The Method of Majority Decision

Abstract

The method of majority decision is defined by: An alternative x is socially at least as good as another alternative y iff the number of individuals preferring x over y is greater than or equal to the number of individuals preferring y over x. The chapter is concerned with the Inada-type necessary and sufficient conditions for transitivity and quasi-transitivity under the method of majority decision.

Appendices

Appendix

3.5 Relationships Among Conditions on Preferences

Let $\mathscr {D}$ be a set of orderings of set S; and let $A = \{x,y,z\} \subseteq S$ be a triple of alternatives. Let L be a linear ordering of the triple A. We define x to be between y and z, denoted by $B_{L}(y,x,z)$, iff $[(yLx \wedge xLz) \vee (zLx \wedge xLy)]$.

3.1.1 3.5.1 Single-Peakedness

Single-Peakedness (SP): $\mathscr {D} \subseteq \mathscr {T}$ satisfies SP over the triple $A \subseteq S$ iff $(\exists \;$a linear ordering L of $A)(\forall \; a,b,c \in A)(\forall R \in \mathscr {D}|A) [aRb \wedge B_{L}(a,b,c) \rightarrow bPc]$. $\mathscr {D}$ satisfies SP iff it satisfies SP over every triple contained in S.

Thus, the satisfaction of single-peakedness by $\mathscr {D}$ over the triple A requires that there be a way of arranging the three alternatives of the triple such that the following holds for every $R \in \mathscr {D}|A$: If an alternative not lying in between (non-between-alternative) is at least as good as the one which lies in between (the between-alternative), then the between-alternative must be strictly preferred to the other non-between-alternative.

Let orderings $\mathscr {D}|A$ satisfy single-peakedness with x between y and z. Then we have: $(\forall R \in \mathscr {D}|A) [xPy \vee xPz]$; consequently, x is not worst in any $R \in \mathscr {D}|A$. Similarly, if single-peakedness holds with y between x and z then y is not worst in any $R \in \mathscr {D}|A$; and if single-peakedness holds with z between x and y then z is not worst in any $R \in \mathscr {D}|A$. Thus, satisfaction of single-peakedness by a set of orderings of a triple implies that there is an alternative in the triple which is not worst in any of the orderings of the set.

The set of all orderings of the triple A in which x is not worst is given by:

$\{xPyPz, zPxPy, xPzPy, yPxPz, xPyIz, xIyPz, zIxPy\}$.

For every ordering in the set, the implication $(yRx \rightarrow xPz)$ holds. Therefore, the set of all orderings of A in which x is not worst satisfies the single-peakedness condition with x between y and z. Similarly, it can be checked that both the set of orderings of A in which y is not worst and the set of orderings of A in which z is not worst satisfy the condition of single-peakedness. This, together with the demonstration of the preceding paragraph, establishes that a set of orderings of a triple satisfies the condition of single-peakedness iff there is an alternative in the triple such that it is not worst in any of the orderings of the set.

3.1.2 3.5.2 Single-Cavedness

Single-Cavedness (SC): $\mathscr {D} \subseteq \mathscr {T}$ satisfies SC over the triple $A \subseteq S$ iff $(\exists \;$a linear ordering L of $A)(\forall \; a,b,c \in A)(\forall R \in \mathscr {D}|A) [bRa \wedge B_{L}(a,b,c) \rightarrow cPb]$. $\mathscr {D}$ satisfies SC iff it satisfies SC over every triple contained in S.

Thus, the satisfaction of single-cavedness by $\mathscr {D}$ over the triple A requires that there be a way of arranging the three alternatives of the triple such that the following holds for every $R \in \mathscr {D}|A$: If the between-alternative is at least as good as one of the non-between-alternatives, then the other non-between-alternative must be strictly preferred to the between-alternative.

Let orderings $\mathscr {D}|A$ satisfy single-cavedness with x between y and z. Then we have: $(\forall R \in \mathscr {D}|A) [yPx \vee zPx]$; consequently, x is not best in any $R \in \mathscr {D}|A$. Similarly, if single-cavedness holds with y between x and z then y is not best in any $R \in \mathscr {D}|A$; and if single-cavedness holds with z between x and y then z is not best in any $R \in \mathscr {D}|A$. Thus satisfaction of single-cavedness by a set of orderings of a triple implies that there is an alternative in the triple which is not best in any of the orderings of the set.

The set of all orderings of the triple A in which x is not best is given by:

$\{yPzPx, zPxPy, zPyPx, yPxPz, yPzIx, zPxIy, yIzPx\}$.

For every ordering in the set, the implication $(xRy \rightarrow zPx)$ holds. Therefore, the set of all orderings of A in which x is not best satisfies the single-cavedness condition with x between y and z. Similarly, it can be checked that both the set of orderings of A in which y is not best and the set of orderings of A in which z is not best satisfy the condition of single-cavedness. This, together with the demonstration of the preceding paragraph, establishes that a set of orderings of a triple satisfies the condition of single-cavedness iff there is an alternative in the triple such that it is not best in any of the orderings of the set.

3.1.3 3.5.3 Separability into Two Groups

Separability into Two Groups (SG): $\mathscr {D} \subseteq \mathscr {T}$ satisfies SG over the triple $A \subseteq S$ iff $(\exists A_{1},A_{2} \subset A)[[A_{1} \ne \emptyset \wedge A_{2} \ne \emptyset \wedge A_{1} \cap A_{2} = \emptyset \wedge A_{1} \cup A_{2} = A] \wedge (\forall R \in \mathscr {D}|A)[(\forall a \in A_{1})(\forall b \in A_{2})(aPb) \vee (\forall a \in A_{1})(\forall b \in A_{2})(bPa)]]$. $\mathscr {D}$ satisfies SG iff it satisfies SG over every triple contained in S.

