The question isn't as much whether he's wealthy, it's more whether he's seen as flaunting his wealth.
I actually disagree with this. While flaunting wealth might itself be off-putting, it does at least imply that the person is conscious of how much wealthier they are than others. But the criticism of Romney is just the opposite: that it shows him as "out of touch", i.e. unaware of the vast difference in wealth between him and the average voter. The fact that the bet offer is seen as an unconscious slip is in fact integral to why it is supposedly damaging: it revealed that he doesn't think of $10,000 as a lot of money. (Rather like how Marie Antoi...
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.