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Math of Voting

Arrow's Impossibility Theorem


Ferris knows a store sells chocolate and vanilla ice cream. He decides before he goes to the store he wants chocolate ice cream. When he arrives he finds out they also have strawberry as a choice. Consequently, he changes his choice to vanilla. This is an illustration of violation of which principle?

  • Unanimity: If everyone prefers \(A\) to \(B\), \(A\) should win.

  • No Dictators: There should not be anyone whose individual preferences always determine who wins.

  • Independence of Irrelevant Alternatives (IIA): Adding extra options should not make existing relations change. That is, if \(A \geq B\), adding option \(C\) should not make \(B \geq A\).

A certain company has measures which come up for a yes or no vote, and the members can each cast a certain number of votes based on the amount of stock they have. If a majority of votes are cast for Yes, the measure passes, otherwise it fails. Suppose the CEO has 9 votes, and each of ten employees has 1 vote. Is this system a dictatorship? (That is, will the CEO's preference always determine who wins?)

Three sets of voters pick between three candidates:

30% prefer A to B to C

40% prefer B to C to A

30% prefer C to A to B

Suppose only the first choice counts, so B wins the election.

However, if C was removed, A would win the election 60% to 40%.

This is an example of violation of which principle?

In the Borda count method of voting, each voters assigns 1 point to their least favorite choice, 2 to their next favorite, 3 to their next favorite, and so on. All points are accumulated and the choice are ranked from highest score to lowest score.

There are two parties, the Square party and Circle party, and 50% of voters always prefer Square candidates and 50% always prefer Circle candidates.

There are two candidates: A from the Square party and Z from the Circle party. An election held with the Borda count would be a tie.

Suppose a new candidate B from the Square party enters the fray. The candidate is less popular than A with everyone; Square party members still would prefer an Square candidate to anyone else. What would happen to the vote?

Voters in a district get to choose between candidates A, B, C, and D. Every voter would rather C win than A. However, in the end it is a tie between the two.

This is an example of violation of which principle?


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