The 2008 presidential nomination season saw 21 Republican primaries and caucuses on Super…

The 2008 presidential nomination season saw 21 Republican
primaries and caucuses on Super Tuesday—February 5, 2008. By that day—only a
month after the Iowa caucus that began the process—more than half of the
Republican contenders had dropped out of the race, leaving only four: John
McCain, Mitt Romney, Mike Huckabee, and Ron Paul. McCain, Romney, and Huckabee
had each previously won at least one state. McCain had beaten Romney in Florida
the week before the big day, and at that point it looked like only the two of
them stood a realistic chance of wining the nomination. In this primary season,
as is typical for the Republican party, nearly every GOP contest (whether
primary or caucus) was winner take-all, so winning a given state would earn a
candidate all of the delegates allotted to that state by the Republican
National Committee. The West Virginia caucus was the first contest to reach a
conclusion on Super Tuesday, as the caucus took place in the afternoon, it was
brief, and the state is in the eastern time zone. News of the result was
available hours before the close of polls in many of the states voting that
day. The following problem is based on the results of that West Virginia
caucus. As we might expect, the caucusers did not all share the same
preferences over the candidates. Some favored McCain, whereas others liked
Romney or Huckabee. The voters also certainly had varied preferences about whom
they wanted to win if their favorite candidate did not. Simplifying substantially
from reality (but based on the actual voting), assume that there were seven
types of West Virginia caucus goers that day, whose prevalence and preferences
were as follows:

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At first, no one knew the distribution of preferences of
those in attendance at the caucus, so everyone voted truthfully. Thus, Romney
won a plurality of the votes in the first round with 41%. After each round of
this caucus, if no candidate wins a majority, the candidate with the least
number of votes is dropped from consideration, and his or her supporters vote
for one of the remaining candidates in the following rounds.

(a) What would the results of the second round have been
under truthful (nonstrategic) voting for the remaining three candidates?

(b) If West Virginia had held pairwise votes among the
four candidates, which one would have been the Condorcet winner with truthful
voting?

(c) In reality, the results of the second round of the
caucus were:

Huckabee: 52%

Romney: 47%

McCain: 1%

 

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