acyclicty and condorcet winners

Acyclicty and Condorcet winners

Grisel Ayllón and Bernardo Moreno

Instituto Tecnológico y de Estudios Superiores de Monterrey

Campus Ciudad de México

E-mail: [email protected]

Departamento de Teoría e Historia Económica, Universidad de Málaga

E-mail: [email protected]

1

Abstract:

Journal of Economic Literature Classi�cation Number: D71.

Keywords:

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1 Introduction

The existence of majority voting equilibria is crucial in many models of the economy where

some of the variables are determined by the political process. Public economics, political

economy, social choice and other strands of economic analysis base their predictions on

the study of equilibria in models of this kind, and yet the existence problems appear even

in the simplest and most stylized situations. The possibility of voting cycles under the

majority rule and unrestricted preference domains is the leading case. It is true that there are

models of economic interest where individual preferences satisfy the type of requirements that

avoid social preference cycles (single-peakedness, single-crossing, value restrictedness etc.).

Under these conditions, a transitive majority rule preference relation is always guaranteed.

However, to guarantee the existence of best alternatives (called Condorcet winners) relative

to the majority rule relation when there a �nite number of alternatives, is su¢ cient that

the majority rule relation does not exhibit strict cycles. Moreover, looking at the conditions

established in the literature, we can �nd examples where the majority rule relation is not

strictly acyclic, that is there are strict cycles for some triple of alternatives, and still �nd a

Condorcet Winner. Thus, we are concerned in how strict cycles can appear in a preference

pro�le and how do they a¤ect the existence of a Condorcet Winner.

In this paper, we look for necessary and su¢ cient conditions to ensure the existence

of best alternatives of the majority rule relation. We start proposing such a condition for

the case in which the preferences are strict and there are 3 alternatives. Before presenting

the condition we bring in our paper an idea already present in the literature, that of a

net preference (introduced by Feld and Grofman, (1986) and that also appear in Gjorgjiev

and Xefteris, 2013). A net preference is the frequency of a preference (say xPiyPiz) minus

the frequency of the opposite preference (zPiyPix), and a positive net preference are those

preferences whose net preference is positive. Given any preference pro�le, we construct the

positive net preference pro�le (that consists only of positive net preferences). A preference

pro�le satis�es our condition if there is an alternative that it is the best alternative for at

least half of the agents with positive net preferences. It is also the case, that for the case of

strict preferences and 3 alternatives, such a condition is also necessary and su¢ cient for the

majority rule relation to be strictly acyclic1.

1Strict acyclicity is related to cycles of strict preferences.

3

In the case or more than 3 alternatives, we �nd that to require the above condition for

every triple of alternatives is su¢ cient for the existence of a best alternative. However, and

although it is also necessary for the strict acyclicity of the majority rule relation, it is not

su¢ cient. A more demanding requirement is needed. A requirement that imposes not only

the above condition for every triple but also that if a majority net alternative x in a triple

is not the majority net alternative in all the triples is because there is another alternative

y that it is a majority net alternative in all triples in which both alternatives, x and y, are

present.

Finally, in the last section of the paper, we deal with indi¤erences. Note that the condi-

tions proposed are based on triples of alternatives. For each triple of alternative, we propose

a procedure to go from any preference pro�le with indi¤erences to a preference pro�le with

indi¤erences that produces the same majority rule relation. We call such a pro�le, a replica

pro�le. The idea is simple, we make a natural extension of preferences taking away the

indi¤erences by duplicating the number of agents of those agents that are not completely

indi¤erent. That is, replacing them with strict preferences in both senses in the binary re-

lationship. This means that if an agent is indi¤erent between two alternatives a and b, then

we eliminate this agent and replace him by two agents where one will prefer a over b and

the other will prefer the opposite, respecting the rest of his strict preferences. And for those

agents that have strict preferences we add a copy of them.

The paper is organized as follows: in the Section 2, we introduce the general notation;

then we present the results regarding acyclicity when there are only three alternatives in

Section 3. Section 4 extends these results to any �nite number of alternatives. Finally, in

Section 5 we extend our results to situations in which we admit indi¤erences.

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2 Preliminaries

Let A be a �nite set of alternatives, S � A any subset of alternatives. Agents�preference

relations are linear orderings on A. Let Pi be the set of all strict preferences2 of agent i. Forany N � N+, Pi 2 Pi, and S � A, let t(Pi; S) be the most preferred alternative of agent iat Pi relative to S. For any N � N+, preference pro�les of the agents in N are elements of

�i2NP ; and they are denoted by PN = fPigi2N . We denote as P Si , the preferences of agenti relative to S � A.The following concept is crucial in all our analysis. For any N � N+, PN 2 �i2NP,

and x 2 S � A; let F x(S; P SN) be the set of agents that have alternative x as their most

preferred alternative in S at pro�le P SN , i.e. Fx(S; P SN) = fi 2 N : xPiy for all y 2 Snfxgg.

Let fx(S; P SN) be number of agents that regard alternative x as the best alternative in S at

pro�le P SN ; namely, fx(S; P SN) = #F

x(S; P SN).

De�nition 1 For any N � N+, and any PN 2 �i2NPi, a 2 A is a majority net alternativerelative to S � A if and only if fa(S; PMS) � #MS

2.

For any N � N+, any PN 2 �i2NPi, and any x; y 2 A, let N(x; y;PN) = fi 2 N : xPiygbe the set of agents that strictly prefer x over y.

