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I saw on Wikipedia when I was looking at the 2024 Republican Party primaries that one of the rules is to receive an outright majority of the votes cast by delegates at the national convention.

What happens if no candidate wins a majority of delegates? Does it go to the candidate who receives a plurality, or is there another round of voting?

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Another round of voting is called until a candidate receives an outright majority vote of the delegates.

See the Republican Party's Rules of the Party, adopted August 24, 2020 and amended April 14, 2022:

  1. d) When at the close of a roll call any candidate for nomination for President of the United States or Vice President of the United States has received a majority of the votes entitled to be cast in the convention, the chairman of the convention shall announce the votes for each person receiving delegate votes cast in accord with their respective state party rules and state law. Before the convention adjourns sine die, the chairman of the convention shall declare the candidates nominated by the Republican Party for President of the United States and Vice President of the United States.

    e) If no candidate shall have received such majority, the chairman of the convention shall direct the roll of the states be called again and shall repeat the calling of the roll until a candidate shall have received a majority of the votes entitled to be cast in the convention.

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    Interesting, so they just vote again with the exact same list of candidates (as opposed to just doing a run-off between the top 2).
    – quarague
    Commented Sep 1, 2023 at 12:13
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    @quarague: if you look back in history, that's been the rule for a long time. Sometimes they had 30+ rounds. en.wikipedia.org/wiki/1880_Republican_National_Convention
    – 264 champagne bottles on ice
    Commented Sep 1, 2023 at 12:19
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    What actually happens is that there is horse trading. Often one of the candidates can "drop out" and ask that their delegates vote for another candidate. After that I think it comes down to the delegates individual choice, although with considerable input from party leaders and the candidates.
    – DJClayworth
    Commented Sep 1, 2023 at 14:41
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    "the 1924 Democratic National Convention holds the record as the longest ever, as divisions within the party concerning Prohibition led to 102 ballots between Alfred E. Smith and William G. McAdoo, before the relatively unknown John W. Davis was chosen as a compromise candidate on the 103rd ballot." en.wikipedia.org/wiki/…
    – ohwilleke ♦
    Commented Sep 1, 2023 at 15:11
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    @ohwilleke I was going to mention that one as an example of how the delegates at a national convention are supposed to just keep voting and voting until there's finally a consensus -- until I saw you had already done so.
    – Lorendiac
    Commented Sep 2, 2023 at 11:01
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@cdjb's answer is better. This is supplementary. Read that first.

What cdjb describes has happened in the actual presidential contest. It's unlikely, but can happen. In 1824 Jackson entered with the most popular and electoral votes, but Adam's was elected presedent.

(Electoral College runs on the same principles, but is governed by different rules that basically do the same.)

Broadly, if you have delegates voting for the winner, require an outright majority to win, and have 3+ candidates, it's common to proceed to multi-round voting.

It's seen as important to have an outright majority to win so nobody can win by splitting their opponents. If Literally Hitler got 20% of the vote, but the rest were evenly split across 5 candidates, going to further rounds of voting might show 80% would prefer Bob over Hitler. These further rounds ensure the victor has more than minority support.

Subsequent rounds may have additional rules, often to cut down on the number of options to help consolidate the vote and get it closer to 50%.

(This is very related to runoff voting.)

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    The simplest rule is to always eliminate the candidate with the least votes. Most people would judge this as "fair" and it guarantees that there are never more than N-1 rounds, given N candidates.
    – Lawnmower Man
    Commented Sep 1, 2023 at 19:21
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    @bharring: There are no completely fair voting systems that allow participants to fully express their preferences in a way that will be considered, but approval voting satisfies the "honesty" criterion. If someone likes Bob better than Charlie, but things Alex is better than both, and David is inferior to both, the voter might achieve the best results by voting for both Bob and Charlie, neither Bob nor Charlie, or for Bob but not Charlie, but would never get a better outcome by voting for Charlie but not Bob, than by one of the other options.
    – supercat
    Commented Sep 1, 2023 at 21:17
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    @Gantendo Arrow's theorem is, succinctly put, that the system is either unfair or a (de facto) dictatorship. Yours would obviously be the latter kind.
    – zibadawa timmy
    Commented Sep 2, 2023 at 2:10
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    @Acccumulation Well what is Gandhi expecting when 90% of the voters prefer not-Gandhi?
    – zibadawa timmy
    Commented Sep 2, 2023 at 5:37
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    @LawnmowerMan Read up on the Condorcet criterion. In either two-candidate subset, Hitler vs Ghandi or Stalin vs Ghandi, Ghandi would win 55% to 45%. 100% of voters agree that Ghandi is at least a decent compromise candidate. I would argue that any election system that does not have Ghandi win in this scenario is manifestly unfair.
    – Douglas
    Commented Sep 2, 2023 at 20:29

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