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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

"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 Simonwithin 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 satisficingin 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.

“[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: SapereAude!):

- What can I know? Whatis 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 mind’s curiosity [mathematics], those arising out of the needs of survival, reproduction or whatever.

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.

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."

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.

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.]
- 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)

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;
- q0Q: a pre-specified initial state of the TM, the state that initiates a computation;
- qQ: 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)::QQ{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, Computable-Uncomputable, 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:

- 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!

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

‘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

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