For any N � N+, PN 2 �i2NPi, and any x; y 2 A we say that x is majority weakly pre-ferred to y (we may also say that x is not defeated by y), xRmy, if and only if N(x; y;PN) �N(y; x;PN) and that x is majority strictly preferred to y (we may also say that x defeates

y), xPmy, if and only if N(x; y;PN) > N(y; x;PN). We refer to Rm as the majority rule

relation.

We can now de�ne the notion of strong Condorcet winner.

De�nition 2 For any N � N+ and S � A, an alternative x 2 S is the strong Condorcetwinner at pro�le PN 2 �i2NPi relative to S if and only if xPmy for all y 2 Snfxg.

If it exists, a strong Condorcet winner is unique and defeates any other alternative in

pairwise majority comparisons. However, there are pro�les PN there is no strong Condorcet

winner but there may be some alternatives that are not defeated by any other alternative

in pairwise majority comparisons.2Most of the paper is devoted to the case in which agents have linear orderings and Section 5 provides

an explanation of how to deal with indi¤erences.

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De�nition 3 For any N � N+ and S � A, an alternative x 2 S is a Condorcet winnerat pro�le PN 2 �i2NPi, if and only if xRmy for all y 2 Snfxg.

Let CW (S; PN) be the set of Condorcet winners for N , relative to S at pro�le PN .

3 Triples of alternatives

In this section, we will focus on situations in which there are only three alternatives, that is

A = fx; y; zg. For any agent i and any Pi 2 Pi, there are six possible linear orderings relativeto A. Among them, we can �nd pairs of preferences that are the opposite of the other. The

identi�cation of such pair of preferences will help us to compute the majority decisions,

as they become irrelevant in the comparison between agents prefering one alternative to

another.

De�nition 4 For any N � N+ any PN 2 �i2NPi and A, we say that agents i; j 2 N have

mutually exclusive preferences Pi and Pj at pro�le PN relative to A if, for any a; b 2 A,aPib() bPja:

Example 1 Let N = f1; 2; 3; 4; 5g and consider the following preference pro�le PN

P1 = P2 P3 P4 P5

z x y y

y z x z

x y z x

Note that agents 3 and 5 have mutually exclusive preference relations.

For any N � N+ , any PN 2 �i2NP ; let MA be the subset of N that contains all

the agents that have non-mutual exclusive preference relations relative to A at pro�le PN

(abusing of the language and when there is no confusion, we refer to MA as the set of non-

mutually exclusive agents at PN relative to A). We refer to PMA as the net pro�le of PN

relative to A and to MA as the net set of agents at PN relative to A.

The reduction of the preference pro�le from PN to PMA will be of great use for our

purposes. Since the set of Condorcet winners relative to A is the same at both pro�les, that

reduction does not make any di¤erence from our point of view. We use Example 1 to clarify

these concepts.

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Example 1 (continuated) Notice that MA = f1; 2; 4g: Hence, PMA is given in the follow-

ing table

P1 = P2 P4

z y

y x

x z

Note that CW (A;PN) = CW (A;PMA) = fzg:

Lemma 1 For any N � N+, any PN 2 �i2NPi, and any a 2 A, if there is some i 2 MA

such that a 6= t(Pi; A), then there exists b 2 Anfag such that bPja for all j 2 MA such that

a 6= t(Pj; A).

Proof. Let N � N+, PN 2 �i2NPi, and a 2 A. Note that if there is only one agent i 2MA

such that x 6= t(Pi; A), the result trivially follows. Suppose that there are at least two agentsi; j 2 MA such that a 6= t(Pi; A), a 6= t(Pj; A). Suppose, to get a contradiction, that theredoes not exist b 2 Anfag with bPia and bPja. Note that alternative a can not be neitherthe best alternative nor the worse alternative for agents i and j. Therefore, without loss of

generality, we have that bPiaPic and cPjaPjb what it is a contradiction to i; j 2 MA, since

agents i and j are mutually exclusive.

Example 2 Let N = f1; 2; 3g and the preference pro�le PN 2 �i2NP be as follows

P1 P2 P3

x y z

y z x

z x y

This preference pro�le illustrates the statement of Lemma 1 as x 6= t(P2; A), zP2x and zP3x.The following two de�nitions present two di¤erent violations of the transitivity of the

majority rule relation. The �rst one happens when the quasi-transitivity of Rm does not

hold and there is a cycle involving a chain of strict preference comparisons (as the one

illustrated in Example 2). We refer to these situations as strict cycles.

De�nition 5 For any N � N+, and any PN 2 �i2NPi, Rm exhibits a strict cycle at PNrelative to A if aPmb, bPmc, and cPma.

7

The second violation of the transitivity of the majority rule relation is stronger. It

happens not only because the quasi-transitivity of Rm is violated but also because there is a

cycle involving a chain that is a combination of strict and indi¤erence preference comparisons.

We refer to these situations as weak cycles.

De�nition 6 For any N � N+, and any PN 2 �i2NRi, Rm exhibits a weak cycle at PN

relative to A if cIma, aPmb, and bPmc .

It is well-known that if there is a �nite set of alternatives and a preference relation is

complete and strictly acyclic, then best choices exist. In the next Proposition, we provide

necessary and su¢ cient conditions that a pro�le of individual preferences must satisfy to

guarantee that the majority rule relation is strictly acyclic and then we also ensure the

existence of Condorcet winners. Such condition if expressed in terms of the net pro�le and

entails the requirement that at least half of the net set of agents regard an alternative as the

most preferred alternative.

