Mathematical Social Sciences 60 (2010) 57–60

Contents lists available at ScienceDirect

Mathematical Social Sciences

journal homepage: www.elsevier.com/locate/econbase

Anonymity, monotonicity, and quota pair systemsJonathan Perry, Robert C. Powers ∗Department of Mathematics, University of Louisville, Louisville, KY 40292, United States

a r t i c l e i n f o

Article history:Received 10 February 2009Received in revised form22 March 2010Accepted 24 March 2010Available online 2 April 2010

JEL classification:D71

Keywords:AnonymityMonotonicitySimple majority ruleAbstention

a b s t r a c t

We introduce the notion of a quota pair system and show that any social choice procedure, where thereare exactly two alternatives and a fixed number of voters, satisfies anonymity and monotonicity if andonly if it is uniquely determined by a quota pair system.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

This paper is concerned with social choice procedures wherethere are exactly two alternatives and a fixed number of voters.Each individual votes for one of the two alternatives or abstains.The procedure that chooses the alternative with the most numberof votes or declares a tie if both alternatives receive the samenumber of votes is called simplemajority rule. In 1952, KennethMayproved that simple majority rule is the only procedure satisfyingthree natural conditions: anonymity, neutrality, and positiveresponsiveness (May, 1952). Anonymity implies that the identitiesof individual voters are not used in determining the social outcomeand neutrality is the requirement that both alternatives should betreated equally. To define positive responsiveness we will say thata change favorable to an alternative x occurs if a voter changes theirvote from the competing alternative y to a tie vote, or if a voterchanges a tie vote to a vote for x, or if a voter changes a vote for yto a vote for x. Positive responsiveness requires that if a procedurechooses x or produces a tie and one or more individuals changetheir vote in favor of x, then the procedure will output x as thesocial outcome. May’s Theorem is a fundamental result in the areaof social choice and has received considerable attention since itfirst appeared in print in 1952. For a small sample of some of theresearch that has been motivated by May’s Theorem see Asan andSanver (2006), Woeginger (2003), and Yi (2005).

∗ Corresponding author. Tel.: +1 502 852 6103; fax: +1 502 852 7132.E-mail addresses: jonperry11@gmail.com (J. Perry), rcpowe01@louisville.edu

(R.C. Powers).

0165-4896/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.mathsocsci.2010.03.005

In 1995, Young et al. (1995)workedwith an extension of simplemajority rule where the social choice is based on a list of quotas{q0, . . . , qn}. For each k in the set {0, 1, . . . , n}, it is assumed thatqk is an integer belonging to the interval

[ n−k+12 , n− k+ 1

]. In

this case, the list of quotas determines a social choice procedureas follows: an alternative x is chosen if it receives at least qk voteswhen exactly k out of the n voters abstain. If neither alternativegets enough votes, then the procedure declares a tie. For example, ifqk =

⌈ n−k+12

⌉for each integer kwith 0 ≤ k ≤ n, then the resulting

procedure is simple majority rule. If qk = n − k + 1 for all k, thenwe get the procedure where the output is always a tie.A list of quotas {q0, . . . , qn} is called a quota system if qk+1 ∈

{qk, qk − 1} for all k ∈ {0, 1, . . . , n}. Young et al. proved that asocial choice procedure is determined by a quota system if andonly if the procedure satisfies anonymity, neutrality, and weakmonotonicity. The condition of weak monotonicity, which wewill simply refer to as monotonicity, is a weakening of positiveresponsiveness. They go on to show, using some nice countingtechniques, that the number of quota systems is given by thebinomial coefficient

(n+1b n2c+1

). Therefore, if n = 5, then there are(

63

)= 20 social choice procedures that satisfy anonymity, neu-

trality, and monotonicity whereas only simple majority rulesatisfies anonymity, neutrality, and positive responsiveness.The resultsmention above highlight the restrictive nature of the

positive responsiveness condition. We go a step further by askingthe following question. What happens if the two alternativesare not treated equally? Dropping neutrality, while still keepinganonymity andmonotonicity, allows one to consider other types ofsocial choice procedures. In particular, Fishburn discusses absolute

58 J. Perry, R.C. Powers / Mathematical Social Sciences 60 (2010) 57–60

special majority functions where, for example, a challenger needsa two-thirds majority to win (Fishburn, 1973). To answer ourquestion, we introduce the notion of a quota pair system andshow that any social choice procedure satisfying anonymity andmonotonicity is uniquely determined by a quota pair system. Inaddition, we use ourmain result to determine the number of socialchoice procedures that satisfy anonymity and monotonicity forn ≥ 2 voters.

