[EM] Monotonicity

Bart Ingles bartman at netgate.net
Tue Jan 15 20:47:21 PST 2002

> The estimate that I ran across was that it those situations would exist
> in about 5% of elections. But IRVie Doug is missing the point. Even if
> it only happens  5% of the time, do we really want a voting system that,
> without a doubt, will sometimes respond oppositely to a voter changing
> his intention about how to vote, due, perhaps to new information that's
> become available.

That 5% figure is probably about right for electoral college/plurality
conflicts as well, where the EC winner defeats the plurality winner (one
such occurrence every 100 years is equal to 4%).  But I don't see many
election reform groups shrugging it off.

In fact non-monotonic situations are probably part of the natural
progression of a party's growth cycle.  Imagine a small but growing
third party (let's call them the neoliberals), who happen to be
ideologically more extreme than the two major parties.

While the party is small, non-monotonicity is not a problem with IRV,
since the party is sure to be eliminated before it can do any damage (or
influence the debate in any meaningful way).  Voters can vote sincerely
without regret, but then this is true for any proposed voting system
other than first-past-the-post.

But suppose the party grows to the point that it begins to garner more
first-choice votes than its ideological neighbor.  Assuming no party has
an outright majority, we have a situation like the following:

 28%      neoliberal, liberal, conservative
 17%      liberal, neoliberal, conservative
 10%      liberal, conservative, neoliberal
 45%      conservative, liberal, neoliberal

Here the neoliberals just begin to outpace the liberals, 28% to 27%.

The only remaining assumption is that the "old" liberal party is divided
in terms of second-choice votes, so that the conservatives can still
out-poll the neo-liberals after the elimination round.  This seems
fairly natural, since the neoliberals have only recently achieved
2nd-place status; by definition the conservatives are still the number
one party in terms of first preferences, so it should be easier for them
to find the few extra 2nd-choice votes needed to win the election.

Now non-monotonicity is obvious.  If all the neoliberals vote sincerely,
the conservatives win.  But a small number of them can vote for anyone
BUT their favorite, causing the conservatives to lose.  Alternatively, a
recount could take votes from the neoliberal camp and award them to the
conservatives, causing the conservatives to lose. 

Or we could have the reverse situation, such as the following:

 26%      neoliberal, liberal, conservative
 17%      liberal, neoliberal, conservative
 10%      liberal, conservative, neoliberal
 47%      conservative, liberal, neoliberal

Here the liberals win the final round, but imagine the CNN reaction when
a recount takes 2% of the vote away from the conservative candidate and
awards it to the neoliberal, causing the conservative to win.

These examples don't seem terribly contrived -- any of the vote
percentages can vary by several points without changing the non-monotone
situation (although the possibility of an actual paradoxical change in
outcome is highest -- approaching 50% -- when the liberals and
neoliberals are in a close race to survive the elimination round).

Bart Ingles

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