Conditions for Social Preference Transitivity When Cycle Involved and A $\hat{O}\mbox{-}\hat{I}$ Framework

We present some conditions for social preference transitivity under the majority rule when the individual preferences include cycles. First, our concern is with the restriction on the preference orderings of individuals except those (called cycle members) whose preferences constitute the cycles, but the considered transitivity is, of course, of the society as a whole. In our discussion, the individual preferences are assumed concerned and the cycle members' preferences are assumed as strict orderings. Particularly, for an alternative triple when one cycle is involved and the society is sufficient large (at least 5 individuals in the society), we present a sufficient condition for social transitivity; when two antagonistic cycles are involved and the society has at least 9 individuals, necessary and sufficient conditions are presented which are merely restricted on the preferences of those individuals except the cycle members. Based on the work due to Slutsky (1977) and Gaertner \& Heinecke (1978), we then outline a conceptual $\hat{O}\mbox{-}\hat{I}$ framework of social transitivity in an axiomatic manner. Connections between some already identified conditions and the $\hat{O}\mbox{-}\hat{I}$ framework is examined.