Download Foundations of Boundedly Rational Choices and Satisficing

Foundations of Boundedly Rational Choices
and Satisficing Decisions
Reviving a Simon Tradition
K. Vela Velupillai
Department of Economics
University of Trento
Via Inama 5
381 00 Trento
Italy
“If we hurry, we can catch up to Turing on the path
he pointed out to us so many years ago.”
Herbert Simon, 1996
"There are many levels of complexity in problems, and
corresponding boundaries between them. Turing computability is an
outer boundary, ... any theory that requires more power than that surely
is irrelevant to any useful definition of human rationality. A slightly
stricter boundary is posed by computational complexity, especially in
its common "worst case" form. We cannot expect people (and/or
computers) to find exact solutions for large problems in
computationally complex domains. This still leaves us far beyond
what people and computers actually CAN do. The next boundary... is
computational complexity for the "average case" .... .. That begins to
bring us closer to the realities of real-world and real-time
computation. Finally, we get to the empirical boundary, measured by
laboratory experiments on humans and by observation, of the level
of complexity that humans actually can handle, with and without
their computers, and - perhaps more important -- what they
actually do to solve problems that lie beyond this strict
boundary even though they are within some of the broader limits.“
Herbert Simon, Letter to the author, 25 May 2000.
Models of Simon and Turing Machine Models
 It was only relatively recently that it became clear to me that I could frame
almost all of Simon’s models within one single, simple, mathematical
framework: The Turing Machine Model. This realization, in turn, came about
quite naturally when it became clear to me that the Models of Behaviour,
Discovery and Hierarchies, in Simon’s works, were underpinned by the concepts
of bounded rationality, satisficing, adapting, induction, abduction, (near)decomposability, simplicity and dynamics. Every one of these concepts were
given algorithmic and, hence, experimentally implementable numerical content.
It was, therefore, natural that I try to study and unify the Models of Simon
within the framework of a Turing Machine Model.
 The only important work by Simon that I have not been able to formalize within
the framework of a Turing Machine Model is his fascinating analysis (together
with Yuji Ijiri) on the size distribution of firms. If I am able to generate the class
of stable distributions and formalize the convolution product (a Faltung sum)
algorithmically, then this part of Simon’s work will also be algorithmic.
Algorithmizing the convolution product is straightforward; algorithmically
generating stable distributions is less easy, but not impossible. My conjecture is
that an analysis of this problem would require the use of Probabilistic Turing
Machines.
A Concise Summary
Formally, the orthodox rational agent's `Olympian' choices [Herbert
A Simon (1983), Reason in Human Affairs, Basil Blackwell,
Oxford. p.19) are made in a static framework. However, a
formalization of consistent choice, underpinned by computability,
suggests satisficing in a boundedly rational framework is not only
more general than the model of `Olympian' rationality; it is also
consistently dynamic. This kind of naturally process-oriented
approach to the formalization of consistent choice can be
interpreted and encapsulated within the framework of decision
problems -- in the formal sense of metamathematics and
mathematical logic -- which, in turn, is the natural way of
formalizing the notion of Human Problem Solving in the NewellSimon sense.
“[T]he Olympian model [of rationality], postulates a heroic man making comprehensive choices in
an integrated universe. The Olympian view serves, perhaps, as a model of the mind of God, but
certainly not as a model of the mind of man.”
Simon, 1983, p.34.
A Double Triptych of Fundamental Concepts
• What is Computability?
• What are Decision Problems?
• What is (Human) Problem Solving?
My staring point, in all my endeavours, has always been a Simonian
modification of the way Kant broke up his central question: What is Man? – into
three ostensibly more circumscribed questions (formulated against the
backdrop of his codification of the battle cry of the Renaissance: Sapere Aude!):
i.
What can I know?
ii.
What must I do?
What is knowledge?
How can we obtain it?
iii.
What may I hope?
Why do we try to obtain it?
Whatever knowledge is, it is algorithmic; therefore we must strive to obtain it
algorithmically; and we obtain it to solve algorithmically formulated problems –
whether they be those of a spontaneous mind’s curiosity [mathematics], those arising
out of the needs of survival, reproduction or whatever.
