choices an introduction to decision theory

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CHOICES


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C H O I C ESA N IN T R O D U C T IO N T OD E C I S I O N T H E O R Y

Michael D.ResnikProfessor of Philosophy

Universityof North Carolinaat ChapelHi l l

University ofMinnesota PressM i n n eap o l i sLondon


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Co p y r igh t 1987 by the Regents of theU nivers ityof Minneso ta .A ll r ights reserved. N o par to f th is publ ica t ionmay bereproduced, s tored in a retr ieval system, ort ransmi t t ed ,inany form or by any means , e lect ronic , mechan ical , pho to-copy ing , record ing , o ro therwise, w i tho ut pr ior wri t tenpermiss ion of thepubl i sher .Published by the Univers i ty of Minnesota Press111T hird A venue S outh, Suite 290, M inneap ol is , M N55401-2520ht tp :/7www .upres s .um n.eduPrinted in the U nited S tates of A merica on acid-free p ap e r

F i f th pr int ing 2000Library of Congress Cataloging-in-PublicationDataResn ik , Michae l D .Choices : an in t roduct ion to decis ion theory.Includes bibl iographies and index.1. Decis ion-making. 2.Stat ist ical dec i s ion .I. Tit le.T57.95.R45 1986 001.53'8 86-11307ISBN 0-8166-1439-3ISBN0-8166-1440-7 (pbk.)

T he U n ive r s i t y of Min n es o t a isan equa l -oppor tun i ty educa to r and emp lo y e r .
http://www.upress.umn.edu/http://www.upress.umn.edu/http://www.upress.umn.edu/http://www.upress.umn.edu/


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In MemoriamBeatrice Salzman Spitzer KatzMary Edwards Holt


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CONTENTS

PREFACE xiACKNOWLEDGMENTS xii iChapter 1Introduction 31-1 What Is Decis ion Theory? 31-2 The Basic Framework 6l-2a Some Philosophical Problems about ProblemSpecifications 81-3 Ce rtainty, Ignorance, and Risk 13l-3a Some Details of Formulat ion 141-4 Decision Trees 171-5 References 19Chapter 2Decisions under Ignorance 21

2-1 Preference Orderings 222-2 The M aximin Rule 262-3 The Minimax Regret Rule 282-4 The Optimism-Pessimism Rule 332-5 The Principle of Insufficient Reason 352-6 Too Many Rules? 382-7 A n A pplication in Social Philosophy:Rawls vs . Harsanyi 412-8 R eferences 44Chapter3Decisions under Risk: Probability 453-1 M aximizing Expected Values 453-2 Probabili ty Theo ry 473-2a Bayes's Theo rem wi thout Priors 553-2b Bayes's Theorem and the V a lueofAdd i tionalInformation 57


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3-2c Statistical Decision Theory and Decisionsunder Ignorance 593-3 Interpretations of Probability 613-3a The Classical View 623-3b The Relat ive Frequency View 653-3c Subjective Views 683-3d Coherence and Condit ional izat ion 753-4 References 79Chapter 4Decisions under Risk: Utility 814-1 Interval U til i ty Scales 814-2 M onetary Values vs. Uti l i t ies 85

4-3 Von Neumann-Morgens tern Ut i l i ty Theory 884-3a Some Comments and Qualif icat ions on theExpected Util i ty Theorem 984-4 Cri t ic isms of Uti l i ty Theory 1014-4a A llais 's Parad ox 1034-4b Ellsberg's Paradox 1054-4c The St. Petersburg Paradox 1074-5 The Predictor Paradox 1094-6 Ca usal Decision Theory 1124-6a Objections and Alternat ives 1154-6b Co ncludin g Rem arks on the Paradoxes 1184-7 References 119

Chapter 5Game Theory 1215-1 The Basic Concepts of Game Theory 1215-2 Tw o-Person Strictly Co mpe ti t ive Games 1275-3 Equ i l ibr ium Strategy Pairs 1295-3a M ixed Strategies 1325-3b Proof of the Maximin Theoremfor Two-by-Two Games 1375-3c A Shortcut 1395-3d O n Taking Chances 1415-4 Two-Person N onzero Sum Games: Fai luresof the Eq uil ib r ium Concept 144

5-4a The Clash of Wil ls 1445-4b The Pr isoner 's Dilemma 1475-4c Other Prisoner 's Dilemmas 1485-4d The Pr isoner 's Dilemma and the Predictor 1505-4e Morals fo r Rat ional i ty and Mora l i ty 1515-5 Coo pera t ive Games 1575-5a Barga in ing G a m e s 159


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5-6 Games with Three or More P layers 1675-7 References 175Chapter6Social Choices 177

6-1 The Problem of Social Choice 1776-2 A rrow's Theorem 1806-2a Arrow's Conditions 1806-2b Arrow's Theorem and Its Proof 1866-3 Majo r i ty Rule 1916-4 Uti l i tar ianism 1966-4a Harsanyi 's Theorem 1976-4b Cri t ique of Harsanyi ' s Theorem 2006-4c Interpersonal Comparisons of Ut i l i ty 2056-5 References 212BIBLIOGRAPHY 215INDEX 219