Thus, the satisfaction of separability into two groups by $\mathscr {D}$ over the triple A requires that there be a way of partitioning the triple into two subsets such that in every $R \in \mathscr {D}|A$ either alternatives belonging to one of the subsets are strictly preferred to alternatives in the other subset or alternatives belonging to the latter subset are strictly preferred to alternatives in the former subset.

Let orderings $\mathscr {D}|A$ satisfy separability into two groups with the two subsets being $\{x\}$ and $\{y,z\}$. Then we have: $(\forall R \in \mathscr {D}|A) [(xPy \wedge xPz) \vee (yPx \wedge zPx)]$; consequently, x is not medium in any $R \in \mathscr {D}|A$. Similarly, if separability into two groups holds with the two subsets being $\{y\}$ and $\{x,z\}$ then y is not medium in any $R \in \mathscr {D}|A$; and if separability into two groups holds with the two subsets being $\{z\}$ and $\{x,y\}$ then z is not medium in any $R \in \mathscr {D}|A$. Thus satisfaction of separability into two groups by a set of orderings of a triple implies that there is an alternative in the triple which is not medium in any of the orderings of the set.

The set of all orderings of the triple A in which x is not medium is given by:

$\{xPyPz, yPzPx, xPzPy, zPyPx, xPyIz, yIzPx\}$.

For every ordering in the set, $[(xPy \wedge xPz) \vee (yPx \wedge zPx)]$ holds. Therefore, the separability into two groups holds with the two subsets being $\{x\}$ and $\{y,z\}$. Similarly, it can be checked that both the set of orderings of A in which y is not medium and the set of orderings of A in which z is not medium satisfy the condition of separability into two groups. This, together with the demonstration of the preceding paragraph, establishes that a set of orderings of a triple satisfies the condition of separability into two groups iff there is an alternative in the triple such that it is not medium in any of the orderings of the set.

3.1.4 3.5.4 Value Restriction

There are two versions of value restriction.

First Version of Value Restriction (VR(1)): $\mathscr {D} \subseteq \mathscr {T}$ satisfies VR(1) over the triple $A \subseteq S$ iff $(\exists$ distinct $a,b,c \in A)[(\forall R \in \mathscr {D}|A)[bPa \vee cPa] \vee (\forall R \in \mathscr {D}|A)[(aPb \wedge aPc) \vee (bPa \wedge cPa)] \vee (\forall R \in \mathscr {D}|A)[aPb \vee aPc]]$. $\mathscr {D}$ satisfies VR(1) iff it satisfies VR(1) over every triple contained in S.

Thus, the satisfaction of VR(1) by $\mathscr {D}$ over the triple A requires that there be an alternative in A such that it is not best in any $R \in \mathscr {D}|A$, or that it is not medium in any $R \in \mathscr {D}|A$, or that it is not worst in any $R \in \mathscr {D}|A$.

The following proposition is immediate.

Proposition 3.1

A set of orderings of a triple satisfies value restriction (1) iff it satisfies at least one of the three conditions of single-peakedness, single-cavedness and separability into two groups.

Proposition 3.2

A set of orderings of a triple violates value restriction (1) iff there is a weak Latin Square.

Proof

Let the set of orderings $\mathscr {D}|A$ over the triple $A = \{x,y,z\}$ contain a weak Latin Square. Without any loss of generality, assume that $\mathscr {D}|A$ contains WLS(xyzx). Then we have: $(\exists R^{s}, R^{t}, R^{u} \in \mathscr {D}|A)[xR^{s}yR^{s}z \wedge yR^{t}zR^{t}x \wedge zR^{u}xR^{u}y]$. x is best in $R^{s}$, y is best in $R^{t}$, z is best in $R^{u}$; x is medium in $R^{u}$, y is medium in $R^{s}$, z is medium in $R^{t}$; x is worst in $R^{t}$, y is worst in $R^{u}$, z is worst in $R^{s}$. Therefore VR(1) is violated. This establishes that if a set of orderings over a triple contains a weak Latin Square then it violates VR(1).              (P3.2-1)

Let $\mathscr {D}|A$ be a set of orderings of the triple A. $\mathscr {D}|A$ would violate VR(1) only if it contains an ordering in which x is best, contains an ordering in which y is best, and contains an ordering in which z is best, i.e., only if $\mathscr {D}|A$ contains $[(xR^{1}yR^{1}z \vee xR^{1}zR^{1}y) \wedge (yR^{2}zR^{2}x \vee yR^{2}xR^{2}z) \wedge (zR^{3}xR^{3}y \vee zR^{3}yR^{3}x)]$. Thus, if $\mathscr {D}|A$ is to violate VR(1), then it must contain at least one of the following eight sets: $(i) \; \{xR^{1}yR^{1}z, yR^{2}zR^{2}x, zR^{3}xR^{3}y\}$, $(ii) \; \{xR^{1}yR^{1}z, yR^{2}zR^{2}x, zR^{3}yR^{3}x\}$, $(iii) \; \{xR^{1}yR^{1}z, yR^{2}xR^{2}z, zR^{3}xR^{3}y\}$, $(iv) \; \{xR^{1}yR^{1}z,$ $yR^{2}xR^{2}z, zR^{3}yR^{3}x\}$, $(v) \; \{xR^{1}zR^{1}y, yR^{2}zR^{2}x, zR^{3}xR^{3}y\}$, $(vi) \; \{xR^{1}zR^{1}y, yR^{2}zR^{2}x, zR^{3}yR^{3}x\}$, $(vii) \; \{xR^{1}zR^{1}y, yR^{2}xR^{2}z,$ $zR^{3}xR^{3}y\}$, $(viii) \; \{xR^{1}zR^{1}y, yR^{2}xR^{2}z, zR^{3}yR^{3}x\}$.

Set (i) forms WLS(xyzx) and set (viii) WLS(xzyx).

