Let's go back to explaining the survey data in which 60 % chose Edwards over Obama, 60 % chose Edwards over Clinton, 60% chose Clinton over Obama and then when asked to cast a one of the three candidates, 45 % chose Clinton, 30 % Obama, and 25% Edwards. single vote for (1) Prove that it is not possible that all voters are rational (have transitive preferences.) (2) Suppose now that the distribution of votes was not 45% chose Clinton, 30% Obama , and 25 % Edwards but 38% Clinton, 32% Obama and 30% Edwards; pairwise data remains unchanged. Prove that it is possible now that all voters are rational (have transitive preferences) and calculate the percentage of votes for each of the six different orderings of the three candidates.

Explanation:  

1) The ratio of Edwards to Obama = 60% : 40%, that is, 3 : 2

The ratio of Edwards to Clinton = 60% : 40%, that is, 3 : 2

Also, the ratio of Clinton to Obama = 60% : 40%, that is, 3 : 2

From the above statements the ratio of Edwards, Clinton and Obama, should reflect as follow in that order  5 : 4 : 3, showing that it is not possible that all voters are rational.

2) If  the votes were to be 38% Clinton, 32% Obama and 30% Edwards,that is, in a ratio of 19 : 16 : 15 respectively, it is rational based on the preceding illustration, but it could have been higher for Edwards to Clinton becuase Edwards had higher votes when compared to Clinton and Obama seprately.  

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