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I C A L L D E C I S I O N T H E O R Y th e collection o f mathematical , logical , andphilosophical theories of decision making by rational individuals take n alone,in competi t ion, and in groups . For me i t includes util i ty theory , game theory ,andsocialchoicetheory. Bywhatever name,it hasbecomeanintegral partofimportant approaches to thephi losophy ofscience, th etheory of rat ionality, andethics. Thedebate between John Raw lsandJohn H arsanyi abouttheproper deci-sion method to usebehind th e "veil of ignorance" isperhaps th e best-known in-stanceof the use of decision theory in philoso phy. But the use of epistemic uti l i tymaximizat ion is equally impo rtant in the phi losophy of science, and the prison-er's dilemma of game theory and Arrow's theorem of social choice theory re-cently have stimulated ethical thought.N ot only is decision theory useful to phi losophers in other fields; it alsoprovides philosophers with philosophical perplexities of its own, such as thewell-known Newcomb's paradox. (I call it the Predictor paradox.) These and ahost of other paradoxes raise serious qu estions for our accepted view of rationa l-i ty , probabi l i ty , and value.Decision theory hasbeen taught largely by stat ist icians, economists, andmanagement scientists. Their expositions reflect their interests in applications ofthe theory to their own domains. My purpose here is to put forward an exposi-tion of the theory that pays particular attention to matters of logical and philo-sophical interest. Thus I present a number of proofs and philosophical commen-taries that are absent from other introductions to the subject. However, mypresentation presupposes no more than e lementary logic and high school al-gebra. In keeping with the introd uctory n ature of the book, I hav e left open manyof the philosophical questions I address .The first four chapters are concerned with decision making by a singleagent. In chapter 1, I set out the standard act-state-outcome approach and thenturninchapter 2 todecision making under ignorance. HereIdiscuss severalofthe well-known rules, suchas the maximin rule and the principle of insufficientreason. In this chapter I also discuss types of scales for measuring uti l i t ies anddemonst ra te wh ic h type o f scale each rule presupposes . With each ru le I givea brief account of itsp hi losophical and pract ica l advantagesa ndd isadvantages .

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PREFACEI then turnto the quest ion of w h i ch ru l e , ifa n y , is the correct one and i l lustratethis wi th an account of the Rawl s -Harsan y i d i spute .Chapters 3 and 4 deal wi th individual decis ions under r i sk. After present-ing the ru l e of expected ut i l i ty max i mi za t i on , I t u r n to the component theor iesassociatedwi thi t probabi l i ty theoryandutil i tytheory.M yaccounto fprobabil-ity begins wi th a development of the probabi l i ty calculus through Bayes ' s the-orem. I then discuss several interpretat ions o f pr ob abi l i ty , pro vin g the D utchBook theorem during the course of my account of subject ive probabi l i ty . Turn-ing to uti l i ty theory , I deve lop the Von Neumann-Morgens te rn approach andprove a representat ion theorem. I take pa ins toexp l a i nhow th i s theorem showsthat al l agents who sat isfy ce r ta in condi t ions of ra t ional i ty m us t have preferenceorder ings that can be represented as i f they maximize expected u t i l i t y . I exp la inw hy this shows that it is rat ional to maximize expected uti l i ty even when oneis taking a one-shot gamble. The discussion of ut i l i ty concludes wi th a reviewofparadoxes Al la i s 's , Ellsberg's, the Predictor , an d the St .Pe te rsburg a ndanexamina t ion of causa l dec i s ion theory .Chap te r 5 isdevoted to game theory wi tha focus on two-person zero sumgames . Idevelop the s tandard accountandprove them axim in theorem fo r two-by- two games. This requires some fancy factor ing but no h igher mathemat ics .I then tu rn to two-person nonze ro sum games and the fa i lure of the eq ui l i br iu mconcept to provide sat i sfactory solut ions. This leads intothe pr isoner ' s d i lemmaand the cha l l enge it poses to the theory o f ra t iona l i ty . I discuss Gauthier ' s at-tempt to show that i t can be rat ional to cooperate w i th one 's p ar tne rs wh en caughtin a pr i soner ' s d i l emma. I pass from th i s to barga in ing games and the so lu t ionsproposed by John Nash and David Gauthier . Thispart of the bookconcludeswith a d i scuss io n of mul t iperson games and coa l it ion theor i es .Chap te r 6 isdevoted to socia l choice the ory . Iprove Arrow 's theorem, a f-te r exp l o r i ngt hena ture of itscondi t ions . Then It u r nto major i ty rule and proveMay's theorem. Last , It u r ntou t i l i t a r i an i smandprove atheorem o f Harsanyi ' s ,wh ic h can be stated dramatical ly as to the effect that in a rational and impar t i a lsociety o f rat ional agents the socia l order ing must be the ut i l i tar ian one . T hebook concludes wi th a cr i t ique of this theorem and a discussion of the problemof in te rpersona l compar i sons of u t i l i ty .A lthoug h I intend this book for use by phi losophe rs (as wel l as socia l andmanagement sc ient i s ts and others) seeking an in t roduc t ion to decis ion theory, Ialso have found i t sui table for a one-semester undergraduate course enrol l ingstudents wi t h a var ie ty of interests and majors . Most of my students have hada semester o f symbol ic log ic , but able s tudents wi th no prev ious work in logich a v edone q u i te wellin the course . T he logical and mathem atical exercises g iventh roughout the book have formed the basis fo r much of the w r i t t en w ork in mycourse, but I have a lso ass igned, wi th good resul ts , a number of phi losophicalessay q u e s t i o n s on the ma te r i a l cove red .

