# Independence of covered alternatives

Independence of covered alternatives is a voting criterion that states that if an alternative X wins an election, and a new alternative Y is added, X will sitll win the election if Y is not in the Uncovered set. Independence of covered alternatives implies ISDA (and hence Smith and the Condorcet criterion) because the uncovered set is a subset of the Smith set.

## Criterion compliances and failures

### Monotonicity

ICA is incompatible with monotonicity for any voting method where a perfect tie can always be broken in favor of a candidate W by adding one or more identical ballots that rank W first. The proof by Forest Simmons is as follows:

The election

```1: B>C>A
1: C>A>B
1: A>B>C
```

is a perfect tie. By the tiebreaking property, the election (call it election 1)

```2: B>C>A
1: C>A>B
1: A>B>C
```

must give B a greater chance of winning than C. Hence, in this election (election 2)

```2: D>B>C
1: B>C>D
1: C>D>B```

B must have a lower chance of winning than in election 1, because all we've done is relabel {A, B, C} in election 1 to {C, D, B} to get election 2.

Now consider election 3:

```2: D>B>C>A
1: C>A>D>B
1: A>B>C>D
```

The uncovered set is {A, B, C}, and eliminating D (which is covered) gives us election 1. So by independence of covered alternatives, D can't win and the other candidates' winning probabilities must be those of election 1. Now raise B on the last ballot to get

```2: D>B>C>A
1: C>A>D>B
1: B>A>C>D
```

Now the uncovered set is {B, C, D}, and eliminating A gives us election 2. By independence of covered alternatives, A can't win and the other winning probabilities must be the same as in election 2. But B's winning probability in election 2 is lower than in election 1. Hence raising B harmed B, which contradicts monotonicity.