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As I understand it if no candidates receives a majority the house of representatives would chose a president among the top three candidates, but what happens in (the unlikely) scenario where there's a tie for third candidate.

The simplest example is with four candidates: if Washington gets 268 votes, Adams 134 votes, Jefferson 68 votes and Madison 68 votes

Also there is a (even more unlikely) scenario when the very top is involved in the tie. If there's a complete tie among more than three top candidates. For example: Washington, Adams, Jefferson and Madison each got 134 votes (and Monroe got 2).

How would it be decided which of the candidates would be eligible for selection by the House of Representatives?

To be clear this question is not the same as "What happens if no candidate reaches 270 electoral votes?" since that only addresses the situation where no candidate gets 270 votes and touches the situation when two candidates gets 269 votes each.

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    @DeplorableNumber9035768 I don't think that is a duplicate since it doesn't answer how it is handled when it's a tie for the third candidate. – skyking Dec 6 '16 at 17:51
  • @DeplorableNumber9035768 I really don't think so at all. I think you totally missed the point. In my contrieved example every elector votes for distinct candidates, that is theres 538 president candidates with one elector vote each. I don't see in that answer how you get the three top candidates among 538 candidates that received equal amount of votes. – skyking Dec 8 '16 at 9:46
  • your 538 candidate scenario was confusing the situation. I have simplified the situation to make it more plausible. The answer to this would likely also answer your 538 unique candidate scenario as well – SoylentGray Dec 8 '16 at 15:17
  • @DeplorableNumber9035768 Still I don't think the other question is a duplicate of this. I don't see how it answers the question raised here. Frankly your example removes some of the essens of my question as it ignored the case where the "top two" candidates are included in the tie (which is IMHO a more interesting case). So I edited the question to include even such scenario. And for plausibility I put no concern in that - I think these scenarios are quite unlikely anyway (but still theoretically possible I think, at least the first two). – skyking Dec 9 '16 at 6:22
  • I get that and have voted to reopen. Feel free to request it be reopened on meta but just needs 1 vote atm. The 538 candidate scenario is impossible with the current laws. The most you could have is 9 candidates with 55 electoral votes and I am not sure the numbers work out to even allow that – SoylentGray Dec 9 '16 at 15:47
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It's unclear. The Twelfth Amendment says:

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...

If the language were "three highest", there would be a case for everyone tied for third being included. But it explicitly says "not exceeding three". I see two possible ways to resolve this, but like with any constitutional crisis, it would very much depend on the specific facts at the time, and the personalities involved.

  1. The House chooses to read it as "the three highest numbers" and allows multiple third place people.
  2. The House says "less is permissible and more is not" and doesn't include any third-place finishers.
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    But solution 2 will not work if for example there are exactly four persons receiving electoral votes, each with the same number of votes exactly. Thus its seems that solution 1 is only possible interpretation. – Joël Dec 6 '16 at 20:06
  • Solution #2 will work because, unless the result actually was a four-way tie, there would be no conflict. Unless they are looking at an actual situation that doesn't work, they are not obligated to safeguard against a scenario that is not happening and will probably never happen. We've never had anyone not reach a majority nor have we had any kind of a tie, so throwing out option #2 for something that will never happen would be a bit on the absurd side. – PoloHoleSet Dec 6 '16 at 21:55
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    @PoloHoleSet "We've never had anyone not reach a majority nor have we had any kind of a tie": Wrong. There was a tie in 1800 (Jefferson and Burr), so that went to the House of Representatives. In 1824, nobody won a majority of electoral votes, so that election also went to the House of Representatives. See history.house.gov/Institution/Origins-Development/…. – phoog Dec 6 '16 at 22:29
  • @phoog - Thank you for that correction. Don't think it undermines my point, at all, but it certainly helps to be factually accurate. – PoloHoleSet Dec 6 '16 at 22:37
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    @PoloHoleSet indeed, the fact that those elections happened does not change the extremely low probability of a four-way tie or any other outcome. Even a previous four-way tie wouldn't. – phoog Dec 6 '16 at 22:46
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In case of a tie for third that is not trivial (all candidates having zero electoral votes), all candidates tied for third are included in the vote in the House. There is no additional tiebreaker. While 538 such candidates is impossible without several changes in the rules (even in Maine and Nebraska, two statewide electors are allocated together), a smaller number of ties like ten is possible.

I suppose that you could have a bunch of faithless electors. But some states require that their electors vote for the candidate on the ballot and won't certify other results. And of course it is unlikely on the face of it. Faithless electors are rare. Only once did more than two electors fail to vote for the selected presidential candidate, and that was because that candidate died.

As a practical matter, the most candidates that have received electoral votes in a single election is five (in 1836). It would be difficult to arrange an election where the top two candidates split all the states with more than three electors and all the states that could choose three electors chose different candidates.

A two-way tie is unlikely. It only happened in 1800 because of the weird way the electors originally cast their votes. The revised system from 1804 on has never had a non-trivial tie. It's not impossible, just unlikely.

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    "all candidates tied for third are included in the vote in the House": do you have anything to support that assertion? It certainly violates the "not exceeding three" clause. – phoog Dec 6 '16 at 22:32

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