Proposition 1 For any N � N+, and any PN 2 �i2NPi, Rm is strictly acyclic relative toA if and only if there exists a majority net alternative relative to A.

Proof. Let N � N+, PN 2 �i2NP, and x 2 A be such that fx(T; PMA) � #MA

2. We

show that Rm is strictly acyclic relative to A. Since fx(T; PMA) � #MA

2, we have that

N(x; y;RMA) � N(y; x;RMA) and N(x; z;RMA) � N(z; x;RMA). Since for any a; b 2 T ,N(a; b;RN) � N(b; a;RN) if and only if N(a; b;RMA) � N(b; a;RMA), we have that xRmy

and xRmz. Therefore, Rm is strictly acyclic relative to A.

Let N � N+; and PN 2 �i2NP, be such that Rm is strictly acyclic relative to A. We

show that for some a 2 A, fa(T;RMA) � #MA

2. Suppose, to get a contradiction, that for

all a 2 A, fa(T;RMA) < #MA

2. By Lemma 1 for all a 2 A, there exists b 2 Anfag such

that N(a; b;RMA) � N(b; a;RMA). Since for all a 2 A, fa(T;RMA) < #MA

2, we have that

N(a; b;RMA) < N(b; a;RMA). Therefore, for all a 2 A, there exists b 2 Anfag such thatbPma. Then, without loss of generality, xPmy, yPmz and zPmx contradicting that Rm is

strictly acyclic.

Corollary 1 For any N � N+, any PN 2 �i2NR, CW (A;RN) 6= ; if and only if for somea 2 A fa(A;RMA) � #MA

2.

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From Proposition 1 it follows that the existence of Condorcet winners is compatible with

the presence of weak cycles. However, if the majority rule relation exhibits a semi-cycle the

best choice is not unique and therefore there is no a strong Condorcet winner as Example 3

shows.

Example 3 Let N = f1; 2; 3; 4g and consider PN as in the following table

P1 P2 P3 = P4

y x z

x z y

z y x

Note that there is no pair of agents having mutually exclusive preferences and that yPmx,

xImz and zPmy. Therefore, we have a weak cycle. Since 2 = f z(A;PN) � #N2= 2, we have

that CW (A;PN) = fzg but there is no strong Condorcet winner.However, Example 1 illustrates that in order to have a strong Condorcet winner we must

strenght the above condition.

Example 1 (continuated) Note that F x(A;PMA) = ;, F y(A;PMA) = f4g, F z(A;PMA) =

f1; 2g. Since f z(A;PMA) = 2 > #MA

2, z is the strong Condorcet winner relative to A at PN .

The next Proposition, shows that the condition in Example 1 is not an accident but the

norm, such a condition is a necessary and su¢ cient condition that a pro�le of individual

preferences must satisfy to guarantee that the majority rule relation has a unique best

choice and then, to ensure the existence of a strong Condorcet winner. The condition if

also expressed in terms of the net preference pro�le and entails the requirement that there

is an alternative that more of half of the net set of agents regard it as the most preferred

alternative.

Proposition 2 For any N � N+ and any PN 2 �i2NPi, a 2 A is a strong Condorcet

winner if and only if for some a 2 T , fa(T; PMT ) > #MT

2.

The proof follows trivially from Proposition 1 and therefore it is omitted.

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4 Beyond triples of alternatives

In Section 4, we have introduced a condition guaranteeing that the majority rule relation is

strictly acyclic when the set of alternatives contains three alternatives. As we have seen in

Proposition 1, the majority rule relation may exibits semi-cycles and this is not an impedi-

ment for the existence of Condorcet winners (although there may not be strong Condorcet

winners). In this section, we study to what extent such results can be generalized to prefer-

ence relations de�ned on any �nite set of alternatives. Along this section, we will assume that

A is any �nite set of alternatives and we refer to T � A as any subset of three alternatives.We �rst extend the de�nition of strict cycle to any number of alternatives greater than

two.

De�nition 7 For any N � N+, and any PN 2 �i2NPi, Rm exhibits a strict cycle at PN oflength k relative to A if for some list of alternatives x1; x2; :::; xk in A, x1Pmx2, x2Pmx3,...,

xk�1Pmxk, and xkPmx1.

We say that Rm is strictly acyclic if Rm does not exhibit strict cycles of any length.

In the next example, we show that strict acyclicity over triples of alternatives does not

guarantee strict acyclicity. In other words, there are preference pro�les for which themajority

rule relation does not exhibit any strict cycle relative to any triple of alternatives T � A, butthe majority rule relation exhibits a strict cycle involving all alternatives in A. Therefore,

there is no Condorcet winner relative to A.

