“Irrelevant” Alternatives and the Presidential Election
The system by which the United States selects its presidents does not possess the property of independence of irrelevant alternatives, for several reasons, even assuming single-peaked preferences.
Consider one case, with 3 candidates– Left, Right, and Ultra-Left: 40% of Americans have the following single-peaked preference ordering: Left > Ultra-Left > Right; 45% of Americans: Right > Left > Ultra-Left; 15% of Americans: Ultra-Left > Left > Right.
Since you only get one vote, if everyone votes truthfully, the Right candidate wins with 45% of the vote. If the Ultra-Left candidate dropped, however, the Left would win with 55% of the vote.
It’s not apparent that this effect swung the most recent election. However, another violation of the independence of irrelevant alternatives, caused by the party system, may have.
Consider the following scenario, with 3 candidates across 2 parties: Left (party A), Right (party B), Center-Right (party B). Just over half of Americans are in party A, and 2/3 of whom all prefer Left > Center-Right > Right and 1/3 of whom prefer Center-Right > Left > Right. The other portion (just under one half) is in Party B, 2/3 of which prefer Right > Center-Right > Left, but 1/3 of which prefers Center-Right > Right > Left. Party B’s primary sends Right to the general election, which they lose to Left. Had Right never ran, Center-Right would have won the primary and then the general election.
We might actually have reason to suspect that such a thing occurred this past election. According to RealClearPolitics, Kasich was beating Clinton by 7.3 points in general polls. Likewise, Sanders was beating Trump by 10 points in general polls (as opposed to Clinton’s admittedly over-estimated 3.2) and beating Kasich by 3.3, such that we might imagine the country’s actual preference ordering to be Sanders > Kasich > Trump > Clinton; had Clinton never run, perhaps we’d have seen a massively different outcome.
http://www.realclearpolitics.com/epolls/2016/president/us/general_election_kasich_vs_clinton-5162.html
http://www.realclearpolitics.com/epolls/2016/president/us/general_election_trump_vs_sanders-5565.html
http://www.realclearpolitics.com/epolls/2016/president/us/general_election_trump_vs_clinton-5491.html
http://www.realclearpolitics.com/epolls/2016/president/us/general_election_kasich_vs_sanders-5817.html