538 has a great article on the role of moderate politicians in congress.

The result of Moderate legislators combining the worst of both parties proposal
The Bayh’s , Lieberman’s and Snowe’s of the world are moderates, in that they muddle together proposals of each side to make bastardized policy proposals that meet in the middle (legislative man-bear-pigs). As long as they remain moderates, patching together existing proposals, they will remain an obstacle, with only the limited utility Schaller [the author of 538] outlines.
What we need are Independents who come up with their own policy solutions rather than just trying to make the best out of the crap the Democrats and Republicans come up with. A majority of voters are registered Independents or “centrists” because they can’t get their mix of policy preferences without supporting both parties. If a candidate or party mirrored their preferences, providing them what they like about the left and what they like about the right, that candidate could rise to a new level of electoral success. There is a massive opportunity for a party to redefine itself, or an independent to eschew party, and provide a “third way”, with its own fully realized policy prescriptions, catering to the median voter.
With increasing political polarization, the door is only opening wider in the middle. Who will be the first to walk though it?
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I agree that centrists in Congress are way over-rated and that combining Democratic and Republican proposals is hardly a good thing simply because you are listening to both sides. We often shouldn’t listen to either side.
However, I disagree with your faith in independent voters and the “median voter”. My hunch is that most independents basically want to take a mix of Democratic and Republican proposals, just like you argued against. There is no reason a priori to think that the mix that independent voters (or “median voters”) choose is better than the mix that centrist politicians choose. In fact, centrist politicians likely already cater to the “median voter”, thereby reflecting a lot of their preferences.
Bottom line: Why is catering to the median voter a good thing? Besides, medians vary in time and across the world. The median isn’t good simply by virtue of being the median.
I think it’s fair to say that I was mixing two points together:
1) My ideal representative of the median voter would propose policies that cater directly to that worldview, rather than cobbling together existing proposals. Right now, we get a democrat supported bill and (occasionally) a republican supported bill. Moderates then set about combining them and/or watering them down. Alternatively, if you had Independents that took the policy initiative, I think you would get better legislation because it would be anchored in the middle, and you can play the parties off of each other so that you end with the least deviation possible.
2) On today’s political continuum, median voters are more alike each other than like either extreme (because the extremes are quite far out there). Our current party structure only caters to the politically insane. On a personal level, I can agree much more with the typical median voter than anyone on the way right or way left. I think less government action is preferable to more, but that some is better than none. I think a lot of existing government action was poorly designed/implemented, which has caused a lot unintended consequences and systemic instability. This opinion is most at home in the middle. The assumption I am making is that an Independent catering to the median can implement a lot of my ideas, while still meeting the median voters principals better than either party, resulting in success. It’s a practical consideration, but taking the side of the median voter (the good and the bad) are the best chance I have at seeing my principals become policies.