Hans Gottinger, Essays on Decision, Information, Computation and Technology

ong>Hansong> W. ong>Gottingerong>

ESSAYS ON

DECISION,

INFOR MATION,

COMPUTATION &

TECHNOLOGY

ong>Hansong> W. ong>Gottingerong>

ESSAYS ON

DECISION,

INFOR MATION,

COMPUTATION &

TECHNOLOGY

ong>Hansong> W. ong>Gottingerong>

STRATEC Munich, Germany

www.stratec-con.net

stratec_c@yahoo.com

gottingerhans@gmail.com

Aggregate Keywords:

Utility, Measurement/Scaling, Preference Orderings, Decisions, Decision-Making under Uncertainty,

Strategy, Game Theory, Statistical Games/Decisions, Mathematical Economics, Information Economics,

Organizations, Behavioral Economics, Bounded/Limited Rationality, Optimal Search, Complexity

Measures, Intelligent Decision Systems, Statistical Expert Systems, Policy Decisions, Regulatory

Decisions, Social/Environmental Decisions.

CONTENTS

HANS W. GOTTINGER

CONTENTS

ESSAYS ON DECISIONS , INFORMATION,

COMPUTATION AND SYSTEMS

Contents ................................................................................................ I

Foreword. .............................................................................................. III

Preface and Introduction ............................................................................... V

1. Preferences, Information and Decision ................................................. 1

1.1 Über die Existenz einer stetigen, reellen Nutzenfunktion

(On the Existence of a continuous, real-valued utility function) ................................... 3

1.2 Methodologische Entwicklungen in der Meßtheorie

(Methodological Developments in Measurement Theory) ....................................... 11

1.3 Existence of a Utility on a Topological Semigroup ............................................... 45

1.4 Conditional Utility ............................................................................. 59

1.5 Foundations of Lexicographic Utility . ........................................................... 69

1.6 Decision problems under uncertainty based on entropy functionals ............................. 79

1.7 Choice and Complexity . ....................................................................... 109

1.8 Computational Costs and Bounded Rationality (Philosophy of Economics, 223-238) ............. 127

1.9 Krohn-Rhodes Complexity on Decision Rules . .................................................. 143

1.10 An Information Theoretic Approach to Large Organizations .................................... 157

1.11 Some Measures of Information arising in Statistical Games ..................................... 169

1.12 Subjective Qualitative Information Structures based on Orderings .............................. 175

1.13 Qualitative Information and Comparative Informativeness ..................................... 205

1.14 On a Problem of Optimal Search .............................................................. 219

2. Intelligent Decision Systems ........................................................ 229

2.1 Intelligent Decision Support Systems (with Peter Weimann) ..................................... 231

2.2 Statistical Expert Systems ...................................................................... 247

2.3 Artificial Intelligence and Economic Modeling .................................................. 261

3. Applications ....................................................................... 269

3.1 Assessment of Social Value in the Allocation of CT Scanners in the Munich Metropolitan Area ... 271

3.2 Adoption Decisions and Diffusion ............................................................. 299

3.3 Choosing Regulatory Options when Environmental Costs are Uncertain ........................ 317

3.4 Dynamic Portfolio Strategies with Transaction Costs ........................................... 331

HANS W. GOTTINGER

CONTENTS

FOREWORD

HANS W. GOTTINGER

FOREWORD

Selected writings of a very prolific author over a long-time period typically open a window both on the

evolution of his thinking and on the changing focusses and fads of the scientific community. ong>Gottingerong>s’s

ong>Essaysong> on Decision, Information, Computation & Technology, which cover roughly the period 1970-2000,

are no exception.

