Optimal Control and Dynamic Games

OPTIMAL CONTROL AND DYNAMIC GAMES

Advances in Computational Management Science

VOLUME 7

Optimal Control

and Dynamic Games

Applications in Finance, Management Science

and Economics

Edited by

RICHARD F. HARTL

University of Vienna,

Austria

and

CHRISTOPHE DEISSENBERG

Université de la Méditerrannée,

Les Milles, France

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 0-387-25804-3 (HB) Springer Dordrecht, Berlin, Heidelberg, New York

ISBN-10 0-387-25805-1 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

ISBN-13 978-0-387-25804-1 (HB) Springer Dordrecht, Berlin, Heidelberg, New York

ISBN-13 978-0-387-25805-8 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 2005 Springer

No part of this work may be reproduced, stored in a retrieval system, or transmitted

in any form or by any means, electronic, mechanical, photocopying, microfilming, recording

or otherwise, without written permission from the Publisher, with the exception

of any material supplied specifically for the purpose of being entered

and executed on a computer system, for exclusive use by the purchaser of the work.

Printed in the Netherlands.

Contents

Part I Applications to Marketing

Richard F. Hartl, Peter M. Kort ...................................

2 Advertising and Advertising Claims Over Time

Charles S. Tapiero ...............................................

Part II Environmental Applications

3 Capital Resource Substitution, Overshooting, and

Sustainable Development

Hassan Bencheckroun, Seiichi Katayama, Ngo Van Long .............

4 Hierarchical and Asymptotic Optimal Control Models

Alain B. Haurie .................................................

5 Common Property Resource and Private Capital

Accumulation with Random Jump

Masatoshi Fujisaki, Seiichi Katayama, Hiroshi Ohta ..................

6 Transfer mechanisms inducing a sustainable forest

exploitation

Guiomar Mart´ın-Herr´an, Mabel Tidball .............................

7 Characterizing Dynamic Irrigation Policies via Green’s

Theorem

Uri Shani, Yacov Tsur, Amos Zemel ...............................

3

19

41

61

77

85

105

XI

Listof Contributors XV

XIX

1 Advertising directed towards existing and new customers

Introduction ...............................................

............................................

......................................................

...

Economic Sustainable Development

Foreword

Contents

Part III Applications in Economics and Finance

8 Volatility Forecasts and the Profitability of Automated

Trading Strategies

Engelbert J. Dockner, G¨unter Strobl ¨ . .............................. .

9 Two-Part Tariff Pricing in a Dynamic Environment

Gila E. Fruchter .................................................

10 Numerical Solutions to Lump-Sum Pension Fund Problems

That Can Yield Left-Skewed Fund Return Distributions

Jacek B. Krawczyk . ..............................................

11 Differentiated capital and the distribution of wealth

Gerhard Sorger ................................................. .

12 Optimal Firm Contributions to Open Source Software

Rong Zhang, Ernan Haruvy, Ashutosh Prasad, Suresh P. Sethi ........

Part IV Production, Maintenance and Transportation

13 The Impact of Dynamic Demand and Dynamic Net

Revenues on Firm Clockspeed

Janice E. Carrillo ............................................... .

14 Hibernation Durations for Chain of Machines with

Maintenance Under Uncertainty

15 Self-Organized Control of Irregular or Perturbed Network

Traffic

Dirk Helbing, Stefan L¨mmer, Jean-Patrick Lebacque ............... .

16 A stochastic optimal control policy for a manufacturing

system on a finite time horizon

Eugene Khmelnitsky, Gonen Singer ................................

17 On a State-Constrained Control Problem in Optimal

Production and Maintenance

Helmut Maurer, Jang-Ho Robert Kimr, Georg Vossen .................

Part V Methodological Advances

18 Reliability Index

A. Bensoussan...................................................

121

141

155

177

197

215

Ali Dogramaci ................................................... 231

239

275

289

311

VIII

Contents

19 The direct method for a class of infinite horizon dynamic

games

Curriculum Vitae - Prof. Suresh P. Sethi ......................

319

335

... 343

Dean A. Carlson, George Leitmann ................................

....................... . ..................... .

