[EM] Thermodynamics

[Kristofer Munsterhjelm]

On 06.06.2022 00:06, Forest Simmons wrote:
> You mention sincere vs insincere voting ... which leads to game theory.
> In game theory most optimal strategies are mixed ... stochastic
> combinations of pure (deterministic) strategies.
> 
> So it is a matter of luck if it turns out that a deterministic strategy
> is optimal.

Yes. I'm just referring to that physics usually doesn't "fight back" the
way strategic voters do :-)

> We think of Approval as a deterministic method, but that's only because
> we have externalized optimal strategy considerations to the (cagey)
> voters and their (mostly gut level) probability estimates.
> 
> Back to multi-winner methods. A rule of thumb for a minimum number of
> seats for good proportional representation is the reciprocal of S=Sum
> (p_i)^2, where p_i is the probability that candidate i would get elected
> by random favorite ballot.

A related connection is to Laakso and Taagepera's effective number of
parties: https://electowiki.org/wiki/Effective_number_of_parties which
describes a party distribution as equivalent to a certain number of
"equally sized" parties. E.g. a dominant-party system may have an ENP
measure of 1.3, signifying that the second party has significantly less
support (or representation) than the first.

This measure is also 1/p_i^2 and has been criticized for being not
uniform enough (i.e. not providing the same results for lots of small
parties as a few large ones). Greene and Bevan suggested an information
entropy measure instead, as I referenced in the Electowiki article.

In addition, for party list PR, Webster's method minimizes the
Sainte-Laguë index of disproportionality, sum over parties i: (seats
given to party i - votes obtained by i)^2/(votes obtained by i). This
is, if I recall correctly, related to the chi-squared test statistic:

x^2 = sum over categories i: (observed counts for i - expected counts
for i)^2/(expected counts for i)

The chi-squared statistic in turn is an approximate G-test. The G-test's
form is:

G = 2 sum over categories i: (observed counts for i) * ln ( observed_i /
expected_i),

which looks vaguely like an entropy term. Wikipedia says it's related to
mutual information, but I know too little about this to comment.
https://en.wikipedia.org/wiki/G-test#Relation_to_mutual_information

(Finally, speaking of random favorite, I wrote a post about a
semiproportional determinization of it for multiwinner, here:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2019-October/002323.html
Its proportionality in the limit, yet possible bad results with only a
few seats shows the limits of PR as lotteries, I think. At least for the
random favorite lottery. But perhaps the idea of eliminating the winner
can be used to extend Plurality-based PR to ranked method PR... or be
used as a component of a Condorcetian multiwinner method.)

-km