[EM] Median Mediated Approval

[Kevin Venzke]

Hi Forest,

> For each candidate X let r(X) be the number of ballots on which X is ranked ahead
> of at least one candidate. This r(X) is sometimes called the "implicit approval"
> of X.
> 
> Also,let f(X) be the number of ballots on which no candidate is ranked ahead of
> X. This f(X) is often called X's  "equal first" or "top" count.
> 
> Elect the candidate X with the greatest value of the sum r(X)+M×f(X), where M is
> the Median of all of the voter suggestions for the number M, restricted to the
> interval between zero and one inclusive. [If there are an even number of
> suggestions, throw in an extra zero so that there will be a well defined median.]
> 
> There are at least three ways to get the voter suggestions for M.
> 1. Voters include suggestions for M in a field provided on their ranked preference
> ballots.

That's rough. It seems like most likely the choice of M decides whether we elect
the r(X) winner or the r(X)+f(X) winner. But within a single round the voter
doesn't know who those are.

> 2. Ranked ballots are counted and the results published for various values of M
> between zero and one. If the value of M actually makes a difference in who wins,
> have the voters vote on the value of M. Otherwise, M is not needed.

That's interesting. I don't think it's that likely that different values of M
are going to offer a lot of different outcomes. But if they do, the resulting
voter strategy could be interesting.

> 3. Elect the CW if there is one, otherwise check to see if M matters. If there is
> no CW, and M does matter complete the election with a call for votes on M.
> 
> It seems like this third version would be ideal for small group settings ... any
> kind of deliberative assembly customarily governed by Robert's Rules.
> 
> Any venue that currently admits runoffs could profit by replacing their current
> system with one of these versions.
> 
> It seems like the third version is an ideal marriage (for generic single winner
> elections) between Condorcet and Implicit Approval. Am I over-looking something?
> 
> How close does it come to satisfying the Favorite Betrayal Criterion?

I have to greatly simplify it, so that it's one round, can be derived from
rankings, and assumes M will either be 0 or 1. (I'm not sure what algorithm can
find the result for every possible value of M.)

So let's say we take the pairwise winner between the r(X) winner and r(X)+f(X)
winner (assuming they differ). That's what I say the M vote represents. In the
Condorcet version, we check Condorcet first.

The Condorcet version is then rather similar in results to Condorcet//TopRankings.

For weak FBC, both versions are better than Condorcet//Approval(implicit).
The Condorcet version is a lot better, when there are three candidates. With
four candidates or without Condorcet, it's not as striking. Neither is as good
as MinMax(WV) etc. (C//A is not actually that great at FBC.)

For compromise (e.g. when you want to have your favorite alone at the top) the
Condorcet version is somewhat worse than C//A while the non-Condorcet version is
a lot worse. (It's probably pretty intuitive that it will be hard to retain your
strict favorite under this method without potentially hurting a compromise.)

Kevin