[EM] Thermodynamics

[Forest Simmons]

El dom., 5 de jun. de 2022 3:06 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> 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.
>

To round out this part of the discussion I should have pointed out that the
global min of entropy is zero, which occurs onlywhen all values (except
one) of p are zero, corresponding to a single winner (n=1) election in this
context.

In the strategy context it would signify a pure/deterministic (as opposed
to mixed) optimal strategy.

>
> 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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>> info
>>
>
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