1. 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

2. Method 1: Plurality Voting (1 person one vote, the choice with the most votes wins) AlternativeVotes X5 Y4 Z2 Voters 1 - 5 (5 voters): X > Y > Z Voters 6 - 9 (4 voters): Y > Z > X Voters 10-11 (2 voters): Z > X > Y

3. 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). AlternativeFirst-placeSecond-placeTotal VotesVotesPoints X5212 Y4513 Z248 Voters 1 - 5 (5 voters): X > Y > Z Voters 6 - 9 (4 voters): Y > Z > X Voters 10-11 (2 voters): Z > X > Y

4. 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

5. 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