# In his book A Mathematician Reads the Newspaper, John Allen Paulos gives the following caricature…

In his book A Mathematician Reads the Newspaper, John
Allen Paulos gives the following caricature based on the 1992 Democratic
presidential primary caucuses. There are five candidates: Jerry Brown, Bill
Clinton, Tom Harkin, Bob Kerrey, and Paul Tsongas. There are 55 voters, with
different preference orderings concerning the candidates. There are six
different orderings, which we label I through VI. The preference orderings (1
for best to 5 for worst), along with the numbers of voters with each ordering,
are shown in the following table (the candidates are identified by the first
letters of their last names)27:

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(a) First, suppose that all voters vote sincerely.
Consider the outcomes of each of several different election rules. Show each of
the following outcomes:

(i) Under the plurality method (the one with the most
first preferences), Tsongas wins.

(ii) Under the runoff method (the top two first
preferences go into a second round), Clinton wins.

(iii) Under the elimination method (at each round, the
one with the fewest first preferences in that round is eliminated, and the rest
go into the next round), Brown wins.

(iv) Under the Borda-count method (5 points for first
preference, 4 for second, and so on; the candidate with the most points wins),
Kerrey wins.

(v) Under the Condorcet method (pairwise comparisons),
Harkin wins.

(b) Suppose that you are a Brown, Kerrey, or Harkin
supporter. Under the plurality method, you would get your worst outcome. Can
you benefit by voting strategically? If so, how?

(c) Are there opportunities for strategic voting under
each of the other methods as well? If so, explain who benefits from voting
strategically and how they can do so.