But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively.
No. You're missing a huge component: values. Political choices are typically between conflicting sets of values or between competing interests of social groups.
Debate rages endlessly between pure capitalists and those who want some socialism thrown in.
What exactly is a "pure capitalist"? This is a strawman frequently erected, but there seems to be a major absence of real people with views that can be described as such.
when we can't take it any more.
Who's "we"? Society is heterogeneous. Some want redistribution right now, some would never want it.
If the average government official is smarter than the average member of the populace, it's potentially a win to have the government make decisions for population members.
Again, no for the same reason. Values are hugely important and you're ignoring them. Besides, government officials have their own incentives which do not necessarily reflect the well-being of the population. And, I hope you have heard that power corrupts..?
Politics as gymnastics for rationalists. No one one Less Wrong is quite sure why politics is a taboo topic or how things got to be that way. What we do think we know is that politics is a great way to bring out the irrationality in people. So why not take advantage of that and use politics as a way to measure rationality? Since politics brings out the most irrationality, it should provide the strongest signal. Since there aren't useful objective metrics of how a political discussion went, we'd have to use subjective judgements by neutral third-party raters, kind of like they do in gymnastics. (In the comment thread for this post, feel free to find fights that you have no dog in, improvise a rationality rubric, and grade participants according to your rubric... let's see how it goes.)
Be a sheep. This is probably the exact opposite of what you were taught in your high school civics class. But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively. It also saves me time, and gives Jane an incentive to put even more time in to carefully considering political issues since she now controls two votes instead of one. Done on a large scale, this could provide an interesting twist on representative democracy. Imagine a directed graph where each node represents a person and an edge is directed from person A to person B if person A is auto-copying person B's votes. There's a government computer system where you can change the person you're auto-copying votes from at any time or override an auto-copied vote with your own personal guess about what's best for society. Other than that, it's direct democracy... all bills are put before all citizens to vote on. Problems this might solve: