who wins?
same votes. three methods. three different winners.
your class of 12 is picking the party food. here are the ballots.
whoever gets the most first-choice votes wins.
pizza wins with just 5 out of 12 first-choice votes. 7 students wanted something else.
eliminate the option with fewest first-choice votes. give those voters' next choice their vote. repeat until someone has a majority.
🌮 tacos eliminated (fewest first-choice votes). those 3 voters had sushi second.
same ballots. completely different winner.
give points by ranking: 2 for first, 1 for second, 0 for third. most points wins.
three methods. three winners. zero ballots changed.
same ballots. three winners.
every method seems fair. every method gives a different answer.
in 1951, kenneth arrow proved this is unavoidable. no voting system with three or more options can satisfy all of these at once:
- if every voter prefers A over B, the group should too
- the A vs B ranking should not depend on how voters feel about C
- no single voter should be able to dictate the outcome
this is arrow's impossibility theorem. every voting system is a tradeoff.
generate random elections and see how often the three methods agree