Let $\mathscr {D}|A$ contain set (ii). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (ii) it contains an ordering in which x is medium, i.e., only if it contains $(zR^{4}xR^{4}y \vee yR^{4}xR^{4}z)$. If $\mathscr {D}|A$ contains $zR^{4}xR^{4}y$, then it contains WLS(xyzx). If $\mathscr {D}|A$ contains set (ii) and $yR^{4}xR^{4}z$, then VR(1) would be violated only if it contains an ordering in which y is worst, i.e., only if it contains $(zR^{5}xR^{5}y \vee xR^{5}zR^{5}y)$. If $\mathscr {D}|A$ contains set (ii), $yR^{4}xR^{4}z$ and $zR^{5}xR^{5}y$, then WLS(xyzx) is contained in it; and if it contains set (ii), $yR^{4}xR^{4}z$, and $xR^{5}zR^{5}y$ then WLS(xzyx) is contained in it.

Let $\mathscr {D}|A$ contain set (iii). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (iii) it contains an ordering in which z is medium, i.e., only if it contains $(yR^{4}zR^{4}x \vee xR^{4}zR^{4}y)$. If $\mathscr {D}|A$ contains $yR^{4}zR^{4}x$, then it contains WLS(xyzx). If $\mathscr {D}|A$ contains set (iii) and $xR^{4}zR^{4}y$, then VR(1) would be violated only if it contains an ordering in which x is worst, i.e., only if it contains $(yR^{5}zR^{5}x \vee zR^{5}yR^{5}x)$. If $\mathscr {D}|A$ contains set (iii), $xR^{4}zR^{4}y$, and $yR^{5}zR^{5}x$ then WLS(xyzx) is contained in it; and if it contains set (iii), $xR^{4}zR^{4}y$, and $zR^{5}yR^{5}x$ then WLS(xzyx) is contained in it.

Let $\mathscr {D}|A$ contain set (iv). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (iv) it contains an ordering in which z is medium, i.e., only if it contains $(yR^{4}zR^{4}x \vee xR^{4}zR^{4}y)$. If $\mathscr {D}|A$ contains $xR^{4}zR^{4}y$, then it contains WLS(xzyx). If $\mathscr {D}|A$ contains set (iv) and $yR^{4}zR^{4}x$, then VR(1) would be violated only if it contains an ordering in which y is worst, i.e., only if it contains $(zR^{5}xR^{5}y \vee xR^{5}zR^{5}y)$. If $\mathscr {D}|A$ contains set (iv), $yR^{4}zR^{4}x$, and $zR^{5}xR^{5}y$ then WLS(xyzx) is contained in it; and if it contains set (iv), $yR^{4}zR^{4}x$, and $xR^{5}zR^{5}y$ then WLS(xzyx) is contained in it.

Let $\mathscr {D}|A$ contain set (v). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (v) it contains an ordering in which y is medium, i.e., only if it contains $(xR^{4}yR^{4}z \vee zR^{4}yR^{4}x)$. If $\mathscr {D}|A$ contains $xR^{4}yR^{4}z$, then it contains WLS(xyzx). If $\mathscr {D}|A$ contains set (v) and $zR^{4}yR^{4}x$, then VR(1) would be violated only if it contains an ordering in which z is worst, i.e., only if it contains $(xR^{5}yR^{5}z \vee yR^{5}xR^{5}z)$. If $\mathscr {D}|A$ contains set (v), $zR^{4}yR^{4}x$, and $xR^{5}yR^{5}z$ then WLS(xyzx) is contained in it; and if it contains set (v), $zR^{4}yR^{4}x$, and $yR^{5}xR^{5}z$ then WLS(xzyx) is contained in it.

Next, let $\mathscr {D}|A$ contain set (vi). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (vi) it contains an ordering in which x is medium, i.e., only if it contains $(zR^{4}xR^{4}y \vee yR^{4}xR^{4}z)$. If $\mathscr {D}|A$ contains $yR^{4}xR^{4}z$, then it contains WLS(xzyx). If $\mathscr {D}|A$ contains set (vi) and $zR^{4}xR^{4}y$, then VR(1) would be violated only if it contains an ordering in which z is worst, i.e., only if it contains $(xR^{5}yR^{5}z \vee yR^{5}xR^{5}z)$. If $\mathscr {D}|A$ contains set (vi), $zR^{4}xR^{4}y$, and $xR^{5}yR^{5}z$ then WLS(xyzx) is contained in it; and if it contains set (vi), $zR^{4}xR^{4}y$, and $yR^{5}xR^{5}z$ then WLS(xzyx) is contained in it.

Finally, let $\mathscr {D}|A$ contain set (vii). $\mathscr {D}|A$ violates VR(1) only if in addition to the orderings in set (vii) it contains an ordering in which y is medium, i.e., only if it contains $(xR^{4}yR^{4}z \vee zR^{4}yR^{4}x)$. If $\mathscr {D}|A$ contains $zR^{4}yR^{4}x$, then it contains WLS(xzyx). If $\mathscr {D}|A$ contains set (vii) and $xR^{4}yR^{4}z$, then VR(1) would be violated only if it contains an ordering in which x is worst, i.e., only if it contains $(yR^{5}zR^{5}x \vee zR^{5}yR^{5}x)$. If $\mathscr {D}|A$ contains set (vii), $xR^{4}yR^{4}z$, and $yR^{5}zR^{5}x$ then WLS(xyzx) is contained in it; and if it contains set (vii), $xR^{4}yR^{4}z$, and $zR^{5}yR^{5}x$ then WLS(xzyx) is contained in it.

