# [EM] Voting by selecting a published ordering

Steve Eppley seppley at alumni.caltech.edu
Sat Apr 22 11:00:47 PDT 2006

```Hi,

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

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.)

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Then count each vote for the non-withdrawn candidate
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

```