The Mathematics of Voting: Paradox, deception, and chaos Marcus Pivato Department of Mathematics, Trent University http://xaravve.trentu.ca/voting.pdf May 8, 2008 Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. ◮ ...Pick one dessert for the dinner party: Apple cobbler, Banana cream pie, Chocolate cake, or Dulce de leche. Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. ◮ ...Pick one dessert for the dinner party: Apple cobbler, Banana cream pie, Chocolate cake, or Dulce de leche. ◮ ...An federal election with four candidates: LiberAl, Bloc Quebecois, Conservative, or New Democratic. Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. ◮ ...Pick one dessert for the dinner party: Apple cobbler, Banana cream pie, Chocolate cake, or Dulce de leche. ◮ ...An federal election with four candidates: LiberAl, Bloc Quebecois, Conservative, or New Democratic. Electorate Profile Preferences # We can describe people’s preferences with a table: A≻B ≻C ≻D 10 A≻C ≻D ≻B 9 A≻D ≻B ≻C 11 B ≻C ≻D ≻A 22 C ≻D ≻B ≻A 23 D ≻B ≻C ≻A 25 Total 100 Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. ◮ ...Pick one dessert for the dinner party: Apple cobbler, Banana cream pie, Chocolate cake, or Dulce de leche. ◮ ...An federal election with four candidates: LiberAl, Bloc Quebecois, Conservative, or New Democratic. Electorate Profile Preferences # We can describe people’s preferences with a table: A≻B ≻C ≻D 10 e.g. this means that 10% of the people prefer A A≻C ≻D ≻B 9 to B, prefer B to C, and prefer C to D. A≻D ≻B ≻C 11 B ≻C ≻D ≻A 22 C ≻D ≻B ≻A 23 D ≻B ≻C ≻A 25 Total 100 Elections (2/84) Suppose people must choose between four alternatives A, B, C, or D. e.g.... ◮ ...Olympic judges must give the gold medal to Argentina, Brazil, Canada, or Denmark. ◮ ...A committee must award a prize to one of four contestants Ariadne, Brynn, Chloe, or Desdemona. ◮ ...Pick one dessert for the dinner party: Apple cobbler, Banana cream pie, Chocolate cake, or Dulce de leche. ◮ ...An federal election with four candidates: LiberAl, Bloc Quebecois, Conservative, or New Democratic. Electorate Profile Preferences # We can describe people’s preferences with a table: A≻B ≻C ≻D 10 e.g. this means that 10% of the people prefer A A≻C ≻D ≻B 9 to B, prefer B to C, and prefer C to D. A≻D ≻B ≻C 11 There is no unanimous favourite. B ≻C ≻D ≻A 22 ≻ ≻ ≻ We must have a vote... C D B A 23 D ≻B ≻C ≻A 25 Total 100 An election gone wrong.... (3/84) Electorate Profile Preferences # A≻B ≻C ≻D 10 A≻C ≻D ≻B 9 A≻D ≻B ≻C 11 B ≻C ≻D ≻A 22 C ≻D ≻B ≻A 23 D ≻B ≻C ≻A 25 Total 100 Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) An election gone wrong.... (3/84) Plurality Vote Preferences # A B C D A≻B ≻C ≻D 10 10 A≻C ≻D ≻B 9 9 A≻D ≻B ≻C 11 11 B ≻C ≻D ≻A 22 22 C ≻D ≻B ≻A 23 23 D ≻B ≻C ≻A 25 25 Total 100 30 22 23 25 Verdict: A wins. Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. An election gone wrong.... (3/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. But most voters prefer any other candidate over A: An election gone wrong.... (3/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. But most voters prefer any other candidate over A: 70% prefer B to A...... An election gone wrong.... (3/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. But most voters prefer any other candidate over A: 70% prefer B to A. 70% prefer C to A...... An election gone wrong.... (3/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. But most voters prefer any other candidate over A: 70% prefer B to A. 70% prefer C to A. 70% prefer D to A. An election gone wrong.... (3/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Consider an election with four candidates A, B, C, and D. (e.g. 10% of the voters prefer A to B, prefer B to C, and prefer C to D.) Clearly A wins the election, with 30% of the vote. But most voters prefer any other candidate over A: 70% prefer B to A. 70% prefer C to A. 70% prefer D to A. How did A win? Splitting the opposition (4/84) A versus B, C, and D Preferences # A≻B B ≻A A≻C C ≻A A≻D D ≻A A≻B ≻C ≻D 10 10 10 10 A≻C ≻D ≻B 9 9 9 9 A≻D ≻B ≻C 11 11 11 11 B ≻C ≻D ≻A 22 22 22 22 C ≻D ≻B ≻A 23 23 23 23 D ≻B ≻C ≻A 25 25 25 25 Total 100 30 70 30 70 30 70 Verdict: B ≻A C ≻A D ≻A Idea: More than 50% of voters despise A, but the ‘anti-A’ vote is ‘split’ between candidates B, C, and D, so A still wins.