This establishes that if a set of orderings over a triple violates VR(1) then it must contain a weak Latin Square.              (P3.2-2)

(P3.2-1) and (P3.2-2) establish the proposition.    $\square$

Second Version of Value Restriction (VR(2)): $\mathscr {D} \subseteq \mathscr {T}$ satisfies VR(2) over the triple $A \subseteq S$ iff $(\exists$ distinct $a,b,c \in A)[(\forall \;$concerned$R \in \mathscr {D}|A)[bPa \vee cPa] \vee (\forall \;$concerned$\; R \in \mathscr {D}|A)[(aPb \wedge aPc) \vee (bPa \wedge cPa)] \vee (\forall \;$concerned$\; R \in \mathscr {D}|A)[aPb {\vee } aPc]]$. $\mathscr {D}$ satisfies VR(2) iff it satisfies VR(2) over every triple contained in S.

Thus, the satisfaction of VR(2) by $\mathscr {D}$ over the triple A requires that there be an alternative in A such that it is not best in any concerned $R \in \mathscr {D}|A$, or that it is not medium in any concerned $R \in \mathscr {D}|A$, or that it is not worst in any concerned $R \in \mathscr {D}|A$.

Proposition 3.3

A set of orderings of a triple violates value restriction (2) iff there is a Latin Square.

The proof of this proposition is omitted as it is similar to the proof of Proposition 3.2.

3.1.5 3.5.5 Dichotomous Preferences

Dichotomous Preferences (DP): $\mathscr {D} \subseteq \mathscr {T}$ satisfies DP over the triple $A \subseteq S$ iff $\sim (\exists$ distinct $a,b,c \in A)(\exists R \in \mathscr {D}|A)[aPbPc]$. $\mathscr {D}$ satisfies DP iff it satisfies DP over every triple contained in S.

Thus, a set of orderings over a triple satisfies DP iff the set does not contain a linear ordering of the triple.

3.1.6 3.5.6 Echoic Preferences

Echoic Preferences (EP): $\mathscr {D} \subseteq \mathscr {T}$ satisfies EP over the triple $A \subseteq S$ iff $(\forall$ distinct $a,b,c \in A)[aPbPc \in \mathscr {D}|A\rightarrow (\forall R \in \mathscr {D}|A)(aRc)]$. $\mathscr {D}$ satisfies EP iff it satisfies EP over every triple contained in S.

Thus, a set of orderings of a triple satisfies EP iff it does not contain an ordering in which the worst alternative of some linear ordering contained in the set is preferred to the best alternative in that linear ordering.²

3.1.7 3.5.7 Antagonistic Preferences

Antagonistic Preferences (AP): $\mathscr {D} \subseteq \mathscr {T}$ satisfies AP over the triple $A \subseteq S$ iff $(\forall$ distinct $a,b,c \in A)[aPbPc \in \mathscr {D}|A\rightarrow (\forall R \in \mathscr {D}|A)(aPbPc \vee cPbPa \vee aIc)]$. $\mathscr {D}$ satisfies AP iff it satisfies AP over every triple contained in S.

Thus, a set of orderings of a triple satisfies AP iff in case there is a linear ordering, say xPyPz, then the set is a subset of $\{xPyPz, zPyPx, zIxPy, yPxIz, xIyIz\}$.³

3.1.8 3.5.8 Extremal Restriction

Proposition 3.4

A set of orderings of a triple satisfies extremal restriction iff it satisfies at least one of the conditions of dichotomous preferences, echoic preferences and antagonistic preferences.

Proof

Let the set of orderings $\mathscr {D}|A$ over the triple $A = \{x,y,z\}$ satisfy ER. The cases of satisfaction of ER over $\mathscr {D}|A$ can be divided into the following mutually exclusive and exhaustive cases:

(i) ER is trivially satisfied if $\mathscr {D}|A$ does not contain any strong ordering. If $\mathscr {D}|A$ does not contain any strong ordering then all three conditions of DP, EP and AP are satisfied.

(ii) There is only one strong ordering, say xPyPz. As all non-strong concerned orderings of a triple belong to both the Latin Squares and ER holds, it must be the case that xRz holds for all non-strong orderings. Therefore it follows that in this case EP holds.

(iii) There are more than one strong orderings. It can be checked that if there are two strong orderings belonging to the same Latin Square, then ER would be violated. From this, it follows that satisfaction of ER implies that there can be at most two strong orderings; and in case the set contains two strong orderings, then one of them would belong to LS(xyzx) and the other to LS(xzyx). Without any loss of generality assume that $\mathscr {D}|A$ contains xPyPz and one strong ordering of LS(xzyx). First consider the subcase when the ordering of LS(xzyx) in $\mathscr {D}|A$ is xPzPy. As all non-strong concerned orderings over a triple belong to both Latin Squares and ER holds, it follows that for all non-strong orderings $R \in \mathscr {D}|A$ we must have both xRz and xRy. Thus in this case, $\mathscr {D}|A$ must be a subset of $\{xPyPz, xPzPy, xPyIz, xIyPz, zIxPy, xIyIz\}$. In all orderings of this set, the best alternative of each of the two strong orderings is at least as good as the worst alternative; consequently, EP is satisfied. Next consider the subcase when the ordering belonging to LS(xzyx) contained in $\mathscr {D}|A$ is yPxPz. As ER is satisfied and all concerned non-strong orderings belong to both Latin Squares, for all non-strong orderings $R \in \mathscr {D}|A$ we must have both xRz and yRz. Thus in this case, $\mathscr {D}|A$ must be a subset of $\{xPyPz, yPxPz, xPyIz, yPxIz, xIyPz, xIyIz\}$. In all orderings of this set, the best alternative of each of the two strong orderings is at least as good as the worst alternative; consequently, EP is satisfied. Finally consider the subcase when the ordering belonging to LS(xzyx) contained in $\mathscr {D}|A$ is zPyPx. As ER is satisfied and all concerned non-strong orderings belong to both Latin Squares, for all non-strong orderings $R \in \mathscr {D}|A$, we must have both xRz and zRz. Thus in this case, $\mathscr {D}|A$ must be a subset of $\{xPyPz, zPyPx, yPzIx, zIxPy, xIyIz\}$. This set satisfies AP. The foregoing establishes that satisfaction of ER implies that at least one of the three conditions of DP, EP and AP holds.              (P3.4-1)

