## Bayesian Regret with PR/multiwinner elections

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:

1. Generate artificial "voters" and "candidates" and "societal options." [Cf, in the old-style single-winner BR methodology, only voters & candidates were generated here.]
2. Each voter, and also each candidate, has "utilities" for each societal option. (Generate them in some randomized fashion. They are real numbers.)
3. Now candidates announce info about their utilities according to some assumed announcement behavior. And voters vote on the candidates, according to some assumed strategic behavior. [Simplest behaviors are "honesty," but others could be tried too.]
4. Now we use the (multiwinner) election system E to elect a subset of the candidates.
5. Now we have a "parliament" (the set of winning candidates).
6. ["stage 2"] the parliament votes, now using a single-winner voting system, on the societal options.
7. Some societal option wins.
8. In view of the winning option, we assess the Bayesian Regret to all the voters (by using utilities from step 2).
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."
9. Repeat steps 1-8 a zillion times to find the mean Bayesian Regret of election method E.

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.)

Update: Billy Tetrud writes first(?) multi-winner BR code (2016): His program, written in javascript, is here and I think it is a good start. Among the (not very many) multiwinner voting methods Tetrud initially implemented, the best one appears to be reweighted range voting.