[EM] Alright, next try. Range voting fix, version 2.

[rob brown]

I'm gonna take another stab at a method that uses the UI of Range Voting,
but tabulates it in a way that makes more sense.  I'll admit, I actually
like the fact that Range Voting collects very "rich" information about voter
preferences, moreso than ranked ballots.  (Of course this is no good if
people have very strong incentive to vote insincerely.  Nor is it good if it
is subject to vote splitting and therefore encourages a two party duopoly,
as it would if significant numbers of people vote sincerely under Warren's
averaging system).

My proposed method of processing Range ballots might not be particularly
practical in large elections, but I submit it as more of a hypothetical.
Maybe it can lead to a more efficient and more deterministic of doing
effectively the same thing, or maybe it could otherwise provide interesting
insights.  Maybe it has some problem (other that those mentioned above) I
haven't thought about.  Or maybe it has been suggested before but I don't
know what keywords to search for....

What I suggest is that, prior to tabulating, each Range ballot be processed
into an Approval ballot.  But, it should be done taking into account optimum
strategy, with knowledge of how others are voting.  You could call this a
"Nash equilibrium seeking" system, as it keeps adjusting Approval ballots
(that is, picking the cutoff point) until a state is reached where no voter
can improve his or her own strategy given that all other people's strategy
stays fixed.  (in reality it is not the voter that is modifying their
strategy, it is the formula that converts the voter's range ballot into an
approval ballot, which can be considered to be operating as an "agent" of
the voter)

Example:  say I rated candidates as follows:

A: 100
B: 75
C: 30
D: 0

Now, the system will make a first try at converting my ballot into an
Approval ballot.  Let's say it uses 50 as the arbitrary cutoff point:

A: yes
B: yes
C: no
D: no

Now a first round is run, tabulating the approval ballots.  The totals comes
out to:

A: 5000
B: 6000 (leading candidate)
C: 1000
D: 500

Now, it can be seen that our conversion of our ballot to Approval did not
pick the best strategy.  It would have been wiser to choose the cutoff point
such that only A was given a thumbs-up, so that A had a better chance of
beating B, which we like less than A.

So the ballot is adjusted, giving only A a yes.  The formula would simply be
that all candidate's that are preferred to the leader are given a yes, and
all that are less preferred are given a no. The leader is given a yes if he
is preferred to the second place candidate, otherwise a no.  (this strategy
is generally recognized as a good one for Appproval voting, see
http://en.wikipedia.org/wiki/Approval_voting#Potential_for_tactical_voting )

All other voter's approval ballots are adjusted as well, if they are
currently non-optimal from a strategy point of view.  Another round is run.

The process is repeated until no one can improve their strategy (i.e. a Nash
equilibrium is found).

I can think of several tweaks to make the system more stable, and have it
better use the Range scores.  For instance, start off with a very low
cutoff, say 20 rather than 50.  Each round, move the cutoff point *toward*
the one determined to be optimum strategy for this voter, maybe by 20%.
This will take more rounds to reach an equilibrium, but would make for less
abrupt changes with each round, giving it better opportunities to find a
stable equilibrium.

I am going to guess a few things:

1) that it will be extremely rare that it does not find a Nash equilibrium
in fewer than 20 rounds
2) that the results of this will be remarkably similar to the results
obtained by some of the better Condorcet methods
3) that there is a means of producing the same results that does not require
processing each ballot multiple times

-rob
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