Example 4 Let N = f1; 2; 3; 4g, A = fx; y; z; wg, and PN 2 �i2NP be as in the followingtable:

P1 P2 P3 P4

w x y z

x y z w

y z w x

z w x y

Note that wPmx, xPmy, yPmz, and zPmw, that is, there is a strict cycle involving all the

alternatives in A and there is no Condorcet winner. However, the majority rule relation is

strictly acyclic relative to any T � A. We now check if the condition introduced in Section

10

3 holds at any subset of three alternatives. Let�s start with T = fx; y; wg. The individualpreference pro�le relative to T is given in the following table

P T1 P T2 P T3 P T4

w x y w

x y w x

y w x y

The set of non-mutually exclusive agents relative to T isMT = N = f1; 2; 3; 4g, F x(T; P TMT ) =

f2g, F y(T; P TMT ) = f3g, Fw(T; P TMT ) = f1; 4g, and fw(T; P TMT ) = 2 =#MT

2. The majority

rule relation is strictly acyclic relative to T but exhibits a weak cycle, yImw, wPmx, and

xPmy.

For T 0 = fx; z; wg, the individual preference pro�le relative to T 0 is given in the followingtable

P T1 P T2 P T3 P T4

w x z z

x z w w

z w x x

The set of non-mutually exclusive agents relative to T 0 isMT 0 = N = f1; 2; 3; 4g, F x(T 0; P T 0MT 0 ) =

f2g, F z(T 0; P T 0MT 0 ) = f3; 4g, Fw(T 0; P T

0

MT 0 ) = f1g, and f z(T 0; P T0

MT 0 ) = 2 =#MT 0

2. The major-

ity rule relation is strictly acyclic relative to T 0 but exhibits a weak cycle, xImz, zPmw, and

wPmx.

For T 00 = fy; z; wg, the individual preference pro�le relative to T 00 is given in the followingtable

P T00

1 P T00

2 P T00

3 P T00

4

w y y z

y z z w

z w w y

The set of non-mutually exclusive agents relative to T 00 isMT 00 = N = f1; 2; 3; 4g, F y(T 00; P T 00MT 00 ) =

f2; 3g, F z(T 00; P T 00MT 00 ) = f4g, Fw(T 00; P T

00

MT 00 ) = f1g, and f y(T 00; P T00

MT 00 ) = 2 =#MT 00

2. The ma-

jority rule relation is strictly acyclic relative to T 00 but exhibits a weak cycle, wImy, yPmz,

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and zPmw.

Finally, for T 000 = fx; y; zg, the individual preference pro�le relative to T 000 is given in thefollowing table

P T000

1 P T000

2 P T000

3 P T000

4

x x y z

y y z x

z z x y

The set of non-mutually exclusive agents relative to T 000 isMT 000 = N = f1; 2; 3; 4g, F x(T 000; P T 000MT 000 ) =

f1; 2g, F y(T 000; P T 000MT 000 ) = f3g, F z(T 000; P T

000

MT 000 ) = f4g, and fx(T 000; P T000

MT 000 ) = 2 =#MT 000

2. The

majority rule relation is strictly acyclic relative to T 000 but exhibits a weak cycle, zImx, xPmy,

and yPmz.

In the light of Example 4, for the majority rule relation to be strictly acyclic it is not

su¢ cient the avoidance of strict cycles at every triple of alternatives. We also have to require

that if a majority net alternative x in a triple is not the majority net alternative in all the

triples in which x is present is because there is another alternative y that it is a majority net

alternative in all triples in which both alternatives, x and y, are present.

??????????

Theorem 1 For any N � N+ and any PN 2 �i2NPi, Rm is acyclic if and only if (1) forevery triple T � A there exists some a 2 T , fa(T; P TMT ) � #MT

2and (2) for any subset of

four alternatives S � A, there exists T � A and a 2 T such that fa(T; P TMT ) >#MT 0

2> 0.

Proof. To be proven.

??????????

Theorem 2 For any N � N+ and any PN 2 �i2NPi, Rm is acyclic if and only if (1) forevery triple T � A there exists some a 2 T , fa(T; P TMT ) � #MT

2and (2) for any T 0 � A

and for some a 2 T 0 such that fa(T 0; P T0

MT 0 ) � #MT 0

2> 0, if there is d 2 A such that

fa(T; P TMT ) <#MT

2for some T = fa; d; xg with x 2 T 0nfag, then fd(T; P TMT ) � #MT

2for

any T = fa; d; xg such that x 2 T 0nfag.

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Proof. Let PN 2 �i2NPi, and Rm be such that (1) and (2) hold. By (1), there is no strict cy-cle involving only three alternatives. Suppose to get a contradiction that Rm exhibits a strict

cycle involving more than three alternatives and take one involving the minimun number of

alternatives, lets say k alternatives. Then, there exists a list of alternatives x1; x2; :::; xk in

A, such that x1Pmx2, x2Pmx3,..., xk�1Pmxk, and xkPmx1. Given that there is no strict cycle

involving less than k alternatives, we have that for T 0 = fx1; xk�2; xk�1g, x1Rmxk�2 andx1Rmxk�1. Since xk�2Pmxk�1,

#MT 0

2> 0. Then, by Proposition 1, fx1(T 0; P T

0

MT 0 ) � #MT 0

2.

Let T = fx1; xk�1; xkg. Note also that xk�1PmxkPmx1 and, since there is no strict cycleinvolving less than k alternatives, xk�1Rmx1. Therefore, fxk(T; P TMT ) <

#MT

2and we get a

contradiction to (2).

