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

Mike Ossipoff dfb at bbs.cruzio.com
Mon Jul 15 05:41:31 PDT 1996

```Hugh R. Tobin writes:

--Mike]

>
> 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

I don't agree with that one. I claim order-reversal is deterred
in Condorcet.

> 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.)

The situation between the would be order-reversers & their intended
victims is such that order-reversal wouldn't be rational, but
could be expected to result in a worse result for the order-reversers
than would have otherwise occurred. As I've said, you can't say
that the Dole voters are strategically sophisticated & inclined
to strategize, and say that the Clinton voters aren't. And the
strategic situation in the game of "chicken" that we're talking
about is completely in the favor of the Clinton voters. They're
the ones whose threat has credibility, the ones who'd suffer the
least if both sides carried out their threats.  And, as if that
weren't enough, the Clinton voters gain additional credibilitly from
the fact that they're the ones defending the Condorcet winner
from cheating, and the Dole voters can't claim any principled
determination to carry out their cheating.

>
> 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

Your error, as I've said, is that you're assuming that the Plurality
voters are strategically sophisticated, well-informed by polls,
and inclined to strategize, but for some reason you're assuming
that the Middle voters aren't.

Sure, if the Plurality voters could count on sincere voting
from their victims, it would be shooting fish in a barrel.
For the reason stated in the previous paragraph, that isn't
a realistic scenario.

> 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.

The argument below is well-put, but it ignores that there will
be non-strategically-intended truncation in Smith//Random. And
when that creates a circular tie, the truncators' candidate, in
a 3-candidate race, will win 1/3 of the time. In Condorcet
that would happen never.

Since, non-strategic truncation is the real problem, not
order-reversal, in the elections in which I've participated,
and will probably be the problem in public elections, it certainly
isn't enough to show that offensive strategy isn't a problem
in Smith//Random, if non-strategic truncation is still
a big problem.

It's like the situation in MPV (Instant Runoff): In Instant
Runoff there's no problem with offensive strategy. But before
you rejoice, I have to break it to you that there's a big
problem _without_ any offensive strategy. Same in Smith//Random.

And if you say that non-strategic truncation isn't going
to happen in a 3-candidate race, I reply that Condorcet
is completely free of any strategy dilemma in a 3-candidate
race, for reasons that I've described in previous letters--
the Middle voters would have no reason to ever vote a 2nd
choice, and, knowing that, no rational person would attempt
order-reversal strategy.

When non-strategic truncation occus in Smith//Random,
it can happen when it would make a circular tie, and
elect the truncators' candidate, and make the opposite
wing regret that they didn't vote Middle in 1st place.
As I said, that's exactly what we want to avoid.

>
> 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

Sorry, but that doesn't help us, the fact that the drastic
give-away defensive strategy by R voters can elect the Condorcet
winner. Of course it can, even in the worst method. But the need
to do that is exactly what we're trying to avoid. That's what
it means to get rid of the LO2E problem.

> 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

I haven't said anything about where Middle is, in the
examples. It doesn't matter, with Condorcet.

Actually, in the Dole, Clinton, Nader example, it's obvious
that Clinton is considerably closer to Dole than to he is
to Nader. This means that, with MPV, the Clinton 2nd place
1st, Dole wins. The LO2E problem is very much still there
in MPV.

> 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

But we do need assume that, while the Dole voters are sophisticated
& well-informed strategizers, the Clinton voters are uninformed
patsies. As I said, the order-reversal strategy situation in
Condorcet is a 2-sided game of chicken, and the Clinton voters
have the upper hand.

You're making unsupportable assumptions about different
sophistication & well-informed-ness of the Dole & Clinton
votes, & you're ignoring the 2-sided nature of the
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)

Again, you talk as if only the Middle voters have a strategy
decision, and as if the order-reversal by the Plurality wing
is a done deal. No, it's a 2-sided situation where both
sides have a threat, and it's a question of which threat
is more credible, and which side has the upper hand.

>
> 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.

Correct. And non-strategic truncation will be very widespread
in many-candidate elections, and Condorcet has Smith//Random
thoroughly beat in that situation.

>
>
> > -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

Again, you're setting up a special situation where the order-reversers
are strategically-sophisticated & well-informed, and their
would-be victims aren't.

> 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.

They make their own candidate prevail by ensuring that the order-reversal
won't take place, due to its expected backfiring result.

>
> > >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.

It has more credibility than the threat of the would be order-reversers
to give the election to the opposite wing.

> >
> > 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

Yes, if the Dole voters are sophisticates & the Clinton voters are
suckers. You can assume any vote configuration that you want to
but the one you propose is unreasonable.

>
> 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

If N is anathma to the C voters, then N will be more than anathma
to the D voters. The threat by the D voters to give the election
to N seems problematic. You aren't looking at both sides to
this threat situation, but are considering the C voters as
having a decision, and the D voters' offensive strategy as a
sure thing.

> 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.

None of us want N to have to do that, which is the problem with
Smith//Random & the compulsory or default false-votes.

>
> 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.

There are features that could be added to Condorcet to further
thwart order-reversal, but since order-reversal is already
so well-deterred, and since added features can make a method
seem complicated to the public, I don't propose these added
features for Condorcet.

If the time ever came when we had a devious electorate and
voters wanted more _deluxe_ anti-order-reversal countermeasures,

>
> >
> > >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

Quite so. And situations where non-strategic truncation occurs, as
it surely will, on a large scale, in many-candidate elections.

> 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.

Well that's the alternative that I advocate. As I said, optional
a can of worms when proposing the method to the public. If people
wanted the optional false contrary half-votes, and if it was what
the public liked, I wouldn't oppose it. But I advise against
proposing it to the publc, because it serves no purpose, in
regards to standards that are met, and because it would lead
to a longer more complicated method, with unnecessary options,
and difficult explanation for why we're proposing it that way.
And it would immediately introduce voters to falsification, something
that otherwise might not occur to people for a long time. We
really don't want the ballot to offer voters the choice to falsify.
Let's not offer it to them officially like that. If they want
to mis-represent their preferences, let them figure it out on
their own.

>
> 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

Solely because they didn't falsify preferences and create a circular
tie when there's a Condorcet winner, and thereby steal the election
from that Condorcet winner? In any case, it wouldn't work anyway,
that falsification, for reasons that I've been discussing. But it
would force other voters to use defensive strategy when they
otherwise wouldn't need to. I don't understand how you consider
that an improvement. Reliably electing the Condorcet winner,
defeating a greater evil without defensive strategy, is something
takes that advantage away. Are you sure you want to do that?

> "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.

No, what it does is: When compulsory or the default assumption,
it creates a need for drastic defensive strategy, a need that
otherwise wouldn't exist in Condorcet's method, a need to
rank a less-liked alternative equal to or over a more-liked
one. And it creates a need for defensive strategy under ordinary
conditions, where Condorcet, without that feature, only needs
defensive strategy when there's order-reversal on an implausibly
large scale.

And when the false-half-votes are an option, you're offering
the voters, officially, on the ballot, the choice to  falsify
their preferences, to cast votes that don't represent a genuine
preference, and you're thereby creating a situation where,
due to falsification, defensive strategy will be needed, if
voters use the option that you offer them.

Though the compulsory or default false-half-votes is worst,

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

--

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