Re: Game Theory Nobel Prize Again and Beyond With PI
From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/23/04
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Date: Mon, 23 Aug 2004 12:43:21 +0000 (UTC)
On 22 Aug 04 17:00:11 -0400 (EDT), Osher Doctorow wrote:
> From the viewpoint of PI, what is so very interesting is that you
>don't have to go beyond Riccati differential equations or systems
>of differential equations to obtain cutting edge research in game
>theory.
Meta-evolutionary techniques are only one aspect of Evolutionary
Programming (EP), which relates in turn to Genetic Algorithms (GA)
and Evolution Strategy (ES). EP appears to have a number of
advantages to GA, and ES is closer to EP than to GA in general.
The Internet has a number of FAQs on EP, including Q1.2: "What's
Evolutionary Programming (EP)"? at www.faqs.org/faqs/ai-faq/genetic/
part/section-3 html or just access it under the above title as
keywords.
An interesting recent paper is "Evolutionary programming using
mutations based on the Levy probability distribution," by Chang-
Yong Lee and Xin Yao of respectively Kongju National University
and U. Birmingham U.K., IEEE Transactions on Evolutionary Computa-
tion, Vol. 8, No. 1, Feb. 2004. The Levy or generalized Levy alpha-
stable distributions relate to Engel and Granger's Nobel Prize in
Economics via the large (in fact, infinite) variances and are of
interest tail region power laws which imply no characteristic length
scale (a property of fractals). The alpha-parameter can tune or
fine tune the pdf, which yields adjustable variation in mutation
which (mutation) is not normal as in the usual EP but Levy type of
which the Cauchy distribution is a major example. Applications cited
by Lee and Yao include turbulence, polymers, biological sysems,
economics. I should mention that a limiting case of the Levy pdf for
alpha --> 2 is the normal distribution and alpha is in general in
(0, 2). For very large y on (-infinity, infinity), the Levy pdf is
approximated by y^(-(alpha + 1)), a power law, and intuitively it
seems a likely conjecture (not mentioned by Lee and Yao) that 2Dx[F]
> fX(x) for x replacing y in the above notation (the authors cited
simply used y in place of x habitually).
The stability of the Levy distribution implies that the sum of Levy
random variables is also Levy, and a few remarks on the relationship
of this to PI are in order. (Bayesian) Conditional Probability or
BCP and Independent Probability/Statistics (IPS) are most emphatically
multiplicative/division rather than additive/subtractive in their
fundamental nature, although they use some of the other operations.
PI is fundamentally additive/subtractive. BCP and IPS were discover-
ed first probably because of their simplicity and use in simplify-
ing formulas in pre-computer eras where the formulas were multiplic-
ative and simplifications were of predominant interest. However,
addition-subtraction is fundamentally in many ways even simpler than
multiplication/division, and in any case the property of retaining
the same distribution under sums goes to the "+" side of the ledger
so to speak. Readers who recall that the gamma distribution is
PI-optimal second only in type to the uniform and beta type
distributions will be happy to recall that the sum of independent
gamma random variables are gamma with the new distribution having
the sum of the first parameters and with the second parameter the
same as the equal second parameter. This also holds for the chi-
square subtype of the gamma distribution (which stays chi-square
under sums with addition of parameters) but not the exponential
subtype (although it stays gamma of course). The normal distribu-
tion of course is preserved under sums both in means and variances.
The uniform distribution doesn't have the reproducing property but
is triangular at least on (0, 1). However, its sums have very
valuable properties in regard to empirical distribution functions.
Moreover, the triangular distribution is arguably quite simple.
Osher Doctorow
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