MA-351 Homework 6

Due at Thursday December 9, 1999, 11h00 (am)
10 bonus points


Solutions may be submitted in person in class, or you may email an ASCII text, html, or postscipt/pdf-formatted document to me (kaltofen@math.ncsu.edu).
Note my office hours on my schedule.

In the plurality-with-elimination group consensus function (social welfare function), the group's ranking is determined as follows: The alternative(s) in the set of alternatives with the least first place votes are ranked last. Then those alternatives are eliminated and the process is repeated. This system is used in Australia. For example, with A = {a,b,c} and t = 8 

Number of votes  4  3  1
========================
1st choice       a  b  c
2nd choice       b  c  b
3rd choice       c  a  a
Since c has 1 first place vote, it is ranked last and eliminated from the group profile. The new rankings are (without revoting): 
Number of votes      4  3  1
============================
new 1st choice       a  b  b
new 2nd choice       b  a  a
Now a and b are tied for first place with 4 votes each. The group profile is (a-b, c). [For more explanation, see the Tannenbaum/Arnold book in the library.]
  1. Give an example of a group profile such that Arrow's Axiom 1 (monotonicity) is violated when using plurality-with-elimination.
  2. Give an example of a group profile such that Arrow's Axiom 2 (independence of irrelevant alternatives) is violated when using plurality-with-elimination.