[EM] Voting by selecting a published ordering

[Steve Eppley]

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    Then count each vote for the non-withdrawn candidate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