2. Results

The set of individuals or voters is given by N = {1, 2, . . . , n}with n ≥ 2. Each individual votes for one out of two alternativesor they abstain. The two alternatives can be identified with1 and −1. The abstention vote is denoted by 0. In this setting, afunction of the form

F : {−1, 0, 1}n → {−1, 0, 1}

is called an aggregation rule or a social choice procedure and ann-tuple R = (R1, . . . , Rn) ∈ {−1, 0, 1}n in the domain of F iscalled a profile. The output F(R) = 0 represents a tie, i.e., neitheralternative is chosen. For any profile R = (R1, . . . , Rn), we letn+(R) = |{i ∈ N : Ri = 1}|, n−(R) = |{i ∈ N : Ri = −1}|, andn0(R) = |{i ∈ N : Ri = 0}|. So n+(R) is the number of individualswho voted for 1, n−(R) is the number of individuals who voted for−1, and n0(R) is the number of individuals who abstained. Ourgoal is to study aggregation rules that satisfy the following twoconditions.Anonymity (A). Given any R ∈ {−1, 0, 1}n and any permutation

Π : N → N , we have F(R1, . . . , Rn) = F(RΠ(1), . . . , RΠ(n)).Monotonicity (MON). For all R, R′ ∈ {−1, 0, 1}n, F(R) ≥ F(R′)

whenever R(i) ≥ R′(i) for i = 1, . . . , n.Here is a simple example of an aggregation rule that satisfies (A)

and (MON).

Example (Absolute (q, `)-majority rule). For any nonnegativeintegers q and ` such that q+ ` ≥ n+ 1 and q, ` ≤ n+ 1, and forany R ∈ {−1, 0, 1}n,

F(q,`)(R) = 1 if n+(R) ≥ q and F(q,`)(R) = −1 if n−(R) ≥ `.

Note that F(q,`)(R) = 0 if and only if n+(R) < q and n−(R) < `.

In order to characterize the class of aggregation rules thatsatisfies (A) and (MON) we introduce the following terminology.A quota pair system based on n voters is a pair of decreasingsequences of integers

q0 ≥ q1 ≥ · · · ≥ qn and `0 ≥ `1 ≥ · · · ≥ `n

such that

(i) 0 ≤ qk, `k ≤ (n+ 1− k) for all k ∈ {0, 1, . . . , n},(ii) qk + `k ≥ (n+ 1− k) for all k ∈ {0, 1, . . . , n},(iii) qk+1 ∈ {qk, qk − 1} and `k+1 ∈ {`k, `k − 1} for all k ∈{0, 1, . . . , n−1}. Observe that if qk = `k for all k, then a quotapair system reduces to a quota system as described by Younget al. (1995).

Example. If n = 2, then n+ 1− k = 3− k and

q0 = 2 ≥ q1 = 1 ≥ q2 = 0 and `0 = 2 ≥ `1 = 2 ≥ `2 = 1

is a quota pair system.

The following theorem establishes a connection between quotapair systems and social choice procedures that satisfy anonymityand monotonicity.

Theorem 1. An aggregation rule f : {−1, 0, 1}n → {−1, 0, 1}satisfies (MON) and (A) if and only if there exists a quota pair system

q0 ≥ q1 ≥ · · · ≥ qn and `0 ≥ `1 ≥ · · · ≥ `n

such that

f (R) = 1 ⇔ n+(R) ≥ qk and n0(R) = k

and

f (R) = −1 ⇔ n−(R) ≥ `k and n0(R) = k.