M . D . R .xii


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ACKNOWLEDGMENTS

A N Y O N E W H O K N O W S the subject wi l l recognize that I owe a large debt tothose w ho have a l ready worked on and wri tten about decision theory . I havedrawn heavily on the material in Luce and Raiffa's Games and Decisions,Raiffa's Decision Analysis, and Sen's Collective Choice and Social Welfare. Iwas a lso for tunate enough to have seen partsof Gau thier 's Morals by Agreementbefore i ts publ icat ion and have included an account of his vie w s on game theory .T he idea for abook o f this sort and the topics to be includedgoesback to acon-versat ion about courses in decision theory that N ed McCl ennen and I had onemorning a lmost ten years ago.Daniel Albert started m y serious interest in decis ion theory when, as aRobert Woods Johnson Cl inical Scholar, he asked me to teach him "the logicof medicine." I have been grateful for his f r iendship and encouragement evers ince. I am a lso grateful for the s t ipend I received as Albert ' s mentor under theClin ica l Scholar ' s program. Duncan MacRae, di rector of the Univers i ty ofNorth Caro l ina Program in Publ ic Pol icy Analysis , included me in a SloanFoundat ion grant , which funded me for a summer to work on this book.Henry Kyburg and Ellery Eells sentm e comments on the book inthei r ca-paci ty as referees for the Un ivers i ty of M inneso ta Press , as did an an on ym ou spoli t ical scientist . They d id m u c h to i mprove the book. Obviously they are notresponsible for any r ema i n i ng faults.I am also grateful for comments , conversat ions, moral support , andeditorial assistance from JamesFetzer, Susan Hale , John H arsan yi , Ph i l ip Ki tch-er, A r t h u r Kuf l ick, Barry Loewer , Cla i re Mi l l e r , Richard Nunan, Jo e Pi t t ,David Resnik, Janet Resnik, Jay Rosenberg , Caro lyn Shear in , Br ian Skyrms,Henry Wes t , andBi l l W r ight . I would also l ike to thank m y undergraduate s tu-dents for quest ions and comments . Thei r s t ruggles to understand the mater ia lforced me to improve m y expos i t ion . I neglected to note most of the i r names ,but one s tands out qui te vividly: Sheldon Coleman, whose react ions caused m eto rewri te several exercises .

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Chapter1INTRODUCTION

1-1. What Is Decision Theory?Decision theory is the product of the jo i n tefforts o f economis ts , ma themat ic ians ,philosophers, social scientists, and stat is t ic ians toward making senseof how in-d iv idua ls and groups make or should make decisions. T he applications ofdeci-sion theory ran ge from v ery abstract speculat ions by phi losophers about ideal lyra t ional agents to pract ical advice from decision analysts t rained inbus in essschools . Research in decision theory isj u s t as varied. Decision theorists wi tha strong mathematical bent prefer to invest igate the logical consequences ofdifferent rules fo r decision making or to explore the mathemat ica l fea tures ofdifferent descriptions of rational behavior. On the other hand, social scientistsin terested in decision theory o ften conduct experimen ts or social su rvey s aime dat d iscovering howreal people (as opposed to "ideally rational agents") actuallybehave in decision-making si tuat ions.Itis thus usual to d iv id e decision theory in to two main branch es: no rma tive(or prescrip t ive) decision theory and descrip t ive decision theory . D escrip t ive de-cisiontheo rists seek to find out how d ecisionsarem a d e t he y invest igate us ordi-nary m ortals; thei r col leagues in norm ative decision theory are supposed to pre-scribe how decisions ought to be made t he y study ideally rational agents. Thisdistinction is somewhat artif icial s ince in fo rm at ion about our actual decision-maki ngbehaviorm ay berelevant toprescrip t ions abouthow decisions shouldbema de. N o sane decision analyst w ould tel l a successful basketball coach that heoughttoconductastatistical survey eve ry timeheconsiders sub sti tuting playe rseven i f an ideal ly rat ional agent act ing as a coach would. We can even imagineconduct ing research with both normative and descriptive ends inmin d : For in-stance,w e might study how expert business execut ives make decisions inordertof ind rules for prescribing how ordina ry fo lk shouldmaketheir busine ssdecis ions.Recent ly some phi losophers have argued that al l branches of the theory ofrational i ty should pay attention to studies by social scientists of related behaviorby h u m a n be ings . Thei r po in t is tha t most prescr ip t ions fo rmula ted in terms ofideal ly rat ional agen ts have l i t t l e or no bear ing on the ques t ion of howh u m a n sshould behave. This is because l og i c i ans , mathemat i c i an s , and p h i l o so p hersusua l ly assume that idea l ly ra t iona l agen ts can acqui re , s to re , an d process un-

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INTRODUCTIONl imited amounts of information, never make logical or mathematical mistakes,and kn o w all the logical consequences of thei r bel iefs. O f co urse , no human sno t even geniuses c om e close to such ideals . This m ay favor studying morereal is t ic models of rat ional agents, but I do not think that w e hav e g ro un d s fordismissingthe usualtheoriesof rationality altogether.The ideals they describe,al though unattainable in practice, sti l l serve to guide andcorrect our th inking.F or example , we know that perfect memories would help us make bet ter deci-s ions . Instead o f set tl ing for poor decisions, we ha ve tr ied to overcome our l imi-tat ions by p rogramming computers which have l arger and better memoriesthan we do to assist us in those tasks whose success depends on stor ing andretr iev ing large quant i t ies of information.