We now show that if Rm is strictly acyclic then for any PN 2 �i2NPi, (1) and (2) aresatis�ed. By Proposition 1, (1) is trivially satis�ed. Suppose without loss of general-

ity, and to get a contradiction, that there exists T 0 = fa; b; cg such that for any x 2 T 0

with fx(T 0; P T0

MT 0 ) � #MT 0

2> 0, and some d 2 A such that fa(T; P TMT ) <

#MT

2for

some T = fa; d; xg with x 2 T 0nfag we have that for some T = fx; d; yg such thaty 2 T 0nfxg, fd(T; P TMT ) <

#MT

2. Since #MT 0

2> 0; it can not be the case that for any

x 2 T 0, fx(T 0; P T 0MT 0 ) � #MT 0

2. We distinguish between two cases:

Case 1 . a is the unique alternative in T 0 such that fa(T 0; P T 0MT 0 ) � #MT 0

2. Let d 2 A be such

that dPmx. We distinguish between two subcases:

Subcase 1.1. aPmb and aPmc. Since for some T = fa; d; xg; where x 2 T 0nfag, fd(T; P TMT ) <

#MT

2, we have that xPmdPmaPmx, contradicting that Rm is strictly acyclic.

Subcase 1.2. aPmb and bPmc. If bPmd then bPmdPmaPmb, we get a contradiction to Rm being

strictly acyclic. If cPmdPmaPmbPmc, we get a contradiction to Rm being strictly acyclic.

Case 2 . a and b are such fa(T 0; P T 0MT 0 ) � #MT 0

2and f b(T 0; P T

0

MT 0 ) � #MT 0

2. Therefore, aImb.

We distinguish among three subcases:

Subcase 2.1. aPmc and bPmc. Note that it cannot be the case that cPmdPmaPmc. Therefore,

bPmdPma and aImb.

Subcase 2.2. aPmc and bImc. Note that it cannot be the case that cPmdPmaPmc. Therefore,

bPmdPma and aImb.

Subcase 2.3. aImc and bPmc. Note that there exist only 3 di¤erent pairs of opposite pref-

erencess: abc=cba, acb=bca, bac=cab. Therefore, any net pro�le is a combination of a repre-

13

sentative preference relation of each of the above pairs of opposite preferences. Since aImb

it cannot be the case that there exist agents with either the following three preference re-

lations (abc; acb; cab) or (cba; bca; bac). Since aImc it cannot be the case that there exist

agents with either the following three preference relations (abc; acb; bac) or (cba; bca; cab).

Also, since bPmc it cannot be the case that there exist agents with the following three pref-

erence relations (cba; acb; cab). There are only two possible admissible preference pro�les:

(abc; bca; bac) and (abc; bca; cab). Assume that the preference pro�le is (abc; bca; bac) and let

n1 be the number of agents exhibiting preference relation abc, n2 those with bca and n3 those

with bac. Since aImb, n1 = n2+n3 and since aImc, n1+n3 = n2. Therefore, n3 = 0. Assume

that the preference pro�le is (abc; bca; cab) and let n1 be the number of agents exhibiting

preference relation abc, n2 those with bca and n3 those with cab. Since aImb, n1 + n3 = n2

and since aImc, n1 = n2 + n3. Hence, n3 = 0. Therefore, in both cases we have that half of

the agents in the net pro�le exhibit preference relation abc and the other half bca. And all

agents prefer alternative b to alternative c. Finally, as cPmd, bPmdPma and aImb.

Thus, in any of the above subcases, bPmdPma and aImb. Since (2) does not hold, there

exists e 2 A such that ePmb and such that for some fe; b; xg = T 00 with x 2 T 0nfbg,f e(T 00; P T

00

MT 00 ) <#MT 00

2. Note that if x = a, aPmePmbPmdPma, contradicting that Rm is

acyclic. If x = c, we have that cPmePmbPmdPma and subcases 2.1 and 2.2 do not hold.

Therefore, only subcase 2.3 can happen, that is, aImc and bPmc. But then, cPmePmb and

bPmc, and we get to a contradiction to Rm being strictly acyclic.

We now present two examples. Example 5 shows that the non-existence of strict-cycles

involving all alternatives does not guarantee the existence of Condorcet winners. Example

6 below illustrates that strict acyclicity of the majority rule relation is not a necessary

condition to guarantee the existence of Condorcet winners.

Example 5. No strict cycle involving all alternatives. Let N = f1; 2; 3; 4; 5g;A = fw; x; y; zg, and PN 2 �i2NP be as in the following table

14

P1 = P2 P3 P4 P5

w x y z

x y z x

y w w y

z z x w

Note that wPmx, xPmy, yPmz, and wPmz, that is, there is not a strict cycle involving all the

alternatives in A. However, the majority rule relation is not strictly acyclic relative to every

triple T � A. We now check if the condition introduced in Section 3 holds at any subset ofthree alternatives. Let�s start with T = fx; y; wg. The individual preference pro�le relativeto T = fx; y; zg is given in the following table

P T1 = PT2 P T3 P T4 P T5

x x y z

y y z x

z z x y

The set of non-mutually exclusive agents relative to T isMT = N = f1; 2; 3; 4; 5g, F x(T; P TMT ) =

f1; 2; 3g, F y(T; P TMT ) = f4g, F z(T; P TMT ) = f5g, and fx(T; P TMT ) � #MT

2. Rm is strictly

acyclic relative to T .

