[EM] Thermodynamics

Forest Simmons forest.simmons21 at gmail.com
Sun Jun 5 15:06:46 PDT 2022

```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.

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.

In general, Sum p_i*r_i is a weighted arithmetic mean of the r values,
where the p values are the normalized weights.

So the given sum S is a kind of mean value of the p values. If there were n
of them, and they were all equal, the mean would be 1/n, so that the rule
of thumb would yield 1/(1/n), that is n, which makes perfect sense.

The same would work for any other kind of weighted mean.  For example the
weighted geometric mean:

G=Prod(r_i^p_i) is a weighted geometric mean of the r values where the p
values are the (,normalized) weights.

If the r vector is a copy of the p vector we get

G=Prod(p_i^p_i)

If we take the log of the reciprocal of G, we get ...

Log(1/G)=-log(Prod(p_i)^p_i), which expands to -Sum(p_i*log p_i), which we
recognize as the Shannon Information/ entropy formula.

A local global max of this entropy occurs when the distribution is uniform,
that is when p_i=1/n.

So it turns out that the rule of thumb formula n=1/S is related to the
Shannon information/Entropy of the favorite candidate lottery
distribution.  In fact, the log of the rule of thumb value is a good
approximation to the Shannon information  ... that is
log(1/S) ~ log(1/G), in general, and the approximation straightens out to
equality if all the p values are equal, or if all are zero except one.

So we begin to see connections between statistical mechanics and the
various distributions that are so ubiquitous in voting methods. These
distributions include mixed strategy distributions, distributions of voters
and candidates in various issue spaces, etc.

Let's keep our eyes open for more connections!

-Forest

El dom., 5 de jun. de 2022 10:44 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 05.06.2022 19:16, Carl Schroedl wrote:
> > As a software guy, the connection I make is to something I have wondered
> > for a while -- whether it is useful to study social choice functions as
> > lossy compression algorithms. I haven't thought it through, but it could
> > be interesting to see if the rate-distortion branch of information
> > theory would apply.
>
> If you're trying to design a method that has the best possible VSE for a
> ranked voting method, then it may be possible to use ideas from vector
> quantization. In a spatial model, the voters rank the candidates
> according to proximity, and then the method finds the winner that's
> closest to the voters using only this information. So it's trying to
> find a vector (n-dimensional point) that's closest, in an Euclidean
> sense, to the distribution of the voters... though unlike ordinary VQ,
> it doesn't know the actual distances, only their ranking.
>
> Similarly, I'd say quota-based proportional representation is like
> clustering. Monroe's method is the most obvious clustering-like PR
> method: you assign each candidate a voter, so that each candidate has
> the same number of voters, and so that the total voter-candidate
> distance is minimized. (One possible objection to Monroe is that it
> doesn't care about what the voter thinks about the composition of the
> rest of the assembly, just his preferred candidate.)
>
> That's for honest voters, though. With strategic voting, the
> "compression method" (clustering method) becomes partly adversarial:
> keep the outcome from degrading too much if some fraction of the votes
> is arbitrarily altered.
>
> -km
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>
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