An aside
Ideas and Growth: Lucas [Economica, February, 2009, p.1]
Effectively computable solutions
"The term 'computing methods' is, of course, to be
interpreted broadly as the mathematical
specification of algorithms for arriving at a
solution (optimal or descriptive), rather than in
terms of precise programming for specific
machines. Nevertheless, we want to stress that
solutions which are not effectively computable
are not properly solutions at all. Existence
theorems and equations which must be satisfied
by optimal solutions are useful tools toward
arriving at effective solutions, but the two must
not be confused. Even iterative methods which
lead in principle to a solution cannot be
regarded as acceptable if they involve
computations beyond the possibilities of presentday computing machines.”
Arrow, Kenneth J, Samuel Karlin & Herbert Scarf (1958), The Nature and Structure of Inventory Problems, in: Studies in the Mathematical Theory of
Inventory and Production, edited by Kenneth J Arrow, Samuel Karlin and Herbert Scarf, Stanford University Press, Stanford, California, p.17.
Russel-Whitehead: PM, Second Edition, p, vi;
"The formalists have forgotten that numbers are needed, not only for doing sums,
but for counting. .... The formalists are like a watchmaker who is so absorbed in
making his watches look pretty that he has forgotten their purpose of telling the
time, and has therefore omitted to insert any works
There is another difficulty in the formalist position, and that is as regards existence.
Hilbert assumes that if a set of axioms does not lead to a contradiction, there must
be some set of objects which satisfies the axioms; accordingly, in place of seeking
to establish existence theorems by producing an instance, he devotes himself to
methods of proving the self-consistency of his axioms. ... Here, again, he has
forgotten that arithmetic has practical uses. There is no limit to the systems of noncontradictory axioms that might be invented. Our reasons for being specially
interested in the axioms that lead to ordinary arithmetic lie outside arithmetic, and
have to do with the application of number to empirical material. This application
itself forms no part of either logic or arithmetic; but a theory which makes it a priori
impossible cannot be right. ....
The intuitionist theory, represented first by Brouwer and later by Weyl, is a more
serious matter. ... The essential point here is the refusal to regard a proposition as
either true or false unless some method exists of deciding the alternative. Brouwer
denies the law of the excluded middle where no such method exists. ... Consequently
large parts of analysis, which for centuries have been thought well established, are
rendered doubtful."
Algorithmic Thinking
"Even those who like algorithms have remarkably little appreciation of the
algorithmic thinking that is required for a
constructive proof. This is illustrated by the nonconstructive nature of
thoroughgoing
many proofs in books on numerical analysis, the theoretical study of practical
numerical algorithms. I would guess that most realist mathematicians are unable
even to recognize when a proof is constructive in the intuitionist's sense.
It is a lot harder than one might think to recognize when a theorem depends on
a nonconstructive argument. One reason is that proofs are rarely self-contained,
but depend on other theorems whose proofs depend on still other theorems.
These other theorems have often been internalized to such an extent that we are
not aware whether or not nonconstructive arguments have been used, or must
be used, in their proofs. Another reason is that
the law of excluded
middle
[LEM] is so ingrained in our thinking that we do not distinguish
between different formulations of a theorem that are trivially equivalent given
LEM, although one formulation may have a constructive proof and the other
not.”
Richman, Fred (1990), Intuitionism As Generalization, Philosophia Mathematica, Vol.5, pp.124-128.
Throwing out Error [
=
Set Theory! ]
"I often hear mention of what must be `thrown out' if one insists
that mathematics needs to be algorithmic. What if one is throwing
out error? Wouldn't that be a good thing rather than the bad thing
the verb `to throw out' insinuates? I personally am not prepared to
argue that what is being thrown out is error, but I think one can
make a very good case that a good deal of confusion and lack of
clarity are being thrown out. .....
How can anyone who is experienced in serious computation
consider it important to conceive of the set of all real numbers as
a mathematical `object' that can in some way be `constructed'
using pure logic? .... Let us agree with Kronecker that it is best to
express our mathematics in a way that is as free as possible from
philosophical concepts. We might in the end find ourselves agreeing
with him about set theory. It is unnecessary.”
