Consider majority voting in a group of size four. There are three objects, called A, B, and C.

Person 1 or P1 has preferences such that A is strictly preferred to B and B is strictly preferred to C, with also A being strictly preferred to C. P2 strictly prefers B to C and C to A, with B being strictly preferred to A also. For P3, we have C strictly preferred to A and A strictly preferred to B with C strictly preferred to B also. Finally, P4 strictly prefers A to B, B to C, and A to C.

1) Show that each individual's strict preference relation is transitive.

2) Show that the majority rule preferences, where tie votes are interpreted as indifference, violate a variation of transitivity. Be sure to clarify what transitivity condition is violated (weak versus strict preference, etc.).