[EM] New BR numbers (preliminary)

[Jameson Quinn]

I've written a new Bayesian Regret simulator in python. I'll open-source
the code, put it on github, and explain exactly how it works later. For
now, I only have a few methods done, but I'll have more tomorrow.

I just wanted to share the first numbers I've gotten. This is an average of
1000 runs with 100 voters and 4 candidates. Each voter has a standard
normal utility for each candidate. Voters are created in clusters in a
Polya/Dirchlet/Husse process, with mutations - that is, a new voter has a
correlation of 0.81 on all utilities with a randomly-selected existing
member of the electorate (or, to be more precise, the electorate plus one
wildcard which, when picked, creates a voter in a new cluster uncorrelated
with any of the existing ones). The initial electorate consists of two
"opposite" voters with a correlation of -0.5 on all utilities. For each
system, honest results are given first, then results in which all voters
use first-order strategy based on the honest results. Instead of straight
BR, I give the average social utility compared to the best (that is, the
inverse of BR); thus, higher numbers (less negative) are better. This was
my first sizable run; it took about 40 seconds to complete.

('Score', -0.020281932241917173, -0.073726013277532365)
('MAV', -0.045652626758973726, -0.052529728966832126)

In other words: this shows that a median method can have better BR than
score voting in some cases. This is because in the median method (here,
MAV), the voters strategize to ensure that their grades for the two
frontrunners straddle the honest top two medians, but they do not
necessarily go all the way to an approval-style ballot. In exceptional
cases, this even leads to a better BR, something that almost never happens
with score.

Note: Adding more candidates makes honest Score look better, because the
variability of the normalization goes down. So 4 candidates is probably the
worst number for honest Score. This effect has very little impact on any of
the other three numbers above, though.

While I find these assumptions reasonable, I realize that others will
disagree. There will be plenty more numbers from where these came from, and
plenty of time to adjust assumptions and argue about which ones are the
most realistic. My main point here is that median methods can, in at least
some circumstances, be better than score voting, even with the same
proportion of strategic voters. I hope to explore the boundaries of those
circumstances further in the days to come.

Jameson
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