Quasi-transitive Individual Preferences

Abstract

This chapter is concerned with the class of neutral and monotonic binary social decision rules and some of its subclasses when individual weak preference relations are reflexive, connected and quasi-transitive rather than orderings. Given that the domain consists of all logically possible profiles of individual reflexive, connected and quasi-transitive weak preference relations, a characterization is provided for the class of neutral and monotonic binary social decision rules. Given that individual weak preference relations are reflexive, connected and quasi-transitive, conditions for quasi-transitivity are derived for the method of majority decision, the class of special majority rules, and the class of social decision rules which are simple games.

Appendices

Appendix

11.6 Notes on Literature

The most important contribution in the context of conditions for quasi-transitivity under the method of majority decision, when individuals have reflexive, connected and quasi-transitive weak preference relations over the set of social alternatives, is that of Inada (1970). In this paper, Inada generalized the conditions of value restriction (2) and of limited agreement for the case of individual weak preference relations being reflexive, connected and quasi-transitive; and showed that the satisfaction over every triple of alternatives of the generalized version of value restriction (2) or of the generalized version of limited agreement or of dichotomous preferences or of antagonistic preferences is sufficient for quasi-transitivity under the method of majority decision, when individual weak preference relations are reflexive, connected and quasi-transitive. These four conditions were combined into a single condition in Jain (1986a). This single condition has been called in this volume as Latin Square partial agreement-Q. Given that individual weak preference relations are reflexive, connected and quasi-transitive, Inada (1970) also showed that satisfaction of at least one of the above four conditions was necessary for quasi-transitivity under the method of majority decision, the term ‘necessary’ having the sense as in Inada (1969). In Jain (2009) it is shown that Latin Square partial agreement-Q completely characterizes the sets $\mathscr {D}$ of reflexive, connected and quasi-transitive weak preference relations over the set of social alternatives such that every profile belonging to $\mathscr {D}^{n}$ yields quasi-transitive social weak preference relation under the method of majority decision, provided the number of individuals is at least five. Complete characterizations of sets $\mathscr {D}$ of reflexive, connected and quasi-transitive weak preference relations over the set of social alternatives such that every profile belonging to $\mathscr {D}^{n}$ yields quasi-transitive social weak preference relation under the method of majority decision, for the cases of 2, 3 and 4 individuals were obtained in Jain (2009).

When individual weak preference relations are reflexive, connected and quasi-transitive, the sufficiency of Latin Square partial agreement-Q for quasi-transitivity under every special majority rule was established in Jain (1986a). Complete characterization of the sets $\mathscr {D}$ of reflexive, connected and quasi-transitive weak preference relations over the set of social alternatives such that every profile belonging to $\mathscr {D}^{n}$ yields quasi-transitive social weak preference relation under social decision rules that are simple games was obtained in Jain (1989).

The characterization of monotonicity and neutrality for the class of binary social decision rules with domain $\mathscr {Q}^{n}$ given in this chapter is from Jain (1996). Characterization of rationality conditions for the class of binary social decision rules with domain $\mathscr {Q}^{n}$ and satisfying the conditions of neutrality and monotonicity is also from Jain (1996).

The sufficiency of value restriction (2) for quasi-transitivity under every neutral and monotonic binary social decision rule, when individual weak preference relations are reflexive, connected and quasi-transitive, was shown in Pattanaik (1970). The proof of the theorem given in this chapter, however, is different.

11.7 Domain Conditions for Acyclicity

Let f be a social decision rule. Unlike transitivity and quasi-transitivity, condition of acyclicity is not defined over triples. Consequently, there is no reason to expect existence of conditions defined only over triples which can completely characterize all $\mathscr {D}$, $\mathscr {D} \subseteq \mathscr {B}$, which are such that all logically possible $(R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}$ give rise to acyclic social $R, R = (R_{1}, \ldots ,R_{n}),$ under f. In fact, if $\mathscr {B} = \mathscr {Q}$, then the subsets $\mathscr {D} \subseteq \mathscr {B}$ which are such that all logically possible $(R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}$ give rise to acyclic social R under the MMD cannot be characterized by a condition defined only over triples as the following theorem shows.

Theorem 11.17

Let f be the method of majority decision; and let $\#S = s \ge 4$ and $\#N = n \ge 2$. Let $\mathscr {D_{Q}} = \{\mathscr {D} \subseteq \mathscr {Q} \mid (\forall (R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}) (R = f(R_{1}, \ldots ,R_{n}) \;\text {is acyclic})\}$. Then, there does not exist any condition $\alpha$ defined only over triples such that $\mathscr {D}, \mathscr {D} \in 2^{\mathscr {Q}} - \{\emptyset \},$ belongs to $\mathscr {D_{Q}}$ iff it satisfies condition $\alpha$.

