The Strict Majority Rule

Abstract

The strict majority rule, also called non-minority rule, is defined by: An alternative x is considered to be socially at least as good as some other alternative y iff a majority of all individuals do not prefer y to x. This chapter provides a characterization of strict majority rule and Inada-type necessary and sufficient conditions for transitivity and quasi-transitivity under the strict majority rule.

Appendices

Appendix

4.6 Interrelationships Among Restrictions on Preferences

In the context of the strict majority rule the following four domain conditions have been found to be relevant.

Extremal Value Restriction (EVR): Let $\mathscr {D} \subseteq \mathscr {T}$ be a set of orderings of S. Let $A = \{x,y,z\}\subseteq S$ be a triple of alternatives. $\mathscr {D}$ satisfies EVR over the triple A iff (i) whenever an alternative is uniquely best in some $R \in \mathscr {D}|A$ then it is not medium in any $R \in \mathscr {D}|A$ unless it is worst also; or (ii) whenever an alternative is uniquely worst in some $R \in \mathscr {D}|A$ then it is not medium in any $R \in \mathscr {D}|A$ unless it is best also. More formally, $\mathscr {D}$ satisfies EVR over A iff $(\forall \; \text {distinct} \; a,b,c \in A) [(\exists R \in \mathscr {D}|A) (aPb \wedge aPc) \rightarrow (\forall R \in \mathscr {D}|A) [(bRaRc \rightarrow cRa) \wedge (cRaRb \rightarrow bRa)]] \vee (\forall \; \text {distinct} \; a,b,c \in A) [(\exists R \in \mathscr {D}|A) (bPa \wedge cPa) \rightarrow (\forall R \in \mathscr {D}|A) [(bRaRc \rightarrow aRb) \wedge (cRaRb \rightarrow aRc)]]$. $\mathscr {D}$ satisfies EVR iff it satisfies EVR over every triple of alternatives contained in S.

Conflictive Preferences (CP): Let $\mathscr {D} \subseteq \mathscr {T}$ be a set of orderings of S. Let $A = \{x,y,z\}\subseteq S$ be a triple of alternatives. Let $\mathscr {D}_{c}$ be the set of concerned orderings in $\mathscr {D}|A$. $\mathscr {D}$ satisfies CP over the triple A iff it is the case there exists a decomposition² $(\mathscr {D}_{1}, \mathscr {D}_{2})$ of the set of concerned orderings in $\mathscr {D}|A$ such that for some distinct $a,b,c \in A$: every ordering $R \in \mathscr {D}_{1}$ is aPbRc and every ordering $R \in \mathscr {D}_{2}$ is cRbPa. More formally, $\mathscr {D}$ satisfies CP over the triple A iff $(\exists \mathscr {D}_{1}, \mathscr {D}_{2} \subseteq \mathscr {D}_{c}) (\exists \; \text {distinct} \; a,b,c \in A)[(\emptyset \subseteq \mathscr {D}_{1} \wedge \emptyset \subseteq \mathscr {D}_{2} \wedge \mathscr {D}_{1} \cap \mathscr {D}_{2} = \emptyset \wedge \mathscr {D}_{1} \cup \mathscr {D}_{2} = \mathscr {D}_{c}) \wedge (\forall R \in \mathscr {D}_{1}) (aPbRc) \wedge (\forall R \in \mathscr {D}_{2}) (cRbPa)]$. $\mathscr {D}$ satisfies CP iff it satisfies CP over every triple of alternatives contained in S.

