Voting Rules: What’s Fair?
Q. What is the most fair voting rule? Suppose three proposals to
spend $15:
•Z=Status Quo (currently, we spend $10 on guns and $5 on butter)
•X= Proposal (peaceniks want to spend $5 on guns and $10 on
butter)
•Y=Amendment on Proposal (health nuts want to spend $10 on
margarine)
And 11 voters with preferences as follows:
•Voters 1 - 5 (5): X p Y p Z
•Voters 6 - 9 (4): Y p Z p X
•Voters 10-11 (2): Z p X p Y
Method 1: Plurality Voting (1 person one vote, the choice
with the most votes wins)
Alternative
Votes
X
5
Y
4
Z
2
Voters 1 - 5 (5 voters):
X>Y>Z
Voters 6 - 9 (4 voters):
Y>Z>X
Voters 10-11 (2 voters):
Z>X>Y
Method 2: Borda Count (each alternative receives “score”
based on rank in each voter's ordering -- here, 2 points if
most-preferred, 1 if second, none if third).
Alternative
First-place
Votes
Second-place
Votes
Total
Points
X
5
2
12
Y
4
5
13
Z
2
4
8
Voters 1 - 5 (5 voters):
X>Y>Z
Voters 6 - 9 (4 voters):
Y>Z>X
Voters 10-11 (2 voters):
Z>X>Y
Method 3: Amendment Procedure (sequentially vote on pairs of
alternatives)
Option 1: vote on type of change first, then on status quo versus change.
Status quo wins.
X vs. Y (X wins, 7-4)
X vs. Z (Z wins, 6-5)
Z is ultimate choice
Option 2: vote on proposal first, then allow amendments to outcome.
Amended proposal wins.
Z vs. X (Z wins, 6-5)
Z vs. Y (Y wins, 9-2)
Y is ultimate choice
Option 3: Vote on amendment first, then on final bill (whether proposal or
amended proposal). Unamended proposal wins.
Z vs. Y (Y wins, 9-2)
Y vs. X (X wins, 7-4)
X is ultimate choice
Strategic voting:
Option 2, amendment voting, but voters 10-11 vote for X even
though they prefer Z.
Why: if X wins, it will beat Y on next vote, yielding a result
these players prefer to result of sincere voting, Y.
Option 2 with sincere voting :
Z vs. X (Z wins, 6-5)
Z vs. Y (Y wins, 9-2), Y is ultimate choice
But with strategic voting:
Z vs. X (X wins, 7-4)
X vs. Y (X wins 7-4), X is ultimate choice