Why centrist voters elect extreme politicians

Andrew Gelman notes that, on a one-dimensional scale of ideological positions,

elected representatives are generally more extreme than voters.

Moreover, geographic polarization of legislators is much more pronounced than that of the voters:

I am not sure what wisdom we are supposed to gain from those facts, but I see some serious problems with the analysis, even if we grant that measuring ideology on a one-dimensional scale is itself unproblematic.
How meaningful is a comparison between voters' survey responses and legislators' votes? I am not asserting a "words vs. deeds" distinction here; voting is a form of speech, especially votes are publicly known, as legislators' votes are. But legislators are actually held responsible for their votes and expected to be consistent. The public, regardless of ideology, tends to like people with strong principles and dislike "flip-floppers". As much as we sometimes think of them as stupid or ignorant, politicians are, on average, unquestionably better informed and have thought more thoroughly about issues than the general public. We should expect their views to be more consistent than voters' views.

Now, there are rarely consistent "moderate" or "centrist" positions on most issues. Political moderates usually hold "left" positions on some issues and "right" positions on others. In George Lakoff's words, they are biconceptuals:

The pressure to dissemble comes from certain commonplace myths about swing voters and the "center." So for starters, let's put to rest the notion of the political or ideological "center"---it doesn't exist. Instead, what we have are biconceptuals---of many kinds.

A more cynical view would be that the "centrists" are inconsistent, confused, or hypocritical. While it would probably be considered unfair to characterize an ordinary private citizen that way, most politicians would very quickly (and perhaps rightly) earn those labels if their voting patterns resembled the survey responses of "centrist" voters.

The next question is why a centrist electorate would elect a polarized legislature. An obvious explanation is the two-party system and electoral districts drawn to put most Congressional seats safely under control of one or the other party. I would agree that those features of our political system increase the polarization of Congress, but the graphs strongly suggest that such an explanation is woefully incomplete.

First, the Senate seems to be no less polarized than the House, and geographically even more so, which belies the safe districts part of the argument. Second, there seems to be very little difference between voters in "battleground states" and "Democratic states", but a vastly different distribution of elected officials, particularly senators. As the states are grouped by the results of Presidential elections, this observation is by no means tautological.

The graphs tell us that very similar electorates consistently elect ideologically very different politicians. A likely explanation is that the survey responses do not measure overall ideological positions accurately. Perhaps voters put significantly different weights on different issues, which makes them prefer different politicians despite having similar (unweighted) average survey responses? That's a conjecture, not a conclusion; the point is that the analysis presented so far is not satisfying and it generates more questions than it answers.