Weakly Conflictive Preferences (WCP): Let $\mathscr {D} \subseteq \mathscr {T}$ be a set of orderings of S. Let $A = \{x,y,z\}\subseteq S$ be a triple of alternatives. $\mathscr {D}$ satisfies WCP over the triple A iff it is the case that: (i) whenever in an $R \in \mathscr {D}|A$ an alternative best in some linear ordering belonging to $\mathscr {D}|A$ is worst, the alternative worst in the linear ordering is best in it; or (ii) whenever in an $R \in \mathscr {D}|A$ an alternative worst in some linear ordering belonging to $\mathscr {D}|A$ is best, the alternative best in the linear ordering is worst in it. That is to say, $\mathscr {D}$ satisfies WCP over the triple A iff $(\forall \; \text {distinct} \; a,b,c \in A)[ (\exists R \in \mathscr {D}|A)(aPbPc) \rightarrow (\forall R \in \mathscr {D}|A) (bRa \wedge cRa \rightarrow cRb)] \vee (\forall \; \text {distinct} \; a,b,c \in A)[ (\exists R \in \mathscr {D}|A)(aPbPc)\ \rightarrow \ (\forall R \in \mathscr {D}|A) (cRa \wedge cRb \rightarrow bRa)]$. $\mathscr {D}$ satisfies WCP iff it satisfies WCP over every triple of alternatives contained in S.

Unique Value Restriction (UVR): Let $\mathscr {D} \subseteq \mathscr {T}$ be a set of orderings of S. Let $A = \{x,y,z\}\subseteq S$ be a triple of alternatives. $\mathscr {D}$ satisfies UVR over the triple A iff (i) there exist distinct $a,b \in A$ such that a is not uniquely medium in any $R \in \mathscr {D}|A$, b is not uniquely best in any $R \in \mathscr {D}|A$, and whenever b is best in an $R \in \mathscr {D}|A$ a is worst in it; or (ii) there exist distinct $a,b \in A$ such that a is not uniquely medium in any $R \in \mathscr {D}|A$, b is not uniquely worst in any $R \in \mathscr {D}|A$, and whenever b is worst in an $R \in \mathscr {D}|A$ a is best in it. More formally, $\mathscr {D} \subseteq \mathscr {T}$ satisfies UVR over the triple A iff $(\exists \; \text {distinct} \; a,b,c \in A) (\forall R \in \mathscr {D}|A) [[(aRb \wedge aRc) \vee (bRa \wedge cRa)] \wedge [aRb \vee cRb] \wedge [bRa \wedge bRc \rightarrow cRa]]\! \vee \! (\exists \; \text {distinct} \; a,b,c \in A) (\forall R \in \mathscr {D}|A) [[(aRb \wedge aRc) \vee (bRa \wedge cRa)] \wedge [bRa \vee bRc] \wedge [aRb \wedge cRb \rightarrow aRc]]$. $\mathscr {D}$ satisfies UVR iff it satisfies UVR over every triple of alternatives contained in S.

Proposition 4.1

A set of orderings of a triple satisfies Latin Square extremal value restriction iff it satisfies extremal value restriction or conflictive preferences.

Proof

Let the set of orderings $\mathscr {D}|A$ over the triple $A = \{x,y,z\}$ violate LSEVR. Then we have: $[(\exists \; \text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t} \in \mathscr {D}|A \cap T[LS(abca)]) (aP^{s}bR^{s}c \wedge cR^{t}aP^{t}b)]$.

a is uniquely best in $aP^{s}bR^{s}c$ and medium in $cR^{t}aP^{t}b$ without being worst; and b is uniquely worst in $cR^{t}aP^{t}b$ and medium in $aP^{s}bR^{s}c$ without being best. Therefore EVR is violated.

Given that $\mathscr {D}|A$ contains $aP^{s}bR^{s}c \wedge cR^{t}aP^{t}b$, it is immediate that CP is violated.

Thus we see that violation of LSEVR implies violation of both EVR and CP, implying that if EVR or CP holds then LSEVR also holds.

Next we show that if both CP and EVR are violated then LSEVR is also violated.

It can be easily checked that the set of orderings $\mathscr {D}|A$ violates CP iff $(\exists \;\text {distinct} a,b,c \in A) (\exists R^{s},R^{t} \in \mathscr {D}|A) [(aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a) \vee (aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c) \vee (aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b) \vee (aP^{s}bP^{s}c \wedge bP^{t}cI^{t}a) \vee (aP^{s}bP^{s}c \wedge cI^{t}aP^{t}b) \vee (aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a) \vee (aP^{s}bI^{s}c \wedge aI^{t}bP^{t}c) \vee (aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a)]$.

If $(\exists \;\text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t} \in \mathscr {D}|A) [(aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a) \vee (aP^{s}bP^{s}c \wedge bP^{t}cI^{t}a) \vee (aP^{s}bP^{s}c \wedge cI^{t}aP^{t}b) \vee (aP^{s}bI^{s}c \wedge aI^{t}bP^{t}c)]$ then LSEVR is violated. Therefore it suffices to consider the remaining four cases.

Consider the case when $\mathscr {D}|A$ contains $(aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c)$. a is uniquely best in $aP^{s}bP^{s}c$ and medium in $bP^{t}aP^{t}c$ without being worst. So one part of the EVR condition does not hold. The other part of EVR condition would be violated iff there is an ordering in $\mathscr {D}|A$ in which a is uniquely worst or an ordering in $\mathscr {D}|A$ in which b is uniquely worst or an ordering in $\mathscr {D}|A$ in which c is medium without being best, i.e., it contains $(bP^{u}cP^{u}a \vee bI^{u}cP^{u}a \vee cP^{u}bP^{u}a \vee cP^{u}aP^{u}b \vee cI^{u}aP^{u}b \vee aP^{u}cP^{u}b \vee aP^{u}cI^{u}b \vee bP^{u}cI^{u}a)$. With the inclusion of the required ordering, in each case LSEVR is violated.

Next consider the case when $\mathscr {D}|A$ contains $(aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b)$. c is uniquely worst in $aP^{s}bP^{s}c$ and medium in $aP^{t}cP^{t}b$ without being best. So one part of the EVR condition does not hold. The other part of EVR condition would be violated iff there is an ordering in $\mathscr {D}|A$ in which b is uniquely best or an ordering in $\mathscr {D}|A$ in which c is uniquely best or an ordering in $\mathscr {D}|A$ in which a is medium without being worst, i.e., it contains $(bP^{u}cP^{u}a \vee bP^{u}cI^{u}a \vee bP^{u}aP^{u}c \vee cP^{u}aP^{u}b \vee cP^{u}aI^{u}b \vee cP^{u}bP^{u}a \vee bI^{u}aP^{u}c \vee cI^{u}aP^{u}b)$. With the inclusion of the required ordering, in each case LSEVR is violated.

When $\mathscr {D}|A$ contains $(aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a)$ then EVR would be violated only if there is an ordering in $\mathscr {D}|A$ in which some alternative is uniquely worst, i.e., only if $\mathscr {D}|A$ contains $(bP^{u}cP^{u}a \vee bI^{u}cP^{u}a \vee cP^{u}bP^{u}a \vee aP^{u}cP^{u}b \vee aI^{u}cP^{u}b \vee cP^{u}aP^{u}b \vee aP^{u}bP^{u}c \vee aI^{u}bP^{u}c \vee bP^{u}aP^{u}c)$. In all cases excepting when the ordering is $(bP^{u}cP^{u}a \vee aP^{u}cP^{u}b)$, LSEVR is violated. When $\mathscr {D}|A$ contains $[aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a \wedge (bP^{u}cP^{u}a \vee aP^{u}cP^{u}b)]$ then EVR would be violated iff an ordering in which a is medium without being worst is included or an ordering in which b is medium without being worst is included or an ordering in which c is uniquely best is included, i.e., $\mathscr {D}|A$ contains $(bP^{v}aP^{v}c \vee bI^{v}aP^{v}c \vee cP^{v}aP^{v}b \vee cI^{v}aP^{v}b \vee aP^{v}bP^{v}c \vee cP^{v}bP^{v}a \vee cI^{v}bP^{v}a \vee cP^{v}aI^{v}b)$. With the inclusion of the required ordering, in each case LSEVR is violated.

