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PARADOXES OCCUR 1992 ELECTION –Bush and Poirot win popular election 2000 Election –Bush II loses popular vote, wins election They happen every day in the rack/stack method used in DoD
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VOTING PARADOXES
AND HOW TO
DEAL WITH THEM
Hannu NurmiUniversity of
TurkuTurku, Finland
VOTING
• Satisfaction and justice in voting outcomes is important
• Every day, somebody is rackin’ and stackin’
• Voting is a way to reach equitable consensus
PARADOXES OCCUR
• 1992 ELECTION– Bush and Poirot win popular election
• 2000 Election– Bush II loses popular vote, wins
election• They happen every day in the
rack/stack method used in DoD
ASSUMPTIONS
• Equal Weight• One Vote Each• Independence (no gaming)• Transitivity (A < B and B < C implies A < C)
• DEFN: An Alternative is one of the choices• NOTATION: a > b means a is prefered to b
PREFERENCE PROFILE
COUNT
3 4 2 7 5 6
1ST A B C A B C
2ND B C A C A B
3RD C A B B C A
WHO WINS?1ST PLACE VOTES
A 3+7 10B 4+5 9C 2+6 8
LAST PLACE VOTESA 4+6 10B 2+7 9C 3+5 8
TOP TWOA 3+2+7+5 19B 3+4+5+6 18C 4+2+7+7 16
A B CA 12 15B 15 12C 12 15
A B CA 0 1B 1 0C 0 1
TOURNAMENT MATRIX
PAIRWISE COMPARISON MATRIXfor 12 voters, B>A (note: nontransitivity)
CONDORSET WINNERS AND LOSERS
• A < B, 13 vs. 8• A < C, 13 vs 8• B < C, 13 vs. 8• But, A wins
plurality vote!• A is the Condorcet
loser– uniformly despised
1 7 7 6A A B CB C C BC B A A
BORDA (1770)
• give k points to last place• give k + a points for second to last• give k + 2a points for third from last• etc.
• Borda never elects the Condorcet loser• Does Not always elect the Condorcet
winner
SUMMED RANKIs the usual bad?
• One (1) point for first place• Two (2) points for second place• etc.
• Sum the point scores• Select the alternative with the
lowest score
ANALYSIS
• Reverse the ranks• k = 1• a = 1
• Always selects the Condorcet winner if it exists
• May select Condorcet loser if it exists
VOTING PARADOXES
• What follows is a set of situations where the vote fails to reflect consensus. Many of these situations are famous.
NO SHOW PARADOX26% 47% 2% 25%
A B B CB C C AC A A B
• Plurality run-off voting• 1st Round: Eliminate C
– A wins in run-off with 51%• Suppose the 47% no-show
– B is eliminated, C subsequently beats A– the 47% get their second choice, not their 3rd
INCONSISTENCY PARADOX
east east east west west west35% 40% 25% 40% 55% 5%
A B C C B AB C B B C CC A A A A B
• Plurality run-off voting in each district• B wins the East in run-off, wins West
outright• Taken as a whole, C beats B in a run-off
ALABAMA PARADOX OF 1881Hamiltonian Apportionment
TOTAL SEATS 299 300
ALABAMA 7.646 7.671TEXAS 9.64 9.672
ILLINOIS 18.64 18.7
ALABAMA SEATS 8 7
• Seats allocated by integer part, remainder allocated by largest fraction remaining
seatsseat
poppop ii
OSTRAGORSKI’s PARADOX
Arises because the following two produce different winners:
1. BEAUTY CONTEST: Each voter votes for the candidate whose stand is closest to his in a majority of issues.
2. ISSUE CONTEST: For each issue, voters pick candidates. The winner is the one winning the majority of issues.
BEAUTY WINNER
A X X X X
B X Y X X
C Y X X X
D Y Y Y Y
E Y Y Y Y
ISSUE WINNER Y Y X
SIMPSON’s REPRESENTATION PARADOX
• Percent who favor higher in the East for both employed and unemployed
• Total percent in favor larger in the West
EAST WEST EAST WEST EAST WEST
EMPLOYED 400,000 90,000 80,000 15,000 20% 17%
UNEMPLOYED 100,000 80,000 50,000 35,000 50% 44%
total 500,000 170,000 130,000 50,000 26% 29%
POPULATIONFAVOR
INITIATIVE