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