When $\mathscr {D}|A$ contains $(aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a)$ then EVR would be violated only if there is an ordering in $\mathscr {D}|A$ in which some alternative is uniquely best, i.e., only if $\mathscr {D}|A$ contains $(aP^{u}bP^{u}c \vee aP^{u}bI^{u}c \vee aP^{u}cP^{u}b \vee bP^{u}aP^{u}c \vee bP^{u}aI^{u}c \vee bP^{u}cP^{u}a \vee cP^{u}aP^{u}b \vee cP^{u}aI^{u}b \vee cP^{u}bP^{u}a)$. In all cases excepting when the ordering is $(aP^{u}bP^{u}c \vee cP^{u}bP^{u}a)$, LSEVR is violated. When $\mathscr {D}|A$ contains $[aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a \wedge (aP^{u}bP^{u}c \vee cP^{u}bP^{u}a)]$ then EVR would be violated iff an ordering in which a is medium without being best is included or an ordering in which c is medium without being best is included or an ordering in which b is uniquely worst is included, i.e., $\mathscr {D}|A$ contains $(bP^{v}aP^{v}c \vee bP^{v}aI^{v}c \vee cP^{v}aP^{v}b \vee cP^{v}aI^{v}b \vee aP^{v}cP^{v}b \vee aP^{v}cI^{v}b \vee bP^{v}cP^{v}a \vee cI^{v}aP^{v}b)$. With the inclusion of the required ordering, in each case LSEVR is violated.

Thus, violation of both EVR and CP implies violation of LSEVR, i.e., if LSEVR holds then it must be the case that EVR or CP holds.

This, together with the earlier demonstration that if EVR or CP holds then LSEVR holds, establishes the proposition.    $\square$

Proposition 4.2

A set of orderings of a triple satisfies weak Latin Square extremal value restriction iff it satisfies at least one of the three conditions of value restriction (1), extremal value restriction and conflictive preferences.

Proof

Let the set of orderings $\mathscr {D}|A$ over the triple $A = \{x,y,z\}$ violate WLSEVR. Then we have: $[(\exists \; \text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A \cap T[WLS(abca)]) (aP^{s}bR^{s}c \wedge bR^{t}cR^{t}a \wedge cR^{u}aP^{u}b)]$.

$aP^{s}bR^{s}c \wedge bR^{t}cR^{t}a \wedge cR^{u}aP^{u}b$ form WLS(abca), therefore VR(1) is violated.

$aP^{s}bR^{s}c \wedge cR^{u}aP^{u}b$ violate LSEVR. Then, In view of Proposition 4.1, it follows that both EVR and CP are violated.

Thus we see that violation of WLSEVR implies violation of all three conditions VR(1), EVR and CP, implying that if VR(1) or EVR or CP holds then WLSEVR also holds.

Next we show that if all three conditions VR(1), EVR and CP are violated then WLSEVR is also violated.

VR(1) is violated by $\mathscr {D}|A$ iff it contains $aR^{s}bR^{s}c \wedge bR^{t}cR^{t}a \wedge cR^{u}aR^{u}b$ for some distinct $a,b,c \in A$. Excepting the cases of $[(aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a \wedge cP^{u}aI^{u}b) \vee (aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a \wedge cI^{u}aP^{u}b) \vee (aI^{s}bI^{s}c)]$, in all other cases WLSEVR is violated.

$(aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a \wedge cP^{u}aI^{u}b)$ violates CP. $\mathscr {D}|A$ containing $aP^{s}bI^{s}c \wedge bP^{t}cI^{t}a \wedge cP^{u}aI^{u}b$ would violate EVR only if an ordering of A in which some alternative is uniquely worst is included, i.e., only if $(bP^{v}cP^{v}a \vee bI^{v}cP^{v}a \vee cP^{v}bP^{v}a \vee aP^{v}cP^{v}b \vee aI^{v}cP^{v}b \vee cP^{v}aP^{v}b \vee aP^{v}bP^{v}c \vee aI^{v}bP^{v}c \vee bP^{v}aP^{v}c)$ is included. In each case, with the inclusion of the required ordering, WLSEVR is violated.

