# Order-Reversal in S//C vs. S//R and other light topics

Hugh R. Tobin htobin at ccom.net
Sun Jul 14 17:17:35 PDT 1996

```Steve Eppley wrote:
>
> There are a couple statements in Hugh's ballot explanation for which
> my understanding differs, so I want to explore them.
>
> Hugh Tobin wrote:
> -snip-
> >In Smith//Random the temptation to strategic voting would be least,
> >because success in creating a circular tie would generate at best a
> >one-third chance of electing one's first choice.
>
> Those odds (1/3) may be enough of a temptation.  It's true there
> would also be a 1/3 chance of electing an even worse candidate, but
> what about the cases where the voter thinks the sincere-Condorcet
> winner is nearly as bad as the worse candidate?  Suppose a voter
> has the following opinions on the three candidates Left, Middle,
> and Right, expressed on a scale of -10 to +10:
>    L= +10, M= -8, R= -10
> And suppose the pre-election poll data indicates M will beat both the
> others pairwise if voters vote sincerely.  If M wins, this voter
> evaluates the outcome as -8.  If there's a random draw, this voter's
> expectation is (1/3 * +10) + (1/3 * -8) + (1/3 * -10) = -8/3.  This
> is much better than -8.  So why shouldn't this voter order-reverse,
> voting L > R > M in hopes of creating a circular tie?
>
> And what will the supporters of the other candidates do?  Some will
> be aware of a lesser of evils dilemma: the supporters of R may feel
> scared to vote sincerely (R > M > L) because of the possibility that
> L will randomly win a circular tie, so they may use the defensive
> strategy of voting M more preferred than their true preference.
>
> It's hard to quantify how often there will be strategic voting
> in Smith//Condorcet and Smith//Random.  To me it looks like
> Smith//Condorcet (and Condorcet) will have the least, not
> Smith//Random, because the "votes against" tiebreaker is such a
> good deterrent.
>

Nobody has come forth to advocate Smith//Random, so here are my thoughts:

I take your points that (a) the middle may not be considered half as good
as the opposite extreme, and (b) that it is hard to quantify how often,
if at all, order-reversal would occur under either system.  Still, it
seems to me that order-reversal strategies would be rational under
Condorcet or Smith//Condorcet across a wider range of plausible
distributions of voter preferences than in Smith//Random.  (This does not
mean I think order-reversal would be common in any of these systems.)

I disagree that the "votes-against" tiebreaker is a good deterrent.
Order reversal will likely occur only if the wing plurality thinks it has
very good polling data, and thus can predict a circular tie if enough of
the plurality ranks the opposite wing second.  In Smith//Condorcet or
plain Condorcet, this same polling data allows the plurality to calculate
whether, if the other voters vote their true preferences, the plurality
wing candidate will win the circular tie.  Thus, the strategy may have a
high probability of success (for example, it could be 9/10 of creating
the tie and 9/10 of winning it once created, or .81).  With a high
probability of success in electing one's first choice, one would employ
the strategy despite a much stronger true preference for the middle
compared to the opposite extreme, compared to the degree of preference
that would make such a strategy rational in Smith//Random, where the
probability of winning the election by order-reversal is always less than
1/3. Moreover, in Smith//Random, in those situations where the
order-reversers would strongly prefer a lottery to electing the middle, a
distribution of voter preferences that will allow them a high likelihood
of getting into the tiebreak seems relatively unlikely, as argued below.

In the example you provide, it seems that the position of middle is much
closer to the opposite wing than to the candidate of the supposed
order-reversers.  In that case, one would expect the middle voters to
rank the opposite extreme (R) second, and that if the L voters do the
same, R will win outright (false Condorcet winner).  The R voters should
not vote M first as a defensive strategy, because that may actually
produce the tie, when otherwise R would have won -- beating M with
insincere help from L voters, and beating L with sincere help from M
voters.  (Even if the R voters could reliably predict that enough M
voters would choose L over R so that L would beat R, and therefore voted
for M, at least this defensive strategy results in the election of the
Condorcet winner).  True, the L voters have much less to lose than they
would if M were closer to their candidate, but I think in most cases they
would have a much greater likelihood of losing before they ever get to
the tiebreak.  The positive expected value that you suggest may be
derived by L voters from getting into the lottery tiebreak must be
discounted for the probability of failing to reach the tiebreak.
For example, if the chance that the tie will be created is only one in
10, and the chance of electing R outright is 9 in 10, then the expected
value of order-reversal is (.9 * -10) + (.1 * -8/3) = -9 4/15. (It is
convenient, I grant you, to be able to choose the probabilities in order
to make one's case).

I conclude that in Smith//Random voters might reverse preferences in a
three-way race only when (a) they have a near-majority; (b) they see
relatively little difference between the two other candidates; and (c)
nonetheless they think there is a significant likelihood that their
candidate will get enough second-place votes from the middle so as to
beat the other wing.  This type of order-reversal would involve less
violent distortion of preferences than the type that I think Ossipoff has
described under Condorcet, where the middle candidate truly in the
middle, or is closer to the order-reversing plurality and therefore would
be much more acceptable to his supporters.  In such a case the wing
plurality, needing substantial support from the middle to avoid being
Condorcet loser, can reasonably expect (and cynically seek to exploit the
fact) that many middle voters will choose their candidate second, over
the other extreme.  We need not assume the somewhat anomalous situation
in your Smith//Random example, where despite the fact that the L voters
all view M being 9/10 of the way to R on the political spectrum,
nonetheless enough M voters will choose L second so that L will beat R.