For T 0 = fw; y; zg, the individual preference pro�le relative to T 0 is given in the followingtable

P T0

1 = P T0

2 P T0

3 P T0

4 P T0

5

w y y z

y w z y

z z w w

The set of non-mutually exclusive agents relative to T 0 isMT 0 = f1; 3; 4g � N , Fw(T 0; P T 0MT 0 ) =

f1g, F y(T 0; P T 0MT 0 ) = f3; 4g, F z(T 0; P T

0

MT 0 ) = ;, and f y(T 0; P T0

MT 0 ) = 2 >#MT 0

2. Rm is strictly

acyclic relative to T 0.

For T 00 = fw; x; zg, the individual preference pro�le relative to T 00 is given in the followingtable

15

P T00

1 = P T00

2 P T00

3 P T00

4 P T00

5

w x z z

x w w x

z z x w

The set of non-mutually exclusive agents relative to T 00 isMT 00 = f1g � N , Fw(T 00; P T 00MT 00 ) =

f1g, F x(T 00; P T 00MT 00 ) = ;, F z(T 00; P T

00

MT 00 ) = ;, and fw(T 00; P T00

MT 00 ) = 1 >#MT 00

2. Rm is strictly


For T 000 = fw; x; yg, the individual preference pro�le relative to T 000 is given in the followingtable

P T000

1 = P T000

2 P T000

3 P T000

4 P T000

5

w x y x

x y w y

y w x w

The set of non-mutually exclusive agents relative to T 000 is MT 000 = N = f1; 2; 3; 4; 5g,Fw(T; P T

000

MT 000 ) = f1; 2g, F x(T; P T000

MT 000 ) = f3; 5g, F y(T; P T000

MT 000 ) = f4g, and there is no alterna-tive a 2 T 000 such that fa(T 000; PMT 000 ) � #MT 000

2: Hence, Rm is not strictly acyclic relative to

T 000. Finally, note that CW (A;PN) = ;.

Example 6 There is a strict-cycle and a Condorcet winner. Let N = f1; 2; 3g;A = fw; x; y; zg, and PN 2 �i2NP be as in the following table

P1 P2 P3

x y z

w w w

y z x

z x y

Note that wPmz, zPmx, xPmy, and wPmy, that is, there isn�t a strict cycle involving all the

alternatives in A. However, the majority rule relation is not strictly acyclic relative to every

triple T � A. We now check if the condition introduced in Section 3 holds at any subset of

16

three alternatives. Let�s start with T = fx; y; wg. The individual preference pro�le relativeto T = fx; y; zg is given in the following table

P T1 P T2 P T3

x y z

y z x

z x y

The set of non-mutually exclusive agents relative to T isMT = N = f1; 2; 3g, F x(T; P TMT ) =

f1g, F y(T; P TMT ) = f2g, F z(T; P TMT ) = f3g, and there is no alternative a 2 T such that

fa(T; P TMT ) � #MT

2: Hence, Rm is not strictly acyclic relative to T = fx; y; zg.


P T0

1 P T0

2 P T0

3

w y z

y w w

z z y

The set of non-mutually exclusive agents relative to T 0 is MT 0 = f1g � N , Fw(T 0; P T 0MT 0 ) =

f1g, F y(T 0; P T 0MT 0 ) = ;, F z(T 0; P T 0

MT 0 ) = ;, and fw(T 0; P T 0MT 0 ) = 1 > #MT 0

2. Rm is strictly



P T00

1 P T00

2 P T00

3

x w z

w z w

z x x


f2g, F x(T 00; P T 00MT 00 ) = ;, F z(T 00; P T

00

MT 00 ) = ;, and fw(T 00; P T00

MT 00 ) = 1 >#MT 00

2. Rm is strictly


For T 000 = fw; x; yg, the individual preference pro�le relative to T 000 is given in the followingtable

17

P T000

1 P T000

2 P T000

3

x y w

w w x

y x y


f3g, F x(T 000; P T 000MT 000 ) = ;, F y(T 000; P T

000

MT 000 ) = ;, and fw(T 000; P T000

MT 000 ) = 1 >#MT 000

2. Rm is strictly


Finally, note that even although Rm is not strictly acyclic we have that CW (A;PN) = fwgand it is the case that for any eT � A such that w 2 eT , fw(eT ; PM eT ) � #M

eT2.

As we show in the following Theorem, the condition that alternative w satis�es in Ex-

ample 6 is not only a su¢ cient but also a necessary condition to guarantee the existence of

Condorcet winner.

Theorem 3 For any N � N+ and any PN 2 �i2NPi, CW (A;PN) 6= ; if and only if forsome a 2 A, and for any T � A such that a 2 T , fa(T; P TMT ) � #MT

2.

Proof. It follows from Proposition 1.

An di¤erent but complementary reading of Theorem 3 suggests that the above condition

can be also interpreted as an alternative de�nition of a Condorcet winner. It is a de�nition

that it not based on comparison among pairs of alternatives but on triple of alternatives and

the number of times that an alternative is the most preferred alternative in a net preference.

Example 7 Computation of Condorcet winners. Let N = f1; 2; 3; 4; 5; 6; 7; 8g; A =fw; x; y; z; sg, and PN 2 �i2NP be as in the following table

P1 P2 = P3 = P4 P5 = P6 P7 = P8

x z w s

s w x y

w s y x

y y s w

z x z z

18

We start the procedure to �nd the Condorcet winner with any triple of alternatives. Let�s

start with T = fs; w; zg. The individual preference pro�le relative to T = fs; w; zg is givenin the following table

P T1 P T2 = PT3 = P

T4 P T5 = P

T6 P T7 = P

T8

s z w s

w w s w

z s z z

The set of non-mutually exclusive agents relative to T is MT = f5; 6g, F s(T; P TMT ) = ;,Fw(T; P TMT ) = f5; 6g, F z(T; P TMT ) = ;, and fw(T; P TMT ) >

#MT

2. Therefore, alternatives s

and z cannot be Condorcet winners and there is only one triple of alternatives to check.