Harold Edwards: Kronecker's Algorithmic Mathematics, The Mathematical Intelligencer, Vol. 31, Number 2,
Spring, p. 14; bold emphases added.
Mathematics is Algorithmic
In mathematics everything is algorithm and nothing is
meaning; even when it doesn't look like that because we
seem to be using words to talk about mathematical
things. Even these words are used to construct an
algorithm.
Wittgenstein, Ludwig (1974), Philosophical Grammar, Basil Blackwell, Oxford, p. 468.
It is in this context that one must recall Brouwer's famous first act of intuitionism, with its uncompromising
requirement for constructive mathematics -- which is intrinsically algorithmic -- to be independent of `theoretical
logic' and to be 'languageless':
"FIRST ACT OF INTUITIONISM Completely
separating mathematics from mathematical language and hence
from the phenomena of language described by theoretical logic,
recognizing that intuitionistic mathematics is an essentially
languageless activity of the mind having its origin in the
perception of a move of time.“
Brouwer, Luitzen E.J (1981), Brouwer's Cambridge Lectures on Intuitionism, edited by D. van Dalen,
Cambridge University Press, Cambridge., p.4.
Kolmogorov’s Visions on the Artificial Intellect of Machines …
"Quite probably, with the development of the modern computing technique it
will be clear that in very many cases it is reasonable to conduct the study of real
phenomena avoiding the intermediary stage of stylizing them in the spirit of the
ideas of mathematics of the infinite and the continuous, and passing directly to
discrete models. This applies particularly to the study of systems with a
complicated organization capable of processing information. In the most
developed such systems the tendency to discrete work was due to reasons that
are by now sufficiently clarified. It is a paradox requiring an explanation that
while the human brain of a mathematician works essentially according to a
discrete principle, nevertheless to the mathematician the intuitive grasp, say, of
the properties of geodesics on smooth surfaces is much more accessible than
that of properties of combinatorial schemes capable of approximating them.
Using the brain, as given by the Lord, a mathematician may not be interested in
the combinatorial basis of his work. But the artificial intellect of
machines must be created by man, and man has to plunge into
the indispensable combinatorial mathematics. For the time being it
would still be premature to draw final conclusions about the implications for the
general architecture of the mathematics of the future.“
Kolmogorov, Andrei Nikolaevich (1983), Combinatorial Foundations of Information Theory and the Calculus of Probabilities,
Russian Mathematical Surveys, Vol. 38, #4, pp. 30-1.
Mathematical Foundations of Algorithmic Thinking
• I am not talking about algorithms for analogue
computing – although the theoretical limits of the
digital computer carry over to its analogue
counterpart.
• I am not referring to the noble practice of numerical
analysis [cf: Blum, Lenore, Felipe Cucker, Michael Shub and Steve Smale (1998),
Complexity and Real Computation, Springer Verlag, New York.]
i.
ii.
Numerical methods as dynamical systems
Dynamical systems as Turing Machines
• I am referring, therefore to:
I. Computability Theory (Recursion Theory)
II. Constructive Mathematics (particularly in its Brouwerian
versions)
What is a Computation?
The Ideal Mechanism for Computing: A Turing Machine
Turing Machine Computation & The Church-Turing Thesis
• A Turing Machine is a quadruple: TM  <Q ,,qo,q>, s.t:
• Q: a finite set of states (configurations) that the TM can enter, one
at a time;
• q0Q: a pre-specified initial state of the TM, the state that
initiates a computation;
• qQ: a pre-specified terminal state of the TM, the state at which a
computation halts;
• : a partial function (i.e., possibly undefined over the whole of its
domain): :QQ{L,R,}
• <0,1,>, which is the symbol set for the computation;
• L: denotes a leftward movement of the ‘controls’ of the TM;
• R: denotes a rightward movement of the ‘controls’ of the TM;
• : denotes no movement by the ‘controls’ of the TM;
• , (q,) is undefined; i.e., if the TM reaches a terminal state
(configuration), then all computation cease;
What is a Decision Problem?
