"Anyway, there is evidence that they work significantly better than chance on untrained people that believe they work. In other words, most of the time for most people."
There is also a lot of evidence that they don't. Or rather: it may be "significantly more than chance", but not enough more to be really useful.
Quote from the first sentence of that first link:
"Most psychologists agree that there is little evidence that polygraph tests can accurately detect lies."
And from the second:
"For federal agencies, the polygraph is a way to get around discrimination laws. There is virtually no appeal you can make if you are failed by a federal polygrapher. The polygraph is a license to abuse power."
And from the conclusion of the third:
"The instrument cannot itself detect deception... false positive rate (innocent persons found deceptive) ranged from 0 to 75 percent and averaged 19.1 percent;"
An average of over 19% false positive rate (government's own figures), and as high as 75%, means the polygraph is effectively useless as a lie detector for any serious purposes. That's a HUGE false positive rate. It simply isn't a basis for punishing someone when there is an almost 20% chance on average that the results are false. And that's just false positives... there are false negatives too.
I repeat: the government knows this, and uses it more as an instrument of intimidation, in order to try to wring confessions out of people, than anything else. Many ex-government-polygraphers -- and subjects of polygraph exams, for that matter -- have told the same story.