Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

A curious thing happened in probability-statistics on the way to "game
theory" and economics. It imitated physics and algebra rather than
philosophy or logic in seeking to explore by using more complicated
mathematics rather than by first looking internally at meanings and
then deciding what to do.

Does this relate to dynamical systems and Probable Correlation? Yes.
Here's how.

Take a look at The History of Economic Thought Website, accessible by
those keywords or at http://cepa.newschool.edu/het/home.htm, by Goncalo
L. Fonseca (Graduate Student in Economics, Johns Hopkins U.) and Leanne
Ussher (Graduate Student in Economics, NYU), in particular the section
"Uncertainty, Information and Games" and the subsection I - Choice
under Risk and Uncertainty, and subsection (6) of that "Riskiness".

There you will discover fascinating definitions of First and Second
Order Stochastic Dominance, and how they relate to characterization of
increasing risk and portfolio allocation and alternative measures of
increasing risk.

First Order Stochastic Dominance is the one that corresponds to my last
posting, where a person prefers prospect F to prospect G, and we say
that F dominates G, F >-1 G or F >-s G or whatever. Let's use the
cdfs with the same letters but with arguments, F(x), G(x), so that this
turns out to be equivalent under certain assumptions to:

1) F(x) < = F(x) for all x

Intuitively, people prefer prospects with much probability mass skewed
toward higher returns rather than toward low returns. This turns out
to be consistent with von Neumann-Morgenstern theory.

However, for risk-averse people, one might consider that how spread out
the returns are in addition to the above will remove more ambiguity,
and this leads to Second Order Stochastic Dominance which turns out to
follow from the First Order Stochastic Dominance but not vice versa.
It involves the integral I(x) = I[G(t) - F(t)]dt > = 0 for all x in [a,
b] where I is the integral from a to x.

Had researchers stopped at First Order Stochastic Dominance, they might
have noticed Probable Correlation. They were in a rush to be fancy
and more complicated, and they didn't notice. It might seem
understandable to go ahead to spread of data, but it also is
understandable to first understand first things first so to speak.

Osher Doctorow

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