Before proving our first result we make a quick observation.The description given in the statement of Theorem 1 can be usedto define the absolute (q, `)-majority rule. Specifically, let qk =min{q, n+ 1− k} and `k = min{`, n+ 1− k} for all k and observethat qk + `k ≥ q+ ` ≥ n+ 1 ≥ n+ 1− k for all k.Proof. Let f be an aggregation rule determined by a quota pairsystem

q0 ≥ q1 ≥ · · · ≥ qn and `0 ≥ `1 ≥ · · · ≥ `n

such that

f (R) = 1 ⇔ n+(R) ≥ qk and n0(R) = k

and

f (R) = −1⇔ n−(R) ≥ `k and n0(R) = k.

Since f (R) is determined by the values n+(R), n0(R), and n−(R) itfollows that f satisfies (A).To show that f satisfies (MON) suppose R and R′ are two profiles

such that R ≥ R′. We want to show that f (R) ≥ f (R′). Assume thatf (R′) = 1 and let k′ = n0(R′). Then n+(R′) ≥ qk′ and so

n+(R) = n+(R′)+ (n+(R)− n+(R′)) ≥ qk′ + (n+(R)− n+(R′)).

If t = min{k′, n+(R) − n+(R′)}, then it follows from (iii) in thedefinition of quota pair system that

qk′ + t ≥ qk′−t .

If k = n0(R), then, since R ≥ R′, k ≥ k′ − t . Using the fact that theq′is are a decreasing sequence we get

n+(R) ≥ qk′ + t ≥ qk′−t ≥ qk.

Since n+(R) ≥ qk it follows that f (R) = 1 and so f (R) ≥ f (R′).Now assume that f (R′) = 0. As above, let k′ = n0(R′) and let

k = n0(R). Then n−(R′) < `k′ and so

n−(R) = n−(R′)− (n−(R′)− n−(R)) < `k′ − (n−(R′)− n−(R)).

If t = min{n− k′, n−(R′)−n−(R)}, then R ≥ R′ forces k ≤ k′+ t . Itfollows from (iii) and the fact that the `′is are a decreasing sequencethat `k ≥ `k′+t ≥ `k′ − t . Observe that `k′ − t ≥ `k′ − (n−(R′) −n−(R)) > n−(R). Therefore, n−(R) < `k and so f (R) ≥ 0 = f (R′).Finally, if f (R′) = −1, then it is automatic that f (R) ≥ f (R′).For the converse, assume that f is an aggregation rule that

satisfies (MON) and (A), we want to show that f is determined bya quota pair system. For each k ∈ {0, 1, . . . , n}, let

qk = min{n+(R)|f (R) = 1 and n0(R) = k}

if {n+(R)|f (R) = 1 and n0(R) = k} 6= ∅; otherwise let qk =(n+ 1)− k. Similarly, let

`k = min{n−(R)|f (R) = −1 and n0(R) = k}

if {n−(R)|f (R) = −1 and n0(R) = k} 6= ∅; otherwise let `k =(n+ 1)− k.If f (S) = 1 for some profile S, then n+(S) ≥ qk with k = n0(S)

by the definition of qk. Suppose S ′ is a profile where n+(S ′) ≥ qkand k = n0(S ′). Choose R such that f (R) = 1, n+(R) = qk andn0(R) = k. So n+(S ′) ≥ n+(R), n0(S ′) = n0(R) and n−(S ′) ≤ n−(R).It follows from (A) and (MON) that f (S ′) ≥ f (R). Thus f (S ′) = 1.It now follows that f (S) = 1 for some profile S if and only if

J. Perry, R.C. Powers / Mathematical Social Sciences 60 (2010) 57–60 59

n+(S) ≥ qk. Similarly, f (S) = −1 if and only if n−(S) ≥ `k wherek = n0(S).To see that we have a quota pair system first observe that 0 ≤

qk, `k ≤ (n + 1 − k) for all k ∈ {0, 1, . . . , n}. So item (i) in thedefinition of quota pair system holds.To show that (ii) holds assume that there exists k ∈

