11B                        Theory of Voting

• Criterion 1: If a candidate gets a majority  (>50%) of the first place votes, he/she should be winner.
• Criterion 2 (Condorcet Criterion): winning candidate should also be winner of pairwise comparisons.
• Criterion 3 (monotonicity): 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.
• Criterion 4 (irrelevant alternatives): Suppose X is the winner, if one or more losing candidates drop from the race, X should still be the winner.

 

 

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

• Suppose the winner is decided by plurality. Analyze whether this choice satisfies the four fairness criteria.


Plurality: A
Crterion 1: does not apply
Criterion 2: Condorcet winner = E not the same as A  NO
 

	A	B	C	D	E
A		B	C	D	E
B			C	D	E
C				D	E
D					E
E

Criterion 3: if new election is held and A is moved higher, A is still the winner.    YES
Criterion 4: if C drops out, then B would win 22 first place votes and supercede A.   NO
 

• Suppose the winner is decided by a single runoff. Analyze whether this choice satisfies the four fairness criteria.


Single runoff:  B
Criterion 1: does not apply.
Criterion 2:   NO
Criterion 3:   YES
Criterion 4:  if A, D, and E drop out , C would win a match between B and C.   NO
 

• Suppose the winner is decided by sequential runoffs. Analyze whether this choice satisfies the four fairness criteria.


Sequential runoffs:  C
Criterion 1: does not apply
Criterion 2:     NO
Criterion 3:     YES
Criterion 4:     if A and D drop out, E would win by sequential runoffs.    NO
 

• Suppose the winner is decided by a Borda count. Analyze whether this choice satisfies the four fairness criteria.


Borda count:  D
Criterion 1: does not apply
Criterion 2:    NO
Criterion 3:    YES
Criterion 4:    NO, if C drops out, E would win a Borda count.
 

• Suppose the winner is decided by pairwise comparison. Analyze whether this choice satisfies the four fairness criteria.

Pairwise comparison: E
Criterion 1: does not apply
Criterion 2:    YES
Criterion 3:    YES
Criterion 4:    YES

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

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.