Finally, the middle voters' only "remedy" in this case is to vote so as
to throw the election to the other extreme -- whose positions may be very
far from acceptable to either the plurality or the middle -- in case the
order-reversal is carried out. (see example below)

So my preference for Smith//Condorcet is based upon the fact that
"least-beaten" (properly defined) seems a fairer and less arbitrary
tiebreaker; my fear that any lottery element may be used by opponents to
deride a reform proposal; and my guess that despite the theoretical
problems of deterring strategic voting under Smith//Condorcet, the
"natural" circular tie would be more common than one engineered by
order-reversal, so that the effect on possible strategies should not
dominate the choice of methods.

> -snip-
>
> As I wrote above, I believe S//R would have more strategic voting
> than S//C, not less, so S//R would be the opposite of a remedy.  If
> there's a candidate who would beat all others pairwise if the voters
> would vote sincerely, S//C will still tend to elect this candidate
> even when another candidate's supporters misrepresent their
> preferences.
>

Would that this were the case.  But I think examples have already been
posted where it is not -- in particular, where if the middle voters
(whose candidate is Condorcet winner) vote their true preferences, and
enough of them prefer the order-reverser's candidate second, that
candidate may win the circular tie.  Moreover, if the middle voters do
not vote their true preferences, then they can assure that the
order-reversal strategy throws the election to the other wing, but cannot
make their own candidate prevail.  See below.

> >9.  In Smith//Condorcet (or plain Condorcet), I am not convinced
> >that opportunities for strategic voting in a 3-way race would be so
> >rare as to be a trivial consideration.  Unless I misunderstand, the
> >"truncation" antidote to order-reversal offered by Mr. Ossipoff is
> >a deterrent that depends on the credibility of a threat by the
> >Condorcet winner that his supporters will vote in a manner (refrain
> >from ranking their true second choice) that cannot help elect their
> >favorite but can help elect their least favored candidate, just to
> >punish the supporters of their second-favored candidate.  This
> >threat may lack credibility.
>
> I've forgotten this scenario; can you post a relevant example?
> Here's the example I remember:
>   The ballots:
>     45: D>N>C   <-- reversal
>     20: C
>     35: N>C>D
>   The pairings:
>     D<C    (45 to 55)
>     C<N    (20 to 80)
>     N<D    (35 to 45)
>   N wins the circular tie using Condorcet (only 45 against in N vs D).
>   The N supporters didn't have to change their votes to counter the
>   D supporters' reversal; the reversal is already deterred by Condorcet.

But suppose that at least 55% of the C voters, 11% of the total, actually
prefer D to N.  (D would not even try order-reversal if D did not believe
something of the sort).  Now if C voters vote sincerely, we get

Ballots:
45: D>N>C   <-- reversal
11: C>D>N
9: C>N>D
35: N>C>D

The pairings:
D<C    (45 to 55)
C<N    (20 to 80)
N<D    (44 to 56)

D is least-beaten, and wins.  The 11 C voters can avoid this only by
truncating or reversing, in either case throwing the election to N, whom
they consider worse than D.  The question, I believe, is whether their
announced intention to do so will suffice to deter enough of the D voters
from reversing so that C will win.  This may depend in part on how many
of the C voters really prefer D, and how strongly, over N.  If N is
believed to be anathema to all 20 C voters, then successful deterrence
seems problematic.  As you suggest in your S//R example, the N voters
could rescue C as Condorcet winner if over 30/35 of them vote C first (at
least if no C voters vote D first out of fear than N will win a circular
tie!), but in practice I suspect N would have to drop out of the race and
get his name off the ballot for this to happen.  I don't think N will do
that (even if the election law and logistics allow it) after all that N
has invested in the race, especially given the possibility that D's
gambit will actually elect N.

Still, when I say that strategic order-reversal under S//C is not a
trivial consideration, this does not mean that I think it is important
enough so that any other method I have seen is preferable -- given that
none are immune from strategy (except "Random", by itself).  It means
only that I think possible modifications of S//C to reduce strategic
opportunities are worth discussing.

>
> >In each pairwise contest between X and Y, count as 1/2 vote for X and
> >1/2 vote for Y an equal ranking of X with Y by a voter, if that voter
> >ranked all other members of the Smith set ahead of X and Y.
> -snip-
>
> This proposal is new to me.  I think we should spend time examining
> it.  It would certainly complicate the definition of the method, and
> considerably increase the time it takes to tally the ballots (since
> it requires an extra pass to calculate the Smith set before the tied
> rankings), but perhaps its properties are important enough to justify
> that.
>

I have addressed this in other postings, which, I fear, dwell too much on
examples where strategic or insincere voting occurs.  I think those
situations are of less importance than how the system works when voters
simply express their actual preferences, and non-preferences, on the
ballot.  Given modern technology the additional computation seems a minor
concern.  I am more concerned about the complications involved in
providing voters an option in regard to half-votes, and related
explanations, when the half-votes are of no significance in the normal
case when we have a Condorcet winner.  In making the proposal I assumed
that there would be a fixed tiebreak rule, not a choice of variations by
each voter, and that the alternative would be counting the equal rankings
as zero in all cases.

The main point of the proposal is to have the tiebreak system conform
more closely to the sincere voter's actual preferences (and
non-preferences) as revealed by his ballot, without the voter having to
understand circular ties or the tiebreak system.  In particular, the
truncating voter in a 3-way race would not want his favorite candidate to
fail to be "least-beaten" soley because he and other supporters of that
candidate truncated rather than voting equal numbers of second
preferences for and against the other two candidates.  Compared to the
"zero option", I think one of its properties is that it reduces a
theoretical incentive for a voter who understands the system, and
believes a circular tie is possible, to rank candidates at random rather
than truncating when he has no preferences between two disfavored
candidates and no reliable information as to which of them is likely to
beat the other.

-- Hugh Tobin

> ---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

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