A nothe r problem with putt ing m uch weight on the d ist inct ion between nor-mat ive and descrip t ive decision theory is that some abstract decision modelshave been in troduced wi t h nei ther normative nor descrip t ive ends in mind. I amthinking of the concept o f rational economic man used in economics. Thishypothet ical being is an ideal ly rat ional agent w hose choices alw ays are the mostl ikely to maximizehis personal profi t . B yappeal ing to a hypothetical society ofrat ional economic men economists can derive laws of supply and demand ando ther im por tan t p r inc ip lesof economic theory. Y etecon omis ts adm i t tha tthe no-t ion of rat ional economic man is not a descrip t ive model . Even people wi th thecoolest heads and greatest business sense failto conform to this ideal . Sometimeswe forget the profit mot ive . Or w e a im a t making aprofi t but miscalculate. Nordo economis ts recommend that we emula te economic m an: M axim iz ing personalprofi t i s not necessari ly the highest good for human beings . Thus the model isintended neither normatively nordescriptively,it is an explanatory idealization.Like phys icists speculat ing about perfect vac uu m s, fr ict ionless surfac es, or idealgases, economists igno re real - l i fe compl icat ions in the hope of erect ing a theorythat wi l l be s imple enough to y ield insights and understanding whi le st i l l apply-ing to the phenomena that prompted i t .F or these reasons I favor d ropp ingthe normative-descrip t ive d ist inct ioninfavor o f atermino logy th at recognizes thegradat ion from exper imenta l and sur-vey research toward the more speculat ive d iscussions of those in terested in ei-ther explanatory or normative ideal izat ions. With the caveat that there is real lya spectrum rather thanahard-and-fas t d iv i s ion ,Ipropose the terms "experimen-tal" and "abstract" to cover these tw o types of decision-theoret ic research.This book wi l l be concerned exclusively w i th abstract decision theory andwil l focus on i ts logical and philosophical founda t ions . Thisdoes not mean thatreaders wi l lf ind nothin ghereof pract ical valu e. Some of the concepts and m eth-ods I wi l lexp o un d are also found inbusiness school textbooks. M y hope istha treaders w i l l come to appreciate the assumptions such textsoften make and someof the perplexities they genera te .An o t h e r impor tan t d iv i s ion w i th i n decision theory is tha t be tween dec i -s ions mad e by i n d i v i d u a l s and those made by g r o u p s . For the p u r p o s e so f thisd iv i s i o n an ind iv idua l need not be a s ingleh u m a n be ing (o r o ther a n i m a l ) . Cor-pora t ions , c lub s , n a t i o n s , s ta tes ,an d un ivers i t i es make d ec i s i o n sas individuals4


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(andcan be held respon sible for them ) when they at tempt to realize some organ-izational goal, such as enha ncing their reputation or be ttering last year's sales fig-ures . Ho we ver , when severa l ind iv idual s whobelongto the same club, corpora-t ion, or unive rsi ty adjudicate differences about gro up goals or pr ior it ies, they areinvolved in making a group decision. W e can i l lus t ra te the difference betweengroup and ind iv idu al decisions by looking at theroleof U nited Statespres idents .By electing a Republ ican rather than a Democrat ic president , voters candecidegeneral social and economic pol icy. Elections are thus one way for United Statescitizens to make group decisions. Once elected th e president also makes deci-sions . They a re in the first instance his ow n choices, to be sure; b ut , insofar as heincorporates thenat ional w i l l , theymay bedecisions of theU nited States aswel l .Thus when Pres iden t Reaganelected toinvade Grena da, this also became adeci-sion of the United States. This was in effect an indiv idual decision by anat ion .W h e n w e turn to game theory we wi l l deal with ind iv idua l dec i sions tha tat first sight look l ike group decisions. Games are decision-making si tuat ionsthat a lway s i n v o lv emore than one indiv idual , but they do not count as groupdecisions because each ind ivid ua l chooses an action w ith the aim of furtheringhis or her own goals. This decision wi l l be based on expecta tions concern inghow other part ic ipants wi l l decide, but , unl ike a group decision, no effort wi l lbe made todevelop a pol icy apply ing to all the part ic ipants. F or example, tw oneighboring department stores are involved in a game when they independentlyconsider havingapost-Christmas sale. Each knows that if one has the sale andthe other does not , th e latter wil l get l i t t le business. Y et each store is forced todecide by i tsel f whi le ant icipat ing what the other wil l do. On the o ther hand , ifthe tw o stores could find a way to choose a schedule for having sales, theirchoice would ordinari ly count as a group dec i s ion . Unfor tunate ly , it is fre-quently difficult to tel l whe ther a given si tuat ion inv olves an in div id ual or agroup decision, o r, when several indiv idualsare choosing, whether they are in-volved in a game or in a group decision.Mo s tof thework in group decision theory hasconcerned the developmentofcommon pol icies for gov erning gro up mem bers andwiththe jus t d istr ibut ion ofresources throughout a group. Indiv idual decision theory, by contrast , has con-centrated on the problem of how ind iv idu als may best further their personal inter-ests, whatever these interests may be. In part icular , indiv idual decision theoryhas , tothis point, m ade noproposals concerning rat ional or ethical ends. Individ-ual decision theory recognizes nodistinction-eithermo ra lo rrational-betweenth egoals of ki l l ing oneself, being a sadist , making a mil l ion dollars, or being amiss ionary . Because of this i t might be possible for ideally rational agents to bebetter oif violatingth epol icies of the groups to which they belong. Some groupdecision theorists have tried todeny this possibi l i ty andhave even gone so far asto offer proofs that it is rational toabideby rationally obtained group policies.It shouldbe n osu rpr i se , then , tha t some phi losop hers hav e become fasci-nated wi th dec i s ion theory . N ot on ly does the theory p romise app l ica t ions tot radi t ionalph i losophica l p rob lemsbut it too isreplete with i ts ownphi losophica lproble m s. We hav e al ready touched on two in at temp ting to draw the d ist inct ions

