[EM] Three Stage Approval Election

Simmons, Forest simmonfo at up.edu
Tue Jun 6 18:26:37 PDT 2006

It's pretty obvious that under Approval you should approve your favorite candidate, and that you should leave unapproved the candidate that you despise the most.  But it isn't always so obvious which of the remaining candidates to approve.
A rule of thumb is to approve the candidate that you would vote for under Plurality along with every candidate that you like better.  This shows that reasonable approval strategy is no harder than Plurality strategy, but Plurality strategy can be hard sometimes, too.
In general, if you can rank the candidates, but cannot assign relative utilities to them, the optimal strategy is this:
For each candidate X, approve X if and only if it is more likely that X will end up in a tie for first place with a candidate that you rank below X than with a candidate that you rank above X.
This strategy maximizes the probability that your ballot will improve the election result from what it would be if you did not vote.
The hard part is getting reliable estimates of those probabilities.
That is the motivation for this Three Stage Approval approach:
1.  Have a randomly chosen ten percent of the voters submit their approval ballots.
This first ten percent of the voters have a disadvantage in that they must rely on polls, expert opinions, and their sincere gut feelings to decide their approval cutoffs.
Before the next ten percent of the voters vote, publish the distribution of approvals, including correlations, from the first ten percent, so that the needed probabilities for optimum approval strategy can be calculated.
2.  The next ten percent of the voters have the advantage of this additional information.
After the second ten percent casts their approval ballots, update the distribution information.
If this posterior distribution predicts a winning candidate with 99.9 percent certainty, then spare the remaining 80% the trouble of voting.
3.  Else allow the remaining voters to vote on the basis of the probabilities computed from this new distribution.
Let W1, W2, and W3 be the winners of the respective stages, i.e. the candidates that received the most approval from the voters in the respective stages.
If  the same candidate wins all three stages, then that candidate is the winner.
Otherwise, have a random number generator pick a number  k  from the set  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with equal likelihood.
If  k=1, then W1 is the winner.  If  k=2, then W2 is the winner.  If  K>2, then W3 is the winner.
This method has the potential of saving lots of voters from trips to the polls, when the race is not close.
If the race is close, then at least 80 percent of the voters will have reliable probabilities on which to base their approval strategies.
Giving W1 and W2 positive probabilities, in proportion to the size of the random sample of voters in the stages that they won, is a way of discouraging the voters in these samples from voting insincerely in an effort to manipulate the probabilities for the final stage.
I'm sure that you can think of many variations on these ideas.  But how's that for starters?
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