$(aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a \wedge cI^{u}aP^{u}b)$ violates CP. $\mathscr {D}|A$ containing $aI^{s}bP^{s}c \wedge bI^{t}cP^{t}a \wedge cI^{u}aP^{u}b$ would violate EVR only if an ordering of A in which some alternative is uniquely best is included, i.e., only if $(aP^{v}bP^{v}c \vee aP^{v}bI^{v}c \vee aP^{v}cP^{v}b \vee bP^{v}aP^{v}c \vee bP^{v}aI^{v}c \vee bP^{v}cP^{v}a \vee cP^{v}aP^{v}b \vee cP^{v}aI^{v}b \vee cP^{v}bP^{v}a)$ is included. In each case, with the inclusion of the required ordering, WLSEVR is violated.

$aI^{s}bI^{s}c$ violates neither CP nor EVR. By Proposition 4.1, a set of orderings of a triple violates both CP and EVR iff it violates LSEVR. Thus, $\mathscr {D}|A$ containing $aI^{s}bI^{s}c$ would violate both CP and EVR iff for some distinct $a,b,c \in A$, it contains $aP^{t}bR^{t}c \wedge cR^{u}aP^{u}b$. $(aI^{s}bI^{s}c \wedge aP^{t}bR^{t}c \wedge cR^{u}aP^{u}b)$ violates WLSEVR.

Thus, violation of all three conditions of VR(1), EVR and CP implies violation of WLSEVR, i.e., if WLSEVR holds then it must be the case that VR(1) or EVR or CP holds.

This, together with the earlier demonstration that if VR(1) or EVR or CP holds then WLSEVR holds, establishes the proposition.    $\square$

Proposition 4.3

A set of orderings of a triple satisfies Latin Square unique value restriction iff it satisfies at least one of the three conditions of value restriction (2), weakly conflictive preferences and unique value restriction.

Proof

Let the set of orderings $\mathscr {D}|A$ over the triple $A = \{x,y,z\}$ violate LSUVR. Then we have: $[(\exists \; \text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A \cap T[LS(abca)]) (aP^{s}bP^{s}c \wedge bP^{t}cR^{t}a \wedge cR^{u}aP^{u}b)]$.

$aP^{s}bP^{s}c, bP^{t}cR^{t}a, cR^{u}aP^{u}b$ are concerned orderings and form LS(abca). Therefore, VR(2) is violated.

$aP^{s}bP^{s}c$ is a linear ordering. In $bP^{t}cR^{t}a$, alternative best in the linear ordering $aP^{s}bP^{s}c$, namely a, is worst, without c, the worst alternative in the linear ordering $aP^{s}bP^{s}c$, being best. Furthermore, in $cR^{u}aP^{u}b$ the worst alternative of the linear ordering $aP^{s}bP^{s}c$ is best without the best alternative of the linear ordering $aP^{s}bP^{s}c$ being worst in it. Thus WCP is violated.

a is uniquely best in $aP^{s}bP^{s}c$, b is uniquely best in $bP^{t}cR^{t}a$, b is uniquely medium in $aP^{s}bP^{s}c$, b is uniquely worst in $cR^{u}aP^{u}b$, and c is uniquely worst in $aP^{s}bP^{s}c$. Thus if UVR is to hold then we must have: (i) a is not uniquely medium in any $R \in \mathscr {D}|A$, c is not uniquely best in any $R \in \mathscr {D}|A$, and $(\forall R \in \mathscr {D}|A) (cRa \wedge cRb \rightarrow bRa)$ or (ii) c is not uniquely medium in any $R \in \mathscr {D}|A$, a is not uniquely worst in any $R \in \mathscr {D}|A$, and $(\forall R \in \mathscr {D}|A) (bRa \wedge cRa \rightarrow cRb)$. In $cR^{u}aP^{u}b$, c is best without a being worst, so (i) cannot hold. In $bP^{t}cR^{t}a$, a is worst without c being best, so (ii) cannot hold. Thus UVR is violated.