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INTRODUCTION

between thevar iou s b ranches of decis ion theory, and wew i l l encountermoreaswe cont inue. However , phi losophers have paid more at tent ion toappl icat ionso fdecis ion theory in ph i los op hy than they hav e to problems wi th in decision th eo ry.T he not ion of a rat ional agent is of p r i mary i mpor tance toph i l o sophy at large.Since Socrates , moral phi losophers have t r ied to show tha t moral act ions are ra-t ionalac t ions , inother word s, thatit is inone'sow n best interestto bemora l . Po-l i t ical phi losophers have s imi lar ly t r ied toestablish that ratio na l agents wil l formju s t societ ies. Such argume nts remained vague unt i l modern dec i s ion theory sup-pl ied precise models of rational i ty and exact principles of socialchoice.It is nowpossib le to formulate wi th near ly mathematical exactness modern vers ions oft rad i t iona l ar gu m en ts in e thics and socia l p hi losophy. The techniques of decis iontheory have a lso suggested new approaches to old ethical and moral problems.Stat i s t ic ians use decis ion theory to prescribe both how to choose betweenhypotheses and how to determine the best act ion in the face of the outcome ofa statistical experiment. Philosophers of science have turned these techniqueswi t h good resul ts on the problems in ra t ional theory choice, hypothesis accep-tance, and ind uct iv e methods. A gain this has led to s igni f ica nt advances in thephi losophy of science.Dec i s ion theory is thus ph i l o soph i ca l l y i mpor tan t as w e l l as i mpor t an t tophi losophy . A f te r we have deve loped more of i ts par t i cu la rs we can d i scusssome of i ts phi loso ph ical app l icat ions and problem s in greater deta i l.

PROBLEMS1. Class i fy the f o l l ow i ng as i n d i v i d u a l o r group decis ions. Which are games?E x p la in yo ur c l ass i f i ca t ions .a . T w o people decide to marry each other .b. The members o f a c lub decide that the annua l dues wi l l be $5.c. The memb ers o f a c lub decide to pay the i r dues .

d. The c i t izens of the Uni ted States decide to amend theC o n s t it u ti o n .e . Tw o gas s ta t ionsdecide to s tar t a pr i ce war .2. If it turned out that everyone believed that 1 + 1=3, would that make it ra-t ional to bel ieve that 1 + 1=3?3. A l thou gh decis ion theo ry declares no goals to be i r ra t io na l in and of them-se lves , do youth ink there are goals nora t ion a l be ing could adopt , fore x a m -p l e , the goa l of ceas ing to be ra t iona l?1-2. TheBasic FrameworkA dec i s i on , w he the r i nd i v i dua lor g roup , invo l vesa choice betweentwo or moreopt ions oracts,each ofw hich wi l l p roduceone of severaloutcomes.F or exam-ple , suppose I have just entered a dark garage that smel ls of gaso l ine . Af te rgrop ing to no a va i l for a l ight swi tch, I consider l ight in g a match, but I hesi ta tebecause Ik n o wthat do i ng som i gh t causean e x p l o s i o n .The acts I amcons i de r -ing are lightamatch,do not lightamatch. As I see it, if I do not l ight them a t c h ,therew i l l beo n l yo n eo u t c o m e , n a m e l y , no explosion results. On theo the r han d ,if I do l ight the m a t c h , tw o o u t c o m e sare p o s s i b l e :an explosion results, no ex-6


