[EM] Compromise allocation of fair share

[Andy Jennings]

Forrest,

On Wed, Dec 15, 2010 at 4:39 PM, <fsimmons at pcc.edu> 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?
 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.

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?

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), and it does
seem very likely to "do the right thing" without rewarding strategy very
much.

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.

Andy
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