# [EM] Two round Condorcet (or maybe 1.01 round)

Forest Simmons forest.simmons21 at gmail.com
Fri Sep 24 20:21:06 PDT 2021

```Here's a slightly different approach ...

If there is no Condorcet winner in the first round, publish the inclusive
pairwise matrix M where the entry in row i and column j is the number of
ballots on which candidate i is ranked ahead of or equal to candidate j.
Note that M(i, i) is just the implicit approval of candidate i.

The random favorite probabilities are also published along with who the
winner would have been under winning votes (wv), margins, or losing votes
(lv) as measures of defeat strength (or lack thereof).

After deliberation the candidates vote (with weights proportional to the
probabilities mentioned above) to determine whether wv, margins, or lv will
be used to resolve a top cycle (should there be one) in the final round.

Note that wv and lv are decisive in their respective choices of B and C,
for the following ballot profile ....

49 C
26 A>B
25 B

whereas margins is ... well ...marginal.

In the case where sincere preferences were 49 C>B instead of 49C, a
majority of 51 would prefer wv since it elects the sincere CW.

In the case where sincere preferences are 25 B>A, a majority of 75 would
prefer that B not win ... and since Plurality prohibits A from winning this
means that those 75 prefer that C win. That majority has power to choose
losing votes as the measure of defeat strength (or lack thereof).

So in both cases a clear majority has the power to defend the sincere CW
... in the first case electing B despite the attack of the minority C
faction, and in the second case to motivate the potential defectors to stay
loyal to A, the sincere CW in that case.

I suspect that in general an appropriate choice among wv, margins, and lv
can greatly facilitate defending a sincere CW against a strategic attack
that would subvert the sincere result if not checked.

Once the voters learn that their insincere ballot manipulations are likely
to be futile if not counter productive, the first round will usually (if
not 99.9% of the time) suffice .... no second poll necessary.

Now let's look at margins in the ballot profile

48 C
28 A>B
24 B

The top cycle is ...

A>B (28 - 24)
B>C (52 - 48)
C>A (48 - 28)

B and C are still the clear respective winners under wv and lv, while the
margins winner is unclear since A defeats B by the same four point margin
as B's defeat of C.

Note that losing votes gives priority to pairwise defeats having the least
opposition to the decision ... the fewest "hard feelings" so to speak...
while winning votes gives priority to pairwise victories having the most
enthusiastic support.

Arguments about which is better based on these considerations are beside
the point ... in a given election the one that best defends the sincere
Condorcet Winner is the one we need!

FWS

El vie., 24 de sep. de 2021 1:39 p. m., Juho Laatu <juho.laatu at gmail.com>
escribió:

>
> > On 23. Sep 2021, at 18.10, Kristofer Munsterhjelm <km_elmet at t-online.de>
> wrote:
> >
> > On 23.09.2021 09:39, Juho Laatu wrote:
> >> What do you think of this rather simple method?
> >>
> >> (1) Arrange a Condorcet election
> >> (2) If there is a a Condorcet winner / no top loop / no tie, elect the
> first round winner
> >> (3) Otherwise arrange a second round, using the same Condorcet variant,
> and elect the second round winner
> >>
> >> The idea is to encourage sincere voting at the first round. And to
> >> reduce the need for defensive strategic voting at the first round,
> >> against strategies that make use of creating a top loop. And to
> >> discourage people that are planning to win by voting strategically.
> >>
> >> Rule (2) could be also such that any first round winner will be
> >> elected, unless there is a formal request to arrange the second round.
> >> The request should maybe be based on a claim that there indeed was
> >> insincere voting that may have influenced the outcome.
> >>
> >> In rule (2) a tie / no Condorcet winner could mean also having
> >> multiple equally strong winners. Maybe we would go automatically to the
> >> second round in this case (instead of picking one of the winners at
> >> random), even if there was no insincere voting.
> >>
> >> Round two uses the same (sub)method as in the first round to remove
> >> any interest of trying to go to the second round because of better
> >> chances of winning with that (sub)method.
> >
> > If you use your first suggestion, then every Condorcet method is equal
> > in the eyes of point 2, so "using the same submethod" doesn't really
> > make sense.
>
> Yes. If the second round will be always held when there is no Condorcet
> winner, then the first round only checks if there is a Condorcet winner.
>
> > I suppose you're referring to the case where step one is
> > e.g. "Arrange a minmax election" and then round two only happens if
> > enough people complain about it.
>
> Yes. The finer details of the first round are relevant only if the
> election may make the final decision already after the first round also
> when there is no Condorcet winner.
