[EM] When wouldn't SODA elect the CW?

Jameson Quinn jameson.quinn at gmail.com
Thu Feb 2 04:35:12 PST 2012

When wouldn't SODA elect the CW?

SODA meets the Condorcet criterion for voted preferences, and a true
[majority/nonmajority] Condorcet winner would be visible in the voted
preferences in a [unique/nonunique] strong Nash equilibrium. So in the
large majority of cases, it would elect the CW.

Here are (the only) two scenarios where I think it might not.

*Scenario 1 (one dimensional pseudo-chicken):*
35: A>B
25: B>A
40: C>>B>A

This is almost the classic chicken dilemma scenario, with a little bit more
one-dimensional logic. But I am *NOT* saying that SODA would fall prey to
the chicken dilemma here. That is to say, unlike Approval and Range, and to
a lesser extent MJ and Condorcet, there is no danger that it will elect C.

What I am saying is that, if candidate C's preference for B over A is weak,
they may choose not to predeclare that preference, in order to be able to
campaign more forcefully against B. Since B's natural preference is A, B
cannot retaliate in kind. Without C's preference, the Condorcet and SODA
winner is A (as with IRV).

Who's the correct winner in this scenario? Definitely not C, but you could
argue either way between B and A. B is the honest Condorcet winner, but
since the preference of the C voters is weak, it may well be that A is the
honest utility winner. (If the C voters' preference for B over A is not
weak, then at least 11 C voters will explicitly approve B, and B will win.)

*Scenario 2 (Weak Condorcet winner):*

In this scenario, B is a relative unknown, and throughout most of the
campaign, A and C focus their attention on each other. After the votes are
counted, the situation is something like the following:
48: A>B
5: B>C
47: C>B

B is the Condorcet winner. "Correct" SODA strategy would be: C doesn't
approve B, A approves B, B wins. But when it's candidate A's turn to assign
delegated votes, they're faced with a concrete choice between B and C. A
realizes that, while B gives voice to important centrist concerns in a way
that speaks to the median voter, that B just doesn't have the experience or
qualifications for the job. So A doesn't approve B.

As I've given the scenario, B's preference C happens to be the weaker of
the two candidates. This will happen about half the time a scenario like
this arises. In this case, B has a choice of whether to give the election
to C or leave A as the winner. Perhaps B can use this power to gain some
centrist concession from the eventual winner.

I think that SODA is getting the right answer in the scenario as portrayed.
Of course, you could in principle vary the details to the point where such
a result would be arguably wrong. One thing is certain though; whatever is
the right winner for the society, both A and C (the major-party candidates,
certainly the winners under plurality) would be unhappy with a B win. Thus,
plurality incumbents will be happy to know that SODA does not necessarily
favor B in this situation.

In other systems – Approval, Range, Condorcet, or even MJ – B could in
principle win. I actually find it pretty implausible that B would win in
any of these systems, but I do think that even the possibility would be
scary to plurality incumbents. Thus, I find the fact that SODA does not
necessarily elect the CW in this scenario to be a net plus for SODA.

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