[EM] PAV and risk functions

[Kristofer Munsterhjelm]

Peter Zbornik wrote:
> Dear Kristofer,
> 
> would the constant relative risk function be of any help for Approval 
> voting?
> 
> F=( s(1)^(r-1)+...+s(n)^(r-1) ) / (r-1).
> 
> s(i) is the number of approved council members that are elected, where 
> 1<=i<=n, n is the number of voters
> r is a coefficient of risk aversion, which determines the rate at which 
> marginal utility of the voter declines with the number of council 
> members awarded to the voter.

That scoring method could be used for PAV (not SAV) style optimization. 
One could create a whole class of PAV style methods this way:

- Define a function f(a, b) -> R, mapping pairs of candidate sets to 
real numbers, where a is the approval ballot and b is the candidate council.
- Voters submit approval ballots v_1 ... v_n
- Using brute force, find the council c so that sum(q = 1..n) f(v_q, c) 
is maximized.

The greedy approximation can be defined in a similar generalized manner, 
but places restrictions upon the kind of f that can work. The greedy 
approximations would also be house monotone, I think, since they work by 
picking one candidate, then another, then another..

> The selection of r determines the behavior of F: 
> If r=1, then F becomes the Bernoulli-Nash social welfare function: 
> log(s(1))+...+log(s(n)) in the limit. As log(0) is minus infinity, this 
> function requires that each voter gets at least one council member, thus 
> it over-represents minorities and insures, that "everyone has their 
> representative" in the council. This function is useless, if all bullet 
> vote for themselves.

Couldn't this be solved in a leximax fashion if only some voters bullet 
vote? That is, an outcome with fewer infinities win over an outcome with 
more no matter what; then if there's a tie between the number of 
infinities, the one with the greatest finite score wins.

> Other uses:
> It seems that F can be used both as a proportionality index and as a 
> majoritan preference index for suitable values of r.
> 
> F can be used to explain why there are different voting systems - they 
> simply have different values of r, i.e. they have different utility 
> functions. For a suitable value of r, block-voting has higher utility 
> and is thus "better" than proportional representation, like STV.

That appears to be similar to my ideas about proportionality and 
majoritarian preference being tradeoffs. For some value of r, people 
would value the latter more than the former. Thus the decision of a 
single value of r would take the shape of multiple curves overlaid on 
the Pareto front, where the curve closest to the origin that still hits 
a method defines the optimal method, somewhat like these economic 
planning examples:

http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys2.gif
http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys3.gif

(Those examples have greater being better, so the front is to the upper 
right rather than lower left in my diagram.)

> F can also be used to measure which voting systems are the "best" and to 
> measure the "distance" or "similarity" between voting systems against 
> some benchmark values of r. 
> It would be cool to see how the chart at 
> http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/ would look 
> like using F with different values of r at the axis. 
> For instance, if r=0, then we would get Bayesian regret.
> F with r=0.5 (or some other value of r between 0<r<1) could maybe be 
> used in the chart instead of the Sainte-Lague proportionality index (SLI).

That would be a series of 1D graphs because the tradeoff would be 
defined by F. I don't see how F could be used as a proportionality 
measure in its own right since my opinion output is fractional, not binary.

To put it another way, say there's a society where the voters vote:
  33% for party A, 15% for party B, 27% for party C, 25% for party D,
and the parliamentary composition is
  41% for party A,  9% for party B, 27% for party C, 23% for party D,

how would you use F to determine how proportional that is? Here, I used 
"party" instead of "opinion" for the sake of simplicity.
The RMSE, GnI, LHI, etc, would all give a proportionality measure output 
when given those inputs.