{0, 1, . . . , n} such that qk + `k 6≥ n + 1 − k. This means thatqk + `k + k ≤ n. Choose a profile R such that n+(R) = qk andn−(R) = n − qk − k. So n0(R) = k. Since n+(R) ≥ qk withk = n0(R) it follows from above that f (R) = 1. On the other hand,since n−(R) = n − qk − k ≥ `k with k = n0(R) it follows thatf (R) = −1. Since R is single valued we get a contradiction. Thusqk + `k ≥ n+ 1− k for all k ∈ {0, 1, . . . , n}.To show that (iii) holds wewill prove that qk+1+1 ≥ qk ≥ qk+1

for all k ∈ {0, . . . , n− 1}. Let k ∈ {0, . . . , n− 1}. To show the firstinequality we consider two cases: qk+1 = n−k and qk+1 < n−k. Ifqk+1 = n− k, then qk+1+1 = n+1− k ≥ qk. If qk+1 < n− k, thenqk+1 + (k + 1) ≤ n. Let R be a profile such that n+(R) = qk+1 andn0(R) = k + 1. Then f (R) = 1 and there exists j such that Rj = 0.Define a new profile R′ such that R′i = Ri for all i 6= j and R

′

j = 1.Notice that R′ ≥ R and so, by (MON), f (R′) ≥ f (R) = 1. Sincef (R′) = 1 and n0(R′) = k it follows that n+(R′) = qk+1 + 1 ≥ qk.To show the second inequality we consider the cases where

qk ≥ (n − k) and qk < n − k. If qk ≥ (n − k), then qk ≥(n+1)−(k+1) ≥ qk+1. Assume that qk < n−k. Then qk+k+1 ≤ n.Let R be a profile such that n+(R) = qk and n0(R) = k. Thenf (R) = 1 and there exists j such that Rj = −1. Define a new profileR′ such that R′i = Ri for all i 6= j and R

′

j = 0. Notice that R′≥ R and

so, by (MON), f (R′) ≥ f (R) = 1. Observe that n0(R′) = k + 1 andthat n+(R′) = n+(R) = qk. Hence qk ≥ qk+1.A similar argument shows that `k+1 + 1 ≥ `k ≥ `k+1 for all

k ∈ {0, . . . , n − 1}. It now follows that q0 ≥ q1 ≥ · · · ≥ qn and`0 ≥ `1 ≥ · · · ≥ `n is a quota pair system and the proof of thetheorem is complete. �

The aggregation rules suggested in Theorem 1 could be appliedto obtain some real-world decisions. We now suggest one possiblescenario where an absolute (q, `)-majority rule is used.A faculty member from a mathematics department retires and

the dean allows the math department to replace this person. Theretired faculty member worked in statistics. There is a strongdesire among a large portion of the faculty that someone shouldbe hired in an area other than statistics. One group wants tohire a topologist and another group wants to hire someone indifferential geometry. The dean of the college feels that consistencyshould be maintained and that the new hire should be in statistics.As a compromise between the dean and the math faculty allparties agree to use an absolute (q, `)-majority voting rule. First,based on the dean’s authority, the default outcome is to hire astatistician. Next, a topologist will be hired if at least two-thirdsof the department can agree on this outcome. Finally, the newhire will be in differential geometry if at least three-fourths of thedepartment can agree on this outcome. To make matters worse,however, the faculty meeting to decide on the hiring area is rightbefore Christmas break. Given the timing of the meeting thereis a strong chance that some faculty members will be absentand so their input is designated as an abstention. The topologyand differential geometry options can be denoted by 1 and −1,respectively. In this case q =

⌈ 23n

⌉and ` =

⌈ 34n

⌉where n is the

number of faculty members in the math department. If the finaloutcome is 0, then the department will hire someone in statistics.There are many aggregation rules satisfying anonymity and

monotonicity. It follows fromTheorem1 that for fix n, there is a oneto one correspondence between the set of social choice proceduresthat satisfy (A) and (MON) and the collection of all quota pairsystems. We now use this correspondence to establish the nextresult.

Theorem 2. The number αn of social choice procedures that sat-isfy (A) and (MON) for n ≥ 2 voters is

αn =

(2n+ 3n+ 1

).

