[EM] "representation score" conundrum

[Jameson Quinn]

A conundrum:

My proposed MSE system is based on a "representation score", in which
(conceptually) each candidate has a certain "power to represent", and that
power is divided evenly among the voters who supported that candidate, with
a certain amount of it wasted on a quota of "phantom" voters.

But what if you have 2 to elect, 2 ballots, and the votes are AB and A? Is
it really fair to say that the A voter is less represented by the winner
set AB than if the other voter had voted just B?

I think there must be a better way to distribute the representation from
each candidate, that does a better job preserving monotonicity.

Try number 1: distribute it as a f(p) share to each voter, where f is a
decreasing function and p is the number of winning candidates that voter
supported, such as 1/p. But with 1/p, adding a vote for a winner to a
ballot would tend to decrease that ballot's satisfaction.

You could use 1/harmonic_sum(p). That's better, but you could still build
artificial cases where adding a vote for a winning candidate would decrease
your representation. In fact, strategically voting for a
universally-approved candidate would probably be a good idea, even if you
don't approve them yourself, because it will reduce your representation,
causing the envy minimizer to want to pull you up more by giving you other
things you want.

You could use the reciprocal of the Euler-Mascheroni
<http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant> formula of
the total proportion of voters who have voted against your winning
candidates. This would mean that voting for a universally-approved
candidate would not reduce your "share" of other candidates; yet it's still
possible to calculate each person's share without knowing satisfaction
scores. I'm sure you could still construct pathologies for this, but it
would be a lot harder, and there's no obvious dominant strategies.

Once you did this, you'd still go on to minimize squared envy against some
fixed point; and sum of squared envy would still be equal to squared bias
(ie, average envy) plus variance (unequal representation).

Note that any such adjustment preserves proportionality, because in the
purely partisan case, each partisan group is homogeneous, so "shares" are
still equal within each group.

This would be no longer a mathematically clean (or "pretty", or "natural")
method, but it is (even) closer to monotonicity, and keeps the nice
advantages.
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