independence of irrelevant alternatives

independence of irrelevant alternatives (IIA) is a criterion that can apply to either of the two components of a voting method. what are those two components?

ballot component: how preferences are translated to votes.
tabulation component: how votes are translated to results.

the ballot component fails IIA for every deterministic (non-random) voting method.

the tabulation component fails IIA for every ordinal (ranked) voting method, but not for cardinal (rated) voting methods such as score voting, approval voting, or majority judgment.

ballot component

suppose a voter supports candidates X and Y, but not Z, in an approval voting election. if Z drops out, that voter will instead only approve X.

or suppose a voter who prefers the green tactically ranks the democrat 1st with IRV. if the democrat drops out, he ranks the green 1st.

tabulation component

suppose we have two electoral scenarios under approval voting.

scenario 1:
30 approve XY
20 approve YZ
10 approve ZX

scenario 2:
30 approve XY
20 approve Y
10 approve X

these two scenarios are identical except that Z is absent from the latter. X wins both scenarios. the tabulation component of approval voting satisfies IIA. but with any ranked voting method, the tabulation component can fail IIA. for example:

scenario 1:
35% XY
33% ZY
32% Y

scenario 2:
65% Y
35% X

Y wins the latter scenario but not the former, even tho the only difference in these scenarios is that Z is absent from the latter.

confusion

confusion arises when these two separate components are consolidated into a holistic voting method check. this has even pervaded the wikipedia entry on IIA.

approval voting, range voting, and majority judgment satisfy the IIA criterion if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. this assumption implies that some voters having meaningful preferences in an election with only two alternatives will necessarily cast a vote which has little or no voting power, or necessarily abstain.

the latter sentence (bolding mine) is mired in confusion. this isn’t about tactical voting per se. it’s entirely possible that the two different approval voting scenarios i cited above represent extremely strategic voters—and the reason their approvals look identical is that their actual preferences, and/or the relative candidate viabilities, are different. the problem with the wiki analysis is that it doesn’t treat the balloting and tabulation components separately, and thus it can only imagine two identical approval scenarios occurring due to voters exercising “absolute approval” independent of the available candidates. this is simply a mistake.

conclusion

the balloting component of approval voting (and every possible deterministic voting method) fails IIA. but the tabulation component of approval voting (and other “cardinal” methods—but not ordinal/ranked methods) satisfies IIA.