# [EM] Thermodynamics

Forest Simmons forest.simmons21 at gmail.com
Sun Jun 5 20:57:13 PDT 2022

```El dom., 5 de jun. de 2022 4:05 p. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> 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 :-)
>

That's what I thought; I was just using it as a segue into another use of
probability in voting theory in the general theme of Robert
Bristow-Johnson's  question.

You are way ahead of me!

>
> > 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
> 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
>
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