[EM] A family of easy-to-explain Condorcet methods

[Daniel Carrera]

On Fri, Jul 2, 2021 at 6:58 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

>
> I'll try to find a better method -- my optimization simulations suggest
> that monotonicity doesn't affect manipulability much, and so there
> should be a DMTBR Smith monotone method out there somewhere. It's just
> very hard to get the pieces to fit.
>
> But in the meantime, if you want to advocate for Pb, go ahead :-)
>

Sure :-)

I just ran an experiment inspired by Darlington (2018). In that paper he
argues that Minimax is the best method because, entirely aside from
Minimax's theoretical limitations, it tends to produce winners closer to
the electorate than other methods. He models voters and candidates as
belonging to a multivariate Gaussian distribution embedded in an
N-dimensional "issue space". Then he finds that Minimax winners tend to
have a lower mean distance to voters than the winners of other methods.

So inspired by that I made a simple simulation: 100 voters and 5
candidates, all drawn randomly from a multivariate Gaussian embedded inside
a 3-dimensional "issue space". Voters rank candidates according to their
distance in issue-space. I ran enough trials to get 10,000 scenarios with
Condorcet cycles. Here's what I found:

-  4,668 trials where Pb and Minimax found the same winner
-  1,906 trials where Minimax found a better candidate than Pb
-  3,426 trials where Pb found a better candidate than Minimax

Again, a candidate is "better" if his mean distance to the voters is lower.

I'm sure that if I change the parameters the values will change. But so far
it really looks like, whatever the theoretical limitations of Pb are (e.g.
clones) in simulated elections it works at least as well as the best
Condorcet methods. Honestly I hadn't expected it to work this well. The way
that Minimax and Ranked Pairs break cycles makes the most sense to me, but
apparently dumb Plurality is a pretty good way to break cycles too.

Cheers,
Daniel
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