That would seem totally in character for Sarah Palin to me (and even more so for Ann Coulter), but then I'm French, so my mental model of Palin may be a bit off.
I don't see it as low class, and I consider "solving" disagreements by betting better than appeals to authority or agreeing to disagree. However it would seem a bit unseemly for someone already in a position of authority (like Thatcher or Merkel), I'm not totally sure why ... maybe because we want to know what they do, not what they think and say, and once they're in power they shouldn't have anything to prove any more. Offering to bet is a way to prove your thinking is correct, which is important for pundits and candidates, but not for those in power.
For those who don't follow politics, Mitt Romney offered to bet Rick Perry $10,000 that Perry had misquoted Romney. (video)
Most political commenters see the move as a gaffe. They claim the bet made Romney look out of touch, because it reminded voters that Romney is rich enough to afford $10,000.
As a believer in prediction markets, I am disappointed in the public's reaction. Romney made a bold move by making his beliefs pay rent. Critics point out that $10,000 is "chump change" for Romney, but Romney still but himself at risk. If he had lost the bet, Perry could have made a production about cashing a $10,000 check from a disgraced Romney. Besides, if money were the issue, Perry could have countered with a non-monetary bet. "Loser has to attend the next debate in a clown suit" or something.
If politicians had to face real consequences every time they made a false statement, they would have a larger incentive to tell the truth. It's a shame Romney's bet probably won't catch on.
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This post is not an endorsement of Mitt Romney or his politics. All I am endorsing is political betting.