Definition 1.6.1 Arrow’s Impossibility Theorem
Arrow’s Impossibility Theoremstates, roughly, that it is not possible for a voting method to satisfy every fairness criteria that we’ve discussed.
Section 1.6 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.
To see a very simple example of how difficult voting can be, consider the election below:
| 5 | 5 | 5 | |
| 1 st choice | A | C | B |
| 2 nd choice | B | A | C |
| 3 rd 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.
In the table below we summarize the various voting methods we have discussed along with the fairness criteria they always satisfy. Note that no methods are guaranteed to satisfy the Irrelevant Alternatives criterion.
| Head-to-head | Majority | Monotonicity | Irrelevant Alternatives | |
| criterion | criterion | criterion | criterion | |
| Plurality | Yes | Yes | ||
| Borda Count | Yes | |||
| Instant Runoff | Yes | |||
| Pairwise Comparison | Yes | Yes | Yes |