The Arrow Theorem and Strategic Misrepresentation Voting

This is the concluding post in the series on the
Arrow impossibility theorem which shows that there
can be no perfect voting system.
Even if you know everyone’s true preferences
for all possible alternatives you cannot find
a system that is guaranteed to find a
reasonable method for determining the best
choice for society. It’s even more complicated
because voters might have an incentive to not
reveal their true preferences. They might try
to manipulate the voting system with strategic
misrepresentation voting.

One of the conditions Arrow lists in the impossibility
theorem is called “the independence of irrelevant
alternatives.” If there are three or more alternatives,
then if people prefer alternative A to alternative B then
making any change to the other alternatives shouldn’t
change their preference between A and B.
You can’t design a voting system where this property
holds without violating some of the other conditions
for a reasonable voting system that will work for all
possible preference patterns.

The implications of this theorem are that people
might have an incentive to calculate that they are
more likely to get what they want by voting
differently from their true preferences.

Suppose there were several candidates and the voting
system simply declared the winner to be whoever gets
the most votes (a plurality, not necessarily a majority).
This is generally not a good system, because the winner
might have a relatively small percent of the votes.
If there were more candidates from party A than there
are from party B, then party A members would become
worried that their vote would be split too many ways
and that would allow a party B member to be elected.
Therefore party A members may not vote their true
preference but instead vote for whichever party A
member seems to be most likely to get the most votes.
In that case we never learn the true preferences of
party A voters.

A more common system would be one with a primary election
so that one candidate from party A and one from party B
advance to the final election. In many cases this system
will work well, but there is a risk of a particularly
nasty type of strategic misrepresentation voting. If there
is only one candidate for party A (perhaps an incumbent
seeking reelection) then party A members may think that
there is no reason to vote in the party A primary and
instead they vote in the party B primary for whichever
candidate they think would be easiest to beat.
We can reduce this problem with a closed primary (open
only to voters of that party) but then you’ve excluded
independents from the nominating process and likely
worsened the centrifigual force problem by pushing candidates
away from the center.

A better option is a top-two primary, where the the two
candidates with the most votes advance to the final election
regardless of party. Washington state held its top-two
primary yesterday.

See

http://myhome.spu.edu/ddowning/fos/ch14app.pdf
for numerous examples about how to deal with the
strategic misrepresentation problem for the
three-stage presidential election proposed in
Freedom, Opportunity, and Security, where
there is a four-candidate semifinal election, and two
candidates advance to the final.

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–Douglas Downing
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