COMSOC Seminar - Brandt and Nehring - Shared screen with speaker view
Franz Dietrich
Ulle Endriss
Best to type your question here
Mark Wilson
QUESTION: main theorem says there exists r, but does it work for every fixed positive r?
Florian Brandl
It works for every fixed positive r provided r is smaller than some value r₀ depending on the distance to a maximal lottery we would like to achieve.
Daniel Eckert
Q: a) Are there any issues with path dependence? b)Process reminds me of genetic algorithms, wrongfully?
Marcus Pivato
Question: The convergence statement is in Cesàro average. So this allows for arbitrarily long excursions away from the limit lottery, arbitrarily far in the future, as long as the time-average converges. Correct? In your statement “the urn distribution is close to a maximal lottery most of the time”, the phrase “most of the time” means a set of times of Cesàro density 1, correct? Final question: is the proof obtained by applying the Perron-Frobenius theorem to obtain the stationary probability measure for a Markov chain defined by the majority margin matrix, plus the Birkhoff ergodic theorem?
Florian Brandl
The existence of a stationary distribution is obtained from the P-F theorem. However, most of the work is determining the stationary distribution.
Marcus Pivato
OK thanks.
Florian Brandl
Close to a maximal lottery most of the time means that given any τ > 0, the distribution in the urn is δ-close to a maximal lottery all but a τ-fraction of the time.
Markus Brill
QUESTION: Didn’t Laslier and Laslier show that “2 is not enough”?
Florian Brandl
Yes, the process can (and with probability 1 will) have long excursions far away from the maximal lottery. However, these excursions happen very rarely.
Marcus Pivato
Can any linear program be coded as a maximal lottery?
QUESTION Can this be interpreted as saying that “fair” deliberation is good to achieve a faire result? Os something similar
Dominik Peters
QUESTION: Do you have any bounds on how long it takes until you start being close to ML often?
Sean Horan
Question — is myopic choice equivalent to sophisticated choice via backward induction?
Florian Brandl
Dominik: One can obtain statements like “given enough iterations, it is likely that the distribution in the urn is close to a maximal lottery”. The number of iterations required for a given distance δ behaves like 1/\delta.
Ulle Endriss
QUESTION from Peter van Emde Boas: Question how fast is the convergence compared to other LP solving methods?
Florian Brandl
Sean: I am not aware of a connection to backward induction. By myopic we mean that voters only consider the effect of their choice on the winner of the current round.
Franz Dietrich
Question: Could we interpret the weights as the importance of making the *correct* judgment on the issue? This would take an epistemic perspective, by assuming that judgments can be (objectively) correct or incorrect.
Marcus Pivato
A better way to think about the weight of issue k is as the importance of agreeing with the majority opinion on issue k.
Marcus Pivato
If issue k has larger weight than issue j, then it is more important to satisfy a majority on issue k than an (equally sized) on issue j.
Marcus Pivato
um… equally sized majority on issue j
Marcus Pivato
This *might* be connected to an epistemic interpretation via some kind of CJT argument. But it is not immediate, because there are multiple issues, voters opinions (and errors) across issues might be correlated, etc.
William Zwicker
QUESTION: Can the dependencies Marcus mentions be handled by placing weights on certain Boolean combinations of issues, rather than restricting weight assignments to individual issues?
Marcus Pivato
Interesting idea. This might work, but it is definitely outside of our framework, because what you suggest would not be “additively separable” across issues, whereas our additive majority rules *are* additively separable.
Marcus Pivato
It is also possible that what you are proposing is equivalent to recoding the JA problem in a larger judgement space where these Boolean combinations are represented explicitly as separate “issues”.
Marcus Pivato
But if you did this recoding, you would probably violate the “thickness” condition which appears on the current slide.
Umberto Grandi
QUESTION: could you give some examples of a "well behaved" domain X (distal and rugged)?
William Zwicker
Thickness means, intuitively, that .. . there are not too many redundancies among issues?
Marcus Pivato
To be precise, it means that there are no linear dependencies in the family of admissible truth value vectors.
William Zwicker
I think that’s a version of “yes.”
Marcus Pivato
A linear dependency would be a fairly strong form of redundancy. Weaker forms of redundancy are OK. For example, the truth values of some issues could be functionally determined by the truth values of other issues, and in general that would not violate thickness.
Franz Dietrich
Question: [Better to state orally. If time permits.]