>
> The method might allow petitions only when there is no Condorcet winner,
> and one presents a scenario and claim (mandatory!) that the outcome of the
> election may have changed because of strategic voting (or there is an exact
> tie). Petitions could be accepted only from the candidates. The idea is
> thus to have second rounds only when they appear to be necessary, i.e. not
> often, and not because people just want to have a second try. The criteria
> for having the second round should reflect this target.
>
> >
> > From a utility perspective, such a method *could* work, because the
> > voters' willingness to petition for a second round is what enables it to
> > distinguish an okay winner from a truly horrible one.
>
> With respect to utilities, it is true that Condorcet focuses on majorities
> and not utilities, and therefore there might be cases where voters have
> some interest in changing the winner to some "higher utility winner". I
> have no good idea on how to accommodate such original sincere preferences,
> but I sort of hope that if one of the candidates or his supporters try to
> fool the system at the first round, people will get angry, and want to
> punish that candidate at the second round, and will rank him lower in a
> large number of ballots (thereby demonstrating changes in candidate
> utilities between the two rounds).
>
> > However, if I were
> > to argue against it, I would probably say that voting isn't narrowly
> > utilitarian to begin with (the well-known paradox of voting), and that
> > the mechanism is complex enough that it might lead to too much of a
> > status quo bias.
> >
> > Or to put it differently, the idea is that there's enough of a
> > barrier/threshold to petitions, then a second round would only happen
> > when the first-round winner is a bad one.
>
> Yes. My target is that people take the first round seriously, and feel
> safe to vote sincerely, since the possibility of the second round protects
> them. As a result almost all elections would not need the second round. The
> second round would be arranged only when the first round "is a bad one"
> (typically because of foul play). Typically a second round would be needed
> only when there is a (quite rare) top loop, and some serious strategic
> voting (burying), or serious fear of it.
>
> > But it's not obvious how this
> > threshold should be adjusted. In one extreme case, you always hold a
> > runoff whenever the Smith set is larger than one. In another, it takes a
> > supermajority to hold the runoff.
>
> Yes, it is important to adjust the second round threshold and criteria
> carefully. My target was that the second round would be arranged only when
> there is a clear risk that the end result changed because of strategic
> voting (or when there is an exact tie that we prefer to resolve this way).
> The best outcome would thus be a method that would on average have about
> 1.001 rounds, and 99% of the votes would be sincere. The threat of second
> round would be there just to keep the first round "civilised".
>
> > I would be more in favor of the former
> > than the latter, particularly if there's some time between the two
> > rounds so that the Smith set candidates have to defend themselves,
> > participate in debates, etc.
>
> And also other candidates than the Smith set candidates. :-)
>
> You could see the second round also only as a technical round where you
> consider the candidates to be already known, and you reserve time only for
> all the strategic discussions and discussions on what went wrong at the
> first round. Also here I'm hoping that the strategic candidates will be
> ranked lower at the second round (but realising that also some false claims
> of strategic voting might appear :-) ).
>
> >
> > Another option I suggested to Forest goes like this:
> >
> > 1. Arrange a Condorcet election with a strategy-resistant method and a
> > method wth good performance under honesty, e.g. Smith,IRV and Ranked
> > Pairs (probably something even better can be constructed).
> > 2. If the winners are the same (which happens whenever there's a CW, but
> > also in some other cases), then that winner wins outright.
> > 3. Otherwise arrange a two candidate runoff between the two winners.
>
> In both approaches it makes sense to use a method with good performance
> under honesty at the first round.
>
> If there was a strategic three candidate cycle at the first round, maybe
> people would reject the candidate whose supporters tried to bury others,
> and it would be just a race between the other two top loop candidates, even
> if also the second round would use a ranked Condorcet method. Using a
> runoff between two candidates would make the second round free of strategic
> voting, but might introduce some strategic voting in the first round. A
> carefully chosen pair of methods at the first round might eliminate
> strategies nicely, but I'm not able to estimate right now, how good the
> combination that you presented would be from this point of view.
>
> The philosophy of the two approaches is different. In your proposal the
> idea seems to be to use two methods to cancel most strategies out, and in
> my proposal to allow people to vote honestly at the first round. There
> would be a chance to vote strategically at the second round if the first
> round fails to be sincere enough. One could of course go on and use three
> rounds, starting from one with ideal performance with sincere votes, and
> then proceeding to some more strategy proof method (or combination of
> methods), and finally to a two candidate runoff. But of course also
> simplicity should have some weight, and this approach might be already too
> complex for most needs.
>
> >
> > Simple, it's not, but the methods could cover each other's weaknesses.