The essays, some of which are in German, are hard to subsume. They span across economics, mathematics,

operations research, expert systems, and then some. They include provocative advances, more standard

efforts, and uncontroversial applications. At their best, they delineate new venues of research that could

significantly impact the way we are doing economics. It has been widely argued, rightfully so in my

opinion, that the imperialistic spread of neo-classical thinking has stuck much of the economic profession

in an increasingly noxious Newtonian vision of the economy together with a very narrow conception of

rational decision-making. As usual with theoretical approaches that struggle to cope with the reality, most

improvements efforts at improvement are purely incremental and do not go beyond patching up the main

deficiencies by twisting standard relationships, adding new one, introducing more and more epicycles while

rejecting any true change in paradigm. Not so ong>Gottingerong>, who did not hesitate to enter terra incognita and

suggest radically new ways of addressing fundamental economic questions.

The first and by far larger section of the book focusses on decision-theoretic problems. Some early works,

published largely in the period 1970-1975, centre about conditions under which preferences can be represented

by a utility function with agreeable properties. The issue was acute at the time but has largely been exhausted

nowadays and arguably stalled rather than promoted our mastering of the very large and most important

class of situations where standard utility functions do not exist. Still, it remains at the very core of neoclassical

economics. A standalone contribution reviews the foundational work in measurement theory that

was carried out at the time by such authors as Krantz, Luce, Pfanzagl, Suppes, and Tversky. The logical

link between utility, decision-making, and measurement is immediate, utility is nothing but a measure of

preferences, and decision-making typically requires measurement. Nonetheless, the theory of measurement

has been largely, and unfortunately, overlooked by the economic profession. To my knowledge, ong>Gottingerong>’s

paper is one of the very few attempts to remedy this and remind economists that much work relevant for

the very core of their discourse is being done outside of their narrow disciplinary circles.

Several later papers in the ong>Essaysong>’ first section are disruptive in their attempt to enrich and transcend the

standard neo-classical framework by introducing concepts and methods from other disciplines, predominantly

information theory, statistical inference, computing, and abstract automata theory. Thus, ong>Gottingerong> suggested

in a 1990 paper using the maximum entropy principle to tackle decision-making under uncertainty. This

was brave – entropy is not easy to interpret, understand or visualize in a socio-economic context. The use of

the concept in economics remains marginal up to now but is the subject of active research in the innovative

fringe and increasingly used in such field as game theory, finance, robust optimization, and organization

theory. Other lines of research suggested in several articles hint to the nowadays very active and promising

efforts in complexity economics and agent-based economics. Other papers combine information theory as

the mathematics of communication and storage of information with more standard economic issues. Here

again, one recognizes some early precursors of complexity and agent-based economics, but also of boundedrationality

models and computational games, among others. Inexplicably, and unfortunately, the ong>Essaysong> do

not include the nice paper on Structure and Complexity in Socio-Economic Systems that ong>Gottingerong> jointly

published with the late Peter Albin in Mathematical Social Sciences 1983.

III

HANS W. GOTTINGER

FOREWORD

The second part of the book is devoted to expert systems, which were the main focus of research in AI in

the 80s. Together with the informatician Peter Weimann, ong>Gottingerong> published towards the end of the decade

a number of papers on the topic. The one reproduced here outlines a flexible shell that allows considering

conditional probabilities and all information available at the time of decision. A contemporaneous paper

with ong>Gottingerong> as sole author discusses in broad terms the concept of a statistical consultation system,

which can help a user with little statistical knowledge to design a proper statistical analysis. Another one,

which sketches an approach for building a ruled-based qualitative model of the macro-economy, has again

some reflexion in recent agent-based economic models. I felt pleased and honoured to find that the author’s

argumentation echoes some idea that G.R. (Robert) Boynton and myself expressed independently in a

1987 paper.

The last section of the ong>Essaysong> is devoted to “applications”, showing that with advancing age ong>Gottingerong> has

been increasingly attracted to less abstract and even to empirical research. It includes a cost-benefits analysis

of PT scanners location and a paper on dynamic environmental regulation under uncertainty. Two other

papers address the questions of optimal dynamic portfolio selection and suggest a novel micro-economic

foundation for diffusion curves.

Reading the ong>Essaysong> requires, for most part, a solid knowledge of mathematics. It is hard, abstract reading.

