[EM] SODA and the Condorcet criterion

[Jameson Quinn]

Here's the new text on the SODA
page<http://wiki.electorama.com/wiki/Simple_Optionally-Delegated_Approval#Criteria_Compliance>relating
to the Condorcet criterion:

It fails the Condorcet
criterion<http://wiki.electorama.com/wiki/Condorcet_criterion>,
although the majority Condorcet winner over the ranking-augmented ballots is
the unique strong, subgame-perfect equilibrium winner. That is to say that,
the method would in fact pass the *majority* Condorcet winner criterion,
assuming the following:

   - *Candidates are honest* in their pre-election rankings. This could be
   because they are innately unwilling to be dishonest, because they are unable
   to calculate a useful dishonest strategy, or, most likely, because they fear
   dishonesty would lose them delegated votes. That is, voters who disagreed
   with the dishonest rankings might vote approval-style instead of delegating,
   and voters who perceived the rankings as dishonest might thereby value the
   candidate less.
   - *Candidates are rationally strategic* in assigning their delegated
   vote. Since the assignments are sequential, game theory states that there is
   always a subgame-perfect Nash equilibrium, which is always unique except in
   some cases of tied preferences.
   - *Voters* are able to use the system to *express all relevant
   preferences*. That is to say, all voters fall into one of two groups:
   those who agree with their favored candidate's declared preference order and
   thus can fully express that by delegating their vote; or those who disagree
   with their favored candidate's preferences, but are aware of who the
   Condorcet winner is, and are able to use the approval-style ballot to
   express their preference between the CW and all second-place candidates.
   "Second place" means the Smith set if the Condorcet winner were removed from
   the election; thus, for this assumption to hold, each voter must prefer the
   CW to all members of this second-place Smith set or vice versa. That's
   obviously always true if there is a single second-place CW.

The three assumptions above would probably not strictly hold true in a
real-life election, but they usually would be close enough to ensure that
the system does elect the CW.

SODA does even better than this if there are only 3 candidates, or if the
Condorcet winner goes first in the delegation assignment order, or if there
are 4 candidates and the CW goes second. In any of those circumstances,
under the assumptions above, it passes the *Condorcet* criterion, not just
the majority Condorcet criterion. The important difference between the
Condorcet criterion (beats all others pairwise) and the majority Condorcet
criterion (beats all others pairwise by a strict majority) is that the
former is clone-proof while the latter is not. Thus, with few enough strong
candidates, SODA also passes the independence of clones
criterion<http://wiki.electorama.com/wiki/index.php?title=Independence_of_clones_criterion&action=edit&redlink=1>
.

Note that, although the circumstances where SODA passes the Condorcet
criterion are hemmed in by assumptions, when it does pass, it does so in a
perfectly strategy-proof sense. That is *not* true of any actual Condorcet
system (that is, any system which universally passes the Condorcet
criterion). Therefore, for rationally-strategic voters who believe that the
above assumptions are likely to hold, *SODA may in fact pass the Condorcet
criterion more often than a Condorcet system*.
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