Warren D. Smith. Nov 2015. UNDER MODIFICATION.

**Abstract.**
We show how some proportional representation (PR) election
ideas dating to Lars Edvald Phragmen 1895,
and Bjarke Dahl
Ebert
in October 2003 can be
written as solving a mixed linear/integer program (MILP) to optimality
to find the "optimal" PR parliament.

Let C=#candidates, V=#voters, W=#winners, 0<W<C<V.
Let **S _{v,c}** be the real-valued

Linear constraints:

Optimization: minimize p by choice of all the variables.
The W election winners then are the c such that e_{c}=1.
The C-W losers are the c with e_{c}=0.

The total number of real variables is VC+1, and of integer variables is C. The total number of 01-interval-containment linear inequality constraints is 2VC+2C. The total number of other linear inequality constraints is V. The total number of linear equality constraints is C+1.

This has formalized a score-using version of Ebert's description of a Phragmen-like system
that is an optimization problem. (Phragmen's original system was not optimizing anything.)
Ebert has his own PR voting system – or at least, it is hoped to be PR,
I have not seen any proof that it is, only incompletely convincing heuristic arguments –
which cannot be formalized in this manner because it would require a *non*linear
objective function. There are many possible nonlinear objective functions.
To explain them, replace the single variable p with V variables p_{v},
replace the first constraint by

and replace the optimization goal by "minimize ∑_{1≤v≤V}F(p_{v})"
where F is some nonlinear function – for example Ebert suggested F(x)=x²
for no particularly convincing reason.
If we use F(x)=x^{μ} in the limit μ→∞ then
we get a more-unique version of the "Phragmen" method I began with.
By "more unique" I mean that the method I described could produce ties and
this limit-method presumably would experience ties less often.

I'd guess this program can usually be solved to optimality in a number of steps of order
2^{C}+(V+C)^{3}.
This is not a theorem, but merely a heuristic runtime estimate,
and not a very convincing one; and perhaps the first + sign should be changed to "multiply"
which would make a rather large difference.

In this 4-candidate 20-voter 2-winner election

10 voters: A=1 B=1 C=1 D=0 10 voters: A=1 B=1 C=0 D=1

both Ebert's L_{2}-norm proposal, and the MILP at top, both would
give p_{v}=1/10 to every voter if the winner-set was AB, but
also if the winner set were CD. I.e. both AB and CD are regarded
by these voting systems as "equally good" winner-duos.
That is foolish since obviously AB is superior.

This same simple election example embarrasses Monroe's election method for the same reason.