[EM] Michael O, who are you talking to or with?

[robert bristow-johnson]

> On 07/21/2024 8:04 PM EDT Michael Ossipoff <email9648742 at gmail.com> wrote:
> 
> 
> No, I didn’t assume that the probability-distribution for states’ populations is uniform.
> 
> I merely assumed uniformity for that distribution *within any particular interval between two whole numbers of quotas*.
> 
> …& it wasn’t so much an *assumption*, as much as part of a useful operational definition for bias. …a bias easily defined, determined & avoided.
> 
> (…explaining why I’d denied an assumption of special conditions.)
> 
> Because in this topic, an interval between two whole numbers of quotas is often mentioned, it needs an abbreviation. I’ll call it an “integer-interval” (ii).
> 
> Obviously it’s within a lower ii that that distribution varies most.
> 
> The 0-1 ii doesn’t count, because , in House-apportionment, each state gets at least one seat.
> 
> It doesn’t count in PR either, because SL specifies .7 (instead of .5) of a quota as the rounding-point in the 0-1 ii.
> 
> That’s to discourage & thwart strategic-splitting. BF’s results are so close to those of SL, that the .7 rounding-point in the 0-1 ii should be used in BF too.
> 
> …& so, the lowest ii in which the operational-definition makes a difference is the 1-2 ii.
> 
> In that ii, the values of the BF & SL rounding-points differ by only about 2%.
> 
> Obviously that % rapidly becomes drastically lower for higher ii s.
> 
> …as also must the variation of the probability-distribution within an ii.
> 
> As for my useful operational-definition of bias:
> 
> 1.
> 
> One justification for it is that the vagaries & continual variations of the probability-distribution for states’ populations isn’t the responsibility of an allocation-rule.
> 
> …as is tacitly, performatively, agreed by every allocation-rule that doesn’t require recalculating an approximation to that distribution at each census.
> 
> 2.
> 
> But what would an alternative to my useful operational definition look like? A mess, that’s what.
> 
> Obviously the probability distribution within each ii could only be *approximated* by a formula based on…what? The historical record of populations in that range?
> 
> Why should old records be assumed relevant to today’s probabilities. We haven’t gotten away from assumptions—a futile goal.
> 
> Or maybe an interpolation or least-squares approximation based on the new populations of the states.
> 
> The operative word there is “approximation”.
> 
> All that extra work for something that’s still only approximate. Exactitude hasn’t been gained.
> 
> Arguably, in a particular ii, the approximation to the probability-distribution, a best-guess, is more realistic than a uniform distribution there.
> 
> But the better likely-accuracy of that guess, doesn’t make it more than a guess.
> 
> I didn’t stumble-upon BF as a useful, feasible avoidance of a usefully operationally defined bias.
> 
> That was the purpose.
> 
> But sure, any other interpretations or interesting aspects that I didn’t know about—yeah I stumbled onto that.
> 
> Bias has very much been part of the merit evaluation & comparison of PR methods, & that was always so with House-apportionment as well.
> 
> So that was my purpose.
> 
> I hadn’t considered the interesting entropy consideration, so yes that was an accidental result.
> 
> Entropy in PR had never occurred to me, but it’s the basis of a number of useful measures of inequality.
> 
> The most relevant 1-number inequality-measure based on summed-aggregation is a generalized entropy called ge(-1).
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