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Study Guides > Mathematics for the Liberal Arts

Which Method is Fair?

Learning Outcomes

  • Determine the winner of an election using preference ballots
  • Evaluate the fairness of an election using preference ballots
  • Determine the winner of an election using the Instant Runoff method
  • Evaluate the fairness of an Instant Runoff election
  • Determine the winner of an election using a Borda count
  • Evaluate the fairness of an election determined using a Borda count
  • Determine the winner of en election using Copeland's method
  • Evaluate the fairness of an election determined by Copeland's method

So Where’s the Fair Method?

At this point, you’re probably asking why we keep looking at method after method just to point out that they are not fully fair. We must be holding out on the perfect method, right? Unfortunately, no. A mathematical economist, Kenneth Arrow, was able to prove in 1949 that there is no voting method that will satisfy all the fairness criteria we have discussed.

Arrow’s Impossibility Theorem

Arrow’s Impossibility Theorem states, roughly, that it is not possible for a voting method to satisfy every fairness criteria that we’ve discussed. To see a very simple example of how difficult voting can be, consider the election below:
5 5 5
1st choice A C B
2nd choice B A C
3rd choice C B A
Notice that in this election: 10 people prefer A to B 10 people prefer B to C 10 people prefer C to A No matter whom we choose as the winner, 2/3 of voters would prefer someone else! This scenario is dubbed Condorcet’s Voting Paradox, and demonstrates how voting preferences are not transitive (just because A is preferred over B, and B over C, does not mean A is preferred over C). In this election, there is no fair resolution. It is because of this impossibility of a totally fair method that Plurality, IRV, Borda Count, Copeland’s Method, and dozens of variants are all still used. Usually the decision of which method to use is based on what seems most fair for the situation in which it is being applied.

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