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

[Jobst Heitzig]

Dear folks,

although Forest's posting comes along so matter-of-factly, let's make it
absolutely clear that it is an

	ENORMOUS MILESTONE!

Why so?

He describes a very SIMPLE, EFFICIENT, and FAIR method which

	REVEALS THE TRUE UTILITY VALUES

of all voters who are rational in the sense of von Neumann and Morgenstern.

The only other known methods which have this revelation property are not
only more artificial and complex but are much less efficient or require
monetary taxes to be paid and destroyed (like the Clarke tax).

Very simple proof that sincere ratings are optimal:

My ratings are only relevant in a specific situation. In this situation
a fall-back lottery has already been determined from all the labels
(thus not dependent on my ratings), and a possible consensus option has
been nominated from the circle on a drawn ballot (thus also
independently from my ratings). If my ratings are relevant, they will
decide between this given fall-back lottery and this given nominated
consensus option, but I will not know beforehand which lottery and which
nominated option they will be (except if I knew all other ballots, which
is impossible in a secret poll). So the only way to make sure that my
ratings will lead to the fall-back lottery when I prefer it over the
nominated consensus option, and that they will lead to the nominated
consensus option when I prefer it to the fall-back lottery, is to give
ratings that reflect my true preferences, in other words, to specify a
set of sincere utility values.

Note that this is not only true in some equilibrium situation but NO
MATTER HOW THE OTHERS VOTE! In other words, it is always a dominant
strategy.

Now, that does not mean, however, that the whole method is
strategy-free, since the other part of the ballot, namely the circle and
the label, are strategic. I may, for example, have incentives to label a
more extreme option as favourite than my true favourite, in order to
lower the expected rating of the fall-back lottery and make a consensus
more probable. However, every such strategic behaviour would be visible
from the ballot since the labelled favourite would not have the highest
rating. That is a very interesting property which I have never seen
before in any method: you have the incentive to vote strategically, but
you cannot hide if you do so!

My guess is that we will soon find a similar method in which a single
voter cannot prevent the consensus completely but only lower its
probability...

Forest: EXCEPTIONALLY WELL-DONE!

Jobst


fsimmons at pcc.edu schrieb:
> A proportionally fair lottery is a lottery method in a which any faction of the
> voters can unilaterally guarantee that their common favorite will be elected
> with a probability proportional to the size of their faction.
> 
> A consensus candidate is any candidate that would be liked at least as much as
> the random favorite by 100 percent of the voters (assuming all voters to be
> rational).
> 
> A consensus lottery is a method that elects consensus candidates with certainty
> (again, assuming rational voters).
> 
> I won't attempt to define "sincere range ballot" here, but the meaning will be
> apparent from this method:
> 
> Ballots are range style (i.e. cardinal ratings).
> 
> Each voter rates the candidates, circles one of the names as a proposed
> consensus candidate, and labels another (or perhaps the same) name as "favorite"
> or "favourite."
> 
> Have I overlooked anything?
> 
> The ballots are collected and the probabilities in the "random favorite" lottery
> are determined.
> 
> These probabilities are used to determine and mark a "random favorite rating
> expectation" on each range ballot.
> 
> A ballot is then drawn at random.
> 
> If the circled name on the randomly drawn ballot has a rating above the "random
> favorite rating expectation," on any ballot (including the one in play), then
> another ballot is drawn, and the indicated favorite of the second ballot is elected.
> 
> Otherwise, the proposed consensus candidate whose name was circled on the first
> drawn ballot is elected.
> 
> That's it.
> 
> Note that any voter has the power to turn the election into "random favorite" by
> giving only one candidate (favorite=consensus) a positive non-zero rating.  But
> whenever that is optimal rational strategy, sincere range yields the same
> expectation, and is therefore optimal, too.
> 
> 
> 
> 
> 
> 
>