IX

Author Index

I am delighted to be invited to give a few remarks at this workshop

honouring Suresh Sethi. He was one of the most hardworking and pro￾lific of any of my (45 or so PhD) students during the course of 42 years

of teaching at GSIA. I would like to discuss our interactions both during

and after the completion of his doctoral thesis. Suresh entered the PhD

program in the fall of 1969 just after I had published a paper on the

application of a new mathematical model called Optimal Control The￾ory which originated in Russia. The paper was called: ”The Optimal

Maintenance and Sale Date of a Machine”. Suresh quickly absorbed

the mathematics on which optimal control theory is base. We wrote a

joint paper called ”Applications of Mathematical Control Theory to Fi￾nance: Modeling Simple Dynamic Cash Balance Problems,” which was

published before the end of 1970. At the same time Suresh wrote nine

additional papers by himself (on topics which I have forgotten). He put

these nine papers together with the dynamic cash balance paper above to

complete his thesis in record time. His PhD was awarded before the end

of 1970. The nine additional chapters in his thesis were also published

by him in subsequent years.

In 1970 it was uncommon for professors to write joint papers with

either their colleagues or with their PhD students. In GSIA we encour￾aged such joint work and other schools have since imitated this practice.

In order to analyze how Suresh has thrived in this environment, I did a

quick count of the number of authors in each of the papers listed in the

Professional Journal Articles section of his vita, obtaining the following

amazing distribution: single author, 37; 2 authors, 96; 3 authors, 113;

4 authors, 46; and 5 authors, 6. Note that three times as many papers

having a single author is about the same as the number of papers having

3 authors; half of the 2 authored papers is about the same as the number

of 4 authored papers, etc. In order to explain how Suresh could have

created an environment in which made these results possible I would like

to discuss some of his personal attributes as follows: (a) his congeniality;

Foreword

(b) his generosity; (c) his breadth of interest; (d) his originality; (e) his

creation of new mathematical applications; and (f) his visibility.

1 Congeniality. As you know, Suresh is very easy to talk to. One of

his favorite questions is, ”What are you working on?” When you

tell him he will respond by giving you hints and suggestions for

directions which you might want to follow in your research on the

paper. If you ask him what he is working on, be prepared to listen

for a couple of hours.

2 Generosity. If you show interest in one of the papers he talks about

and you make a suggestion for furthering it, he may invite you to

become a coauthor, and assign you a promising direction in which

to look for additional results. On the other hand, if he likes what

your problem is, he may suggest that he become a coauthor of

your paper. If you say yes to either of these suggestions, then be

prepared to have him knock on your door a few months later and

ask, ”How you are getting along with our joint problem.”

3 Breadth of Interest. In 1970 mathematicians maintained strict con￾trol over the kinds of applications which were favourably received

in their journals: only mathematical models employing ordinary

differential equations or partial differential equations, and applied

only to applications involving either physics or engineering prob￾lems. What would they say to paper number 5 in Part (ii) Finance

and Economics by M. Gordon and S. Sethi, ”Consumption and In￾vestment When Bankruptcy is Not a Fate Worse Than Death.”

Also what would they say to paper 6 in Part(iii) Marketing by E.

Haruvey, A. Prasad, and S. Sethi, ”Harvesting Altruism in Open

Source Software Development.” By skimming through his vita, you

can see that Suresh knows no bounds on the use of various kinds of

theoretical areas such as mathematics, statistics, economics, etc.

to analyze a wide range of new application areas.

4 Originality. Let me list a few of the new applications areas that

appear in his papers: optimal cattle ranching; stochastic manufac￾turing systems; choosing robot moves in a robotic cell; risk aversion

behavior in consumption/investment problems; scheduling of the

injection process for golf club head fabrication lines; peeling layers

of an onion, an inventory model with multiple delivery modes and

forecast horizons; etc. Obviously he enjoys choosing humorous ti￾tles for his papers, but each paper contains a serious analysis of an

actual real life application.

