Equilibrium in Approval Voting and Repeated Balloting

[Forest Simmons]

As has been pointed out the existence of Nash (or any other kind of )
equilibria doesn't mean that cycling won't occur in repeated balloting.

In game theory optimal strategies often require randomization.

Use of an optimal strategy that requires randomization doesn't guarantee
that you won't regret your choice on any individual play of the game. It
just guarantees that in the long run you will come out ahead
statistically.

In the case of perfect information there would be no advantage to repeated
balloting.

When information is imperfect, repeated balloting may bring the voters
closer to the perfect information case, but insincere voting up until the
last round of repeated voting can frustrate that goal.

Cumulative repeated approval voting mitigates that possibility because the
results of insincere voting carry over to the next round, creating a
disadvantage for the insincere voter.

When it becomes apparent that the last round is at hand (because the quota
is within reach) then enough accurate information is available to make a
near optimal strategy play on the last round.

If the approval leaders have been cycling up until the last round, then
the optimal strategy will be a mixed strategy, i.e. one requiring
randomization.

Well, that's the way I understand the situation. Correct me if I'm wrong
on any account.

Voters don't like randomization unless they are the ones doing it,
otherwise the best method (if feasible) would be to have the voters submit
their sincere utilities and have a computer privy to all of the ballots
find for each of them their optimal (possibly mixed) strategy and execute
it for them (using a suitable random number generator in the mixed case)
by submitting the resulting approval (or whatever) ballot for them.

Cumulative Repeated Approval Balloting comes approaches this ideal without
randomizing for the voters.

Forest