Proof

Let condition $\alpha$ defined only over triples be such that $\mathscr {D}, \mathscr {D} \in 2^{\mathscr {Q}} - \{\emptyset \},$ belongs to $\mathscr {D_{Q}}$ iff it satisfies condition $\alpha$. Let $S = \{x,y,z,w,t_{1}, \ldots , t_{s-4}\}$. Consider $\mathscr {D} = \{(xPy, yIz, xIz; \;x,y,zPwPt_{1}P \ldots Pt_{s-4}),$ $(yPz, zIx, yIx;\; x,y,zPwPt_{1}P {\ldots } Pt_{s{-}4})\}$. It is immediate that the MMD yields acyclic R for every $(R_{1}, {\ldots } ,R_{n}) \in \mathscr {D}^{n}$; and consequently it follows that $\mathscr {D} \in \mathscr {D_{Q}}$. As condition $\alpha$ is defined only over triples, it follows that $\mathscr {D}$ must be satisfying $\alpha$ over every triple of alternatives. Therefore it follows that if $A \subseteq S$ is a triple and $(\exists \;\text {distinct}\; a,b,c \in A) [\mathscr {D}|\{a,b,c\} = \{(aPb \wedge bIc \wedge aIc), (aIb \wedge bPc \wedge aIc)\} \vee \mathscr {D}|\{a,b,c\} = \{aPbPc, aIbPc\}]$ then $\mathscr {D}$ would satisfy $\alpha$ over A.

Now consider the following $(R_{1}, \ldots ,R_{n})$.

$(xP_{1}y, yI_{1}z, zP_{1}w, wI_{1}x, xI_{1}z, yI_{1}w;\; x,y,z,wP_{1}t_{1}P_{1} \ldots P_{1}t_{s-4})$

$(\forall i \in N - \{1\})(xI_{i}y, yP_{i}z, zI_{i}w, wP_{i}x, xI_{i}z, yI_{i}w;\; x,y,z,wP_{i}t_{1}P_{i} \ldots P_{i}t_{s-4})$.

The R yielded by the MMD for the above configuration is: $(xPy, yPz, zPw, wPx, xIz, yIw;$ $x,y,z,wPt_{1}P \ldots Pt_{s-4})$, which violates acyclicity. Now, for every triple of alternatives $A \subseteq S$ we have $(\exists \;\text {distinct}\; a,b,c \in A)[\{R_{i}|A {\mid } i \in N\} \subseteq \{(aPb, bIc, aIc), (bPc, cIa, bIa)\}] \vee (\exists \;\text {distinct}\; a,b,c \in A)[\{R_{i}|A \mid i \in N\} \subseteq \{aPbPc, aIbPc\}]$. Therefore, it follows that either $\{(aPb \wedge bIc \wedge aIc), (aIb \wedge bPc \wedge aIc)\}$ or $\{aPbPc, aIbPc\}$ must be violating $\alpha$, contradicting the earlier conclusion that both of these sets satisfy $\alpha$. This contradiction establishes the theorem.$\square$

From the above theorem the following corollary follows immediately.

Corollary 11.1

Let f be the method of majority decision; and let $\#S = s \ge 4$ and $\#N = n \ge 2$. Let $\mathscr {D_{A}} = \{\mathscr {D} \subseteq \mathscr {A} \mid (\forall (R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}) (R = f(R_{1}, \ldots ,R_{n}) \;\text {is acyclic})\}$. Then, there does not exist any condition $\alpha$ defined only over triples such that $\mathscr {D}, \mathscr {D} \in 2^{\mathscr {A}} - \{\emptyset \},$ belongs to $\mathscr {D_{A}}$ iff it satisfies condition $\alpha$.

If the number of individuals is greater than or equal to 11 then the sets $\mathscr {D}\subseteq \mathscr {T}$ which are such that all logically possible $(R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}$ give rise to acyclic social R under the MMD, however, can be characterized by a condition defined only over triples. The following theorem can easily be proved.

Theorem 11.18

Let $\#S \ge 3$ and $\# N = n \ge 11$. Let $\mathscr {D} \subseteq \mathscr {T}$. Then the method of majority decision f yields acyclic social $R, R = f(R_{1}, \ldots ,R_{n}),$ for every $(R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}$ iff $\mathscr {D}$ satisfies the condition of Latin Square partial agreement.²

As far as the remaining cases are concerned, there is no uniformity among them. In some cases it can be shown that no condition defined only over triples can be a characterizing condition while in some other cases it is possible to formulate a characterizing condition defined only over triples. For instance, it can be shown that if the number of individuals is four then there does not exist any condition defined only over triples which can characterize the sets of orderings which invariably give rise to acyclic social R. On the other hand the validity of Theorem 11 can be shown for $n = 9$ as well.³