[EM] Heitzig consensus and brinkmanship

Kristofer Munsterhjelm km_elmet at t-online.de
Wed Jul 29 10:44:59 PDT 2020

On 16/07/2020 02.02, Kevin Venzke wrote:
> Hi Kristofer,
> Le mardi 14 juillet 2020 à 18:40:19 UTC−5, Kristofer Munsterhjelm
> <km_elmet at t-online.de> a écrit :
>> I was thinking about the possibility of using the mechanism to direct a
>> government or organization to act in a minmax manner: one that intend to
>> make the worst off best off, rather than improve the condition of the
>> median voter.
> Just to interject quickly. To my mind these two things are (naively)
> the same, and if results matched the preference of the median voter
> you would have a good thing. What I expect instead, with two factions
> fighting over who can capture a majority, is that the factions don't
> propose (or don't enact) the median position. They want the vote of
> that position, and those voters can come to the victory party, but
> they won't be in control.

Doesn't the pizza election show that these are not the same? Suppose the
utilities are:

7 voters: Pepperoni 9, Mushroom 8
3 voters: Pepperoni 0, Mushroom 9

The median voter prefers pepperoni. But a minmax outcome is the one that
leaves the worst-off voter best off, and that's mushroom. In this case,
the Heitzig consensus fails to deliver minmax, because the 70%
supermajority prefers a random ballot to mushroom. But if it's
two-sided, say:

6 voters: Pepperoni 9, Mushroom 8, Anchovies 0
2 voters: Anchovies 9, Mushroom 8, Pepperoni 0
2 voters: Anchovies 0, Mushroom 9, Pepperoni 0

then the outcome of a random ballot is 0.6 * Pepperoni + 0.2 * Anchovies
+ 0.2 * Mushroom. The expected score is thus:

To the group of 6 pepperoni voters: 7.0
To the group of 2 anchovy voters: 3.4
To the group of 2 mushroom voters: 1.8

and everybody prefers mushroom to this, so it's in everybody's interest
to choose mushroom as the consensus. Hence the minmax option wins, but
in a majoritarian election method or a strategic Range election,
Pepperoni wins.

>> From what I remember, Jobst and Forest were originally
>> trying to find a method to avoid a majority dictatorship, so my idea is
>> in a way to consistently take that to its logical conclusion. If the
>> state or the organization must pay attention to every voter, or to a
>> supermajority of them, then it can't afford to leave some of them
> badly off.
>> But if it's to be used as a part of normal operating procedure, then it
>> has to resist strategy to some degree, and it can't take the whole
>> organization or state down with it at the first sign of trouble. So if
>> the brinkmanship scenario is a problem, then either the mechanism has to
>> be augmented to stop it being a problem, or the assembly has to somehow
>> be able to keep the peace enough that politics will never become that
>> contentious to begin with.
> It seems like a tall order...

Yes. I don't know of any other mechanisms that come as close as it does
to implementing minmax, so it would be really nice if it could be made
to work.

>> Yes, that is a possibility - that a way out is to make the consensus
>> option at least as good on expectation as the roll of the dice,
>> discounted by whatever risk aversion exists.
>> That's an important point, I think. The consensus option doesn't have to
>> be extremely good. For it to be chosen, it just has to be preferred to
>> rolling the dice by everyone. If it's barely better, that's still good
>> enough to make it pass.
> I think that may be true (if we rule out, as I say, a value to being 
> perceived as unwilling to compromise), but I wonder how often such a
> consensus option could be expected to exist? I picture the math of it
> very simply but it seems like it should be nearly a wash.
> When you say "to make the consensus option at least as good" do you
> envision some kind of mechanism that could actually improve what the
> consensus option is? Or maybe, easier to imagine: a rule that imposes 
> some kind of universal penalty if consensus isn't achieved. A forfeiture of 
> office seems like the most obvious.

I was thinking that the random ballot outcome can be quite bad. E.g. the
expected scores in the two-sided pizza election:

6 pepperoni voters: 7.0
2 anchovy voters: 3.4
2 mushroom voters: 1.8

If someone blinks (e.g. an anchovy voter mistakenly doesn't set mushroom
as the consensus option), then the outcome is not particularly good for
society as a whole. All we *really* need is the expectation of the
lottery to be less than the consensus option, while resisting strategy.
So in some ideal world, the expected value for the fair lottery would be
something along the lines of

6 pepperoni voters: 8 - epsilon
2 anchovy voters: 8 - epsilon
2 mushroom voters: 9 - epsilon

and even for very small epsilon, it would still be preferable to choose
the mushroom consensus option. But how to implement such a lottery, much
less in a strategy-resistant manner, I have no idea.

In a way, it's like the concept of MAD: if you give every voter his
personal doomsday button to push if he doesn't get a satisfactory
outcome, then minmax will happen if it's at all achievable. However, the
outcome should consensus be impossible is truly horrible. The better the
system can be in the "no consensus" case while leaving consensus
preferable, the better the method is.

The other side of that coin is what I said in the earlier post: if the
consensus option is always at least as good as a random ballot, then
it'll always be chosen. So making the structure around the mechanism
conducive to finding a consensus would also help.

We'd have to be careful that the alternative to consensus isn't biased,
though. "Forfeiting one's office" might well be, just like "status quo
prevails" is.

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