You know, that's an interesting question.
I thought about it some... and I don't think there's a difference.
Competition can be fierce in some liberal parts of the country. California, for instance -- there are a lot of very liberal men, who are very competitive. But conservative men can be extremely competitive too.
Their competitiveness can take different flavors. Liberal men are more likely to try to compete by making it seem like you are their intellectual or ethical inferior. Conservative men are more likely to try to compete by making it seem like you are their physical or cultural inferior. Both can be very competitive, though. Although neither necessarily is.
I've spent a lot more time in liberal cities than I have in conservative areas. But just among liberal locales, I can say that places with a higher male:female ratio tend to be a lot more competitive (e.g., 'Man Francisco' and 'Man Diego' feature much more competitive males than New York and Washington, D.C.). I'd suspect it's the same in conservative country too.