# [EM] SODA and the Condorcet criterion

Jameson Quinn jameson.quinn at gmail.com
Fri Aug 5 12:45:18 PDT 2011

```2011/8/5 <fsimmons at pcc.edu>

> 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.
>

Well.... almost. See below.

>
>
> 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.
>

That is a key point. The other aspect of SODA is that it allows candidate A
to change their rankings after they see candidate B's rankings. In the rules
on the SODA page, it is deliberately left vague how many recursions of that
are possible, or what the exact rules are there. One possible rule would be
some form of binding "I'll prefer you if you'll prefer me" declaration, to
avoid recursion. The point is that, however the rule works formally, that no
candidate will ever get caught by surprise, and so any candidate can make a
credible threat: I'll truncate you if you truncate me.

That is not to say that all inter-candidate preferences would be mutual.
Just that if both sides agreed that mutual preference was appropriate, as in
a true case of near-clones, there could be no "sneak attacks" of truncation.
In other cases, truncation threats between candidates would be almost
certainly inneffective... "OK, go ahead and truncate me, I don't like you
anyway.". The candidate making the threat is either weaker (in which case
they have no reason to make it, because they'll never get the transferred
votes) or stronger (in which case they have no reason because they'll never

So, I'm satisfied that SODA has enough safeguards against over-truncation by
candidates, which helps resolve the "chicken problem".

However. SODA still does not completely eliminate this problem. Individual
voters, by voting explicitly non-delegated bullet votes, still can truncate,
if they realize it works. That is much less likely, because a "lazy" voter
will delegate by default. After all, If Fsimmons does not see this strategy
possibility, how many normal voters will? Still, I must admit, it's
possible.

I've thought of ways to resolve it, but I don't see any easy, simple ones.
It is absolutely not an option to keep voters from casting non-delegated
votes which were originally delegated to another) are weighted by D/(D+U),
where D is that candidate's delegated total and U is that candidate's direct
undelegated approval total. This does a good job at fixing the
voter-truncation chicken problem - but it makes the system badly
nonmonotonic. Any candidate who received more second-hand than first-hand
votes would have their final total reduced for each direct approval they'd
gotten! So you could fix the fix, ensure so that second-hand votes beyond
D+U were weighted fully... but by now, you could certainly no longer call
the system SODA, it would become CODA.

So I think the best thing to do is just ignore this vestigal chicken
problem.

>
> 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).
>

Ugh. The big problem with this is that approval-style votes for a candidate
will be, by definition, from voters who disagree with that candidate's
actual ordering. Also, as a small group, it would be very vulnerable to
hijacking, at little cost.

>
> 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.
>

Yes, I sympathize with the goal. But I can't see how to achieve it without
inventing CODA.

JQ

>
> 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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