From the 12th Amendment of the Constitution (emphasis mine)

The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.

It is relatively well known that if no Presidential candidate receives a majority of electoral votes, the House of Representatives elects the President.

I have always looked at this as a barrier to a multi-party Presidential election, because with 3 or more strong candidates, the odds of receiving a majority of electoral votes drops dramatically. Even in an election with a clear "winner" (say 45%-30%-25%), the election still goes to the House.

If the entire House was elected every four years, this would not be an issue, as the the party of the "winner" in the hypothetical election above would likely also win the greatest number of seats in the House and would be elected there. But because this isn't the case (and possibly due to gerrymandering), there is no guarantee that the "winner" will make any significant gains and be elected in the House, and the electorate and/or strong third party candidates themselves choose not to take that chance.

Is this a common idea regarding US Presidential elections, or is there a logical mistake I'm making somewhere but just don't see it?

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If the entire House was elected every four years,

The entire House is elected every four years. In fact, the entire House is up for election every two years. It's in the Senate that only a third is up in any particular election.

Even in an election with a clear "winner" (say 45%-30%-25%), the election still goes to the House.

First, I wouldn't describe that as an election with a clear winner. 55% of the vote is against the clear leader. I also find this confusing in that we usually don't talk about percentages of electoral votes. But the majority requirement only applies to electoral votes. Note that in three presidential elections in a row from 1992 to 2000, no candidate won a majority of the popular vote. The electoral majority requirement was met each time.

Second, I actually think that this makes things easier for third parties. Consider the current election. If you are a Never Clinton Republican, you can vote for Gary Johnson more comfortably. If Johnson wins enough states to keep either candidate from getting a majority, then the House can still prevent Clinton from winning. It reduces the spoiler effect of the House rather than increasing it.

The last time that the election actually went to the House, it wasn't the leader who won. He was regarded as a dangerous demagogue and the other candidates banded together to defeat him. So there was no disadvantage from the third and fourth candidates running. They still were able to prevent the leading vote getter from winning with a minority of the vote.

The larger problem with third party candidates is that they prevent a preferred but inferior candidate from winning in a state. For example, if all the Ralph Nader votes in Florida in 2000 had gone to Al Gore instead, Gore would have won Florida and the election. Even though Nader had no electoral votes. This is why many propose switching to a ranked method like Instant-Runoff voting (IRV) or one of the Condorcet-compliant methods. They better capture secondary preferences.

The most significant obstacle to viability of a third party candidate comes principally from the "winner take all" system for selection of the electors that has been chosen in almost every state. This is not inherently part of the electoral college as set forth in the Constitution, but is not forbidden either. Since a third party candidate can gain no electoral votes without a plurality in any given state, third party movements have had little success in building a continuous voice and establishing a long term presence. They have tended to have (and bear in mind the above obstacles also present a barrier to statistically valid data here) a relatively small following or to be single candidate oriented (like Nader & Perot).

In addition to the winner-take-all per-state electoral college rules (that @Brythan mentions), the per-state "get on the ballot" rules discourage the creation of a viable third party. In each state, the two major parties have a clear path to get their candidate on the ballot. However, in each state, the rules are different. Those rules are a formidable barrier to getting a "third party" candidate on the ballot in enough states to be viable.

There's also a chicken and egg problem. In order to create a third party, you really need both congressional representation and a viable Presidential ticket. Without a good presidential candidate, it's hard to rally the troops around a new party. Without congressional representation, it's very hard for a viable third party candidate to win (because if the candidate is truly viable, then the election will likely be settled in the house (and the VP election in the senate, where it takes three election cycles to get a full compliment of members).

There were big changes in 1860, but that was a special year - it was when the country nearly broke apart permanently and when the festering issue of slavery really came to a head.

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