Let T 0 = fx; y; wg, the individual preference pro�le relative to T 0 is given in the followingtable

P T0

1 P T0

2 = P T0

3 = P T0

4 P T0

5 = P T0

6 P T0

7 = P T0

8

x w w y

w y x x

y x y w

The set of non-mutually exclusive agents relative to T 0 is MT 0 = f1; 2; 3; 4g, F x(T 0; P T 0MT 0 ) =

f1g, F y(T 0; P T 0MT 0 ) = ;, Fw(T 0; P T

0

MT 0 ) = f2; 3; 4g, and fw(T 0; P T0

MT 0 ) = 3 >#MT 0

2. Then, the

unique Condorcet winner is alternative w, that indeed is a strong Condorcet winner.

We �nish this Section, presenting the corresponding result for the existence of a strong

Condorcet winner.

Theorem 4 For any N � N+ and any PN 2 �i2NPi, a 2 A is a Strong Condorcet Winnerrelative to A if and only if for any triple T � A such that a 2 T , fa(T; PMT ) > #MT

2.

Proof. It follows from Proposition 2

5 Existence of indi¤erences

In the precedent sections, we have restricted ourselves to situations in which the preferences

of the agents are linear orderings but in this section we relax such assumption. Let R be the

19

set of complete, re�exive, and transitive orderings on A and Ri � R be the set of admissible

preferences for agent i. For any N � N+, a preference pro�le, denoted by RN = (R1; ::; Rn);is an element of �i2NRi. As usual, we denote by Pi and Ii the strict and the indi¤erence

part of Ri, respectively. We denote as RTi , the preferences of agent i relative to T � A andas RT

i , the set of all preferences over T .

For any agent i and any T = fx; y; zg � A, there are 13 possible preferences of i on T . Inparticular there is one preference relation showing complete indi¤erence. Before introducing

the main idea in this section, we provide the following de�nition.

De�nition 8 For any T = fx; y; zg � A, we say that agent i is a unconcerned agent atRi 2 Ri relative to T , if xITyIT z.

In what follows, we will show that indi¤erences are not an impediment for the analysis

and the extension of the previous results. To simplify the notation, for any N � N+, any

T � A and any RTN 2 �i2NRTi , let I

TRN= fi 2 N : aIib and either cPia or aPic for a; b; c 2 Tg

be the set of agents that are indi¤erent between exactly two alternatives relative to T .

De�nition 9 Let N � N+, T = fx; y; zg � A and RTN 2 �i2NRTi be such that I

TRN6= ;. We

de�ne the replicated preference pro�le of RTN , RT

N[NT as (1) for any j 2 NnITRN thereexist j 2 N and j0 2 NT , such that R

T

j = RT

j0 = RTj , (2) for any i 2 ITRN there exist i 2 N

and i0 2 NT , such that aPT

i b, and [cPT

i a if and only if cPTi a] and bP

T

i0a, and [cPT

i0a if and

only if cP Ti0 a].

That is, for any N � N+ and any T = fx; y; zg � A, the replica preference pro�le

duplicate the number of agents; for the agents with indi¤erences between two alternatives,

the replica will preserve the strict binary relation and will break down the indi¤erence aIib

by creating two individuals fi; i0g with the strict binary relations aP ib and bP i0a, and forthe rest of agents an exact replica is created:

Example 8 Let N = f1; 2; 3; 4; 5; 6g; T = fx; y; zg � A; and consider RTN as in the followingtable

RT1 = RT2 RT3 = R

T4 RT5 RT6

z x y yz

y z xz x

x y

20

We now construct the replicated preference pro�le ofRTN , RT

N[NT whereN[NT = f1; 10; 2; 20; 3; 30;4; 40; 5; 50; 6; 60g as follows

R1 = R10 = R2 = R20 R3 = R30 = R4 = R40 R5 R50 R6 R60

z x y y y z

y z x z z y

x y z x x x

The remarkable feature that allow us to work with or without indi¤erences is given in

the following lemma.

Lemma 2 For any triple T � A, any N � N+, and any RN 2 �i2NRi such that ITRN 6= ;,N(a; b;RN) � N(b; a;RN) if and only if N [N 0(a; b;R

T

N[NT ) � N [N 0(b; a;RT

N[NT ) where

RT

N[NT is the replicated preference pro�le of RTN .

Proof. Let N � N+, RN 2 �i2NRi and T � A be such that ITRN 6= ;. Let RT

N[NT be the

essentially replica preference pro�le of RTN . Note that for any a; b 2 T , if k 2 N(a; b;RN) thenk 2 N(a; b;RTN) and k0 2 N(a; b;R

T

NT ). Also, for any i 2 ITRN such that aIib, we have thati 2 N(a; b;RTN) and i0 2 N(b; a;R

T

NT ). Finally, the unconcerned agents at RN relative to T ,

does not play any role at all when comparing alternatives in T . Therefore, for any a; b 2 T ,N [NT (a; b;R

T

N[NT ) � N [NT (a; b;RT

N[NT ) if and only if N(a; b;RN) � N(a; b;RN).Lemma 2, tells us that there is no loss of generality in assuming pro�les of preferences

where all agents have strict preferences. The computation of the social binary relation

becomes easier and the results of the previous sections hold when for any triple of alternatives

we consider the subset of N that contains all the non-mutual exclusive pairs of agents given

a replicated preference pro�le and exclude the unconcerned agents. We clarify this idea in

the following example.

Example 9 Let N = f1; 2; 3g; A = fw; x; y; zg, and RN 2 �i2NR be as in the following table

R1 = R2 R3 R4

xw y z

yz wz w

x x

y

21

Let�s start with T = fx; y; wg. The individual preference pro�le relative to T = fx; y; zg isgiven in the following table

RT1 = RT2 RT3 RT4

xw y w

y w x

x y

We now construct the replicated preference pro�le ofRTN , RT

N[NT whereN[NT = f1; 10; 2; 20; 3; 30; 4; 40gas follows

RT

1 = RT

2 RT

10 = RT

20 RT

3 = RT

30 RT

4 = RT

40

x w y w

w x w x

y y x y

And the set of non-mutually exclusive agents at RT

N[NT relative to T isMT = f10; 20; 4; 40g �N [NT , F x(T;R

T

MT ) = ;, F y(T;RTMT ) = ;, Fw(T;RTMT ) = f10; 20; 4; 40g, and fw(T;RTMT ) =

4 > #MT

2. wPmx, xImy and wPmy; Rm exhibits neither strict nor weak cycles relative to T .


RT0

1 = RT0

2 RT0

3 RT0

4

w y z

yz wz w

y

We now construct the replicated preference pro�le of RT0

N , RT 0

N[NT 0 where N [ NT 0 =

f1; 10; 2; 20; 3; 30; 4; 40g as follows

RT 0

1 = RT 0

2 RT 0

10 = RT 0

20 RT 0

3 RT 0

30 RT 0

4 = RT 0

40

w w y y z

y z w z w

z y z w y

And the set of non-mutually exclusive agents at RT 0

N[NT 0 relative to T 0 isMT 0 = f1; 2; 10; 4g �N[NT 0, Fw(T 0; R

T 0

MT 0) = f1; 2; 10g, F y(T 0; RT0

MT 0) = ;, F z(T 0; RT0

MT 0) = f4g, and fw(T 0; RT0

MT 0) =

22

3 > #MT 0

2. wPmy, yImz and wPmz; Rm exhibits neither strict nor weak cycles relative to T 0.


RT00

1 = RT00

2 RT00

3 RT00

4

xw wz z

z x w

x


N , RT 00


f1; 10; 2; 20; 3; 30; 4; 40g as follows

RT 00

1 = RT 00

2 RT 00

10 = RT 00

20 RT 00

3 RT 00

30 RT 00

4 = RT 00

40

x w w z z

w x z w w

z z x x x

And the set of non-mutually exclusive agents atRT 00

N[NT 00 relative to T 00 isMT 00 = f10; 20; 3; 4g �N[NT 00, Fw(T 00; R

T 00

MT 00 ) = f10; 20; 3g, F x(T 00; RT00

MT 00 ) = ;, F z(T 00; RT00

MT 00 ) = f4g, and fw(T 00; RT00

MT 00 ) =

3 > #MT 00

2. wPmx, xImz and wPmz; Rm exhibits neither strict nor weak cycles relative to

T 00.

For T 000 = fx; y; zg, the individual preference pro�le relative to T 000 is given in the followingtable

RT000

1 = RT000

2 RT000

3 RT000

4

x y z

yz z x

x y


N , RT 000


f1; 10; 2; 20; 3; 30; 4; 40g as follows

RT 000

1 = RT 000

2 RT 000

10 = RT 000

20 RT 000

3 = RT 000

30 RT 000

4 = RT 000

40

x x y z

y z z x

z y x y

23

The set of non-mutually exclusive agents relative to T 000 is MT 000 = f1; 2; 4; 40g � N [NT 000 ,

F x(T 000; RT 000

MT 000 ) = f1; 2g, F y(T 000; RT000

MT 000 ) = ;, F z(T 000; RT000

MT 000 ) = f4; 40g, and fx(T 000; RT000

MT 000 ) =

f z(T 000; RT 000

MT 000 ) = 2 = #MT 000

2. xPmy, yImz and xImz; Rm exhibits neither strict nor weak

cycles relative to T 000.

Finally, note that alternative w is such that fw(eT ;R eTM eT ) > #M

eT2, and by Theorem 3 we

have that w 2 CW (A;RN). Also, since the condition in Theorem 1 is satis�ed, we have the

majority rule relation is strictly acyclic.

ReferencesBlack, D. (1958): The Theory of Committees and Elections. Cambridge University Press,

Cambridge, MA

Gjorgjiev R. and Xefteris, D. (2013). "Transitive Supermajority Rule Relations", Mimeo

Scott L. Feld and Grofman, B. (1986). Partial Single-Peakedness: An Extension and

Clari�cation. Public Choice, V ol:51; No:1; pp:71� 80Sen, A. and Pattanaik, P. K. (1969): "Necessary and Su¢ cient Conditions for Rational

Choice under Majority Decision", Journal of Economic Theory 1 : 178� 202

24

acyclicty and condorcet winners

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