[EM] Compromise allocation of fair share

[Andy Jennings]

Forrest,

I'm trying to make sure I understand exactly what the "Ultimate Lottery"
methods are.

So the "Ultimate Lottery" singlewinner method is:

1. Voters submit homogeneous functions of p1,p2,...,pn
2. Choose the configuration (p1,p2,...,pn) which maximizes the product of
all voters' functions
3. Use a lottery that elects candidate i with probability pi.
(Ideally we would solve the maximization problem over the space of all
possible p1,p2,...,pn which sum to 1.  If that's not possible we can allow
people to submit possible outcomes and just choose the maximum one out of
all the submissions.)

And the "Ultimate Lottery" multiwinner method is:

1. Voters submit homogeneous functions of p1,p2,...,pn
2. Choose the configuration (p1,p2,...,pn) which maximizes the product of
all voters' functions
3. Entity i gets voting power pi in the parliament.
(We can restrict the space we're considering so no more than M entities get
seated, or we can just consider the whole space and seat anyone with
positive voting power.)

Is this correct?

Andy



On Wed, May 18, 2011 at 11:51 AM, <fsimmons at pcc.edu> wrote:

> ----- Original Message -----
> From: Andy Jennings
> Date: Wednesday, May 18, 2011 7:14 am
> Subject: Re: [EM] Compromise allocation of fair share
> To: fsimmons at pcc.edu
> Cc: election-methods at lists.electorama.com
>
> > Forrest,
> >
> > On Wed, Dec 15, 2010 at 4:39 PM, wrote:
> >
> > > I would like to modify my proposal for a new kind of list PR method.
> > >
> > > 1. Voters submit ballots indicating their favorite parties.
> > These ballots
> > > are used to find the standard list PR allocation of N seats by
> > some standard
> > > method. We call this allocation the Fallback allocation.
> > >
> > > 2. All interested entities submit as many allocation
> > nominations as
> > > desired. We require that these allocate whole numbers of
> > seats to the
> > > parties, with the total number of seats equal to the same
> > number N in all
> > > cases. Let the letter S represent the set of these nominated
> > allocations,> with the Fallback allocation included,
> > >
> > > 3. Voters also submit ballots indicating their criteria for
> > ordering the
> > > members of the set S. Let Beta be the set of these ballots.
> > For example one
> > > voter could say that of any two allocations she prefers the
> > one which gives
> > > the least number of seats to party P7.
> > >
> > > 4. Voters also submit ballots in the form of functions that are
> > > homogeneous of degree one for the purpose of contrilling their
> > share of the
> > > allocation in the optimization step below. Call this set of
> > ballots H, for
> > > "homogeneous."
> > >
> > > 5. Eliminate all of the members of S that do not Pareto
> > dominate the
> > > Fallback allocation according to the ballots in the set Beta.
> > Let S' be the
> > > subset of non-eliminated members of S.
> > >
> > > 6. Let S'' be the set of all members of S' that are not
> > Pareto dominated
> > > by some other member of S'.
> > >
> > > 7. Allocate the seats to the various parties P1, P2, ... etc.
> > in accord
> > > with the allocation p=(p1, p2, ...) from S'' that maximizes
> > >
> > > the product (over f in H) of f(p) .
> > >
> > >
> >
> > Is this still your current proposal for a Ultimate Lottery-based
> > PR method?
>
> Not "proposal" so much as an example of where you end up if you take
> fairness,
> expressivity, and manipulation resistance to their logical conclusion in
> one
> grand method.
>
> > I have some questions about it.
> >
> > 1. Is it really necessary for voters to choose a "homogeneous of
> > degree one"
> > function to broker their evaluation of S* in step 7? Wouldn't
> > the voters'
> > functions be evaluated (in step 7) only on the simplex
> > p1+p2+...+pn=1? Why
> > not let each voter specify an arbitrary function on the simplex?
> > If a
> > homogeneous function is needed for proofs, then you can use the unique
> > homogeneous function generated by the values on the simplex.
>
> Good point!
>
> >
> > 2. Why must the voters give two different schemes for evaluating the
> > outcomes? Shouldn't each voter's function, f, be enough to
> > create their
> > ordering of the ballots in step 2?
>
> Two reasons: (1) The functions in H might not be capable of enough
> expression
> for some voters, and (2) if the ballots of Beta were used for anything
> beyond
> the Pareto (i.e. unanimous) decisions, there would be incentives for
> insincere
> ranking.
>
> Essentially, all of the allocations in S'' are Pareto tied improvements on
> the
> fallback allocation, and the Ultimate Lottery step based on the ballots in
> H is
> used to break the tie.
>
> >
> > 3. Are the Pareto elimination steps in 5 and 6 necessary? It
> > seems that
> > Pareto domination would be very rare so steps 5 and 6 would
> > hardly ever do
> > anything. Even if a Pareto dominated option made it through
> > steps 5 and 6,
> > it seems like it could never win in step 7. (Assuming each
> > voter's f
> > function is compatible with their rank-ordering scheme.)
> >
> > I'm really starting to like the simpler system where every voter
> > submits a
> > linear utility function and we choose the allocation that
> > maximizes the
> > product of the utilities. It is completely invariant to any voter
> > re-scaling their utility function (though not to translation),
> [this is a property that is preserved by homogeneous functions]
> > and it does
> > seem very likely to "do the right thing" without rewarding
> > strategy very
> > much.
>
> As in range voting the method has no need for insincere strategy, but it is
> not
> strategy proof.  Use of max and min functions (which are also homogeneous)
> makes
> for easier, more robust strategy.
>
> If we wanted to restrict to some subset of H, instead of linear
> combinations of
> the p_i , I would take combinations generated by three operation: (1)
> multiplication of p_i's by positive numbers, (2) minimization, and (3)
> maximization.  It sees to me that this restriction on H would be
> strategically
> equivalent to no restriction, just as restricting Range to Approval is
> strategically equivalent to no restriction in the single winner case.
>
> >
> > I'm still trying to understand the extra layers of complexity,
> > including the
> > consequences of allowing non-linear utility functions, and why
> > they are
> > necessary.
>
> I hope that my remarks help you see where these complexities came from.
>
> And you are right, this is way too complex for a public proposal.
>
> I think that RRV applied to parties (repetitions allowed) would be a much
> better
> public proposal: let N be the number of seats. Do N steps of RRV applied to
> the
> voters' ratings of the parties, keeping the elected parties in the mix.  If
> a
> party wins k times, then it gets k seats.
>
> >
> > Andy
> >
>
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