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

[fsimmons at pcc.edu]

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.