[EM] The Small Voting Machine (was Strong FBC)

[Forest Simmons]

Alex Small has resurrected the idea of an ideal voting machine that would
be the election method equivalent of the Carnot Engine of thermodynamics
or the Universal Turing machine of computer science.

He suggested as input the (nominally) sincere, ranked preference ballots
of the voters.

The machine invokes a game theory oracle to supply the best approval
strategy for each player based on a complete knowledge of all the
submitted ballots.  Then the machine implements these optimal strategies
on behalf of the players (voters) to see who wins.

Rob LeGrand's citing of a version of the dictator theorem led to a
reminder that in general we can expect the optimal strategies to be
stochastic mixtures of pure strategies, so that the SVM would not be a
deterministic machine, i.e. repeating the input rankings will not always
yield a repetition of the winning candidate.

At first this might seem like a big defect, but if you just consider it as
a kind of temporal version of proportional representation for supporters
of the Smith set candidates, it doesn't seem like such a bad feature after
all.

Adam pointed out that this composite game might still require strategy to
optimize one's expected utility. In other words, it might not always be
best to input your sincere ranking into the Small Voting Machine.

At first I thought I could prove otherwise. Consider the following
(defective) heuristic argument, and see if you can spot the biggest hole:

Suppose I submit an insincere ranking with hopes of getting better
results.  The oracle supplies the best strategy for that fake ranking, and
the SVM implements it on my behalf.

Since this strategy is optimal for my fake ranking, it is probably
sub-optimal for my true ranking. I cannot intervene with some other
strategy based on my true preferences, because the SVM implements the
oracle supplied strategies automatically.

It's like trying to fake left, when I intend to throw right, and then
being constrained to throw left, after all.

Well, the main hole that I see in this argument is that it doesn't take
into account the changed strategies that the oracle will supply for the
other voters.

In the analogy, perhaps my strongest throw is to the right, but it is
better to throw left, because that plays on a big weakness of my
opponents.

Admittedly, the difference between my true and my fake preferences
shouldn't affect the other voters' optimal strategies very much if there
are lots of voters, but remember, the SVM is supposed to be an ideal
machine.

Furthermore, the effect could be multiplied by block voting of large
factions.  In particular, if an approximation to the SVM were used in a
proxy runoff, there would be one voting block per candidate.

So Adam's concern is valid both theoretically and practically.

I don't want to make this installment too long, so I will give my proposed
modification of Alex's machine in another message.

Forest




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