Let $\mathscr {D}|A$ violate ER. Then $\mathscr {D}|A$ contains a strong ordering of the triple, and an ordering belonging to the same Latin Square as the one to which the strong ordering belongs and in which the alternative worst in the strong ordering is strictly preferred to the alternative best in the strong ordering. Without any loss of generality assume that $\mathscr {D}|A$ contains xPyPz and an ordering belonging to LS(xyzx) with zPx, i.e., yRzPx or zPxRy. Each of $\{xPyPz, yRzPx\}$ and $\{xPyPz, zPxRy\}$ violates DP as there is a strong ordering; violates EP as there is a strong ordering and an ordering in which the alternative best in the strong ordering is not at least as good as the alternative worst in the strong ordering; and violates AP as there is a strong ordering, namely xPyPz, but we do not have for every $R \in \mathscr {D}|A$, $(xPyPz \vee zPyPx \vee xIz)$. Thus, violation of ER implies that all three conditions of DP, EP and AP are violated.             (P3.4-2)

(P3.4-1) and (P3.4-2) establish the proposition.    $\square$

The way EP and AP have been defined here, they trivially hold over a triple if the set of orderings over the triple does not contain a strong ordering of the triple. Thus, DP implies EP as well as AP. Consequently, it follows that, in view of Proposition 3.4, the following proposition also holds.

Proposition 3.5

A set of orderings of a triple satisfies extremal restriction iff it satisfies echoic preferences or antagonistic preferences.

3.1.9 3.5.9 Taboo Preferences

Taboo Preferences (TP): $\mathscr {D} \subseteq \mathscr {T}$ satisfies TP over the triple $A \subseteq S$ iff $xIyIz \notin \mathscr {D}|A \wedge (\exists$ distinct $a,b \in A)(\forall R \in \mathscr {D}|A)(aRb)$. $\mathscr {D}$ satisfies TP iff it satisfies TP over every triple contained in S.

Thus, a set of orderings of a triple satisfies TP iff the set does not contain the unconcerned ordering and there exists a pair of distinct alternatives $a,b \in A$ such that in every ordering of the set a is at least as good as b.

3.1.10 3.5.10 Weak Latin Square Partial Agreement

Proposition 3.6

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

Proof

Let WLSPA hold over the set of orderings $\mathscr {D}|A$ of triple $A = \{x,y,z\}$.

(i) If $\mathscr {D}|A$ does not contain a weak Latin Square then VR(1) is satisfied; as a set of orderings of a triple violates VR(1) iff the set contains a weak Latin Square.

(ii) Suppose $\mathscr {D}|A$ contains one of the two weak Latin Squares, say LS(xyzx), and does not contain the other one. This implies that:

$\mathscr {D}|A$ does not contain the ordering xIyIz.              (P3.6-1)

As $\mathscr {D}|A$ does not contain WLS(xzyx) we have: $\sim [(\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A)(xR^{s}zR^{s}y \wedge zR^{t}yR^{t}x \wedge yR^{u}xR^{u}z)]$. Without any loss of generality assume that $\mathscr {D}|A$ does not contain any ordering R such that xRzRy. This implies that we must have:

$(\forall R \in \mathscr {D}|A)[(xRz \rightarrow yPz) \wedge (zRy \rightarrow zPx)]$.              (P3.6-2)

Consequently, we have:

$(\forall R \in \mathscr {D}|A)[(xRyRz \rightarrow xRyPz) \wedge (zRxRy \rightarrow zPxRy)]$.              (P3.6-3)

If no $R {\in } \mathscr {D}|A \cap T[LS(xyzx)]$ is strong then (P3.6-3) implies that $(\forall R {\in } \mathscr {D}|A)(yRx)$, and therefore, in view of (P3.6-1), TP is satisfied.

Next consider the case when some ordering belonging to $\mathscr {D}|A \cap T[LS(xyzx)]$ is strong. This strong ordering cannot be xPyPz in view of zPxRy belonging to $\mathscr {D}|A$, otherwise WLSPA would be violated. This strong ordering cannot be zPxPy either in view of xRyPz belonging to $\mathscr {D}|A$, otherwise WLSPA would be violated. Thus, if an $R \in \mathscr {D}|A \cap T[LS(xyzx)]$ is strong, it has to be yPzPx. Once again, it follows that: $(\forall R \in \mathscr {D}|A)(yRx)$, and therefore, in view of (1), TP is satisfied.

(iii) Finally consider the case when $\mathscr {D}|A$ contains both the weak Latin Squares. If no ordering belonging to $\mathscr {D}|A$ is strong, then DP holds, and consequently, ER is satisfied.

Suppose $\mathscr {D}|A$ contains a strong ordering, say, xPyPz.

WLSPA holds over $\mathscr {D}|A$ $\wedge$ $\mathscr {D}|A$ contains both the weak Latin Squares $\wedge$ $\mathscr {D}|A$ contains xPyPz $\rightarrow$ $(\forall R \in \mathscr {D}|A)[(yRzRx \rightarrow yRzIx) \wedge (zRxRy \rightarrow zIxRy)]$              (P3.6-4)

(P3.6-4) implies that $\mathscr {D}|A$ does not contain any of yPzPx, yIzPx, zPxPy, zPxIy              (P3.6-5)

If xPyPz is the only strong ordering in $\mathscr {D}|A$ then (P3.6-5) implies that $(\forall R \in \mathscr {D}|A)(xRz)$. Therefore, EP holds, and consequently, ER is satisfied.

Next suppose that, in addition to xPyPz, $\mathscr {D}|A$ contains another strong ordering. In view of (P3.6-5), it follows that this strong ordering must belong to WLS(xzyx).

First consider the case when this other strong ordering contained in $\mathscr {D}|A$ is xPzPy.

