Is a referendum with three options the solution?

(This is an exam exercise I gave to my MEBA Master students, but here I show the solutions -the exam was yesterday; any similarity with real facts is pure coincidence)

A community of 100 citizens  has three groups: Icons (40 individuals), Fires (35) and Econs (25). Using the data in the next table, which shows the ranking of preferred alternatives among three options (I, F and E) for each type of voter,

a)  Is there a stable winner in votes with two alternatives? Yes, F is the stable winner (so, the Condorcet paradox is absent from this problem).

b) Are collective preferences transitive? Yes, F defeats I, I defeats E and F defeats E (F is a Condorcet winner).

c) In which of the three groups would we find the median voter? Among the Fires. If we organize the population in one segment from the most pro-I to the most pro-E, a Fire has half of the population at one side and half of the population at the other.

d) Under which rules does and does not the median voter’s option  win in votes with three alternatives?

With three options, the median voter only wins if the voting rules are based on the Borda count, that is, if each voter gives points to the three alternatives in descending order of preference. This is assuming that all voters understand the rules, do not change their mind, understand the alternatives, all of them do vote, and they do not do it strategically. If a voter can choose only one of the three options, the winner will be I, which is not the option preferred by the median voter.

The exercise does not take into account externalities of the decision on other populations or commitment problems associated to each of the options (for example, option I may expropriate previous investments in human capital, settling decisions or professional options).

Table

	Icons (40)	Fires (35)	Econs (25)
First Option	I	F	E
Second Option	F	I	F
Third Option	E	E	I