[EM] Maximal lotteries (was :Re: The critical importance of Precinct Summability.)

[Kristofer Munsterhjelm]

On 2024-08-08 17:53, Closed Limelike Curves wrote:
> I suspect the uncovered set might be slightly better because it's a 
> close approximation of the bipartisan set that isn't too hard to 
> explain. Maximal lotteries also have some very nice strategy-resistance 
> properties.

You've referred to the maximal lottery and strategic voting before. I'd 
like to know in more detail what you're referring to by your statements.

In the Monroe post, you said:

> If group strategy was a reasonable model of voters, it wouldn't
> matter which electoral system we picked, because the outcome would
> always be maximal lottery.
What do you mean? Do you mean that:

- Every (non-strategyproof) method has a unique Nash equilibrium per 
election under group strategy, whose expected outcome is that election's 
maximal lottery,

- Every method has the maximal lottery as one of its Nash equilibria,

- The relation between group strategy and the maximal lottery only holds 
for some methods, and these methods have the ML as their unique Nash 
equilibrium,

- As above, but "as one of its Nash equilibria".

or something else entirely?

> On this topic and the lack of focus on proportional representation 
> mentioned elsewhere, I think it would be super useful to have some kind 
> of strongly-summable PR algorithm. ElectoWiki claims Ebert's method is 
> summable 
> <https://electowiki.org/wiki/Summability_criterion#Multi-winner_generalizations_and_results>, but the link is broken and Ebert has some big issues (e.g. negative response and Pareto inefficiency).

I tried to look at the forum through IA and I couldn't find any mention 
of strong summability; it must have been in a later post on that thread, 
which IA hasn't archived. So at best that needs a {{cn}}.

To my knowledge, whether Droop proportionality is compatible with strong 
summability is still open. I have a very broad idea of how one might 
resolve summability questions like this, but I would need some 
mathematical primitives that I'm not sure how to construct.

-km