WLSPA holds over $\mathscr {D}|A$ $\wedge$ $\mathscr {D}|A$ contains both the weak Latin Squares $\wedge$ $\mathscr {D}|A$ contains xPzPy $\rightarrow$ $(\forall R \in \mathscr {D}|A)[(yRxRz \rightarrow yIxRz) \wedge (zRyRx \rightarrow zRyIx)]$              (P3.6-6)

(P3.6-6) implies that $\mathscr {D}|A$ does not contain any of yPxPz, yPxIz, zPyPx, zIyPx              (P3.6-7)

(P3.6-5) and (P3.6-7) imply that $\mathscr {D}|A \subseteq \{xPyPz, xPzPy, xPyIz, xIyPz, xIzPy, xIyIz\}$. Therefore, EP holds, and consequently, ER is satisfied.

Next consider the case when this other strong ordering contained in $\mathscr {D}|A$ is yPxPz.

WLSPA holds over $\mathscr {D}|A$ $\wedge$ $\mathscr {D}|A$ contains both the weak Latin Squares $\wedge$ $\mathscr {D}|A$ contains yPxPz $\rightarrow$ $(\forall R \in \mathscr {D}|A)[(xRzRy \rightarrow xRzIy) \wedge (zRyRx \rightarrow zIyRx)]$              (P3.6-8)

(P3.6-8) implies that $\mathscr {D}|A$ does not contain any of xPzPy, xIzPy, zPyPx, zPyIx              (P3.6-9)

(P3.6-5) and (P3.6-9) imply that $\mathscr {D}|A \subseteq \{xPyPz, yPxPz, xPyIz, xIyPz, yPxIz, xIyIz\}$. Therefore, EP holds, and consequently, ER is satisfied.

Finally consider the case when this other strong ordering contained in $\mathscr {D}|A$ is zPyPx.

WLSPA holds over $\mathscr {D}|A$ $\wedge$ $\mathscr {D}|A$ contains both the weak Latin Squares $\wedge$ $\mathscr {D}|A$ contains zPyPx $\rightarrow$ $(\forall R \in \mathscr {D}|A)[(xRzRy \rightarrow xIzRy) \wedge (yRxRz \rightarrow yRxIz)]$              (P3.6-10)

(P3.6-10) implies that $\mathscr {D}|A$ does not contain any of xPzPy, xPzIy, yPxPz, yIxPz              (P3.6-11)

(P3.6-5) and (P3.6-11) imply that $\mathscr {D}|A {\subseteq } \{xPyPz, zPyPx, yPzIx, zIxPy, xIyIz\}$. Therefore, AP holds, and consequently, ER is satisfied.

This completes the proof of the assertion that satisfaction of WLSPA implies satisfaction of at least one of the three conditions of value restriction (1), taboo preferences and extremal restriction.

Let $\mathscr {D}|A$ violate WLSPA. Without any loss of generality, assume that $\mathscr {D}|A$ contains WLS(xyzx) involving a strong ordering, say xPyPz, and also contains an ordering belonging to LS(xyzx) with zPx. That is to say, $(\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A)[(xP^{s}yP^{s}z \wedge yR^{t}zP^{t}x \wedge zR^{u}xR^{u}y) \vee (xP^{s}yP^{s}z \wedge yR^{t}zR^{t}x \wedge zP^{u}xR^{u}y)].$              (P3.6-12)

$\mathscr {D}|A$ contains $R^{s},R^{t},R^{u}$ such that $(xP^{s}yP^{s}z \wedge yR^{t}zR^{t}x \wedge zR^{u}xR^{u}y)$ implies that VR(1) is violated.              (P3.6-13)

$xP^{s}yP^{s}z {\in } \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (xPy) \wedge (\exists R \in \mathscr {D}|A) (yPz) \wedge (\exists R \in \mathscr {D}|A) (xPz)$              (P3.6-14)

$yR^{t}zR^{t}x \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (yPx \vee xIyIz)$              (P3.6-15)

$zR^{u}xR^{u}y \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (zPy \vee xIyIz)$              (P3.6-16)

(P3.6-12) $\rightarrow (\exists R \in \mathscr {D}|A) (zPx)$              (P3.6-17)

$xIyIz \in \mathscr {D}|A$ implies that TP is violated.              (P3.6-18)

From (P3.6-14)-(P3.6-17) it follows that if $xIyIz \notin \mathscr {D}|A$ then also TP is violated.              (P3.6-19)

(P3.6-18) and (P3.6-19) establish that TP is violated.              (P3.6-20)

From (P3.6-12) it follows that: $(xP^{s}yP^{s}z, yR^{t}zP^{t}x \in \mathscr {D}|A) \vee (xP^{s}yP^{s}z, zP^{u}xR^{u}y \in \mathscr {D}|A)$              (P3.6-21)

(P3.6-21) implies that ER is violated.              (P3.6-22)

(P3.6-13), (P3.6-20), and (P3.6-22) establish the assertion that if a set of orderings of a triple satisfies at least one of the three conditions of value restriction (1), taboo preferences and extremal restriction, then it satisfies WLSPA.    $\square$

3.1.11 3.5.11 Limited Agreement

Limited Agreement (LA): $\mathscr {D} \subseteq \mathscr {T}$ satisfies LA over the triple $A \subseteq S$ iff $(\exists$ distinct $a,b \in A)(\forall R \in \mathscr {D}|A)(aRb)$. $\mathscr {D}$ satisfies LA iff it satisfies LA over every triple contained in S.

Thus, a set of orderings of a triple satisfies LA iff there exists a pair of distinct alternatives $a,b \in A$ such that in every ordering of the set a is at least as good as b.

3.1.12 3.5.12 Latin Square Partial Agreement

Proposition 3.7

A set of orderings of a triple satisfies Latin Square partial agreement iff it satisfies at least one of the three conditions of value restriction (2), extremal restriction and limited agreement.

