By Warren D. Smith. Idea invented about 2005.
At first I thought "Bayesian Regret" (mathematical definition) methodology was only going to be usable for comparing single-winner election methods. In multi-winner elections many other issues come to the fore besides just "utility of the winner," e.g. how well the winners work together, how well they "represent" the public, etc. (For example, one voter in a country of this ilk told me he intentionally voted against his favorite party because he did not want to them to win too big, i.e, to get too-many seats.)
However, it later occurred to me (and this idea has never been tried or tested, as yet) how to compare multiwinner methods. The technique is "two stage" (and that is the key idea). Explanation:
Actually, more realistic is probably to hold several single-winner societal option votes, among the options in chunk A, chunk B, and chunk C respectively – e.g. A=options about abortion policy, B=options about border with Mexico, C=options about health care, etc. – find the winning A-option, the winning B-option, and the winning C-option; and then assess the BR for the voters of these 3 simultaneous winning options. Call that "chunking."
Note, BR(E) depends not only on E but also on all the "knobs on the side of the simulator" namely the assumed voter- and candidate-behavior models, the number of candidates and seats and voters and societal options (and chunks), which single-winner system is employed in the "second stage," and the utility generator.
Hopefully (but it is unclear whether this hope will come true!) some election method E will turn out to be robustly superior to the others you try. "Robustly" means "pretty independently of all those knob settings."
That's basically what (fortunately) happened with single-winner BR, with range voting.
I invented this idea about 2005. As of January 2009 still nobody has ever programmed the computer to do it yet. (Some people have said they'd do it, but all bailed out before completing the project.)
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