Hi,
A few weeks ago I posted a message about this simple voting method:
Before election day, each candidate publishes an ordering
of all the candidates.
On election day, each voter selects one candidate.
Treat each vote as if the voter had expressed the ordering
published by the candidate she selected.
My thanks to all who took time to consider it and/or discuss it. (I found time
to read all
the responses for only about a week, unfortunately.)
I'd like to add two more comments to the discussion:
1. Several people commented that candidates would tend to rank most others at
the bottom
(or, equivalently, would omit most candidates from their orderings). Someone
wrote that
candidates would rank only themselves. Someone else wrote that they'd rank
only members
of their own party. I believe this issue is very important; the value of the
method
depends on candidates being willing, typically, to rank compromise candidates
over those
they (and their supporters) sincerely believe are worse. So, can we more
explicitly
discuss the incentives to rank more and the incentives to rank few?
I think the incentives on the candidates will depend on how the votes are
tallied. In my
message a few weeks ago, I didn't specify the procedure with which to tally the
voters'
orderings, other than to point out that my favorite algorithm, MAM, could be
used. One
nice property of MAM is that it satisfies the Truncation Resistance criterion,
which is
relevant to this discussion.
I believe the following tallying procedure is the simplest one that is
worthwhile:
After election day (but before the winner is calculated)
publish the vote totals.
Then allow each candidate some time (a few days?) to choose
whether to withdraw from contention.
Then count each vote for the non-withdrawn candidate
who is highest in the ordering published by the
candidate selected by the voter.
In other words, Plurality Rule with a withdrawal option. (Note that the
withdrawal option
mitigates a problem caused by the Electoral College in U.S. presidential
elections, making
it reasonable for multiple candidates to run for president without fear of
being spoilers.)
Given this procedure, would candidates have sufficiently strong incentives to
rank
compromise candidates over worse candidates?
Here's an example to consider. Suppose the candidates for some office are
Gore, Bush, and
McCain. Let's take it for granted that each candidate will rank himself
topmost. Assume
that before election day Bush publishes the ordering "Bush over McCain over
Gore." Assume
that the day after the election these vote totals are published:
Bush selected by 35,000,000 voters
McCain selected by 20,000,000 voters
Gore selected by 45,000,000 voters
Bush sees that if no candidate withdraws, Gore will be elected. Bush also sees
that if he
withdraws and McCain does not, the 35M who voted for him would have their votes
count for
McCain, which would give McCain a total of 55M votes, electing McCain. Would
Bush
withdraw? Clearly there's an incentive for him to do so; after all, his
ordering ranked
McCain over Gore and it would look quite peculiar to observers if he allowed
Gore to be
elected.
But let's add another detail to the example: Assume McCain had published the
ordering
"McCain over Bush over Gore." If Bush could persuade McCain to withdraw, Bush
would be
elected. Both Bush and McCain have an incentive to see Gore defeated, having
both ranked
Gore at the bottom, and presumably they will strike a deal that elects one of
them and
defeats Gore.
Let's add another detail: Assume Gore had published the ordering "Gore over
McCain over
Bush." Gore knows Bush and McCain can strike a deal that would elect one of
them and have
an incentive to do so. Assuming Gore's ordering was sincere, the best he can
hope for is
for McCain to be elected. Gore can accomplish this by withdrawing. McCain
finds himself
in the driver's seat: Bush prefers McCain over Gore, and Gore prefers McCain
over Bush. I
would expect McCain to be elected.
How would Gore be better off if he had ranked only himself? That would strip
him of his
power to affect the outcome by withdrawing. It might cost him some votes, too,
from
voters who prefer Gore over McCain over Bush and want to do what they can to
defeat Bush.
The same incentive for Gore to rank "Gore over McCain over Bush" appears to
also hold in
the case where Bush ranks only himself and McCain ranks Bush over Gore. In the
case where
McCain ranks only himself and Bush ranks McCain over Gore, it appears not to
matter what
Gore does, since Bush would withdraw and McCain would win.
Regards,
Steve
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