Proof

Let LSPA hold over the set of orderings $\mathscr {D}|A$ of triple $A = \{x,y,z\}$.

(i) If $\mathscr {D}|A$ does not contain a Latin Square then VR(2) is satisfied; as a set of orderings of a triple violates VR(2) iff the set contains a Latin Square.

(ii) Suppose $\mathscr {D}|A$ contains one of the two Latin Squares, say LS(xyzx), and does not contain the other one. As $\mathscr {D}|A$ does not contain LS(xzyx) we have: $\sim [(\exists$ concerned $R^{s},R^{t},R^{u} \in \mathscr {D}|A)(xR^{s}zR^{s}y \wedge zR^{t}yR^{t}x \wedge yR^{u}xR^{u}z)]$. Without any loss of generality assume that $\mathscr {D}|A$ does not contain any concerned ordering R such that xRzRy. This implies that we must have:

$(\forall$ concerned $R \in \mathscr {D}|A)[(xRz \rightarrow yPz) \wedge (zRy \rightarrow zPx)]$.             (P3.7-1)

Consequently we have:

$(\forall$ concerned $R \in \mathscr {D}|A)[(xRyRz \rightarrow xRyPz) \wedge (zRxRy \rightarrow zPxRy)]$.              (P3.7-2)

If no $R {\in } \mathscr {D}|A \cap T[LS(xyzx)]$ is strong then (P3.7-2) implies that $(\forall R \in \mathscr {D}|A)(yRx)$, and therefore LA is satisfied.

Next consider the case when some ordering belonging to $\mathscr {D}|A \cap T[LS(xyzx)]$ is strong. This strong ordering cannot be xPyPz in view of zPxRy belonging to $\mathscr {D}|A$, otherwise LSPA would be violated. This strong ordering cannot be zPxPy either in view of xRyPz belonging to $\mathscr {D}|A$, otherwise LSPA would be violated. Thus, if an $R \in \mathscr {D}|A \cap T[LS(xyzx)]$ is strong, it has to be yPzPx. Once again, it follows that: $(\forall R \in \mathscr {D}|A)(yRx)$, and therefore LA is satisfied.

(iii) Finally consider the case when $\mathscr {D}|A$ contains both the Latin Squares. If no ordering belonging to $\mathscr {D}|A$ is strong then ER is satisfied.

Suppose $\mathscr {D}|A$ contains a strong ordering, say, xPyPz.

LSPA holds over $\mathscr {D}|A$ $\wedge$ $\mathscr {D}|A$ contains both the Latin Squares $\wedge$ $\mathscr {D}|A$ contains xPyPz $\rightarrow$ $(\forall$ concerned $R \in \mathscr {D}|A)[(yRzRx \rightarrow yPzIx) \wedge (zRxRy \rightarrow zIxPy)]$              (P3.7-3)

(P3.7-3) implies that $\mathscr {D}|A$ does not contain any of yPzPx, yIzPx, zPxPy, zPxIy              (P3.7-4)

$\mathscr {D}|A$ contains both the Latin Squares $\rightarrow (\exists \;$ concerned $\; R \in \mathscr {D}|A)(zRyRx)$              (P3.7-5)

(P3.7-4) $\wedge$ (P3.7-5) $\rightarrow (\exists R \in \mathscr {D}|A)(zPyPx)$              (P3.7-6)

(P3.7-6) $\rightarrow (\forall \;$ concerned $\; R \in \mathscr {D}|A)[(yRxRz \rightarrow yPxIz) \wedge (xRzRy \rightarrow xIzPy)]$              (P3.7-7)

(P3.7-7) implies that $\mathscr {D}|A$ does not contain any of yPxPz, yIxPz, xPzPy, xPzIy              (P3.7-8)

(P3.7-4) $\wedge$ (P3.7-8) $\rightarrow \mathscr {D}|A \subseteq \{xPyPz, zPyPx, yPzIx, zIxPy, xIyIz\}$.              (P3.7-9)

(P3.7-9) implies that ER is satisfied. This completes the proof of the assertion that satisfaction of LSPA implies satisfaction of at least one of the three conditions of value restriction (2), limited agreement and extremal restriction.

Let $\mathscr {D}|A$ violate LSPA. Without any loss of generality, assume that $\mathscr {D}|A$ contains LS(xyzx) involving a strong ordering, say xPyPz, and also contains an ordering belonging to LS(xyzx) with zPx. That is to say, $(\exists \;$concerned$\; R^{s},R^{t},R^{u} \in \mathscr {D}|A)[(xP^{s}yP^{s}z \wedge yR^{t}zP^{t}x \wedge zR^{u}xR^{u}y) \vee (xP^{s}yP^{s}z {\wedge } yR^{t}zR^{t}x \wedge zP^{u}xR^{u}y)].$              (P3.7-10)

$\mathscr {D}|A$ contains concerned $R^{s},R^{t},R^{u}$ such that $(xP^{s}yP^{s}z \wedge yR^{t}zR^{t}x \wedge zR^{u}xR^{u}y)$ implies that VR(2) is violated.              (P3.7-11)

$xP^{s}yP^{s}z \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (xPy) \wedge (\exists R {\in } \mathscr {D}|A) (yPz) \wedge (\exists R \in \mathscr {D}|A) (xPz)$              (P3.7-12)

concerned $yR^{t}zR^{t}x \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (yPx)$              (P3.7-13)

concerned $zR^{u}xR^{u}y \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (zPy)$              (P3.7-14)

(P3.7-10) $\rightarrow (\exists R \in \mathscr {D}|A) (zPx)$              (P3.7-15)

(P3.7-12)-(P3.7-15) imply that LA is violated.              (P3.7-16)

From (P3.7-10) it follows that: $(xP^{s}yP^{s}z, yR^{t}zP^{t}x \in \mathscr {D}|A) \vee (xP^{s}yP^{s}z, zP^{u}xR^{u}y \in \mathscr {D}|A)$              (P3.7-17)

