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badger comments on Polymath-style attack on the Parliamentary Model for moral uncertainty - Less Wrong
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My reading of the problem is that a satisfactory Parliamentary Model should:
Since bargaining in good faith appears to be the core feature, my mind immediately goes to models of bargaining under complete information rather than voting. What are the pros and cons of starting with the Nash bargaining solution as implemented by an alternating offer game?
The two obvious issues are how to translate delegate's preferences into utilities and what the disagreement point is. Assuming a utility function is fairly mild if the delegate has preferences over lotteries. Plus,there's no utility comparison problem even though you need cardinal utilities. The lack of a natural disagreement point is trickier. What intuitions might be lost going this route?
I think there's a fairly natural disagreement point here: the outcome with no trade, which is just a randomisation of the top options of the different theories, with probability according to the credence in that theory.
One possibility to progress is to analyse what happens here in the two-theory case, perhaps starting with some worked examples.
Alright, a credence-weighted randomization between ideals and then bargaining on equal footing from there makes sense. I was imagining the parliament starting from scratch.
Another alternative would be to use a hypothetical disagreement point corresponding to the worst utility for each theory and giving higher credence theories more bargaining power. Or more bargaining power from a typical person's life (the outcome can't be worse for any theory than a policy of being kind to your family, giving to socially-motivated causes, cheating on your taxes a little, telling white lies, and not murdering).
In the set-up we're given the description of what happens without any trade -- I don't quite see how we can justify using anything else as a defection point.
I think the the Nash bargaining solution should be pretty good if there are only two members of the parliament, but it's not clear how to scale up to a larger parliament.
For the NBS with more than two agents, you just maximize the product of everyone's gain in utility over the disagreement point. For Kalai-Smodorinsky, you continue to equate the ratios of gains, i.e. picking the point on the Pareto frontier on the line between the disagreement point and vector of ideal utilities.
Agents could be given more bargaining power by giving them different exponents in the Nash product.
Giving them different exponents in the Nash product has some appeal, except that it does seem like NBS without modification is correct in the two-delegate case (where the weight assigned to the different theories is captured properly by the fact that the defection point is more closely aligned with the view of the theory with more weight). If we don't think that's right in the two-delegate case we should have some account of why not.
The issue is when we should tilt outcomes in favor of higher credence theories. Starting from a credence-weighted mixture, I agree theories should have equal bargaining power. Starting from a more neutral disagreement point, like the status quo actions of a typical person, higher credence should entail more power / votes / delegates.
On a quick example, equal bargaining from a credence-weighted mixture tends to favor the lower credence theory compared to weighted bargaining from an equal status quo. If the total feasible set of utilities is {(x,y) | x^2 + y^2 ≤ 1; x,y ≥ 0}, then the NBS starting from (0.9, 0.1) is about (0.95, 0.28) and the NBS starting from (0,0) with theory 1 having nine delegates (i.e. an exponent of nine in the Nash product) and theory 2 having one delegate is (0.98, 0.16).
If the credence-weighted mixture were on the Pareto frontier, both approaches are equivalent.