In the Florida State Senate, there are some actions that can only be taken with a three-fifths or two-thirds majority. There are 40 members of the state senate, and 3/5 of that would be 24. However, when we require a simple majority, we'd expect it to be not 20, but 21 votes. At the same time, in congressional government, it takes 60 - not 61 - of 100 senators to shut down a filibuster.

Is the case that when we ask for a majority vote we really mean floor(n/2) + 1 and when we ask for a three-fifths or two-thirds vote we really mean ceil(3*n/5) or ceil(2*n/3)?

If that's not the case, what is the pattern?

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    Not entirely sure of the problem, as it seems pretty straight forward and you've answered your own question. A simple majority is greater than 50%, so in that case 21 votes. A two-thirds or three-fifths majority means that 2/3 or 3/5 members must vote for the measure to succeed. Think of it as the target number to reach. – user7754 Jun 21 '16 at 3:34
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    Also note that it is sometimes written as "present and voting" That is if a quorum is present (consider 21 as a quorum in this case). Amajority would then be 11 members actually voting. It could also mean that those abstaining would not be counted in the calculation. The 3/5 could be 13 people. – sabbahillel Jun 21 '16 at 18:32
  • @sabbahillel, that's a good call. Let's say that for the purposes of this question though, we're only talking about a three-fifths majority of all voting members, present or not. – Everyone_Else Jun 21 '16 at 18:35
  • @Someone_Else I added to the front of the post. – sabbahillel Jun 21 '16 at 18:56

Regardless of the size of the majority, the formula is always ceil(n*frac). As Thomo said, "think of it as the target number to reach." Since you can't reach a number with fractions of people, you always round up to the next whole person. This will differ from the floor(n*frac)+1 value when n is perfectly divisible by the denominator.

For example:

  • Simple Majority:
    • frac = 1/2 -> floor(n/2)+1, ceil(n/2)
    • For a 40-person body: floor(40/2)+1 = 21, ceil(40/2) = 20 (potential for ties)
    • For a 55-person body: floor(55/2)+1 = 28, ceil(55/2) = 28
  • 2/3 Supermajority:
    • frac = 2/3 -> floor(2n/3)+1, ceil(2n/3)
    • For a 40-person body: floor(40*2/3)+1 = floor(80/3)+1 = 27, ceil(40*2/3) = ceil(80/3) = 27
    • For a 55-person body: floor(55*2/3)+1 = floor(110/3)+1 = 37, ceil(55*2/3) = ceil(110/3) = 37
  • 3/5 Supermajority:
    • frac = 3/5 -> floor(3n/5)+1, ceil(3n/5)
    • For a 40-person body: floor(40*3/5)+1 = floor(120/5)+1 = 25, ceil(40*3/5) = ceil(120/5) = 24
    • For a 55-person body: floor(55*3/5)+1 = floor(165/5)+1 = 34, ceil(55*3/5) = ceil(165/5) = 33

The only edge case is when a vote requires a simple majority and the body has an even number of people. In this case, an even split (50%) is not a majority (which is defined as more than half), so you need to "round up" to the next whole person in order to constitute an actual majority. ceil() by itself doesn't capture this if you use 1/2 as your fraction, but that's because 1/2 is "tie vote" and not "majority". Instead, you need to use something like 1/2.1, 10/21, 1000000000000/2000000000000.00000000000001, or really anything else which is greater than 1/2.

Also, note that any body where a tie vote is possible will have some mechanism to resolve ties - usually it's either by explicitly giving tie-breaking power to someone (who may not have been involved in the original vote, such as the US Senate and the VP) or by giving someone's vote an inherent +.5 (so the result is always something like 20.5 to 20).

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  • So, in this particular case, the three-fifths majority of 40 would actually require 25 votes? – Everyone_Else Jun 21 '16 at 12:35
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    @Someone_Else - Ummmm.... y'know, now I'm not sure. Let me look into that and get back to you. I was kindof tired when I wrote this, and I might have been off. – Bobson Jun 21 '16 at 12:41
  • Also note that it depends onif the requirement is written as "present and voting" or "of the whole". – sabbahillel Jun 21 '16 at 18:29
  • @Someone_Else - Alright, I revised the answer. I was definitely off before, and it's actually much closer to what sabbahillel has in their answer. – Bobson Jun 22 '16 at 19:52

Note that a simple majority must be 1 more than 50% so that the vote does not equal a tie. You were mistaken as to the meaning in that case. Thus, a simple majority of a 40 vote chamber must be 21 votes. There are some situations in which a odd number of votes is cast so that the difference is one vote, while in an even number of votes, the minimum difference is two votes.

There are some situations in which the by-laws require a minimum difference of two votes. For example, in ancient Israel, the high court (of 71 justices) was not allowed to convict in a capital case unless the majority for conviction was by at least two votes. A vote of 37 - 34 could convict, while a vote of 36 - 35 could not even though it was a majority (of 1) for guilty.

In many cases, it often means present and voting. This means that many votes require a certain minimum casting ballots (a quorum) and the actual number of votes counted will be used to determine the "majority" or the minimum number required in a vote of some special type (such as the 3/5 majority that you mention). This means that if some members abstain, they will not be counted and the majority will be that of the votes actually counted.

Since the percentage is the minimum required for an action, the result will be the fewest votes that are above the actual (fractional) vote counted. This means that the best pattern would be ceil(x) because that takes into account that the the case when the number of votes is exactly divisible. For example .6(40) = 24 so that floor() == ceil(). However, the U.S. Senate 2/3 majority must be 67 votes because the exact fraction of 66.66666 means that floor(200/3) would be less that the minimum amount.

Each body has its own rules as to what is considered a quorum, either a specific fixed number or a percentage of the total number of members (usually a majority).

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  • floor(.6n)+1 and ceil(.6n) are not the same in the case when n is divisible by 5, which is the case that I am concerned about. What number is required for action in this case? – Everyone_Else Jun 21 '16 at 19:07
  • @Someone_Else I added the case to make it clearer – sabbahillel Jun 21 '16 at 19:16
  • So, you're saying that ceil(fraction*n) votes are required, unless fraction=1/2, in which case it's n/2 +1? – Everyone_Else Jun 21 '16 at 19:19
  • Sorry to be pedantic about this, I'm just trying to learn about the corner cases – Everyone_Else Jun 21 '16 at 19:20
  • @Someone_Else yes, because the definition of majority means more than a tie – sabbahillel Jun 21 '16 at 19:31

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