>
> Yes. Have you checked if there are strategic possibilities, trying to push
> the election to the second round, trying to vote so that some appropriate
> two candidates would go to the second round? The most difficult case might
> be one with a sincere three candidate top loop. (Too difficult for me to
> get a grip of that right now. :-) )
>
> > Taking this reasoning further, it should be possible to design "minmax
> > robust methods" that return two candidates so that the VSE after a
> > honest two-candidate runoff is the best possible, no matter the level of
> > strategy.
>
> From majority perspective, assuming a natural top loop of three
> candidates, picking two of them (or some others) for the second round must
> be hard. (VSE = Voter Satisfaction Efficiency)
>
> >
> > Well, in theory. Actually designing such a method with desirable
> > properties (monotonicity, etc) would be much harder :-)
>
> Yes. Strategies are complex, and having two methods trying to fight
> together against various combinations of strategies, with different
> combinations of defences, makes it even more complex. You could say that I
> took the easy way out when proposing an approach where we just hope that
> people are sensible enough and do not use strategies, under the protection
> of the second round, under the threat of having to vote also at the second
> round, having their strategic plans revealed to other voters already at the
> first round, and paying a price if voters don't like their strategic games,
> and fearing that voters will bullet vote some better and more sincere
> candidate, or at least not vote for the strategists, or have some working
> counter strategies at the second round. :-)
>
> In some societies people might be more inclined to use strategies and fear
> strategies of others more than in some other societies. The approach that I
> proposed hopefully makes it one step easier to vote honestly in both cases.
> In the best case it would be just a standard Condorcet election with no
> need to ever resort to the second round.
>
> BR, Juho
>
>
> >
> > -km
>
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>

El 24 sep. 2021 1:39 p. m., "Juho Laatu" <juho.laatu at gmail.com> escribió:

> On 23. Sep 2021, at 18.10, Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:
>
> On 23.09.2021 09:39, Juho Laatu wrote:
>> What do you think of this rather simple method?
>>
>> (1) Arrange a Condorcet election
>> (2) If there is a a Condorcet winner / no top loop / no tie, elect the
first round winner
>> (3) Otherwise arrange a second round, using the same Condorcet variant,
and elect the second round winner
>>
>> The idea is to encourage sincere voting at the first round. And to
>> reduce the need for defensive strategic voting at the first round,
>> against strategies that make use of creating a top loop. And to
>> discourage people that are planning to win by voting strategically.
>>
>> Rule (2) could be also such that any first round winner will be
>> elected, unless there is a formal request to arrange the second round.
>> The request should maybe be based on a claim that there indeed was
>> insincere voting that may have influenced the outcome.
>>
>> In rule (2) a tie / no Condorcet winner could mean also having
>> multiple equally strong winners. Maybe we would go automatically to the
>> second round in this case (instead of picking one of the winners at
>> random), even if there was no insincere voting.
>>
>> Round two uses the same (sub)method as in the first round to remove
>> any interest of trying to go to the second round because of better
>> chances of winning with that (sub)method.
>
> If you use your first suggestion, then every Condorcet method is equal
> in the eyes of point 2, so "using the same submethod" doesn't really
> make sense.

Yes. If the second round will be always held when there is no Condorcet
winner, then the first round only checks if there is a Condorcet winner.

> I suppose you're referring to the case where step one is
> e.g. "Arrange a minmax election" and then round two only happens if
> enough people complain about it.

Yes. The finer details of the first round are relevant only if the election
may make the final decision already after the first round also when there
is no Condorcet winner.

The method might allow petitions only when there is no Condorcet winner,
and one presents a scenario and claim (mandatory!) that the outcome of the
election may have changed because of strategic voting (or there is an exact
tie). Petitions could be accepted only from the candidates. The idea is
thus to have second rounds only when they appear to be necessary, i.e. not
often, and not because people just want to have a second try. The criteria
for having the second round should reflect this target.

>
> From a utility perspective, such a method *could* work, because the
> voters' willingness to petition for a second round is what enables it to
> distinguish an okay winner from a truly horrible one.

With respect to utilities, it is true that Condorcet focuses on majorities
and not utilities, and therefore there might be cases where voters have
some interest in changing the winner to some "higher utility winner". I
have no good idea on how to accommodate such original sincere preferences,
but I sort of hope that if one of the candidates or his supporters try to
fool the system at the first round, people will get angry, and want to
punish that candidate at the second round, and will rank him lower in a
large number of ballots (thereby demonstrating changes in candidate
utilities between the two rounds).

> However, if I were
> to argue against it, I would probably say that voting isn't narrowly
> utilitarian to begin with (the well-known paradox of voting), and that
> the mechanism is complex enough that it might lead to too much of a
> status quo bias.
