# Local independence of irrelevant alternatives

A criterion weaker than IIA proposed by H. Peyton Young and A. Levenglick is called **local independence from irrelevant alternatives** (LIIA).^{[1]}
LIIA requires that both of the following conditions always hold:

- If the option that finished in last place is deleted from all the votes, then the order of finish of the remaining options must not change. (The winner must not change.)
- If the winning option is deleted from all the votes, the order of finish of the remaining options must not change. (The option that finished in second place must become the winner.)

An equivalent way to express LIIA is that if a subset of the options are in consecutive positions in the order of finish, then their relative order of finish must not change if all other options are deleted from the votes. For example, if all options except those in 3rd, 4th and 5th place are deleted, the option that finished 3rd must win, the 4th must finish second, and 5th must finish 3rd.

Another equivalent way to express LIIA is that if two options are consecutive in the order of finish, the one that finished higher must win if all options except those two are deleted from the votes.

LIIA is weaker than IIA because satisfaction of IIA implies satisfaction of LIIA, but not vice versa.

Despite being a weaker criterion (i.e. easier to satisfy) than IIA, LIIA is satisfied by very few voting methods. These include Kemeny-Young and ranked pairs, but not Schulze.

Just as with IIA, LIIA compliance for rating methods such as approval voting, range voting, and majority judgment require the assumption that voters rate each alternative individually and independently of knowing any other alternatives, on an absolute scale calibrated prior to the election. Such an assumption would imply that there exist elections where, although a voter has slight differences in preference, that voter would rate them all equal if required to cast a vote.

## Implications

LIIA and majority together imply Condorcet, Smith, and independence of Smith-dominated alternatives.

If a method passes majority, then in an election with only two candidates, the winner must pairwise beat or tie the loser. LIIA thus requires that if X finishes directly ahead of Y in the election method's outcome, then X must pairwise beat or tie Y. Since a Condorcet winner pairwise beats everybody else, it follows that he must finish first.

Furthermore, it's impossible for a candidate outside of the Smith set to finish ahead of one inside, because by definition every candidate in the Smith set pairwise beats every candidate outside of it.

Finally, independence of Smith-dominated alternatives follows from that all the Smith candidates finish ahead of every non-Smith candidate. By LIIA, eliminating all the non-Smith candidates must not change the outcome.

## References

- ↑ Young, H. Peyton (1995).
*Equity: In Theory and Practice*. Princeton University Press. ISBN 0-691-04464-3.