But it is also a rewarding lecture, for a wide circle of senior academics and graduate students alike. In

addition to providing valuable insights beyond the mainstream wisdom it sheds light on the way science

advances. Not only by taking the safe way of incremental improvement in accepted ideas, but also and

more dangerously by combining ideas, by reaching across disciplines, in the hope that some of these efforts

will bloom and survive.

Prof. Dr. Christophe Deissenberg, Luxembourg

April 26, 2018

PREFACE AND INTRODUCTION

HANS W. GOTTINGER

ong>Essaysong> on Decisions , Information, Computation and Systems

PREFACE AND INTRODUCTION

Not only in economics, psychology and business but generally in most human activities, decisions and decision

making, whether deliberate or routine, play a crucial part in the human pursuit of happiness, prosperity,

individual and collective satisfaction. A precursor of decision-making , strategy and planning, dates back

to the classical Greeks and Chinese up to modern European times. The conceptual tools developed with

game theory, statistical decision theory, operations research, systems analyis/engineering and management

science all originated in the twentiest century and are fast expanding in the digital world with intelligent

decision systems. Applications proliferate in health care (medical decision-making), logistics, transportation,

industrial and public services facilitated through enhanced tools involving artificial intelligence (AI),

machine learning, human-machine interactions, and machine-to-machine cooperations (Internet of Things).

Within economics the behavioral foundations emerged from game theory with links to

competition theory and policy, competitiveness, organizations and teams up to managerial economics

with decision analysis of multiple objectives, risk and uncertainty. New subdisciplines have emerged such

as information economics, computational economics, behavioral and experimental economics.The latter

got recent prominence through Nobel Prizes in economics for D.Kahneman (2002) and R. Thaler (2017).

The collection of selected essays to follow emphasizes some limited foundational, theme related key issues

subsumed under the title. We proceed partly in chronological order which also moves from more theoretical,

conceptual themes to further application. All are categorized in three parts. At the end a brief bibliography

points to relevant related literature.

1. Preferences, Information and Decisions

1.1. Über die Existenz einer stetigen, reellen Nutzenfunktion (On the Existence of a continuous,

real-valued utility function)

The article pursues a simple representation of binary relations , preference-indifference or strict preference,

on a bundle of objects X (commodity bundles) by a numerical function (utility) on the real line. For its

representation it only uses ordering properties compatible for continuous functions on the real line. This

distinguishes itself from G. Debreu’s representation theorems (Debreu [1]) as he uses

ordered topological spaces and structural topological properties later extended to continuity properties of

Paretian utility (Debreu [2]). The reason behind is that basic rational preferences can be shown for “ordinal

utility functions“ in deterministic settings (or consumer choice theory) without taking recourse to more

aadvanced topological tools. The latter lead to more elegant representations but less intuitive economic

interpretations.

1.2. Methodologische Entwicklungen in der Messtheorie (Methodological Developments in

Measurement Theory)

This piece serves as a limited survey on measurement theory following a comprehensive , seminal treatment

by Pfanzagl [3]. It complements issues of measurement of utility theory as used in economic and psychology

research. We show the similarity in the structure of measurement theory to that of utility theory.

It lends itself to (a) algebraic metric operations , (b) axiomatic models, (c) order relations linking various

entities, (d) topology on ordered sets and (e) transformations of empirical relational systems on numerical

scales.

1.3 Existence of a Utility on a Topological Semigroup

Here we go beyond ordinal utility of Sec. 1.1. and explore the connection between cardinal and expected

utility theory. Again the work of G. Debreu [4]

has been seminal. To prove existence for an additive utility representation we need specific algebraic

HANS W. GOTTINGER

PREFACE AND INTRODUCTION

assumptions in addition to those of the order topology commonly found in an orderd to topological semigroup

(tsg). The use of an order topology on a semigroup suggests a wide variety of embeddings of preference

orderings into real numbers through dimensional transformations. That is, a tsg would allow a collection

of metrics (beyond additivity) to be transformed into real numbers to “cardinalize“ utilities. Utilities on

tsgs would fit as one category of several algebraic-topological constructs to clasif diverse preference orders

and utilities (Vind [5]).