XII

Foreword

5 Visibility. Besides looking at his publications and working papers,

it is possible to measure the extent of the influence of Suresh’s work

on the fields of Operations Research, Engineering, Economic, etc,

by looking at his professional activities which include talks pre￾sented at various meeting, invited talks at universities, member￾ship meeting locations in societies, etc. Let S be the set of all

these locations that Suresh has attended together with all of the

possible such locations that he has not yet attended. If we plotted

each of his travels over the years on a globe of the earth they might

resemble what is called ergodic (random) motion. A theorem in

ergodic theory states that if you let ergodic motion continue long

enough, each of the locations in S will be visited with probability

one. I propose the following Suresh Ergodic Theorem: If you go to

any location in S and wait long enough at that location, you will

meet Suresh with probability one.

I would like to wish Suresh Sethi a very happy sixtieth birthday, and I

look forward to keeping up with his future publications.

Gerald L. Thompson

Professor of Systems and Operations Research Emeritus

Tepper School of Business at Carnegie Mellon, Pittsburgh

[email protected]

XIII

List of Contributors

Hassan Bencheckroun

McGill University

[email protected]

Masatoshi Fujisaki

University of Hyogo

[email protected]

A. Bensoussan

University of Texas at Dallas

[email protected]

Richard F. Hartl

University of Vienna, Department

of

Business Studies, Vienna, Austria

[email protected]

Janice E. Carrillo

University of Florida

[email protected]

Ernan Haruvy

The University of Texas at Dallas

[email protected]

Dean A. Carlson

Mathematical Reviews

[email protected]

Alain B. Haurie

Logilab-HEC, University of

Geneva

[email protected]

Christophe Deissenberg

Les Universités à Aix en Provence

[email protected]

Dirk Helbing

Dresden University of Technology

[email protected]

Engelbert J. Dockner

University of Vienna, Department of

Finance

[email protected]

Seiichi Katayama

Kobe University, RIEB

[email protected]

Ali Dogramaci

Department of Industrial Engineering,

Bilkent University

[email protected]

Eugene Khmelnitsky

Dept. of Industrial Engineering,

Tel-Aviv University

[email protected]

Gila E. Fruchter

Bar-Ilan University

[email protected]

Jang-Ho Robert Kim

Westfälische Wilhelms-Universität

Münster, Institut für Numerische

und Angewandte Mathematik

[email protected]

List of Contributors

Peter M. Kort

Tilburg University, Department

of Econometrics and Operations

Research & CentER,

Tilburg and University of Antwerp,

Department of Economics

[email protected]

Helmut Maurer

Westfälische Wilhelms-Universität

Münster, Institut für Numerische

und Angewandte Mathematik

[email protected]

Jacek B. Krawczyk

Victoria University of Wellington

[email protected]

Hiroshi Ohta

Kobe University, GSICS

[email protected]

Stefan Lämmer

Dresden University of Technology

[email protected]

Ashutosh Prasad

The University of Texas at Dallas

[email protected]

Jean-Patrick Lebacque

Institut National de Recherche sur

les Transports er leur Sécurité

(INRETS)

[email protected]

Suresh P. Sethi

The University of Texas at Dallas

[email protected]

George Leitmann

University of California at Berkeley

[email protected]

Uri Shani

Department of Soil and Water

Sciences, The Hebrew University

[email protected]

Ngo Van Long

McGill University

[email protected]

Gonen Singer

Dept. of Industrial Engineering,

Tel-Aviv University

[email protected]

Guiomar Martín-Herrán

Departamento de Economía

Aplicada, Universidad de Valladolid

[email protected]

Gerhard Sorger

University of Vienna

[email protected]

XVI

List of Contributors

Günter Strobl

University of North Carolina,

Department of Finance

[email protected]

Yacov Tsur

Department of Agricultural Eco￾nomics and Management, The

Hebrew University

[email protected]

Charles S. Tapiero

ESSEC, France

[email protected]

Georg Vossen

Westfälische Wilhelms-Universität

Münster, Institut für Numerische

und Angewandte Mathematik

[email protected]

Gerald L. Thompson

Tepper School of Business at

Carnegie Mellon, Pittsburgh

[email protected]

Amos Zemel

University of the Negev

[email protected]

Mabel Tidball

INRA LAMETA, 2 Place Viala,

Montpellier

[email protected]

Rong Zhang

Chongqing University

[email protected]

XVII