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INTRODUCTIONplosion results. M y decision is notclear-cut because it is not certain thatan ex-plosion w i l l resul t i f I l ight the m atch. T hat w i ll depend o n the am ount and d istr i -but ion of the gasol ine vapor in the garage. In other words, the outcome of myact wil l depend on the state of the environment in which the act takes p lace.A s this example i l lustrates, decisions involve three components acts,states,andoutcomes,wi th the latter being ordinari ly determined by the act andthes ta te und er whichi t takes place. (Some outcomes are certain, nomat ter wh atthestate o r act . F or instance, thatIwi l lei ther l iveth ro ughthe day or d ied u r i n git is certain whether or not I l ight the match and regardless of the presence ofgasol ine in the garage.) In decision theory we also construe the term "state"ina very broad sense to i n c lud enon phys ical as wel l asphy s ica l cond i tions . If youanda fr iend bet on the correct s tatemen t of the fund am ental theorem of calculus,the outcome (yo ur winn ing or losing) is determined by a m athe m atical s tate, thatis , by whether your vers ion rea l ly fo rmula tes the fundamenta l theorem.When analyzing adecision problem,thedecisionanalyst(whomay be thedecision maker himself) must determine there levan t set ofacts, states, andout-comes for character izing the decision problem. In the match example , the actdo not lightamatchm ight have been specified fur ther asusea flashlight, returnin anhour,ventilate thegarage,al l of w hich invo lve doing som ething other thanl ighting the match. S imi lar ly , I might have descr ibed the outcomes d if ferent lybyus in gexplosion (n o damage), explosion (light damage), explosion (moderatedamage), explosion (severe damage). Final ly , the states in the example couldhave been analyzed in terms of the gasol ine- to-ai r rat io in the garage. This intu rn could g ive r i se to an infinity of states, since there are infini te ly ma ny ra t iosbetween zero andone. A s Ihave descr ibed the examp le , ho wev er , the re levan tacts, states, and outcomes are best taken as the simpler ones. We can representth is ana lys i s in a decision table. (See table 1-1.)

1-1 StatesExp los iveG as Level N onexp los iveLight a MatchActs Do Not L ighta Match

In general , a decision table contains a row corresponding to each act , a columnfo r each state, and anen t ry ineach square correspond ingto theoutcome for theact of that row and state of that co lumn.Suppose w e change the match example .N o w Iw a n ttocause an explosionto scare some f r iends who are wi th me. But I am a prac t ica l j o k e r , not am u r -d e r e r , so Iw a n tthe e x p l o s i o nto be n o n d a m a g i n g . T h e n in a n a l y z i n g this deci-sion p ro b l em w e w o u l d be forced to b r e a k d o w n the exp lo s iv e o u tco me in todamaging Inondamaging. But the ma g n i t u d e of the explosion w o u l ddepend on

7

Explos ion N o Exp lo s io n

N o Explos ion N o Explos ion


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INTRODUCTIONthe amount of vapor in the garage, so our new analysis would also require adifferent d iv i s ionof the environment in to states. This might y ield a decision ta-ble such as 1-2.1-2

X

StatesA m o u n t of Gas PresentY

ActsL igh t

Do No t L igh t

N o Explosion

No Exp los ion

ExplosionN o DamageN o Explosion

Explos ionDamageNo Exp lo s io n

In specifying a set of acts, states, and outcomes, or in d r a w i n ga decisiontable, w e determine aproblem specification. A mo ra l to be d r a w n from thematch exam pleisthat several p roblem specif icat ionsm aypertain to thesamede-cision si tuat ion. In such cases the dec i s ion analys t must de termine the p roperspecification or specif icat ions to apply . This i s a problem in appl ied decision the-ory yet i t may be absolutely crucial . In 1975, government heal th experts con-ducted ane labora te ana lys i s p r io r to deciding to issue the swin e inf luenza vac-cine to the general publ ic . B ut according to newspaper accounts, they simplynever considered the outcome that actual ly r e su l t edt ha t the vaccine wouldpara lyze a n u m b e rof people. Thus they failed to use the proper problem speci-fication in making thei r decision.For a problem specification to bedef ini te andcom plete, its states must bemutual ly exclusive and exhaust ive; that is, one and only one of the states mustobtain. In the match example, i t would not do to specify the states as no vapor,some vapor, much vapor, since the second two do notexclude each other. Light-ing a match under the middle state might o r m i g h t no t cause an exp los ion . O nthe o ther hand , i f we used the states no vapor, much vapor, w e would neglec tto cons ider what happens when there is some but not much vapor .Sec uring m utua l ly exclusive state specif ications may require carefula n a l y -s is ,but we can easi ly guarantee exhau st iveness by adding to a listo f nonexh aus-t ive states thed escrip t ion none of thepreviousstatesobtain. Like the cover an-swer "noneof the above" used to complete mul t ip le-choice quest ions, this easymo v e can lead a decision analyst to ove rlook relevant possibi l i t ies. Perhapssomething like thiswas at work in the swine fluvaccinecase.l-2a. Som e Philosophical P roblems about Problem SpecificationsSelecting a problem specif icat ion is really an issue that arises inap p ly in g deci -sion theory . W ewi l l eschew such problems inthis book an d, hence forth, assum etha twe are d ea l i n gwithproblem specif icat ions whose states are mutua l lyexclu-sive ande x h a u s t i v e .Y etthere are severa l in teres t ing phi losophica l i ssues re la tedto the choice of p ro b l em spec if ica t ions . Three mer i t ment ion here .

The f i rst concerns the proper descrip t ion of states. Any decision problem8

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INTRODUCTION

+$5-$2

-$2+$3

involvessome outcomes thedecision m aker regard s asbetter than others. O ther-wisetherewo uldbe nocho ice wor th m aking. Thusanydec i s ion mightb e speci-fied in terms of the state descriptions things turn outwell, they do not. Suppose,fo r example, that you are offered a choice between bet t ing on the American orNat ional League teams in the Ai l -Star Game. A winning bet on the AmericanLeague pays $5 and one on the Nat ional League pays $3. A loss oneither betcosts $2 . We would usual ly represent this decision problem with table 1-3.