"By a decision procedure for a given formalized theory T we understand a
method which permits us to decide in each particular case whether a given
sentence formulated in the symbolism of T can be proved by means of the
devices available in T (or, more generally, can be recognized as valid in T). The
decision problem for T is the problem of determining whether a decision
procedure for T exists (and possibly for exhibiting such procedure). A theory T
is called decidable or undecidable according as the solution of the decision
problem is positive or negative.“
Tarski, Alfred, (in collaboration with) Andrzej Mostowski and Raphael M. Robinson (1953), Undecidable Theories, North-Holland
Publishing Company, Amsterdam p.3; italics in the original.
A decision problem asks whether there exists an algorithm to decide whether a
mathematical assertion does or does not have a proof; or a formal problem does
or does not have a solution. Thus the characterization must make clear the
crucial role of an underpinning model of computation; secondly, the answer is
in the form of a yes/no response. Of course, there is the third alternative of
`undecidable‘ – ‘don’t know’ - too.
Remark: Decidable-Undecidable, Solvable-Unsolvable, ComputableUncomputable, etc., are concepts that are given content algorithmically.
Computational complexity theory and decision problems
The three most important classes of decision problems that almost characterise
the subject of computational complexity theory, underpinned by a model of
computation, are the P, NP and NP-Complete classes.
Concisely, but not quite precisely, they can be described as follows:
i.
P defines the class of computable problems that are solvable
in time bounded by a polynomial function of the size of the
input;
ii. NP is the class of computable problems for which a solution
can be verified in polynomial time;
iii. A computable problem lies in the class called NP-Complete
if every problem that is in NP can be reduced to it in
polynomial time.
 Why are these ‘definitions’ imprecise?
 Because ‘Solvable’, ‘verifiable’ and ‘reduced’ need to be made
precise in terms of allowable methods!
Computability vs. Constructivity
• Acceptability – or not – of the ChurchTuring Thesis.
• Validity of the Law of the Excluded Middle
• Computing by the Ideal Computing Mechanism
vs. the Constructions by an Ideal
Mathematician
• Varieties of Constructive Mathematics vs.
Higher Recursion Theory
• Indeterminacy of Finite Mechanisms vs.
Choice Sequences
‘Impossibility’ (!!) of Bounded Rationality
In his fascinating and, indeed, provocative and challenging chapter, titled What
is Bounded Rationality (cf: Gigerenzer, Gerd & Reinhard Selten (2001; editors),
Bounded Rationality: The Adaptive Toolbox, The MIT Press, Cambridge,
Massachusetts., chapter 2, p.35), Reinhard Selten first wonders what bounded
rationality is,a nd then goes on to state that an answer to the question `cannot
be given' now:
"What is bounded rationality? A complete answer to this
question cannot be given at the present state of the art.
However, empirical findings put limits to the concept and
indicate in which direction further inquiry should go."
In a definitive sense - entirely consistent with the computational underpinnings
Simon always sought - I try to give a `complete answer' to Selten's finessed
question. I go further and would like to claim that the `limits to the concept'
derived from current `empirical findings' cannot point the direction Simon
would have endorsed for `further inquiry' to go - simply because current
frameworks are devoid of the computable underpinnings that were the
hallmark of Simon's behavioural economics.
Paradoxes of Computation Universality
• What is computation universality
• What is a universal computer
• A boundedly rational satisficing agent solving
decision problems is capable of universal
computation
• The Olympian agent is not capable of universal
computation
• A necessary condition for rational behaviour is
capability of universal computation.
• The Impossibility of Inferring Rational Behaviour
• The Algorithmic Unsolvability of Decision Problems
– i.e., Undecidabiliy
“You may also be interested in the
evidence … of our paper that the learned
man and the wise man are not always the
same person. Of course, this has been known
for a long time, but it is nice to have such
definite evidence against the pedant.”
Herbert Simon to Bertrand Russell, 9 September, 1957
“I am also delighted by your exact
demonstration of the old saw that wisdom
is not the same thing as erudition.”
Bertrand Russell to Herbert Simon, 21 September, 1957