[EM] Electing Cabinets/Executive Committees

[Kristofer Munsterhjelm]

On 2024-03-06 04:26, Closed Limelike Curves wrote:
> I assume there's methods for this, but I don't know the search terms: 
> say I want to proportionally elect an inhomogeneous committee, like a 
> Cabinet or a set of executive officers. What methods handle this?

Steven Brams considers this exact problem in the book "Mathematics and 
Democracy" (Chapter 9, "Allocating Cabinet Ministries in a Parliament"). 
He first gives the following algorithm:

For i = 1...n:
	- Let the "current party" be the party next in line according to 
Sainte-Laguë based on votes in the most recent parliamentary election.[1]
	- Ask the current party which of the remaining positions it would like 
to claim (e.g. PM, minister of defense, etc.)
	- Assign the chosen position to the party.

He then shows that this method, although proportional, is nonmonotone in 
the sense that sometimes a party might want to be asked later. So he 
introduces a "trading step" where at each i, the current party may ask 
some other party if it wants to go first instead. He finally shows that 
this does not completely eliminate tactical voting problems, but that 
"some of the problems of cabinet selection can be ameliorated if not 
solved" (section 9.8).

If you don't have parties, you might want to look into the matrix vote: 
https://en.wikipedia.org/wiki/Matrix_vote. I'm not aware of any 
generalization to methods other than Borda; the obvious multi-way Range 
method[2] is not proportional.

-km

[1] Section 9.3. discusses the different divisor methods, and Brams 
argues that whether you'd prefer D'Hondt or Sainte-Laguë is a matter of 
preference - whether you think large-party bias is worth it for 
stability or not. IMHO, the way to solve kingmaker problems with party 
list parliamentarism is to use a slightly consensus-biased ranked 
method. If that method is house monotone, then it could be used as a 
replacement here. (Note that this is not the same as using D'Hondt, due 
to parallels to center squeeze if only first preferences are used.)

[2] Each voter rates each candidate for each position, then the method 
selects candidates so that the sum of chosen candidate-position pairs is 
maximized. This can be done by linear programming.