6

Consider an election in which one voter's preference order was Alice > Bob > Carol. If she were to change her preference order to Carol > Alice > Bob, there are 9 possibilities as to what could happen:

  1. Alice wins in both cases; her change had no effect
  2. Bob wins in both cases; her change had no effect
  3. Carol wins in both cases; her change had no effect
  4. Alice won before but Bob wins now
  5. Alice won before but Carol wins now
  6. Bob won before but Alice wins now
  7. Bob won before but Carol wins now
  8. Carol won before but Alice wins now
  9. Carol won before but Bob wins now

I know that the monotonicity criterion requires that cases 8 and 9 above can't happen. I can't figure out which criterion would also require that cases 4 and 6 can't happen either. At first it sounded similar to IIA, but that looks like it's just about adding and removing candidates to and from the entire race, not what happens when individual voters update their preferences. Does this criterion have a common name? If so, what is it?

0

1 Answer 1

3

The wikipedia article you link to has a heading called "The many forms of IIA" in which it has the formulation:

The social preferences between alternatives x and y depend only on the individual preferences between x and y.

4 and 6 violate this wording of IIA: whether Alice or Bob wins depends on the preferences between Alice and Carol.

Even the wording "adding or removing a candidate shouldn't affect the ranking of the other candidates" bars 4 and 6. Consider the following four cases:

(I) Carol is in the race, and the voter's preference is Alice > Bob > Carol.
(II) Carol in in the race, and the voter's preference is Carol > Alice > Bob.
(III) Carol is not in the race, and the voter's preference is Alice > Bob > Carol.
(IV) Carol is not in the race, and the voter's preference is Carol > Alice > Bob.

If we combine your case 4 with my cases, then we see that Alice wins in (I) and Bob wins in (II). So who wins in (III) and (IV)?

Suppose Alice wins in (IV). But if we start with (IV) and add Carol to the race, then we go to (II), and Bob wins. So adding the irrelevant alternative Carol changed who won. This shows that if Alice wins in (IV), IIA is violated. A similar argument shows that if Bob wins in (III), that also means IIA is being violated. But whatever election rules we have, they can't distinguish between (III) and (IV), so those two cases should be the same. That is, either Alice wins in both (III) and (IV) (in which case IIA is violated because Alice wins in (IV)) or Bob wins in both (III) and (IV) (in which case IIA is violated because Bob wins in (III)). Thus, IIA prohibits cases 4 and 6.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .