Ron and Veronica need to choose a name for their newborn daughter. After giving the matter much thought, they have narrowed the list of possible names to four: Abigail, Iris, Irene, and Olga. They now must choose one name from this list. Each parent ranks the four names in order of preference. Given each of the following decision rules, determine whether it is a social choice function.

If yes, determine whether it is monotonic, and whether it is manipulable. Justify your answers. (If a rule is monotonic, provide a direct proof that it satisfies the property of monotonicity. If a rule is manipulable, provide an example showing how it may be manipulated.)

(a) If both parents select a name as being the most preferred, that name is chosen. Otherwise, if there is only one name that both parents rank within their top two most-preferred names, that name is chosen. Otherwise, Abigail is the chosen name.

(b) If both parents select a name as being the most preferred, that name is chosen. Otherwise, if there is only one name that both parents rank within their top two most-preferred names, that name is chosen. Otherwise, the parents toss a coin to determine the name of their daughter, with Iris chosen if the coin shows heads, and Olga chosen if the coin shows tails.

(c) If Ron most prefers the name Irene, that name is chosen. Otherwise, the name that Veronica most prefers is chosen.

(d) As a first step, the name that each parent ranks last by preference is removed from the list under consideration. This leaves two or three names in contention. In the next step, the name that each parent ranks last by preference from among the two or three remaining names is removed from the list under consideration. This leaves zero, one, or two names. If two names remain, the process of removing the name that each parent least prefers from the list under consideration is repeated again. If only one name remains, that name is chosen. Otherwise, Olga is chosen as the child's name.