Learning Outcomes
- Apply Arrow’s Impossibility Theorem
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
It is not possible for a voting method to satisfy all four fairness criteria.
Recall
A voting method is said to satisfy a fairness criterion if every possible election using the method will be fair according to that criterion. For a method to violate a fairness criterion,
there needs to be just a single election using the method whose outcome is unfair with respect to this criterion.
Here is a table summarizing the voting methods we have discussed and which fairness criteria they do or do not meet.
| Condorcet Criterion | Majority Criterion | Monotonicity Criterion | IIA Criterion | |
| Plurality | can violate | satisfy | satisfy | can violate |
| Single Run-off | can violate | satisfy | can violate | can violate |
| IRV | can violate | satisfy | can violate | can violate |
| Borda Count | can violate | can violate | satisfy | can violate |
| Pairwise Comparisons | satisfy | satisfy | satisfy | can violate |
Example
Here is a very simple example of how difficult voting can be. Consider the election described by the following preference schedule.
| 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, Pairwise Comparisons, 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.
This is the end of the Voting Theory Reading for the Final Project.
