[EM] Simulations with social welfare functions

[Jobst Heitzig]

Dear Forest!

You wrote:
> I wonder how Bucklin would fare in your simulations?  

I will be able to do further simulations on Monday.

> Or how about
> the quartile variation of Bucklin in which the "bar" is lowered
> simultaneously on the range style ballots until at least one
> candidate is rated above the bar on at least 75 percent of the
> ballots.  

My immediate guess is that it will perform somewhere between Approval
and Range Voting.

> If this yields more than one winner, then eliminate winners
> by random ballot until there is only one left.

This is unlikely in my simulation setting since the ratings are real
numbers and candidates' and voters' positions are drawn from a standard
normal distribution. (By the way, should I use a different distribution ?)

> I like Liberal Fair Choice (LFC).  It works well in the example that
> Kevin Venzke always worried about:
> 
> 49 C 
> 24 B  (but sincere is B>A>C)
> 27 A>B>>C
> 
> No candidate is strongly defeated since the approval order is B>C>A,
> but more than 51/2 voters prefer C to B, and more than 51/2 prefer A
> to B, and more than 49/2 prefer A to C.
> So the winner is chosen by random ballot.

Right. And had the 24 voted B>>A>C or B>A>>C instead, the solution would
be the same.

> What if we modified LFC by using MinOf2 to pick the winner from among
> the candidates that are not strongly defeated?

Interesting. But although that might increase the performance in terms
of social welfare still a bit further, I doubt that it is worth the
effort since MinOf2 requires range-style ballots while LFC requires only
rankings with approval cutoffs!

> Or how about going beyond Gini by using MinOf3,

By which you mean: draw 3 ballots at random and elect the candidate
whose minimum range value on these ballots is largest?

> or by using MinOf(K)
> for the largest K that distinguishes a winner, 

Here I don't understand what you mean by "distinguish a winner"...

> i.e. using random
> ballot to eliminate all but one candidate?

Do you mean: draw ballots at random and each time eliminate the
lowest-ranked candidate until only one remains?

> Could we modify LFC for the multiwinner case by relaxing the
> definition of "strongly defeated" appropriately?

Perhaps, but I must say that I never thought about multi-winner methods
thoroughly. My impression is that for electing a multi-seat
representative body, something like Delegable Proxy would be my choice.

Yours, Jobst