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Posts tagged with Vilfredo Pareto

A sensible partial order ranking countries in the 2008 Olympics
—made by the guy who wrote PuTTY
I love posets because of situations like this. Not everything in life is completely one-dimensional, but that’s not to say it’s 100% disorderly either! Mathematics has some pretty good ideas for how to rationally label the world, sometimes.
  
Tatham’s description of the dilemma of ranking countries absolutely on 3 criteria reminds me of
multicriteria modelling
Arrow’s impossibility theorem
various multicriteria bargains or decisions to be made in regular life, like
which house to buy
which job to take
which city to move to
which président to elect
which school to attend
which product to recommend to a customer
all of which, if they’re quantifiable at all, usually consist of multiple independent, uncombinable factors (Would you like your président to be humane, or good with the economy? Well, I’d ideally like both… Would you like your employer to not beat you, or to pay you a living wage? Um, again, ideally both…). In the most naïve approach are combined with a linear weighting. (A concave weighting with positive cross-partials should be more sensible.)
As Tatham notes, using the magic of irrational numbers it’s possible to guarantee uniqueness of a ranking—but would the ranking be any good?
Well given that we know K Arrow’s Impossibility result, maybe we should just reduce our expectations. Instead of trying to squeeze a 2-ton elephant into a miniskirt made for Kate Moss, maybe we should relax the requirements and just hope to get a poset. That can be more doable.
It’s an application of Vilfredo Pareto’s genius idea.

A sensible partial order ranking countries in the 2008 Olympics

—made by the guy who wrote PuTTY

I love posets because of situations like this. Not everything in life is completely one-dimensional, but that’s not to say it’s 100% disorderly either! Mathematics has some pretty good ideas for how to rationally label the world, sometimes.

  

Tatham’s description of the dilemma of ranking countries absolutely on 3 criteria reminds me of

  • multicriteria modelling
  • Arrow’s impossibility theorem
  • various multicriteria bargains or decisions to be made in regular life, like
  • which house to buy
  • which job to take
  • which city to move to
  • which président to elect
  • which school to attend
  • which product to recommend to a customer

all of which, if they’re quantifiable at all, usually consist of multiple independent, uncombinable factors (Would you like your président to be humane, or good with the economy? Well, I’d ideally like both… Would you like your employer to not beat you, or to pay you a living wage? Um, again, ideally both…). In the most naïve approach are combined with a linear weighting. (A concave weighting with positive cross-partials should be more sensible.)

As Tatham notes, using the magic of irrational numbers it’s possible to guarantee uniqueness of a ranking—but would the ranking be any good?

Well given that we know K Arrow’s Impossibility result, maybe we should just reduce our expectations. Instead of trying to squeeze a 2-ton elephant into a miniskirt made for Kate Moss, maybe we should relax the requirements and just hope to get a poset. That can be more doable.

It’s an application of Vilfredo Pareto’s genius idea.

(Source: godplaysdice.blogspot.com)


hi-res




Central Limit Theorem
A nice illustration of the Central Limit Theorem by convolution.in R:
Heaviside <- function(x) {      ifelse(x>0,1,0) }HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )HHHH <- convolve(HH, rev(HH),   type = "open"   )HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )etc.
What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.
This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.
(To see this another way, think of the combinatorics.)

Central Limit Theorem

A nice illustration of the Central Limit Theorem by convolution.

in R:

Heaviside <- function(x) {      ifelse(x>0,1,0) }
HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )
HHHH <- convolve(HH, rev(HH),   type = "open"   )
HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )
etc.


What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.

This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.




NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.

(To see this another way, think of the combinatorics.)


hi-res