1-3 Am er ican League W ins N at ional League W insBet Amer i canBet National

Given that you are not a to tal ly loyal fan of ei ther league, this way of lookingat the choice would lead you to choose between th ebets on the basis of howprobable you thought the American League to win. (Later w e wi l l see thatyoushould bet on the Am erican League i f you think i ts chances are bet ter tha n 5 in12.) But suppose you use table 1-4 instead. Then you would s implybet A m e r -1-4 I W in M y Bet I Lose ItBet Amer i canBet Na t i o n a l

ican on the grounds that that bet pays bet ter . Y oum ight ev en a rgue toy o urse l fas fo l l o ws :I wi l l e i ther win or lose. If I w in , be tt in g A mer i can is bet ter , andif I lose, my bet does not mat t e r . Sow ha tev er hap p en s , I do at least as wel lbybet t ing Amer ican .T heprincip le i l lustratedinthis reasoning isca l l edthedominanceprinciple.W e say tha t an ac t A dom inates another ac t B i f , in a s ta te-by-s ta te compar ison,A y ields outcomes tha t are at least as good as those y ielded byBand in some statestheyareevenbetter. The dominan ce principle tells us to rule out dominated acts.If there is an act that dominates al l o thers, the pr in cip le has uschoose it.H o w e v e r , w ecanno t a lw ays re lyon thed o m i n a n c ep r i n c ip l e a sane x a m -p lefrom the d isarm am ent debate dem onstrates. D oves argu e that d isarm am ent ispreferable whether or not a war occurs. For, they claim, i f there is no war andw ed i sarm, more funds w i l lbeava i l ab leforsocial prog rams ; iftherei s a war andw e disarm, wel l , bet ter R ed than dead. This produces decision table 1-5.

1-5 W a r N o W a rA r mDisarm

+$5+$3

-$2-$2

DeadRed

Status QuoImp ro v ed Society

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INTRO DUCTIO NGiv e n the view that i t i s bet ter to be Red than dead, dis arm ing do min ates .Hawks need not quest ion anything that has t ranspi red. They can s implyrespond that disarming makes i t v i r tual ly cer ta in that the other s ide wi l l a t tackus but that cont inu in g to a rm makes w ar very u n l ike ly . Doves ha ve no t takenth is into account . They have not considered that in th i scase the act we chooseaffects the probabi l i t ies of the s ta tes . The example shows that the dominanceprinciple appl ies only when the acts do not affect the probabi l i ty of the s ta tes .T he same problem arises in the bet t ing example. T he p robab i l i ty o f w i n -n ingvar i es wi ththe betchosen . Soreasonin g according to thedom inance pr inc i-ple does not apply to the choice between bet t ing American or N a t i ona l .On e m ight think that a l l that i s wr on g wi th the bet t ing exam ple i s a mis ap-pl icat ion of the dominance pr inciple , but Ithink there is a deeper problem here.Itis acaseof anillegitimateproblemspecification. In anydecision table that usesstates such as / win, o r things turn outwell, we cansubst i tutethe state descrip-tion/ make the right choice withouthaving tochangetheoutcomesor the effectof the various acts on the states. But i t is surely pointless to use a decision tablewi th the s ta te headings / make the right choice, Ifail to make the right choiceto make that very choice. If you a l ready knew the right choice, w hy bother set-t ing up a decis ion table? A ctua l ly , th is i s a bi t f l ippant . The real pro blem is thatthedesignat ion of the term "right choice" varies with theact. If I bet A mer i can ,I make the r ight choice i f and only i f the American League w i n s . Correspond-ingly fo r be t t ing Nat iona l .S o thephrases / make the rightchoice,Ifail to maketh e right choice cannot serve asstate descriptions, since theydo no tpickout oneand the same state no mat te r what the act. The same point obviously appl ies todescr ipt ions such as / winor things turn out well.It is nota lways cl ear , howev er , whenas ta te desc r ipt ionisprop er . Suppose,fo r example, thatyou are t ryingto choose between going to lawschool andgo ingtobusiness school . It istempt ing to use state descriptions such asIam a successoropportunities aregood,but amoment 's thought should convinceyoutha t theseare var i an tsof I make the right choice and , thus , improper . Unfor tuna te ly thereis no a lgor i thm for determining whether a s ta te descr ipt ion i s proper .Nor are there a lgor i thms for deciding whether a set of states is re l evant .Suppose again that you are deciding between law school and business school .The states th e rainfall is above average for the next three years, it is not areplainly i r re levant to your dec i s ion . But how about there is an economic depres-sion threeyearsfrom now, there isnot? Perhaps lawyers wi l l do wel l dur ingadepression whereas business school graduates wi l l remain unemployed; thosestates might then be re l evant to cons ider .I wi l l leave the problem of state descriptions to proceed to another one,which a l so involves the choice o f problem speci f icat ions. In this instance, how-ever , the state descriptions involved may be entirely proper and re l evant . To i l -lustrate the prob lem I have in mi nd let us suppose that you w a n t to buy a carwho se asking pr ice i s$4,000. How much shou ld you b id? Before you can an-s we r that q u e s t i o n , y o u m ust cons ider severa l a l te rn a t ive b id s , say , $3,000,$3,500, an d$4,000.T hose three b ids wo uld genera teathree-ac t prob lem spec i -10


