If Obama runs to the center and wins he may well stay there, as Bill Clinton pretty much did (they thought they had the perfect stealth lefty in Bill Clinton).
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Clinton ran right and ruled left for the first two years as he give hillary the lead. the disasterous failure of her health care plan helped give the republicans an opening under newt to take over congress for the first time in decades. subsequently clinton signed a lot of republican legislation that he bitterly fought against. later he would claim credit for the same legislation.