Median Mediated Approval
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.
-
Voters include suggestions for M in a field provided on their ranked
preference ballots. -
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. -
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?
-Forest
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.
- 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.
- 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.
- 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