[EM] Cumulative Repeated Approval Balloting for DSV

[Forest Simmons]

The goal of maximizing voting power while minimizing manipulability seems
to be an elusive will 'O wisp.

However, in committees and other small groups one option is repeated
balloting.  For example, approval ballots can be repeated N times or until
the ballots stabilize, whichever comes first.

One problem with this method is that when there is no Condorcet Winner,
the ballots may cycle radically, and the N_th time cutoff more or less
randomly chooses a member of the cycle.  When approaching the N_th
balloting, various factions may try to trick others with feigned mutual
support right up to but not including the last ballot. Or on the other
hand, they may stubbornly bullet up through the penultimate balloting
hoping that they will be used as a lesser evil compromise on the last vote
by other factions.

In other words, repeated approval balloting can be a game of chicken or
prisoners dilemma or high stakes poker.  Repeated plurality balloting is
worse, yet that is what Lorrie Cranor used as her main example of Declared
Strategy Voting (DSV).

It seems to me that it would add stability and remove some of the high
stakes poker feeling if the approval totals were accumulated for each
candidate, and that the win would go to the first candidate to reach
total approval of, say, K*M, where K is the number of candidates and
M is the number of voters.

After each balloting the cumulative approval totals of the candidates
would be shown in a bar graph, with previous levels marked by color
changes, and the K*M goal clearly marked as well.

The voters would have a good enough idea of the progression to make a
reasonable decision of how far down to approve in the next balloting.
Drastic last minute changes in strategy would not be enough to make up for
the cumulative trend, so approval increments would not gyrate wildly.

How does DSV relate to this?

The proposed version of DSV (Cumulative Repeated Approval Balloting
Declared Strategy Voting or CRAB_DSV) would take as input each voter's
relative utilities for the various candidates in the form of some kind of
CR ballot, say the Five Slot Grade Ballot (possibly with plus and minus
options for more resolution).

The first balloting would be on the basis of each voter approving each of
his/her above mean utility (i.e. above average grade) candidates. The
subsequent approvals would be made with decision theoretic methods applied
to the grade ballots in conjunction with the candidates' approval
statistics, i.e. the statistics described by the bar graphs mentioned
above.

I believe that CRAB_DSV would overcome most if not all of the
disadvantages of the version of DSV that Ms. Cranor devoted most of her
attention to in her dissertation.  In particular, it has the stability of
her stochastic ballot by ballot version and the repeatability of her
non-random repeated balloting version.

Since Cranor's DSV was based on plurality, it bore a certain similarity to
IRV, though less manipulable. Imagine doing repeated plurality balloting.
Compare that with repeated approval balloting, and finally compare that
with cumulative repeated approval balloting, and I think you can see a
definite progression towards stability.

Cumulative repeated plurality balloting would add some stability to
Cranor's version, but it seems to me it would be the stability of getting
into a rut, and being unable to move to a better rut, whereas approval
doesn't penalize tentative testing of the waters in various directions in
search for a better equilibrium.

CRAB_DSV would allow voters to submit sincere grade ballots without fear
of regret.  The decision theoretic calculations automatically optimize the
approval votes according to the information that becomes available at the
end of each round of repeated balloting.  Sophisticated voters have no
advantage over the naive.

If they think they do, they can limit their grades to A's and F's, for
example, and the DSV calculations will automatically cast each of their
ballots with all A's translated as approvals and all F's translated as
non-approvals.

The method is not summable in polynomial size data structures, but an
approximation can be done in O(K^3) where K is the number of candidates.

That's enough for now.

Forest