1.4 Conditional Utility

Here I consider a situation where the utility of a risky/uncertain prospect in a Von Neumann-Morgenstern

utility context is affected by an “extraneous chance mechanism“, sort of stochastic shock through changes

of states of nature which additionally affects its utility valuation. Putting it in Savage’s system [6] of

decision acts would yield the difference between prior utility (non conditionalized) and posterior utility

(conditionalized) as a revealed measure of the value of information (VI) provided one would know about

the extraneous chance mechanism.

1.5 Foundations of Lexicographic Utility

A common preference order on the choice of sure or random prospects would be open to tradeoff or

compensatory choices reflected simply in microeconomic standard indifference diagrams. Lexicographic

utility can no longer be represented as a real-valued function but is multidimensional on a vector space.

In between we could think of mixed preferences, lexicographic or compensatory, or lexicographic subject

to 2nd, ... , nth order fixed constraints making a feasible preference set. Also stochastic preferences over

lexicographic choices allow lexicographic tradeoff structures.

1.6 Decision Problems under Uncertainty based on Entropy Functionals (Theory and Decision,

1990)

In this paper it is shown how various criteria of optimal decisions under uncertainty relate to the entropy

function known from classical information theory.

Of particular interest is the “Expected Utility of Perfect Information“(EUPI) being closely linked to Shannon’s

information measure. This gives rise to other decision theoretic notions such as expected opportunity loss,

payoff relevant information emerging from statistical decision analysis . They are conceptually applied

to optimality at equilibrium in many person games as well as to specific types of economic organizations

such as teams.

1.7 Choice and Complexity

Here we relate human choice processes to computation and machines. We move from the notions of

effective computability, effective algorithms to computational complexity with respect to computable

relations identified as preference relations generating choice processes. In the center of observations will

be a basic model of a social choice machine (SCM). The basic features from a SCM would be threefold:

(i) characterizing computational rationality as a sort of bounded or limited human/machine rationality. (ii)

computational rationality being bound by the computational complexity of the choice process, (iii) rational

choice processes being restricted by the computational difficulty of effectively realizing rational choice

functions. Exploring recursive computational functions in recursive topological spaces yield a description

of effective computability and corresponding complexity. They end up being “simulated“ by sequential

finite state machines as Turing machines. The complexity number of a Turing machine simulating a choice

function is the minimal length of the program which simulates this machine.

1.8 Computational Costs and Bounded Rationality

The bound will be achieved by the fact that computations are not costless, that is they use procedures that

require the use of scarce resources . In such situations decision may use simple heuristics or “rules of

thumb“ to reduce the cost of computation. A device to measure computational bounds could be finite state

sequential machines or Turing machines. Some applications relate to the construction of aggregation of a

consumer price index up to decentralized resource allocation in the theory of the firm.

PREFACE AND INTRODUCTION

HANS W. GOTTINGER

1.9 Krohn-Rhodes Complexity on Decision Rules

This is designed as an updated review of algebraic complexity of Krohn-Rhodes [7] and its connection to

bounded rationality properties of H. Simon [8], its intrinsic application to chess-playing programs, heuristics

and problem solving with interface issues of economics to computer and management science.

1.10 An Information-Theoretic Approach to Large Organizations

Here we look at the interaction of decision , information and performance in large economic organizations

and challenges that would arise from issues of bounded rationality treated in previous sections. Decision

theoretic and computable models of organizations have been advanced by March and Simon [9] and

Marschak and Radner [10]. I build processing tasks in terms of a “machine model“ that face payoff relevant

information and complexity limits.

1.11 Some Measures of Information arising in Statistical Games

Payoff relevant information with respect to an expected utility/loss function arises from statistical decision

functions embedded in game theory (Blackwell and Girshick [11]). They serve as value of information

(VI) provided by experiments. In a best sense VI, positive or negative, is the amount that payoff-relevant

information adds to or reduces from the payoff function associated with a decision in a statistical game. I

propose various measures of information emerging from statistical decision theory that reflect the economic

aspects of usefulness of information (based on some kind of utility or loss function) rather than the original

physical/engineering viewpoint of transmitting and controlling information flows through a large (noisy

or noiseless) communication channel (Shannon and Weaver [12]). Thus it creates an informational metric

for payoff functions in terms of an economic value of information.

