Theory of Voting

•  1 (Majority Criterion)If a candidate gets a majority  (>50%) of the first place votes, he/she should be winner.
•  2 (Condorcet Criterion): winning candidate should also be winner of pairwise comparisons.
•  3 (Monotonicity Criterion): Suppose X is the winner and suppose that in another election some voters are able to rank X higher, with no change for the other candidates, then X should still win.
•  4 (Irrelevant Alternatives Criterion): Suppose X is the winner, if one or more losing candidates drop from the race, X should still be the winner.
•  (5 (Pareto Condition): if everyone prefers candidate 1 over candidate 2, then candidate 2 should not be among the winners of the election.This is subsumed by the Condorcet Criterion.)

Example 1 again:

 

First	A	B	C	D	E	E
Second	D	E	B	C	B	C
Third	E	D	E	E	D	D
Fourth	C	C	D	B	C	B
Fifth	B	A	A	A	A	A
	18	12	10	9	4	2

The plurality winner is A.

• Is the Majority Criterion satisfied?
• Is the Condorcet Criterion satisfied?
• Is the Irrelevant Alternatives Criterion satisfied? I.e. can you find a losing candidate X, such that when we drop X, this would change the plurality outcome?


The single runoff winner is B.

• Is the Majority Criterion satisfied?
• Is the Condorcet Criterion satisfied?
• Is the Irrelevant Alternatives Criterion satisfied?


The sequential runoff winner is C.

• Is the Majority Criterion satisfied?
• Is the Condorcet Criterion satisfied?
• Is the Irrelevant Alternatives Criterion satisfied?


The  Borda count winner is D.

• Is the Condorcet Criterion satisfied?
• Is the Irrelevant Alternatives Criterion satisfied?


The Condorcet winner is E.

• Is the Majority Criterion satisfied?
• Is the Condorcet Criterion satisfied?
• \Is the Irrelevant Alternatives Criterion satisfied?

Problem 1:  For the following preference schedule find the winner by Borda count:

A	B
B	C
C	A
11	10

Is there a Majority winner of first place votes?

Problem 2: determine the Condorcet winner. Is the Irrelevant Alternatives Criterion satisfied?

First	A	A	E	C	D
Second	E	C	B	B	B
Third	C	D	A	A	A
Fourth	D	E	C	D	E
Fifth	B	B	D	E	C
	1	1	1	1	1

Arrows Impossibility Theorem:  No voting system can satisfy all four fairness criteria in all cases.

A weak version of Arrows Impossibility Theorem:
Given an odd number of voters and 3 or more candidates and an election. Then there is no voting system that satisfies both the Condorcet Criterion and the Irrelevant Alternatives Criterion and that produces at least one winner.

Proof: Given the election:

First	A	B	C
Second	B	C	A
Third	C	A	B
	1	1	1

A is a non winner, so drop B, then A is still a non winner.
B is a non winner, so drop C, then B is still a non winner.
C is a non winner, so drop A, then C is still a non winner.
So no winner.

Approval voting:  might give fairer results more often than traditional methods.

Ballot:

candidate	approve	don't approve
A	X
B	X
C		X
D	X

• Suppose that the candidates A and B have moderate political positions, while candidate C is quite liberal. Voter opinions about the candidates are as follows:
• 28% want A as their first choice, but would also approve of B.
• 29% want B as their first choice, but would also approve of A.
• 1% want B as their first choice, and approve neither of A nor of C.
• 42% want C as theri first choice, and approve of neither A nor B.
a)    If all voters could only vote for their first choice, which candidate would win by plurality?
b)    Which candidate wins by an approval vote?

• Imagine that a small company has four share holders who hold 26%, 26%, 25% and 23% of the company's stock. Assume that votes are assigned in proportion to share holding. Also assume that decisions are made by strict majority vote. Explain, why although each individual holds roughly one-fourth of the stock, the individual with 23% holds no effective power in voting.