HW XXXXXXXXXXImpossibility Theorem
XXXXXXXXXXDoes the Borda rule satisfy A
ow’s Unanimity axiom? Does it satisfy
Independence? Justify your answer.
XXXXXXXXXXThe purpose of this question is to illustrate how electoral outcomes are
less sensitive to the voting rule when voters share an underlying common
interest. Suppose that the true welfare ranking of policies , ??, and ?? is ?? ≻ ?? ≻
??, but that votes’ opinions of the welfare ranking may differ from this.
Specifically, any voter has a probability 4
15
of believing that ?? ≻ ?? ≻ ??,
probabilities 3
15
each of believing ?? ≻ ?? ≻ ?? or ?? ≻ ?? ≻ ??, probabilities 2
15
each
of believing ?? ≻ ?? ≻ ?? or ?? ≻ ?? ≻ ??, and a probability 1
15
of believing that ?? ≻
?? ≻ ??. Suppose further that there are three voters, Alice, Betty, and Cedric,
whose opinions are statistically independent of one another.
a. (+8) If the three voters vote sincerely using plurality rule (
eaking a
three-way tie, if necessary, with equal probability), how likely are they to
choose ??? How likely are they to choose ??? How likely are they to
choose ???
. (+6) If the voters vote sincerely using the Borda rule, what are the
expected scores for ??, ??, and ???
HW XXXXXXXXXXImpossibility Theorem