[EM] Electing Cabinets/Executive Committees

[Ted Stern]

See https://en.wikipedia.org/wiki/Method_of_Equal_Shares

and its associated website, https://equalshares.net/

On Mon, Mar 11, 2024 at 10:23 AM Closed Limelike Curves <
closed.limelike.curves at gmail.com> wrote:

> Just realized: for legislatures or orgs with proportional budgeting, this
> can be done by VCG mechanism! Legislators "pay" for cabinet positions using
> their share of the budget. Those who spend more voting power on cabinet
> picks will have less voting power when it comes time to pass a budget.
> (Downside: this is definitely complicated, even if it's probably a big
> improvement in terms of efficiency.)
>
> On Thu, Mar 7, 2024 at 8:00 AM Closed Limelike Curves <
> closed.limelike.curves at gmail.com> wrote:
>
>> Very helpful, thank you! I'll try and see if there's anything from the
>> fair division or cake-cutting literature on this.
>>
>> On Wed, Mar 6, 2024 at 3:01 AM Kristofer Munsterhjelm <
>> km_elmet at t-online.de> wrote:
>>
>>> 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.
>>>
>> ----
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