Foundations of Boundedly Rational Choices and Satisficing Decisions

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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

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"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.
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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
  Simons 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 Simons 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 Simons 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.

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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
  Newell-Simon sense.
• The 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.

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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!)
• What can I know? ? What is knowledge?
• What must I do? ? How can we obtain it?
• 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 minds curiosity
  mathematics, those arising out of the needs of
  survival, reproduction or whatever.

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An asideIdeas and Growth Lucas Economica,
February, 2009, p.1
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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
  present-day 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.

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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
  non-contradictory 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."

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Algorithmic Thinking

• "Even those who like algorithms have remarkably
  little appreciation of the thoroughgoing
  algorithmic thinking that is required for a
  constructive proof. This is illustrated by the
  nonconstructive nature of 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.

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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.

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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.

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Kolmogorovs 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.

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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.
• Numerical methods as dynamical systems
• Dynamical systems as Turing Machines
• I am referring, therefore to
• Computability Theory (Recursion Theory)
• Constructive Mathematics (particularly in its
  Brouwerian versions)

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What is a Computation?

• The Ideal Mechanism for Computing A Turing
  Machine

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Turing Machine Computation The Church-Turing
Thesis

• A Turing Machine is a quadruple TM ?
  ltQ,?,qo,q?gt, 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,?
• ??lt0,1,?gt, 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

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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 dont know
  - too.
• Remark Decidable-Undecidable,
  Solvable-Unsolvable, Computable-Uncomputable,
  etc., are concepts that are given content
  algorithmically.

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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
• P defines the class of computable problems that
  are solvable in time bounded by a polynomial
  function of the size of the input
• NP is the class of computable problems for which
  a solution can be verified in polynomial time
• 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!

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Computability vs. Constructivity

• Acceptability or not of the Church-Turing
  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

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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.

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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

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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