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Section5.5Independence of Irrelevant Alternatives and Arrow's Theorem

SubsectionThe Independence of Irrelevant Alternatives Criterion (IIA)

The Independence of Irrelevant Alternatives Criterion (IIA)
  • The Independence of Irrelevant Alternatives (IIA) criterion states that election results should not change if a losing candidate is left out.
  • To check whether IIA is violated, we must remove a losing candidate, keeping the voters' preferences the same otherwise, and then find the winner of the new election. If the winner of the new election is different than the old election, then IIA was violated.

Let's use this example to show how all of our voting methods violate IIA.

Example5.31
Number of voters
Rankings 6 5 4 2
1st choice A D C B
2nd choice B B D C
3rd choice D C B D
4th choice C A A A
Table5.32

SubsectionArrow's Impossibility Theorem

Arrow's Impossibility Theorem

Any voting method that uses voters' ranked choices to decide an election with a single winner will violate at least one of the four criteria we've discussed: majority, monotonicity, Condorcet, or IIA.

Majority Criterion Monotonicity Criterion Condorcet Criterion IIA Criterion
Plurality Method Satisfies Satisfies Violates Violates
Borda Count Method Violates Satisfies Violates Violates
Instant Runoff Voting (IRV) Method Satisfies Violates Violates Violates
Pairwise Comparison Method Satisfies Satisfies Satisfies Violates
Table5.33Fairness criteria satisfied by each voting method

This map is based on the one found here. It has been modified by combining similar systems into common categories and making the colors easier to distinguish.

Figure5.34Electoral systems around the world

The map in Figure5.34 shows the systems used to elect the lower house of each country's legislature. You can see a larger version if you right-click and select view image. Note that the Borda count is included in the other systems category (green), but is used in only one of those countries: the Pacific island nation of Nauru. It is also used, however, to elect two seats in the parliament of Slovenia that are reserved for the Hungarian and Italian ethnic minorities.