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Posted By: Timothy Chow
Date: Saturday, 8 June 2024, at 12:12 p.m.

In Response To: OT problem regarding a failed vote that required super-majority approval (Jim Stutz)

Jim Stutz wrote:

(As always, an incomplete answer is a wrong answer.)

Nice problem (assuming I've interpreted it correctly). Here's a wrong answer. At least three additional yea votes are needed. To see this, suppose that b voters were present and a voted yea. Since the motion failed by two votes, it follows that (a + 1)/b < 2/3, or

3a + 3 < 2b.

Suppose toward a contradiction that two more yea voters had attended, and that that would have been sufficient to push the yeas over 2/3. Then (a + 2)/(b + 2) ≥ 2/3, or

3a + 6 ≥ 2b + 4.

But if we subtract 3 from both sides of this inequality, we get

3a + 3 ≥ 2b + 1,

which contradicts our previous inequality.

Notice that under some circumstances, even three additional yea voters would not be enough; e.g., if a = 0 and b = 2 then four additional yea voters would be needed.

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