[EM] secret ballots and proxy voting

[Bayle Shanks]

Typically when large numbers of people are voting you'd like to have
secret ballots so that the Mafia can't buy votes or intimidate people
and also so that people feel free to make unpopular choices.

However, when the people voting are representing others, you often
want to publish who voted for what so that the constituents can use
the past voting records of their representatives to decide whether to
vote for them in future elections.

In a proxy voting system, where voters can allow other voters to vote
for them 'by proxy', and particularly in a transitive proxy voting
system, where the proxies can be re-proxied (e.g. Alice can give a
proxy to Bob who can give both Alice's proxy and his own to Caroline),
you want to satisfy both these objectives.

You want everyone's vote to be secret, because you don't want the
Mafia to intimidate them or buy their votes, and you want unpopular
outcomes to be feasible. 

But you also want everyone's votes to be
public, because you don't want to give your proxy to someone who says
they'll do one thing with your proxy and then actually does another,
without you ever knowing.

One fear is that the Mafia will say, 'You'd better give me your proxy
or you'll be punished'. I think you can probably fix that by not
giving proxy holders very precise information on how many proxies they
hold, when they were given, or who gave them.

Even if each person casts their own vote secretly, but can see which
way their own proxied vote was vast, the Mafia just has to
secretly ally with a small number of proxy givers in order to see
which way the proxied votes are being cast (note that even if the
system let the vote caster know whose proxies they hold, they
don't know which proxy-ers are allied with the mafia).

One idea is just to say, if you accept proxies your votes are
public, otherwise they are secret. This essentially reduces the
transitive proxy system to ordinary voting however because
it provides no way to have proxy holders who can cast proxied votes in
a way that the Mafia can't control.

Here's an idea I had to deal with this problem.

Give each person two ballots: a secret ballot and a public ballot.

Everyone can see which way they vote their public ballot. If they hold
proxies from others, the proxies' secret ballots follow their secret
ballot and the proxies' public ballots follow their public ballot. The
originators of the proxies don't ever find out which way their secret
ballots were cast.

To tally the vote, for each candidate, you sum the secret ballots for
that candidate, then you sum the public ballots for that candidate,
then you multiply these two sums together, then you take the square
root.

After transforming sums in this manner, you can use most existing
voting methods to determine the winner.

For instance, if there are five voters and two candidates, and they
vote like this:

PUBLIC BALLOT
      CANDIDATE
VOTER    A       B
1	1	 0
2	1	 0
3	1	 0
4	0	 1
5	0	 1

SECRET BALLOT
      CANDIDATE
VOTER    A       B
1	 1	 0
2	 0	 1
3	 0	 1
4	 0	 1
5	 0	 1

then the public ballot tally for A is 3, the secret ballot tally for A
is 1, the public ballot tally for B is 2, the secret ballot tally for
B is 4; the combined tally for A is sqrt(3 + 1) = 2, the combined
tally for B is sqrt(2 + 4) = 2.45.


Virtues:
* you can use your secret ballot to express your true preference
* however, if you care about influencing the election, you can have
the most impact if your secret ballot matches your public ballot. So
there is at least some incentive not to lie about what you plan to do
if you accumulate proxies.

I expect that what would happen is that the Mafia would
be limited to corrupting public ballots (and people lying about what
they are doing with their proxies to attract proxies from the
opposition party would be limited to corrupting secret
proxied ballots). If the Mafia can only reach a subpopulation of
voters, then that subpopulation will effectively have less weight,
because the multiplication of the public and the secret tallies
effectively downweights voters who cast their public and secret
ballots differently.

A slightly different approach would be to provide only public ballots,
but in addition a way for each person to secretly submit a request to
ignore their public ballot while counting votes. So now the Mafia can
effectively prevent you from voting but they can't do anything more.
You'd have to find a way to implement this so that the Mafia can't use
the same information that allows the vote-counters to match ballots to
ignore requests to find out how you voted, however.

Thoughts? Other solutions?

thanks,
  bayle