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:

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%