Advances in Spatial Science

Transcription

1 Advances in Spatial Science Editorial Board Manfred M. Fischer Geoffrey J.D. Hewings Peter Nijkamp Folke Snickars (Coordinating Editor) For further volumes:

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3 Sven Erlander Cost-Minimizing Choice Behavior in Transportation Planning A Theoretical Framework for Logit Models 123

4 Professor Sven Erlander Department of Mathematics Linköping University Linköping Sweden ISSN ISBN e-isbn DOI: / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: estudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Front view Staircase to second floor, Administration Building, Linköping University Side view Painting by Oscar Reutersvärd transformed into 3-dimensional sculpture by Björn Kruse and Tomas Ohlsson Perspectives of Science v

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7 Preface In the Administration building at Linköping University we have one of Oscar Reutersvärd s Impossible Figures in three dimensions. I call it Perspectives of Science. When viewed from a specific point in space there is order and structure in the 3-dimensional figure. When viewed from other points there is disorder and no structure. If a specific scientific paradigm is used, there is order and structure; otherwise there is disorder and no structure. My perspective in Transportation Science has focused on understanding the mathematical structure and the logic underlying the choice probability models in common use. My book with N. F. Stewart on the Gravity model (Erlander and Stewart 1990), was written in this perspective. The present book stems from the same desire to understand underlying assumptions and structure. It investigates how far a new way of defining Cost-Minimizing Behavior can take us. It turns out that all commonly used choice probability distributions of logit type log linear probability functions follow from cost-minimizing behavior defined in the new way. In addition some new nested models appear. The new way of defining cost-minimizing behavior is the following: Cost-minimizing behavior obtains if the likelihood (probability) of any independent N-sample of observations is a decreasing function of the average cost of the N- sample. From this definition we have a basic formulation that can be falsified or tested by observations, thus satisfying the Popper criterion of the scientific status of a theory by its falsifiability, or refutability or testability (Popper 1963). The new way of defining cost-minimizing behavior is closely related to Tony Smith s Cost-Efficiency Principle (Smith 1978b, 1983, 1988). In several books Amartya Sen, who received the Prize in Economic Sciences in Memory of Alfred Nobel in 2007, has investigated what should be meant by freedom of choice in deterministic cases. He writes. The foundational importance of freedom of choice may well be the most far-reaching substantive problem neglected in standard economics (Sen 1988). In this book I introduce a measure of freedom of choice in the probabilistic case related to the Shannon (1948) measure of how much choice is involved. The standard approach in deriving choice probability functions is the Additive Random Utility Maximizing (ARUM) approach developed by Ben-Akiva (1974), Williams (1977), McFadden (1974, 1978a, 1981), and others. The standard approach vii

8 viii Preface uses composite cost expected achieved perceived cost as a welfare measure. This interpretation of the composite cost is not available in my approach. However, the interesting thing is that the same log sum formula comes out as a welfare measure combining cost and freedom of choice. This explains some of the strange behavior of the composite cost in limiting cases for the value of the parameter. This book is intended for PhD students and researchers in Transportation Science, Transportation Planning, Regional Science, Regional Planning, Economics and Optimization. The reader should have a background in mathematics, optimization, probability, statistics, and economics, equivalent to a M Sc in Industrial Engineering and Management (I-linjen) at Linköping University. The style is mathematical with propositions and proofs. Technical details have been collected in an Appendix. The book has grown out of my interest in and occupation with traffic planning models over many years. Many people have contributed to my knowledge and understanding of Transportation Science. I wish to thank D.E. Boyce, M. Florian, L.G. Mattsson, T.E. Smith and N.F. Stewart, for many interesting and inspiring discussions. In particular I wish to acknowledge the profound influence the ideas of Tony Smith have had on my thinking ever since we met and I read his pioneering paper on efficiency in My special thanks go also to Neil Stewart for the great fun we had in writing our book on the gravity model. I also wish to thank the faculty, in particular S. Danielsson and U. Hjorth, and former PhD students, P.Å. Andersson, K.O. Jörnsten, T. Larsson, J.T. Lundgren, M. Patriksson and S. Schele, at the Department of Mathematics, Linköping University. J.T. Lundgren joined me in writing early versions of a part of this book. Linköping, Sweden May 2010 Sven Erlander Linköping University

