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

Juho Laatu juho.laatu at gmail.com
Fri Sep 24 13:39:22 PDT 2021

> 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

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