Turnout thresholds violate monotonicity

I thought I'd share some number-crunching that I've just prepared for a lecture. Since this year's Democratic Theory students were disrupted by industrial action, even before lockdown, I figured that I'd use the strike ballots as a topical example.

By law, a strike ballot needs 50% turnout. Here are the results from October 2019 ballots. (If the link doesn't work, there's a redirect: tinyurl.com/ucu2019.)

Let's focus on the LSE:

	Turnout	Yes	No
Actual	241/602 (40%)	206 (86%)	33 (14%)
Possible	361/602 (60%)	206 (57%)	153 (43%)

The actual result (top row) meant that the motion failed, owing to insufficient turnout, despite having 86% support from those who voted.

However, this means that had an extra 120 people voted against the motion - the possibility illustrated on the bottom line - it would have passed. (Actually, around 60 would have been enough to hit the 50% turnout threshold - and it could have been as many as 172 without changing the majority outcome.)

It's perverse that more people voting against something makes it happen. However, this isn't unique to strike ballots. A similar situation arose in a 2018 referendum in (the Former Yugoslav Republic of) Macedonia (BBC; Wikipedia).