We see that the violation of LSUVR implies violation of all three restrictions of VR(2), WCP and UVR. This establishes that if any of the three restrictions of VR(2), WCP and UVR is satisfied then LSUVR is also satisfied.

Next we show that if $\mathscr {D}|A$ violates all three restrictions of WCP, VR(2) and UVR, then it violates LSUVR.

It can be easily checked that the set of orderings $\mathscr {D}|A$ violates WCP iff $(\exists \;\text {distinct} a,b,c \in A) (\exists R^{s},R^{t} \in \mathscr {D}|A) (aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a)\ \vee \ (\exists \;\text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A) (aP^{s}bP^{s}c \wedge bP^{t}cI^{t}a \wedge cI^{u}aP^{u}b) \vee (\exists \;\text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t},R^{u},R^{v} \in \mathscr {D}|A) (aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b \wedge cI^{u}aP^{u}b\, \wedge \, cP^{v}bI^{v}a)\, \vee \, (\exists \text {distinct} \; a,b,c \in A) (\exists R^{s},R^{t},R^{u},R^{v} \in \mathscr {D}|A) (aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c \wedge cI^{u}aP^{u}b \wedge aP^{v}cI^{v}b)$.

First consider the case when $\mathscr {D}|A$ contains $aP^{s}bP^{s}c$ and $bP^{t}cP^{t}a$ for some distinct $a,b,c \in A$. $\mathscr {D}|A$ would violate UVR iff (i) there is an ordering in $\mathscr {D}|A$ in which a is uniquely medium, or (ii) there is an ordering in $\mathscr {D}|A$ in which c is uniquely best and there is an ordering in $\mathscr {D}|A$ in which b is uniquely worst, or (iii) there is an ordering in $\mathscr {D}|A$ in which c is uniquely best and there is an ordering in $\mathscr {D}|A$ in which b is worst without a being best, or (iv) there is an ordering in $\mathscr {D}|A$ in which b is uniquely worst and there is an ordering in $\mathscr {D}|A$ in which c is best without a being worst, or (v) there is an ordering in $\mathscr {D}|A$ in which b is worst without a being best and there is an ordering in $\mathscr {D}|A$ in which c is best without a being worst. That is to say, $\mathscr {D}|A$ which includes $aP^{s}bP^{s}c$ and $bP^{t}cP^{t}a$ would violate UVR iff it includes $[(bP^{u}aP^{u}c \vee cP^{u}aP^{u}b) \; \vee \; [(cP^{u}aP^{u}b \vee cP^{u}aI^{u}b \vee cP^{u}bP^{u}a) \wedge (cP^{v}aP^{v}b \vee cI^{v}aP^{v}b \vee aP^{v}cP^{v}b)]\; \vee \;[(cP^{u}aP^{u}b \vee cP^{u}aI^{u}b \vee cP^{u}bP^{u}a) \wedge (cP^{v}aP^{v}b \vee cP^{v}aI^{v}b)]\, \vee \, [(cP^{u}aP^{u}b \vee cI^{u}aP^{u}b \vee aP^{u}cP^{u}b)\, \wedge (cP^{v}aP^{v}b \vee cI^{v}aP^{v}b)] \vee [(cP^{u}aP^{u}b \vee cP^{u}aI^{u}b)\wedge (cP^{v}aP^{v}b \vee cI^{v}aP^{v}b)]]$. Excepting the cases when we have $[(aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a \wedge bP^{u}aP^{u}c) \vee (aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a \wedge cP^{u}bP^{u}a \wedge aP^{v}cP^{v}b)]$, in all other cases LSUVR is violated. $(aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a \wedge bP^{u}aP^{u}c)$ does not violate VR(2); VR(2) would be violated only if $\mathscr {D}|A$ includes $[(\text {concerned} \; cR^{v}aR^{v}b) \vee (\text {concerned} \; aR^{v}cR^{v}b \wedge \text {concerned} \; cR^{w}bR^{w}a)]$. With the required inclusion, in each case LSUVR is violated. $(aP^{s}bP^{s}c \wedge bP^{t}cP^{t}a \wedge cP^{u}bP^{u}a \wedge aP^{v}cP^{v}b)$ does not violate VR(2); VR(2) would be violated only if $\mathscr {D}|A$ includes $(\text {concerned} \; cR^{v}aR^{v}b \vee \text {concerned} \; bR^{v}aR^{v}c)$. With the required inclusion, in each case LSUVR is violated.