1.12 Subjective Qualitative Information Structures based on Orderings

In this essay I reverse the qualitative relation “not more probable than“ as a “primitive“ of probability to

information in review of Savage’s [6] introduction to subjective probability. This is an attempt to axiomatize

subjective information as a conceptual precursor to generating subjective probabilities on corresponding

(information induced) events. The primitive relation “not more informative than“ is constructed on the basis

of order topopologies by the Hungarian mathematician A. Cszaszar called topogenous structures. I then

show that semi-topogeneous information structures (also named experiments) have a natural mapping on

subjective probability structures of the Savage type. With the more recent advance of artificial intelligence,

machine learning and big data potential applications can be foreseen that such qualitative information

structures could be machine-generated in terms of qualitative orderings.

1.13 Qualitative Information and Comparative Informativeness

This essay provides a conceptual qualification , refinement and expansion of the previous Section 1.12 and

a diversification into various directions.

1.14 On a Problem of Optimal Search

A simple search as a classical opertaions research (OR) problem (Stone [13]) is treated as a sequential

statistical decision problem and involves some optimal stopping. Given a sequential decision problem, in

order to find the best decision (policy) now is whether to stop and make a decision or to go on and take

another observation it is desirable to know the best decision in the future., Consequently, the search for an

optimal decision should not proceed according to chronological time but in reverse order to work backwards

in time since the present optimum involves the future optimum. This is incorporated in the principle of

dynamic programming (Bellman [14]). In a grid type search, with T [(pk, N] formally denoting the minimum

average number of comparisons of cells per successful search, given N cells and prior distribution (pk) on

k trials. Then the search procedure starts with the selection of a cell for the first comparison . T [(pk, N] is

subject to the formalism of dynamic programming.

VII

HANS W. GOTTINGER

PREFACE AND INTRODUCTION

2. Intelligent Decision Systems

2.1 Intelligent Decision Support systems

We describe intelligent decision systems as a prototype of a decision technology that subjects itself to

computerization, therefore opening itself up to computational tools such as artificial intelligence techniques

through expert systems, neural networks and machine learning. The “intelligence“ and computational parts

have increased tremendously though the Internet over the last 30 years (ong>Gottingerong> [15]) but the underlying

statistical decision models are still valid and ramifications in application areas are widely perceived.

2.2 Statistical Expert Systems

On the interface of machine generated “Big Data“ and proper statistical treatment an advice giving program

such as an expert system or decision support system suggests a targeted range of statistical tools and expert

judgements for data analysis involving classification and regression analysis, decision trees, variable selection

and econometrics. The fast computational generation of online data could also activate built-in-intelligent

mechanisms of classifying, categorizing, visualizing, aggregating diverse unstructured data types thus

allowing data analytic tools for statistical metrics – useful for statistical inference and decisions.

2.3 Artificial Intelligence and Economic Modelling

Here is one of the early attempts to explore artificial intelligence/expert system techniques for micromacro

economic models. It discusses some methodological issues in implementing those tools connecting

influence diagrams with qualitative reasoning on graphs and data fusion as similar lines of modeling have

been pursued in AI based “qualitative physics“ modeling.

3. Applications

3.1 Assessment of Social Value in the Allocation of CT Scanners in the Munich Metropolitan

Area

This is an illustrative case study of a synthetic benefit/risk/cost analysis of an allocation problem of a medical

technology that calculates the social value as a decision criterion. The empirical inputs are based on this

specific case, the value judgements on “value of life“ as given in the mid 1980s and the medical technology

parameters at that time. The methodology used for this case may be possibly adapted to a comparative case

of resource allocation decision at any time for other “technology assessment“ purposes with appropriate

impacts and overall checked by sensitivity analysis on the major input variables.

