Risk and Uncertainty (Utility, Decision)
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University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Spring 1985 RISK AND UNCERTAINTY (UTILITY, DECISION) PAUL SNOW University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation SNOW, PAUL, "RISK AND UNCERTAINTY (UTILITY, DECISION)" (1985). Doctoral Dissertations. 1455. https://scholars.unh.edu/dissertation/1455 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. 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Zeeb Road Ann Arbor, Ml 48106 8521501 Snow, Paul RISK AND UNCERTAINTY University of New Hampshire Ph.D. 1985 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1985 by Snow, Paul All Rights Reserved I RISK AND UNCERTAINTY BY Paul Snow A.B., Harvard College, 1973 S.M., Harvard University, 1975 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements fo r the Degree of Doctor of Philosophy in Engineering May, 1985 ALL RIGHTS RESERVED © 1985 Paul Snow This dissertation has been examined and approved. , Dissertation director, Charld's K. Taft Professor of Mechanical Engineering *rt W. Corel 1 ^ Professor of Mechanical Engineering Eugene< . Freuder Associate Professor of Computer Science Filson H. Glanz Associate Professor of Electrical Engineering JeWreyfetSbhl Assistant Professor of Business Administration Date ACKNOWLEDGEMENTS Thank you to William M. Golding of the Governor's Energy Office (John H. Sununu, Governor and Dennis A. Hebert, Director) for giving me work with fle xib le hours. Thanks also to Joyce Donohue for her patience while I wrote this. Of course, I am grateful to the faculty of the dissertation committee for volunteering th e ir time on a project remote from the ir everyday professional concerns. Special thanks to Charles K. Taft, who was generous with his time and with what material support was his to give. TABLE OF CONTENTS ACKNOWLEDGEMENTS ......................................................................................... iv ABSTRACT................................................................................................................vi CHAPTER PAGE I. THE RECEIVED THEORY OF EXPECTED UTILITY .............................................. 1 The Von Neumann-Morgenstern Axioms ................................................ 1 The Kelly Rule .......................................................................................... 14 The Friedman-Savage Problem ........................................................ 20 II. THE GAMBLER'S RUIN AND EXPECTED UTILITY.............................................28 Exponential U tility and Ruin Constraints .................................... 28 Choice and the Probability of Ruin ...................................................34 Power Law U tility and Ruin C onstraints ...........................................44 I I I . UTILITY-LIKE DECISION RULES ................................................................. 56 Power Law Buying and Selling Price R ules .......................................56 The Buying Price and the Independence Axiom ............................ 65 Bounded U tilitie s and Ruin Constraints ........................................ 72 IV. DECISIONS INVOLVING GOALS ................................................................... 82 Ruin in Two-Barrier Random Walks .................................................... 82 Goals and the A lla is Problem ...............................................................88 Bands, Barriers and the Dividend Problem .................................... 97 V. UNCERTAINTY AND PARTIAL RISK...................................................................104 Received Theories of Decisions under Uncertainty ................ 104 Uncertain Mixtures of Lotteries and Maximin ............................ 113 The Ellsberg Problem ....................................................................... 123 Conclusions .............................................................................................. 135 LIST OF REFERENCES............................................................................................ 136 ABSTRACT RISK AND UNCERTAINTY by PAUL SNOW University of New Hampshire, May, 1985 The conventional theory of decision making under risk relies on axioms that reflect assumptions about people's subjective attitudes towards wealth. The assumptions are unverifiable, and the axioms are too restrictive. They forbid some decision rules that a plausibly rational decision maker ("DM") could find useful. A new method, at once less re strictive and less dependent upon subjective assumptions, motivates expected u t ilit y techniques by assuming that a DM wishes to place an upper lim it on the probability of ruin. All bounded-above, increasing functions defined over a suitable domain can serve as u t ilit y functions. Now, DM can evaluate lotte ries according to th e ir buying prices, useful i f one plans to withdraw capital from risk. DM can rigorously distinguish "once in a lifetime" lotteries from ordinary gambles; that's helpful when facing Allais' problem. Also, DM can exploit partial knowledge of state probabilities without choosing arbitrary point estimates for all uncertain odds. That helps to resolve problems that combine elements of both risk and uncertainty, like Ellsberg's. The new method allows DM to do everything permitted under the old axioms, and more besides, with fewer and less ambitious assumptions. CHAPTER I THE RECEIVED THEORY OF EXPECTED UTILITY The von Neumann-Morgenstern Axioms Suppose a decision maker ("DM") faces a choice among two or more gambles. A gamble or lo tte ry is a set of outcomes and an associated set of probabilities. The outcomes are mutually exclusive and exhaustive. An outcome is either a capital transfer to (or from) DM, or another lottery, or a choice among lotteries. Note that a transfer of wealth for certain may, when convenient, be viewed as a lottery with a single outcome whose probability is one. Assume that DM knows a ll of the gambles that are available when a choice must be made. Assume also that DM knows a ll of the possible outcomes in each available gamble. Except in Chapter V, DM w ill also know a ll of the probabilities. Although lotte ries may be continuous random processes, the examples pursued here w ill generally be discrete lo tte rie s, binomial or multinomial. Von Neumann and Morgenstern [98] proposed a system of axioms that counsel DM to make one's choice in the following way. DM should choose some increasing function U() . The domain of JJ(J_ shall be amounts of money, either changes in wealth (sometimes called the "prizes" of the lottery) or else the total wealth DM w ill attain after the lo tte ry is completed. Whether one uses total wealth or changes in wealth is a matter of convenience. For each outcome of each gamble, DM evaluates the chosen function. Then, for each gamble, DM computes the average of the U() values 1 2 offered by the gamble. One selects fo r play that gamble with the highest expected value of U() . The U() function is called a " u t ilit y function". The advice given by the von Neumann and Morgenstern axioms is succinctly summarized as "maximize