[EM] Anyone got a good analysis on limitations of approval and range voting?

[Raph Frank]

On Tue, Nov 10, 2009 at 8:11 PM, Jobst Heitzig <heitzig-j at web.de> wrote:
> Hello Kristofer,
>> Assume (for the sake of simplicity) that we can get ranked information
>> from the voters. What difference would a SEC with Random Pair make, with
>> respect to Random Ballot?
>
> This sounds interesting, but what exactly do you mean by Random Pair?
> Pick a randomly chosen pair of candidates and elect the pairwise winner
> of them? I will think about this...

Presumably, it means that the voter submits 2 ballots, a ranking and a
nomination for the 2nd round?

Clearly, your rankings should be honest, as it is only looked at once
the 2 candidates have been decided.

However, your nomination would have to be made tactically.

It would require that the voter decide the probability of the
candidate they nominate winning.

If you nominate the condorcet winner, then the odds of your candidate
winning the second round is 100%, as no other candidate can possibly
beat him..

However, if you nominate an extremist, then your nomination is almost
certain to fail, as he will lose to virtually any other candidate.

If the voter distribution is symmetric (and voter utility is
symmetric) around a central point, then the nominated candidate who is
closest to the centre will win.

If each voter nominates their favourite, then you best strategy is to
nominate the the candidate which maximises

f(distance)*utility

f(distance) is the fraction of the nominations that nominate
candidates further away than that distance from the centre.

f(0) is automatically 1 and f(most extremist candidate's distance) is
automatically 0.  Also, f(d) is a monotonic decreasing function.

Thus, when considering 2 candidates of near equal utility, you should
nominate the candidate nearest the centre.

However, if all voters do that, then most of the nominations will
start to be clustered near the centre.  This means that the voters
should nominate candidates even closer to the centre.

I.e. if f(d) = 0.1, then you would have to prefer that candidate at
least 10 times better than the condorcet winner in order to nominate
him.

I think the effect could very easily end up being that the condorcet
winner normally wins.

It could also be implemented in 2 formal rounds.  In the first round,
each voter votes for 1 candidate.  2 candidates are picked at random,
using random ballot.

Those 2 candidates then proceed to the run off.  This might even make
people accept random ballot.  The problem that a candidate with 1%
support could get to be President is eliminated.  (Unless it happens
twice in 1 election.)