Independence of Irrelevant Alternatives

According to the independence of irrelevant alternatives (IIA) criterion, the winner of an election should not depend on the presence or absence of less prefered candidates. Although there are several variations in the implimentation of IIA, RankChoiceVoting.jl tests for violations of IIA by removing subsets of lossing candidates.

Below, we will illustrate how the instant runoff system can violate the IIA criterion. Consider the preference profile of $n=100$ voters for candidates $C = \{a,b,c\}$. As the code block below shows, canidate $a$ is the winner.

using RankChoiceVoting
system = InstantRunOff()
data = [[:a,:b,:c],[:b,:c,:a],[:b,:a,:c],[:c,:a,:b]]
counts = [37,22,12,29]
rankings = Ranks(counts, data)
evaluate_winner(system, rankings)

1-element Vector{Symbol}:
 :a

However, when $b$ is eliminated, $c$ is now the winner.

system = InstantRunOff()
data = [[:a,:c],[:c,:a]]
counts = [49,51]
rankings = Ranks(counts, data)
evaluate_winner(system, rankings)

1-element Vector{Symbol}:
 :c

Usage

The following examples illustrate various uses of the I criterion.

Satisfies

We can see which systems are guaranteed to satisfy the IIA criterion by calling satisfies with the independence of irrelevant alternatives criterion object.

using RankChoiceVoting
criterion = IIA()
satisfies(criterion)

VotingSystem[]

The results above show that none of the supported rank choice voting systems in the package are guaranteed to satisfy IIA.

Example

In the next example, we will illustrate how to check whether the Bucklin system violates the IIA criterion for a specific preference profile. Consider the following preference profile of $n=100$ voters for $m=3$ candidates.

ranks = [[:a,:b,:c],[:b,:c,:a],[:b,:a,:c],[:c,:a,:b]]
counts = [37,22,12,29]
rankings = Ranks(counts, ranks)

Ranks
┌────────┬──────────────┐
│ Counts │ Ranks        │
├────────┼──────────────┤
│ 37     │ [:a, :b, :c] │
│ 22     │ [:b, :c, :a] │
│ 12     │ [:b, :a, :c] │
│ 29     │ [:c, :a, :b] │
└────────┴──────────────┘

Now, create an Bucklin voting system object:

system = Bucklin()

Bucklin()

In the code block below, we can determine the winner of the election with the function evaluate_winner as follows:

evaluate_winner(system, rankings)

1-element Vector{Symbol}:
 :a

In the next code block, we will use the function satisifies to determine whether the Bucklin system complies with the IIA criterion for the preference profile above.

criterion = IIA()
satisfies(system, criterion, rankings)

false

The results indicate that the Bucklin voting system violates IIA for this preference profile.