, using the symmetry of the formulae w.r.t. x and
y we have: φ(x) N φ(y) ⇒ v1(x) b v1(y) or [v1(x) =v1(y) and (v2(x) b v2(y)], which proves the theorem.
It is interesting to evaluate the maximal number of equivalency classes for that weak order
depending on n.
It can be proved that this problem is equivalent to the problem of evaluation of number of
unordered samples of size n from k elements with repeating, where k = 3, n — the number of
agents, i.e. the maximal number of equivalency classes is equal to:
ðn þ 2Þ! ðn þ 2Þðn þ 1Þ
Cnþ3−1 ¼
n
¼ :
n!d2! 2
ðn þ 2Þðn þ 1Þ
This means that the maximal value of the function φ is equal to .
2

5. 110 F. Aleskerov et al. / Mathematical Social Sciences 53 (2007) 106–110
3. Concluding remarks
The model of threshold aggregation had first been stated in (Aleskerov and Yakuba, 2003).
Similar axioms were formulated in that paper and similar theorem was proved using the notion of
vector closeness in terms of some pseudometric. So, in this work the direct proof of the re-
presentation theorem (with slightly different set of axioms) has been obtained.
In Sholomov (2004) it was shown that almost all the results about the properties of order
relation in Rn can be obtained for the spaces of arbitrary dimension with three-graded ranking.
However, on our knowledge the problem of axiomatic aggregation of such rankings has not even
been stated.
It is interesting to extend the model obtained here for the individual rankings with more than
three grades. In this case many versions of non-compensatory threshold axiom arise. Similar
problems can be seen in the problem of freedom of choice (Sen, 1988) and manipulation over set-
wise preferences (hyper-relations) (Ôzyurt and Sanver, 2004; Pattanaik and Xu, 1998).
Acknowledgement
We are very thankful to Professors Rafig Agaev, Pavel Chebotarev and Remzi Sanver for many
helpful comments.
References
Aleskerov, F., Yakuba, V., 2003. On one method of aggregation of ranking of special style. II International Conference on
Control Problems, ICS RAS, Moscow, p. 116.
Hillinger, C., 2004. Voting and the cardinal aggregation of judgements. University of Munich Discussion Paper.
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Theory 26 (1), 115–128.
Ôzyurt S, Sanver M.R., 2004. A general impossibility result on strategy proof social choice hyperfunctions. Working
paper, Istanbul Bilgi University.
Pattanaik, P., Xu, Y., 1998. On preferences and freedom. Theory and Decision 44, 173–198.
Sen, A., 1988. Freedom of choice: concept and content. European Economic Review 32, 269–294.
Sholomov, L.A., 2004. Logical methods for studying relations in criterial spaces with arbitrary ordinal scales. Automation
and Remote Control 65 (5), 700–709.