The rhyming triplets, Boris, Horace, and Maurice, are the membership committee of
the very exclusive Dead Poets Society. The final item on their agenda one morning is
a proposal that Alice should be admitted as a new member. No mention is made of
another possible candidate called Bob, so an amendment to the final item is proposed.
The amendment says that Alice’s name should be replaced with Bob’s. The rules for
voting in committees call for amendments to be voted on in the reverse order to which
they are proposed. The committee therefore begins by voting on whether Bob should
replace Alice. If Alice wins, they then vote on whether Alice or Nobody should be
made a new member. If Bob wins, they then vote on whether Bob or Nobody should be
made a new member. The figure below represents the order in which voting takes place.
It also shows how the three committee members rank the three possible outcomes.
(a) Who will win the vote if everybody just votes according to their rankings?
(b) Why should Horace switch to voting for the candidate he likes least at the first
vote?
(c) What happens if everybody votes strategically? (Hint: Focus on equilibria in
(weakly) dominant strategies.