3.2 Adoption Decisions and Diffusion (Swiss Journal of Economics and Statistics, 1991)

The article applies the use of decision modelling under risk/uncertainty to describe the decision-making

process of a firm in its drive of technology adoption and the diffusion of innovation. It shows that adoption

decisions are inheretly linked to risk behavior of firms.

3.3 Choosing Regulatory Options when Environmental Costs are Uncertain

This application area is concerned with the potential of public policies against long-term climate change.

A model of optimal statistical decisions is used to determine the value of information (VI) on restricting

greenhouse gas emissions against choosing effective regulatory measures (including carbon taxes) for

implementation of emission control. Since a balancing process evolves over time with collecting of

information the appropriate algorithmic handling is through dynamic programming (as in Secs. 1.11 and

1.14). A crucial parameter in the evaluation of the decision model is the “critical probability“ or threshold

to determine the differential cost of delaying additional emission restrictions when such restrictions will

be necessary in later periods as against the cost of imposing additional restrictions now that later prove

unnecessary. Alternatively, a “critical probability“ could also determine when the costs of stringently

restricting emissions are small in the future relative to the foregone benefits of limiting emissions in the

VIII

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HANS W. GOTTINGER

current period. Though composed at a time when climate treaties have only started to be discussed in the

Kyoto Protocol, the method would fit to a scaled-up application of the Paris Climate Accord (PCA,2016).

3.4 Dynamic Portfolio Strategies with Transaction Costs (Journal of Policy Studies, 2005)

Here I explore a portfolio investment choice model , in financial economics, with a mixed single risky/

riskless asset. to maximize the investor’s expected utility at terminal wealth assuming different forms and

size of transaction costs in trading of assets.

The vintage collection of essays over forty years pursues a path from the foundations of decision theory,

its mathematical representations and conceptual ramifications to aspects of information, computation,

complexity and intelligent decision systems.

Special application areas cover policy analysis of health care delivery, technology adoption in innovating

firms and industries, , and when to induce effective and efficient dynamic regulatory control in policies

toward environmental damage containment of climate change processes and optimal investment decisions

with constraints.

ong>Hansong> W. ong>Gottingerong>, Feb. 2018

References

[1] Debreu, G., “Representation of a Preference Ordering by a Numerical Function“, in Decision Processes,

R.M. Thrall, C.H. Coombs and R.G. Davis, eds., Wiley: New York 1954, 159-165

[2] Debreu, G., “Continuity Properties of Paretian Utility“, International Economic Review 5, 1964, 285-293

[3] Pfanzagl, J., Theory of Measurement, Physica Verlag: Würzburg-Wien 1968

[4] Debreu, G., “Topological Methods in Cardinal Utility Theory“, in Mathematical Methods in the Social

Sciences, K.J.Arrow, S. Karlin and P. Suppes, eds., Stanford Univ. Press: Stanford,Ca., 1960,16-26

[5] Vind, K., Independence, Additivity and Uncertainty (Studies in Economic Theory),

Springer: New York 2003

[6] Savage, L.J., The Foundations of Statistics, Wiley: New York 1954

[7] Rhodes, J., Application of Automata Theory and Algebra, World Scientific: Singapore 2010

[8] Simon, H., Models of Bounded Rationality, Vol. 2, MIT Press: Cambridge,Ma. 1982

[9] March, J.G. and H. Simon, Organizations, Wiley: New York 1958

[10] Marschak, J. and R. Radner, The Economic Theory of Teams, Yale Univ. Press: New Haven,Cn. 1972

[11] Blackwell, D. and M.A. Girshick, Theory of Games and Statistical Decisions, Wiley: New York 1955

[12] Shannon, C.E. and W. Weaver, The Mathematical Theory of Communication, The Univ. of Illinois

Press: Urbana, Il. 1949

[13] Stone, L.D., Theory of Optimal Search, Academic Press: New York 1975

[14] Bellman, R., Dynamic Programming, Princeton Univ. Press: Princeton,NJ 1957

[15] ong>Gottingerong>, H.W., Internet Economics-Models,Methods and Management, Bentham Science: London.

