[EM] Choosing leaders in a legislature

[Andy Jennings]

What would be the ideal way to choose leaders in a legislature?

In the Arizona house and senate, for example, once our legislators are
elected, the majority party caucuses to choose the leadership.  Assuming
the Hotelling model, let's say they end up choosing the median legislator
on their half of the political spectrum.  It follows that the legislature
will be led by someone from about the 25th percentile on the political
spectrum.  Then, if the other party gains control, the leadership will
swing to the 75th percentile on the political spectrum.  Wouldn't it be
much better we could force the leadership to be near the median of elected
legislators?

In the Arizona senate, for example, which has thirty members, the majority
party may have as few as seventeen members and the caucus could be
controlled by nine, or thirty percent of the senate.  I'm not sure, but I
have to imagine that this is common.

Here are some ways to force the leadership near the median (assume a
legislative body of 31 members and just two parties, for now):

1. "The majority party shall nominate 16 senators for president and the
minority party shall choose among them."

2. "The minority party shall nominate 16 senators for president and the
majority party shall choose among them."

I think either of those would tend to choose a president near the median.
 Is one better than the other?  Is it possible to extend to multiple
parties?  Without forcing them to form a majority coalition first?

Say there are three parties.  Should each party, in turn, eliminate ten
people?  Does the largest party or the smallest go first?

Or should the first party eliminate 15 and then the other parties choose
among the remaining 16 via approval voting or something?

Other ideas?

~ Andy
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