[EM] A Proportionally Fair Consensus Lottery for which Sincere Range Ballots are Optimal

[Jobst Heitzig]

Hello Forest,

> Most of the credit should be yours; in fact, the proof and all of the
> ingredients are yours.  I hurried to post the message this morning, because I
> was sure that you were going to beat me to it!  I would certainly believe you if
> you said that you had already thought of the same thing but didn't have time to
> post the message before I did.

Well, I don't. I was never before thinking of asking separately some
additional information from the voters which is already contained in the
ratings part of their ballot (namely what the favourite is), in order to
transfer the strategic incentives away from the ratings to this separate
information. And that is exactly the genial part!

Now that the general technique is clear, we can easily derive a lot of
similar methods, also some that only incorporate a very small amount of
chance, for those who don't like chance processes in voting methods.


The general technique is this:

Ask for ratings and some additional information. Use the additional
information and to determine -- independently from all ratings -- two
possible winners or winning lotteries, at least one of which must be a
lottery of at least two options in which the probabilities can vary.
Then use the ratings to decide between these two possibilities in some
monotonic way (e.g., using unanimity as in your proposal, or some
qualified majority, or even Random Ballot, or whatever). Then
strategy-freeness in the ratings part follows from the fact that they
are only used in a monotonic binary choice between lotteries which are
not known before.


For example:

Method "Range top-3 runoff" (RT3R)
===================================

1. Each voter separately supplies
   a "nomination" range ballot and a "runoff" range ballot.

2. From all "nomination" ballots, determine
   the options A,B,C with the top-3 total scores a>b>c.

3. Let L be the lottery in which B wins with probability
   p = max(0,(2b-a-c)/(b-c))  and C wins with probability 1-p.

4. Let q be the proportion of "nomination" ballots
   on which the lottery L has an expected rating
   below the rating of A on that ballot.

5. Option A wins if,
   on at least the same proportion q of all "runoff" (!) ballots,
   the lottery L has an expected rating
   below the rating of A on that ballot.
   Otherwise B wins with probability p and C wins with probability 1-p.


Notes:

p is so designed that it can take all values between 0 and 1 but will be
the larger the lower c is, in order to get a large expected rating of
the final winner. Of course, the formula for p could be modified in all
kinds of ways.

The expected total "nomination" rating of the final winner is at least

	a - 2(a-b),

so if the race between A and B is close (i.e., a-b is small), we have
quite an efficient outcome. On the other hand, if A clearly beats B, it
will win with a high probability since the true proportion of voters who
prefer A to B and C is probably larger than can be seen from the
strategically used nomination ballots.


What do you think?

Yours, Jobst