9 Contents 1 Logit Models for Spatial Interaction: Background Introduction Cost-MinimizingBehavior IntuitiveGravityModels and Most ProbableState Approach User Equilibrium in a Network Econometric Models of Probabilistic Choice Luce s AxiomaticDerivation ARUM Additive Random Utility Maximization Approach StructuredorNested LogitModels TransportationProblemin LinearProgramming Lagrangian Methods of Deriving Logit Models Welfare Measures Empirical and Policy Relevance of the New Paradigm EmpiricalRelevance Policy Relevance Behavioral Foundations of Spatial Interaction Models Basic Ideas Cost-Minimizing Behavior and Equilibrium Probability Models Freedomof Choice Cost-MinimizingBehavior The Simple (Multinomial) Logit Model Exhibits Cost-MinimizingBehavior Cost-MinimizingBehaviorImplies the Logit Model Welfare Measure GraphicalTest Some ParticularDiscrete Choice Models Equilibrium Choice of Origin,Destination androute Choice of Origin,Destination,Mode androute ix

10 x Contents 3.13 Comments Notes About Notation Part I Cost-Minimizing Behavior: Constant Link Costs 4 Logit Models for Discrete Choice Preliminaries The Simple (Multinomial)Logit Model Formal Derivation of the Simple (Multinomial) Logit Model The GeneralLogit Modelfor Cost-MinimizingBehavior AxiomaticDerivationsof Logit Models Axioms forcost-minimizingbehavior Axioms forpayoff-maximizingbehavior Axioms forarum Derivation ARUM Derivation of the Simple (Multinomial) Logit Model Properties of the Expected Achieved Perceived Utility GeneralizedExtremeValue Model Extensions Comments onthe Cost Function DifferentInterpretationsof the Same Model Cost-Minimizing Behavior for one Decision Maker Making N Repeated Decisions Comments Notes Some Particular Logit Models Introduction Stochastic Route Choice The Multi-Attribute Discrete Choice Model GeneralizedCost The GravityModel for TripDistribution The Gravity Model for Trip Distribution with Several Cost Attributes Structured(Nested)Logit Models The StructuredLogit Model: The Joint LogitModel The StandardNested Logit Model The StandardARUM Nested Approach The LogitModel with IndividualCost Values Socioeconomic Factors Commentsand Extensions Notes... 86

11 Contents xi 6 Welfare, Benefit and Freedom of Choice Introduction AchievementMeasure Freedomof Choice: Preliminaries Freedomof Choice Measure Freedom of Choice in the Probabilistic Case Welfare Measures Welfare Measurefor the Simple Logit Model NumericalIllustrations Welfare Measurefor the GeneralLogit Model Welfare Measuresfor some ParticularModels Welfare Measurefor the Stochastic Route Choice Model Welfare Measure for the Multi-Attribute Case Welfare Measurefor Structured(Nested) Logit Models Welfare Measurefor GravityModel for TripDistribution Welfare Measures for Models with Socioeconomic Factors Extensions ExtendedBenefit Measure A Lower Bound on Observed Negative Entropy Value of Time Comments Revealed Freedom of Choice, Diversity of Choice, Flexibility of Choice Freedomof choice:advantageandachievement Entropyand Freedomof Choice Welfare andbenefit Measures Notes Graphical Tests of Cost-Minimizing Behavior in Logit Models Introduction Testing for Cost-Minimizing Behavior in the Simple Logit Model GraphicalTest of Cost-MinimizingBehavior Asymptotic Distribution of the Observed NegativeEntropy SimulationExperiments Figures Final Remarks Multi-Attribute Discrete Choice Models GraphicalTest of Cost-MinimizingBehavior StructuredLogitModels GraphicalTest of Cost-MinimizingBehavior Commentsand Extensions Notes...122

12 xii Contents Part II Equilibrium 8 Equilibrium Introduction Smith s Cumulative Cost Function and Equilibrium Route Choice Continuous Approximation Choice of Origin,Destination androute Most ProbableTrip Pattern Most ProbableFlow Continuous Approximation Choice of Origin,Destination,Mode androute Continuous Approximation Commentsand Extensions Notes Appendix Representation Theorem for Cost-Minimizing Probability Distributions Likelihood and Entropy of a Sample MaximumEntropy Maximum Likelihood References...149