If $\mathscr {D}|A$ contains $(aP^{s}bP^{s}c \wedge bP^{t}cI^{t}a \wedge cI^{u}aP^{u}b)$ then LSUVR is violated.

If $\mathscr {D}|A$ contains $(aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b \wedge cI^{u}aP^{u}b \wedge cP^{v}bI^{v}a)$ then it would violate UVR iff an ordering in which a is uniquely medium or an ordering in which b is uniquely best or an ordering in which b is best without a being worst is included, i.e., $[bP^{w}aP^{w}c \vee cP^{w}aP^{w}b \vee bP^{w}cP^{w}a \vee bP^{w}cI^{w}a \vee bI^{w}aP^{w}c]$ is included. In all cases other than that of $(aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b \wedge cI^{u}aP^{u}b \wedge cP^{v}bI^{v}a \wedge cP^{w}aP^{w}b)$ LSUVR is violated. $(aP^{s}bP^{s}c \wedge aP^{t}cP^{t}b \wedge cI^{u}aP^{u}b \wedge cP^{v}bI^{v}a \wedge cP^{w}aP^{w}b)$ does not violate VR(2). VR(2) would be violated iff $(\text {concerned} \; bR^{k}cR^{k}a \vee \text {concerned} \; bR^{k}aR^{k}c)$ is included. With the inclusion of the required ordering LSUVR is violated.

If $\mathscr {D}|A$ contains $(aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c \wedge cI^{u}aP^{u}b \wedge aP^{v}cI^{v}b)$ then it would violate UVR iff an ordering in which c is uniquely medium or an ordering in which a is uniquely worst or an ordering in which a is worst without c being best is included, i.e., $[aP^{w}cP^{w}b \vee bP^{w}cP^{w}a \vee bI^{w}cP^{w}a \vee cP^{w}bP^{w}a \vee bP^{w}cI^{w}a]$ is included. In all cases other than that of $(aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c \wedge cI^{u}aP^{u}b \wedge aP^{v}cI^{v}b \wedge aP^{w}cP^{w}b)$ LSUVR is violated. $(aP^{s}bP^{s}c \wedge bP^{t}aP^{t}c \wedge cI^{u}aP^{u}b \wedge aP^{v}cI^{v}b \wedge aP^{w}cP^{w}b)$ does not violate VR(2). VR(2) would be violated iff $(\text {concerned} bR^{k}cR^{k}a \vee \text {concerned} \; cR^{k}bR^{k}a)$ is included. With the inclusion of the required ordering LSUVR is violated.

Thus if all three restrictions of VR(2), WCP and UVR are violated then LSUVR must be violated.

This together with the earlier demonstration that violation of LSUVR implies violation of all three restrictions of VR(2), WCP and UVR, establishes the proposition.    $\square$

4.7 Notes on Literature

The characterization of non-minority rule given here is based on Jain (1994). The condition which ensures transitivity under the non-minority rule was formulated by Fine (1973). Fine’s analysis is essentially equivalent to establishing Theorem 4.2 given here. The derivation of transitivity condition for the case of odd number of individuals is based on Jain (1989). The condition for quasi-transitivity was first derived in Jain (1984); in it, it was shown that the union of VR(2), WCP, and UVR is an Inada-type necessary and sufficient condition for quasi-transitivity under the non-minority rule.