Proof. For any quota pair system q0 ≥ q1 ≥ · · · ≥ qn and`0 ≥ `1 ≥ · · · ≥ `n there is a corresponding sequence of orderpairs of integers

(c0, d0), (c1, d1), . . . , (cn+1, dn+1)

where c0 = d0 = 0, ck = qn+1−k − `n+1−k, and dk = qn+1−k +`n+1−k − k for k = 1, . . . , n+ 1. Item (ii) in the definition of quotapair system is equivalent to di ≥ 0 for i = 0, . . . , n + 1 and item(iii) is equivalent to

(ck+1, dk+1) ∈ {(ck − 1, dk), (ck + 1, dk), (ck, dk + 1), (ck, dk − 1)}

for k = 0, . . . , n. Observe that any sequence (c0, d0), (c1, d1), . . . ,(cn+1, dn+1) where c0 = d0 = 0, di ≥ 0 for i = 0, . . . , n + 1, and(ck+1, dk+1) ∈ {(ck − 1, dk), (ck + 1, dk), (ck, dk + 1), (ck, dk − 1)}for k = 0, . . . , n, can be identified with a walk of length n + 1starting at the origin and at each step moving one unit either tothe left, right, up, or down and still remain in the upper half-plane. It is not hard to see that the correspondence mentionedhere that associates integer lattice walks with quota pair systemsis invertible. Therefore, the number of quota pair systems for n ≥ 2voters is the same as the number of integer lattice walks of lengthn + 1 that start at the origin and stay in the upper half-plane.The number of such walks is known to be the binomial coefficient(2n+3n+1

)(Guy, 2000). �

Observe thatα2 = 35,α3 = 126,α4 = 462, andα5 = 1716. To givea sense of these numbers, recall from the introduction that for fivevoters there are 20 social choice procedures that satisfy anonymity,neutrality, and monotonicity. Comparing the latter with α5 showsa significant increase in the number of social choice procedures inthe case of 5 voters when neutrality is dropped. This increase iseven more dramatic for larger values of n.The results of this paper highlight the restrictive nature of

neutrality under the assumptions of anonymity and monotonicity.Of course, other combinations of conditions can be studied. Forexample, it is shown in Perry and Powers (2008) that the numberof aggregations rules that satisfy anonymity and neutrality is

3

⌊n2+2n+1

4

⌋and the number of aggregation rules that satisfy just

anonymity is 3n2+3n+2

2 .The aggregation rules presented in this paper are related to

votingmethods where the notion of abstention plays an importantrole. The referee of this paper pointed out that our model ofvoting is close in spirit to the models presented in Felsenthaland Machover (1997), Freixas and Zwicker (2003, 2009), andRubinstein (1980). In fact, an anonymous (j, k) voting rule definedin Freixas and Zwicker (2009) corresponds to an aggregation rulesatisfying (MON) and (A) in the case where j = k = 3. As a futureproject, it would be interesting to establish a strong connectionbetween the work presented here and the model developed inFreixas and Zwicker (2003, 2009).

Acknowledgement

Partially supported by University of Louisville SROP grant.

References

Asan, G., Sanver, M.R., 2006. Maskin monotonic aggregation rules. EconomicsLetters 75, 179–183.

Felsenthal, D.S., Machover, M., 1997. Ternary voting games. International Journal ofGame Theory 26, 335–351.

60 J. Perry, R.C. Powers / Mathematical Social Sciences 60 (2010) 57–60

Fishburn, P.C., 1973. The Theory of Social Choice. Princeton University Press,Princeton, NJ.

Freixas, J., Zwicker, W.S., 2003. Weighted voting, abstention, and multiple levels ofapproval. Social Choice and Welfare 21, 399–431.

Freixas, J., Zwicker, W.S., 2009. Anonymous yes–no voting with abstention andmultiple levels of approval. Games and Economic Behavior 67, 428–444.

Guy, R.K., 2000. Catwalks, sandsteps, and Pascal pyramids. Journal of IntegerSequences 3, #00.1.6.

May, K.O., 1952. A set of independent necessary and sufficient conditions for simplemajority decision. Econometrica 20, 680–684.

Perry, J., Powers, R.C., 2008. Aggregation rules that satisfy anonymity and neutrality.Economics Letters 100, 108–110.

Rubinstein, A., 1980. Stability of decision systems under majority rule. Journal ofEconomic Theory 23, 150–159.

Woeginger, G., 2003. A new characterization of themajority rule. Economics Letters81, 89–94.

Yi, J., 2005. A complete characterization of majority rules. Economics Letters 87,109–112.

Young, S.C., Taylor, A.D., Zwicker, W.S., 1995. Counting quota systems: acombinatorial question from social choice theory. Mathematics Magazine 68,331–342.