The mathematics of voting The paradoxes of democracy

Voting theory Voting theory is the mathematical treatment of the process by which democratic societies or groups resolve conflicting opinions of the members of the group into a single choice of the group. A vote is an expression of a voter's preference about the outcome of an election.

Arrow’s impossibility theorem In 1952, the mathematical economist Kenneth Arrow discovered a remarkable fact: For elections involving three or more candidates, there is no consistently fair democratic method for choosing a winner. In fact, Arrow proved that a method for determining election results that is democratic and always fair is a mathematical impossibility.

Fundamental terms Ballot A ballot is a record of how a voter voted. Majority A candidate with a majority of the votes has more than half of the votes. Plurality A candidate with a plurality of the votes has more votes than any other candidate.

Fundamental terms Preference Ballot It is a ballot in which a voter is asked to rank all the candidates in order of preference Voters Place AdamBettyColinDaveEdithFrank 1 st ADAABD 2 nd CCCCAC 3 rd BBBBCB 4 th DADDDA

Fundamental terms Preference Schedule A preference schedule is a table which summarizes the results of all the individual preference ballots for an election. Amount of voters Place st ADB 2 nd CCA 3 rd BBC 4 th DAD

The Plurality Method For the Plurality Method, the candidate with the most first place votes wins. The winner does not have to receive a majority of the first place votes!

Example The mayor of Smallville is being chosen in an election. There are four candidates: Paul, Rita, Sarah and Tim. 500 registered voters cast their preference ballots. The results are summarized in the preference schedule below:

Example Amount of voters Place st PTTS 2 nd RRRR 3 rd SSPP 4 th TPST

The Plurality Method Who is the winner by the Plurality Method? First place votes: Tim: 220 Sarah: 150 Paul: 130 Rita: 0

The Plurality-with-elimination Method Plurality with Elimination is carried out in rounds. After each round of voting the candidate with the fewest first place votes is eliminated. When only two candidates remain in a round, the candidate with the most votes wins the election. After each round cross out name of eliminated candidate from the preference schedule and recount the first-place votes!

The Plurality-with-elimination Method Who is the winner by the Plurality-with- elimination Method? Since Tim was eliminated in third round, the only remaining candidate is Sarah.

The Borda Count Method For this method, each candidate gets 1 point for each last place vote received, 2 points for each next-to-last point vote etc. to N points for each first place vote. The candidate with the largest point total wins the election. N(…) + N-1(…)+ … + 2(…) + 1(…) =

The Borda Count Method Who wins the election using the Borda Count Method? Rita: 1500 Sarah: 1200 Tim: 1160 Paul: 1140

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