HANS W. GOTTINGER

PREFACE AND INTRODUCTION

PREFERENCES,

INFOR MATION

AND DECISION

1.1 ÜBER DIE EXISTENZ EINER STETIGEN, REELLEN NUTZENFUNKTION

HANS W. GOTTINGER

1.1 ÜBER DIE EXISTENZ EINER STETIGEN, REELLEN NUTZENFUNKTION

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1.1 ÜBER DIE EXISTENZ EINER STETIGEN, REELLEN NUTZENFUNKTION

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1.1 ÜBER DIE EXISTENZ EINER STETIGEN, REELLEN NUTZENFUNKTION

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1.1 ÜBER DIE EXISTENZ EINER STETIGEN, REELLEN NUTZENFUNKTION

290 H.-vV. ong>Gottingerong>: Eine stetige, reelle Nutzenfunktion

Doch w 2 P y P w 1 impliziert, wegen w 2 I x 2 , w 1 I x 1 und S. 4, x 2 P

h (y) P x 1 , und das bedeutet lf (y) - f (x) 1 < e. Da x e X und e > 0 willkürlich

gewählt werden können, gewinnen wir die Stetigkeit von f.

Literatur

[l] C. Caratheo dory: Vorlesungen über reelle Funktionen. Leipzig und

Berlin: B. G. Teubner, 1939.

[2] G. De b r e u: Theory of Value. Cowles Foundation for Research in Economics

(Monogr. 17), New York: John Wiley & Sons, 1959.

[3] H. Sonnenschein : The Relationship between Transitive Preference

and the Structure of the Choice Space. Econometrica 33 (1965), S. 624-634.

[4] H. Wold (in Verbindung mit L. Jureen): Demand Analysis.

New York: John Wiley & Sons, 1953.

[5] T. Y ok o y am a: On Uniformity and Continuity Conditions in the

Theory of Consumer's Choice. Osaka Economic Papers 3 (1954), S. 29-35.

Anschrift des Verfassers: Dr. ong>Hansong>-Werner ong>Gottingerong>, Research Associate,

Department of Economics, University of California, Berkeley, California 94 704,

USA.

Prlnted In A ustrla

1.2 METHOLOGISCHE ENTWICKLUNGEN IN DER MESSTHEORIE

HANS W. GOTTINGER

Methodologische Entwicklungen in der Meßtheorie':")

Von ong>Hansong> - Werner Gott in g er, München

'') Eberhard Fels (1924-1970) zum Gedenken.

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1.2 METHOLOGISCHE ENTWICKLUNGEN IN DER MESSTHEORIE

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1.3 EXISTENCE OF A UTILITY ON A TOPOLOGICAL SEMIGROUP

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1.3 EXISTENCE OF A UTILITY ON A TOPOLOGICAL SEMIGROUP

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1.3 EXISTENCE OF A UTILITY ON A TOPOLOGICAL SEMIGROUP

1.4 CONDITIONAL UTILITY

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1.4 CONDITIONAL UTILITY

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1.4 CONDITIONAL UTILITY

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1.4 CONDITIONAL UTILITY

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1.4 CONDITIONAL UTILITY

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1.4 CONDITIONAL UTILITY

1.5 FOUNDATIONS OF LEXICOGRAPHIC UTILITY

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1.5 FOUNDATIONS OF LEXICOGRAPHIC UTILITY

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1.6 DECISION PROBLEMS UNDER UNCERTAINTY BASED ON ENTROPY FUNCTIONALS

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1.7 CHOICE AND COMPLEXITY

HANS W. GOTTINGER

Mathematical Social Sciences 14 (1987) 1-17

North-Holland

CHOICE AND COMPLEXITY

ong>Hansong> W. GOTTINGER

The University of Maastricht (RU), Institute of Management Science, PO Box 591, Maastricht,

The Netherlands

and

Department of Systems Engineering, University of Virginia, Charlol/esville, VA 22901, U.S.A.

