[EM] hybrid 2-stage and 1-stage Bayesian Regret for comparing multiwinner election systems

[Kristofer Munsterhjelm]

Warren Smith wrote:
> So perhaps the best or simplest way to resolve the inherent conflict
> (in multiwinner voting systems) between the two goals of (a) good
> representation and (b) good winner quality, is... to accept it.
> 
> Let the Hybrid Bayesian Regret of a multiwinner voting system be a
> LINEAR COMBINATION of 1-stage and 2-stage regret.  Call the weights W1
> and W2.

You could also simply plot the results in 2D, setting the x coordinate 
to be 1-stage and y coordinate to be 2-stage. That would be similar to 
my proportionality-regret diagrams. You could then find the Pareto front 
and let whoever may desire to do so, draw a curve or a line on it to 
find the optimum according to whatever combination they want.

> 2-stage BR is explained here:
> http://www.rangevoting.org/BRmulti.html

A note on this: I also think there's a checks-and-balances dynamic 
within the parliament and between the parliament and the voters. In the 
former, members of different parties (or members of the same party, if 
elected by party-neutral methods) keep each other from veering off 
towards corruption; in the latter, the voters can see, by the actual 
bargaining being done in parliament, whether some representatives are 
acting against their wishes, and if so, which representatives are. These 
effects would not, I think, be reflected by 2-stage BR. A more 
sophisticated 2-stage BR might use game theory to try to account for 
such things, but exactly how, I'm not sure.

> 1-stage regret is simply the sum (over all voters and all winners) of
> utilities for the winners for the voters -- subtracted from same thing
> with the optimum winners who would have maximized this 1-stage utility
> (this subtraction makes it a "regret" rather than a "utility").
> 
> You may object "the weights in this linear combination are arbitrary!"
> 
> That's a valid objection.  However, I could counter that maybe that is
> the way things really are.  This is just yet another "knob" on the
> side of the "Bayesian Regret machine" -- others include #candidates,
> #winners, utility-generation model, voter-behavior model, etc.

All your other knobs are input knobs, no? The #candidates, #winners, 
utility generation models and parameters, etc., specify what kind of 
data is run through the voting methods (and subsequent utility 
determination function), whereas the weight-knob is an adjustment of the 
(combined) utility function itself.

So it would seem to be a quite different sort of knob. Adjusting it 
would be similar to adjusting 1-stage BR between, say, mean utility and 
worst utility (maximin).