Jean-Charles de Borda (1733-99), a contemporary of Condorcet in France, argued for a democratic system that deviates

Question:

Jean-Charles de Borda (1733-99), a contemporary of Condorcet in France, argued for a democratic system that deviates from our usual conception of majority rule. The system works as follows: Suppose there are M proposals. Each voter is asked to rank these €” with the proposal ranked first by a voter given M points, the one ranked second given (M ˆ’1) points, and so forth.3 The points given to each proposal are then summed across all voters, and the top N proposals are chosen€”where N might be as low as 1. This voting method, known as the Borda Count is used in a variety of corporate and academic settings as well as some political elections in countries around the world.
A: Suppose there are 5 voters denoted 1 through 5, and there are 5 possible projects {A,B,C,D,E} to be ranked. Voters 1 through 3 rank the projects in alphabetical sequence (with A ranked highest). Voter 4 ranks C highest, followed by D, E, B and finally A. Voter 5 ranks E highest, followed by C, D, B and finally A.
(a) How does the Borda Count rank these? If only one can be implemented, which one will it be?
(b) Suppose option D was withdrawn from consideration before the vote in which voters rank the options. How does the Borda Count now rank the remaining projects? If only one can be implemented, which one will it be?
(c) What if both D and E are withdrawn? 3There exist other versions of Borda€™s method€” such as assigning (M ˆ’1) points to the top ranked choice and leaving zero for the last ranked. For purposes of this problem, define the method as it is defined in the problem.

Table 28.3: Borda Count Implies A Wins and C comes in Third
(d) Suppose I get to decide which projects will be considered by the group and the group allows me to use my discretion to eliminate projects that clearly do not have widespread support. Will I be able to manipulate the outcome of the Borda Count by strategically picking which projects to leave off?
B: Arrow€™s Theorem tells us that any non-dictatorial social choice function must violate at least one of his remaining four axioms.
(a) Do you think the Borda Count violates Pareto Unanimity? What about Universal Domain or Rationality?
(b) In what way do your results from part A of the exercise tell us something about whether the Borda Count violates the Independence of Irrelevant Alternatives (IIA) axiom?
(c) Derive again the Borda Count ranking of the five projects in part A given the voter preferences as described.
(d) Suppose voter 4 changed his mind and now ranks B second and D fourth (rather than the other way around). Suppose further that voter 5 similarly switches the position of B and D in his preference ordering€”and now ranks B third and D fourth. If a social choice function satisfies IIA, which social rankings cannot be affected by this change in preferences?
(e) How does the social ordering of the projects change under the Borda Count? Does the Borda Count violate IIA?