Communicated by F.W. Roush

Received 6 May 1986

An attempt is made to propose a concept of limited rationality for choice functions based on

computability theory in computer science.

Starling with the observation that it is possible to construct a machine simulating strategies of

each individual in society, one machine for each individual's preference structure, we identify

internal states of this machine with strategies or strategic preferences. Inputs are possible actions

of other agents in society, thus society is effectively operating as a social choice machine. The

main result states that effective realization of choice functions is bound by the 'complexity of

computing machines'. Given a certain social choice machine, this complexity is simply the length

of the shortest program which simulates this machine.

Key words: Limited rationality; computability; cognitive science; complexity; social choice.

1. Introduction

Ever since choice theory has established itself as part of economic theory and

mathematical economics there have been attempts to axiomatize it on the basis

of set theory and topology. To the extent that 'human rationality' and 'human

problem-solving' has been taken as an anchor point for constmcting 'artificial intelligence'

it would be natural to model human choice processes by computational

procedures and by representations of computational theory.

In economic theory the problem of representation of rational choice or rational

decision mles has obtained primary attention. The realizability of such representation,

however, in terms of computational viability has so far been neglected .. In

economic theory, with the exception of Simon's path-breaking work, the matter of

effective computability of choice and decision mies has essentially been confined to

the problem of costliness of mies. As A. Rubinstein (1985) pointed out, economists

have found it 'difficult to embed the procedural aspects of decision-making in formal

economics models'. As Lewis (1985) has put it: ' .. .if rationality is constrained by

effective computability within the framework of recursive functions ... , the notion

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1.8 COMPUTATIONAL COSTS AND BOUNDED RATIONALITY

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228 H. W. ong>Gottingerong> / An information theoretic approach

Zh=inf{z;}. Let Z1,=0 and for i-:;:.h Jet z;=z;+(Zhl(n-1)). Then by (b)

f(z1, ... , z) v1. So by minimality f(z1, ... , z) = v1. This is a contradiction since

(z1, ... , Zn ) had a maximum number of zeros among its coordinates. So some Zi are

zero\ By (c) we now apply this same argument to the function Uk(m -1) where

No= {J}. Then we find that at least two of the Zi must be zero. By continuing this

process we find that all but one of the Zi must be zero.

5. The input machine

The input machine collects certain inputs Xj from sources outside or inside the

organiztion, and converts them in a one-to-one fashion into a form that the output

machine can process them. The collection and conversion of Xj will in general

require a certain processing time also, for which the notation tY> was introduced.

(The superscript (i) will henceforth be omitted, for notational simplicity.)

The allowances for the complexity of an input processing task are different from

those of an output machine, in fact, they appear to have no counterpart on the output

machine.

According to a !arge volume of psychometric data, the processing time ('the

reaction time') for an input symbol Xj varies with the probability with which the

symbol arrives. The input machine in other words, somehow quickly accumulates

statistical evidence concerning the relative frequency with which the various Xj are

received and then adapts its processing times accordingly. Symbols that occur rarely

are processed more slowly and those that come up frequently are disposed of

quickly. There are, in fact, indications that the variation of t i with the probability P i

is roughly logarithmic, i.e.

lj = foj - Cj log Pj J (4)

but this observation does not seem to be uniformly acepted by experimental psychologists.

Under these circumstances it may be appropriate to define load dependence for

input machines in a way that is roughly analogous to Definition 1 for output

machines, but includes eq. (4) as a special possibility. In such a case the analogy

should further make plausible allowance for the complexity of alternate and parallel

processing tasks. The size of an input alphabet can be quite !arge. lt may be appropriate

to associate the notion of the complexity of the task of input processing with

the numbers of input symbols, and the probability of their occurrence in roughly the

same way in which this notion was associated with the number of destinations (or of

permutations) for the output machine. The qualitative analogy that suggests itself

here would then be this: an input processing task would be the easier, the smaller the

number n of symbols in the output alphabet, and if n remains the same the task

should become easier if the frequency of the processing is increased.

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