The characterization of monotonicity and neutrality for the class of binary social decision rules with unrestricted domain and satisfying Pareto-indifference given in this chapter is from Jain (1988). Characterization of transitivity for the class of binary social decision rules with unrestricted domain and satisfying the Pareto-criterion was obtained in Jain (1977) and for quasi-transitivity in Guha (1972) and Blau (1976). The characterization of acyclicity for the class of social decision rules with unrestricted domain and satisfying the conditions of neutrality, monotonicity and the Pareto-criterion is from Jain (1977). The maximal sufficient conditions for transitivity for the class of binary, neutral and monotonic rules, and for its subclass satisfying the Pareto-criterion, were derived in Jain (1987). Sufficiency of value restriction (2) for quasi-transitivity under every neutral and monotonic binary social decision rule, and sufficiency of limited agreement for quasi-transitivity under every neutral and monotonic binary social decision rule satisfying the Pareto-criterion, were established in Sen and Pattanaik (1969). The proofs given here of these two theorems are, however, different from those given in Sen and Pattanaik (1969).
In this chapter the class of neutral and monotonic binary social decision rules has been discussed. A closely related class is that of Pareto-transitive binary social decision rules. A social decision rule \(f: \mathbb {D} \mapsto \mathscr {C}\) is Pareto-transitive iff \((\forall (R_{1}, \ldots ,R_{n}) \in \mathbb {D}) (\forall x,y,z \in S) [(xRy \wedge (\forall i \in N)(yP_{i}z) \rightarrow xPz]\). In Salles (1975) it is claimed that for the class of Pareto-transitive binary social decision rules an Inada-type necessary and sufficient condition for transitivity is that over every triple of alternatives value restriction (1) or cyclical indifference (CI) holds. \(\mathscr {D}\) satisfies CI over a triple of alternatives A iff \((\exists \; \text {distinct} \; x,y,z \in A) [(\mathscr {D} \mid A \subseteq \{xPyIz, yPzIx, zPxIy\}) \vee (\mathscr {D} \mid A \subseteq \{xIyPz, yIzPx, zIxPy\})\). We show in what follows that: (i) The satisfaction of (VR (1) or CI) over every triple of alternatives is not an Inada-type necessary condition for transitivity for the class of Pareto-transitive binary social decision rules. (ii) The satisfaction of (VR (1) or CI) over every triple of alternatives is not a sufficient condition for transitivity for the class of Pareto-transitive binary social decision rules. (iii) There does not exist an Inada-type necessary and sufficient condition for transitivity for the class of Pareto-transitive binary social decision rules.
An Inada-type necessary and sufficient condition for transitivity partitions the set of all nonempty subsets of \(\mathscr {T}\) into two subsets \(\mathscr {T}^{T}_{1}\) and \(\mathscr {T}^{T}_{2}\) such that: for every \(\mathscr {D} \in \mathscr {T}^{T}_{1}\), every \((R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}\) yields transitive social R; and for every \(\mathscr {D} \in \mathscr {T}^{T}_{2}\), there exists a \((R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}\) which yields intransitive social R. As MMD defined for an odd number of individuals yields transitive social R for every \((R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}, \mathscr {D} = \{xPyPz, zPyPx, yPzIx, zIxPy\}\), it follows that if there exists an Inada-type necessary and sufficient condition for transitivity for the class of Pareto-transitive binary social decision rules then it must be the case that \(\{xPyPz, zPyPx, yPzIx, zIxPy\} \in \mathscr {T}^{T}_{1}\). Like the MMD defined for an odd number of individuals, the strict majority rule defined for an odd number of individuals is also a Pareto-transitive binary social decision rule. Consider the strict majority rule defined for \(S = \{x,y,z\}; n = 2k + 1, k \ge 5\). Let \((R_{1}, \ldots ,R_{n}) \in \mathscr {D}^{n}, \mathscr {D} = \{xPyPz, zPyPx, yPzIx, zIxPy\}\), be such that: \(n(xP_{i}yP_{i}z) = k \wedge n(zP_{i}yP_{i}x) = k-1 \wedge n(y_{i}PzI_{i}x) = 1 \wedge n(z_{i}IxP_{i}y) = 1\). We obtain: \([n(xP_{i}y) {=} k+1 \wedge n(yP_{i}x) = k \wedge n(yP_{i}z) = k+1 \wedge n(zP_{i}y) = k \wedge n(xP_{i}z) = k \wedge n(zP_{i}x) = k-1]\), which results in \([xPy \wedge yPz \wedge xIz]\), violating transitivity. Therefore, if there exists an Inada-type necessary and sufficient condition for transitivity for the class of Pareto-transitive binary social decision rules then it must be the case that \(\{xPyPz, zPyPx, yPzIx, zIxPy\} \in \mathscr {T}^{T}_{2}\). As \(\{xPyPz, zPyPx, yPzIx, zIxPy\}\) cannot belong to both \(\mathscr {T}^{T}_{1}\) and \(\mathscr {T}^{T}_{2}\), it follows that for the class of Pareto-transitive binary social decision rules there does not exist any condition whatsoever that is an Inada-type necessary and sufficient condition for transitivity.
