# [EM] SODA and the Condorcet criterion

fsimmons at pcc.edu fsimmons at pcc.edu
Fri Aug 5 12:01:13 PDT 2011

```Jameson,

as you say, it seems that SODA will always elect a candidate that beats every other candidate majority
pairwise.  If rankings are complete, then all pairwise wins will be by majority.  So at least to the degree
that rankings are complete, SODA satisfies the Condorcet Criterion.

Also, as I mentioned briefly in my last message under this subject heading, SODA seems to completely
demolish the "chicken" problem.

48 A
27 C>B
25 B>C

Candidates B and C form a clone set that pairwise beats A, and in fact C is the Condorcet Winner, but
under many Condorcet methods, as well as for Range and Approval, there is a large temptation for the
25 B faction to threaten to truncate C, and thereby steal the election from C.  Of course C can counter
the threat to truncate B, but then A wins.  So it is a classical game of "chicken."

Some methods like IRV cop out by giving the win to A right off the bat, so there is no game of chicken.
But is there a way of really facing up to  the problem?  i.e. a way that elects from the majority clone set
by somehow diffusing the game of chicken?

The problem is that in most methods both factions must decide more or less simultaneously.  However,
if the decisions can be made sequentially, then the faction that "plays" first can safely forestall the
chicken threat of the other.  That's one reason that it makes sense for SODA to have the candidates
play sequentially, and to have the strongest candidate of a clone (or near clone) set go before the other
candidate or candidates in the clone set.

Since DAC is designed to pick out the strongest candidate in the plurality winner clone set, it is a
natural for setting the order of play (in the sophisticated version of SODA).

Another way to solve the chicken problem is to not allow truncations.  But in SODA it seems essential
to allow the candidates to truncate.  However there is a pressure  for the candidates to not truncate too
high up in the rankings; if they do, they lose credibility with their supporters, so fewer of them will
delegate their approval decisions to them.

Since having complete rankings helps both in chicken and with regard to the Condorcet Criterion, it
might be worth using the implicit order in the approval ballots of the supporters of candidate X to
complete X's rankings by using that implicit order to rank the candidates truncated by X (or otherwise
ranked equal by X).

This would discourage X from too much truncation, and would make it more likely that the true CW was
elected in the (usual?) case where there is one.

Forest

> From: Jameson Quinn
> To: EM
> Subject: [EM] SODA and the Condorcet criterion
> Here's the new text on the SODA
> page> Delegated_Approval#Criteria_Compliance>relatingto the Condorcet
> criterion:
> It fails the 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
> Condorcetcriterion (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
> .
>
> 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
> Condorcetsystem (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
> Condorcetcriterion more often than a Condorcet system*.

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