(P3.7-17) implies that ER is violated.              (P3.7-18)

(P3.7-11), (P3.7-16), and (P3.7-18) establish the assertion that if a set of orderings of a triple satisfies at least one of the three conditions of value restriction (2), limited agreement and extremal restriction, then it satisfies LSPA.    $\square$

3.6 Notes on Literature

The first result providing a condition under which an alternative will necessarily exist that will be able to defeat every other alternative in a majority vote, i.e., a Condorcet winner, is due to Black (1948). Black showed that if there exists a way of arranging alternatives from left to right such that every individual’s preferences are single-peaked and the number of individuals is odd then there will exist an alternative that will defeat every other alternative in a majority vote. Arrow (1951) formalized and slightly generalized Black’s condition of single-peakedness. Arrow showed that if there exists a linear (strong) ordering of all alternatives such that for each individual i, $xR_{i}y$ and B(x, y, z) together imply $yP_{i}z$, and if the number of individuals is odd, then the social R generated by the MMD is transitive.

If there exists a strong ordering of all alternatives such that, for each individual i, $xR_{i}y$ and B(x, y, z) together imply $yP_{i}z$, it is immediate that for every triple of alternatives there would exist a strong ordering such that, for each individual i, $xR_{i}y$ and B(x, y, z) together would imply $yP_{i}z$. But the converse is not true. That is, even if there exists a strong ordering over every triple of alternatives such that, for each individual i, $xR_{i}y$ and B(x, y, z) together imply $yP_{i}z$, there may not exist a strong ordering of all alternatives such that, for each individual i, $xR_{i}y$ and B(x, y, z) together imply $yP_{i}z$. This can be seen by the following example.

Example 3.1

Let $S = \{x,y,z,u\}; N = \{1,2,3,4\}$.

$R_{1}: xP_{1}yP_{1}zP_{1}u$

$R_{2}: zP_{2}xP_{2}yP_{2}u$

$R_{3}: xP_{3}zP_{3}uP_{3}y$

$R_{4}: uP_{4}xP_{4}zP_{4}y$

In this case, there exists no strong ordering of four alternatives for which all four orderings can be expressed as single-peaked preferences. But it can easily be verified that the assumption of single-peaked preferences is satisfied for every triple of alternatives.              $\lozenge$

Although Arrow assumed single-peakedness of individual preferences over all alternatives, his proof used only single-peakedness over every triple of alternatives. Thus, Arrow’s proof is valid for the more general theorem. This was first pointed out by Inada (1964). In this paper, Inada also formulated single-cavedness and separability conditions and showed that, given that the number of individuals is odd, each of them is sufficient to guarantee transitivity under the MMD. He also introduced the condition of dichotomous preferences and showed that it was sufficient for transitivity under the MMD regardless of the number of individuals.

The three conditions of single-peakedness, single-cavedness and separability were combined into a single condition of value restriction [referred to as the first version of value restriction in this text] by Sen. He also pointed out that single-peakedness, single-cavedness and separability conditions over a triple were equivalent, respectively, to requiring that there is some alternative that is not worst in any ordering over the triple; that there is some alternative that is not best in any ordering over the triple; that there is some alternative that is not medium in any ordering over the triple.

The most important contribution in the context of conditions for transitivity under the MMD is that of Inada (1969). In this paper, Inada introduced three more conditions of echoic preferences, antagonistic preferences and taboo preferences. He showed that echoic preferences and antagonistic preferences were sufficient conditions for transitivity under the MMD regardless of the number of individuals. He also showed that, if there were no restrictions on the number of individuals, then if a set of orderings of a triple violated all three conditions of dichotomous preferences, echoic preferences and antagonistic preferences, then for some assignment of individuals over these orderings the social R generated by the MMD would be intransitive. In other words, if there were no restrictions on the number of individuals, then apart from dichotomous preferences, echoic preferences and antagonistic preferences, no other conditions were to be found which could ensure transitivity under the MMD. He also showed, given the number of individuals to be odd, sufficiency of taboo preferences for transitivity under the MMD. For the case of odd number of individuals, Inada showed that if any set of orderings violated all of the conditions of VR(1), taboo preferences, dichotomous preferences, echoic preferences and antagonistic preferences, then for some assignment of individuals over these orderings the social R generated by the MMD would be intransitive. Thus, Inada’s 1969 paper established (i) conditions which completely characterize all sets of orderings of a triple which invariably give rise to transitive social R under the MMD, when there are no restrictions on the number of individuals; and (ii) conditions which completely characterize all sets of orderings of a triple which invariably give rise to transitive social R under the MMD, when the number of individuals is odd.

The three conditions of dichotomous preferences, echoic preferences and antagonistic preferences were combined into a single condition of extremal restriction by Sen and Pattanaik (1969). They also showed that the union of the second version of value restriction, limited agreement and extremal restriction completely characterizes the sets of orderings of a triple that invariably give rise to quasi-transitive social preferences under the MMD.

The three conditions of second version of value restriction, limited agreement and extremal restriction were combined into a single condition of Latin Square partial agreement in Jain (1985). The conditions of the first version of value restriction, taboo preferences, dichotomous preferences and antagonistic preferences were combined into a single condition of Weak Latin Square partial agreement in Jain (2009).

As the notion of a social decision rule is defined for a specified set of individuals, strictly speaking, one should make a distinction among MMDs defined for different numbers of individuals. Kelly (1974) pointed out that the characterization of sets of orderings of a triple which guarantee quasi-transitivity or acyclicity depends on the number of individuals. From this perspective, the union of three conditions of second version of value restriction, limited agreement and extremal restriction characterizes the sets of orderings of a triple that invariably give rise to quasi-transitive social preferences under the MMD only for five or more individuals, and not for the cases of three and four individuals. The conditions for quasi-transitivity for the cases of three and four individuals were established in Jain (2009).