>
> Or to put it differently, the idea is that there's enough of a
> barrier/threshold to petitions, then a second round would only happen
> when the first-round winner is a bad one.

Yes. My target is that people take the first round seriously, and feel safe
to vote sincerely, since the possibility of the second round protects them.
As a result almost all elections would not need the second round. The
second round would be arranged only when the first round "is a bad one"
(typically because of foul play). Typically a second round would be needed
only when there is a (quite rare) top loop, and some serious strategic
voting (burying), or serious fear of it.

> But it's not obvious how this
> threshold should be adjusted. In one extreme case, you always hold a
> runoff whenever the Smith set is larger than one. In another, it takes a
> supermajority to hold the runoff.

Yes, it is important to adjust the second round threshold and criteria
carefully. My target was that the second round would be arranged only when
there is a clear risk that the end result changed because of strategic
voting (or when there is an exact tie that we prefer to resolve this way).
The best outcome would thus be a method that would on average have about
1.001 rounds, and 99% of the votes would be sincere. The threat of second
round would be there just to keep the first round "civilised".

> I would be more in favor of the former
> than the latter, particularly if there's some time between the two
> rounds so that the Smith set candidates have to defend themselves,
> participate in debates, etc.

And also other candidates than the Smith set candidates. :-)

You could see the second round also only as a technical round where you
consider the candidates to be already known, and you reserve time only for
all the strategic discussions and discussions on what went wrong at the
first round. Also here I'm hoping that the strategic candidates will be
ranked lower at the second round (but realising that also some false claims
of strategic voting might appear :-) ).

>
> Another option I suggested to Forest goes like this:
>
> 1. Arrange a Condorcet election with a strategy-resistant method and a
> method wth good performance under honesty, e.g. Smith,IRV and Ranked
> Pairs (probably something even better can be constructed).
> 2. If the winners are the same (which happens whenever there's a CW, but
> also in some other cases), then that winner wins outright.
> 3. Otherwise arrange a two candidate runoff between the two winners.

In both approaches it makes sense to use a method with good performance
under honesty at the first round.

If there was a strategic three candidate cycle at the first round, maybe
people would reject the candidate whose supporters tried to bury others,
and it would be just a race between the other two top loop candidates, even
if also the second round would use a ranked Condorcet method. Using a
runoff between two candidates would make the second round free of strategic
voting, but might introduce some strategic voting in the first round. A
carefully chosen pair of methods at the first round might eliminate
strategies nicely, but I'm not able to estimate right now, how good the
combination that you presented would be from this point of view.

The philosophy of the two approaches is different. In your proposal the
idea seems to be to use two methods to cancel most strategies out, and in
my proposal to allow people to vote honestly at the first round. There
would be a chance to vote strategically at the second round if the first
round fails to be sincere enough. One could of course go on and use three
rounds, starting from one with ideal performance with sincere votes, and
then proceeding to some more strategy proof method (or combination of
methods), and finally to a two candidate runoff. But of course also
simplicity should have some weight, and this approach might be already too
complex for most needs.

>
> Simple, it's not, but the methods could cover each other's weaknesses.

Yes. Have you checked if there are strategic possibilities, trying to push
the election to the second round, trying to vote so that some appropriate
two candidates would go to the second round? The most difficult case might
be one with a sincere three candidate top loop. (Too difficult for me to
get a grip of that right now. :-) )

> Taking this reasoning further, it should be possible to design "minmax
> robust methods" that return two candidates so that the VSE after a
> honest two-candidate runoff is the best possible, no matter the level of
> strategy.

>From majority perspective, assuming a natural top loop of three candidates,
picking two of them (or some others) for the second round must be hard.
(VSE = Voter Satisfaction Efficiency)

>
> Well, in theory. Actually designing such a method with desirable
> properties (monotonicity, etc) would be much harder :-)

Yes. Strategies are complex, and having two methods trying to fight
together against various combinations of strategies, with different
combinations of defences, makes it even more complex. You could say that I
took the easy way out when proposing an approach where we just hope that
people are sensible enough and do not use strategies, under the protection
of the second round, under the threat of having to vote also at the second
round, having their strategic plans revealed to other voters already at the
first round, and paying a price if voters don't like their strategic games,
and fearing that voters will bullet vote some better and more sincere
candidate, or at least not vote for the strategists, or have some working
counter strategies at the second round. :-)

In some societies people might be more inclined to use strategies and fear
strategies of others more than in some other societies. The approach that I
proposed hopefully makes it one step easier to vote honestly in both cases.
In the best case it would be just a standard Condorcet election with no
need to ever resort to the second round.

BR, Juho

>
> -km

----
Election-Methods mailing list - see https://electorama.com/em for list info
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