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INTRODUCTIONfication. But is this the best specif icat ion for your problem? How about usinga wider range ofbids? O r bids wi th smaller increments? Thoseare clearly rele-v a n t questions and they bear on which decision table you ul t imately use. Nowif we think of the various answers to these quest ions as giv ingrise to differentdecisiontables (e.g., thethree-bidtablefirstmentioned, another with thebids$2,500,$3,000, $3,500, $3,750, etc.) , choosing th e best problem specificationamounts tochoosingbetween decision tables. We are thus invo lvedi n a second-order decision, that is, a decision about decision problem specif icat ions. W e canapply decision theory to this decision too. For example, if the best act accordingto any of the sp ecifications und er consideration isbid$4,000,it would not m atterwhichtable you choose. A nd if al l the tables use thesamestates, we can combinethem into one big table whose rows consist of all the bids used in any of thesmal lertables and whose column s are the states in quest ion. The best bid accord-ing to that table w il l be the best for yo ur decisio n. If the states va ry too, the solu-t ion is not so clear . B ut that is a technical problem fo r decision theory. Let uscont inue wi th the phi losophical problem. We have now formulated a second-order decision problem concerning the choice o f tables for your o r ig ina l b id -dingp rob lem. But ques t ions may ar ise concerning our choice of a second-orderproblem specif icat ion. Should w e hav e cons idered o ther first-order tables wi thaddi t ional bids or other sets of states? Should w e have used different methodsfo r eva luat ing the acts in our f irst-order tab les? Approaching these ques t ionsth rough decision theory w i l l lead us to generate a set of second-order tables andat tempt to p ick thebest of these to use . But now we hav ea th i rd -order dec i s ionprob l em. A n inf in i te regress of dec i s ion p roblems is off and r u n n i n gT he difficultyherecan be putsuccinctly byobservingthatwheneverw e ap-ply decision theory w e must make some choices: At the least , we must p ick theacts, states, and outcomes to be used in our p rob lem spec i f ica t ion .But if we usedecis ion theorytomake thosechoices , w em us t makeyetano therse t ofchoices.This does not show that it is impossible to app ly dec i s ion theory . But i tdoes show that to avoid an inf inite regress of decision analyses anyapp l ica t ionof the theory mu stbe based ul t imatelyon choices thataremad e wi tho u titsbene-fi t . Let us cal l such decisions immediate decisions. Now someone might objec tthat insofara sdecision theory de fines rat ional decisio n mak ing , only those deci-s ions mad e wi th itsbenef i t should coun t as ra t iona l . Thu s imme dia te dec i s ionsare not r a t i o n a l , and because a ll dec i sions depend ul t im ate lyon these, nodeci-s ions are ra t ional .

Should we give up d ecision theory? I think not . The object ion I ha ve ju strehearsed a ssum es that decision theo ry has cornered the marke t on rat ional deci-sion m akin g , that a dec i s ion made wi thouti ts benefits is i r ra t ional . In fac t , it isfrequent ly i r ra t ional to use decision theory; the costs in t ime or mo n ey may betooh igh.If anan gry bear ischasing yo u, itwo uldn otmake senseto usedec i s iontheory top i c k w h i c htree to c l i m b . On the o the r hand , i t is not a l w ays r a t i ona lto m a k e an i m m e d i a t e dec i s ion e i t h e r . Y ou w o u l d not (or shou ld no t ) choosey o u r career , co l l ege , o r p rofess iona l schoo l wi thout weighing the p ro san dc o n sof a fewa l t e r n a t i v e s .

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INTRODUCTION

But then how do we decide w h e n to do a decis ion analys isand w h e n tomake an immediate decis ion? Well , we do not do i t on a case-by-case basis .Each t ime I see a car coming at me in my lane, I do not ask myse l f , Shou ld Ido a decis ion analys is todecide between braking, pul l ing off the road, or con-t inuing as I am or should I make an immediate decis ion? I f I d id I would havebeen kil led years ago. (I would a lso have trapped myself in an infinite regress.)Instead, I follow an unstated policy of letting my "gut reactions" make thechoice. A nd that isp l a in lythe ra t iona l th ing to d o, since a sober, healthy, andexper ienced dr iver usual ly does the r ight thing in such s i tuat ions.W e l ivebymany policies that telluswhenw eshould makeanim mediate decis ionandwhena decis ion analys is of some kind is required. Some of those policies are morerational than others; they lead in general to better l ives. This means that i t maybe appropr ia te from time to time to reassess one or more of our policies. Ofcourse, decis ion theory may help with that task.A persistent skepticmight object thatnow weneedapolicy forreassessingpolicies, and anotherregress o f decisions is in the offing. But I wil l leave thematter as it stands.A final pre l im inary phi losophical issuei si l lustrated by th i sfanciful e x a m -p l e . Baker and Sm ith, competi tors in the oi l busi nes s , are both considering leas-ing an oi l f ie ld. Baker hires a decision analy st to advise him , and S mith decidesto base hisdecision on the fl ip of aco in . T hedecis ion analyst obta ins extensivegeological surveys, spends hours reviewing Baker 's balance sheets, and f inal lyconcludes that the r isks are so great that Baker should not bid on the lease atall . Letting the fl ip of a coin decide for him, Smith pays dearly for the lease.Y e t , toeveryone's surpr i se , a year later he f inds one of the largest o il reservesin his s ta te and makes pi les of money. Something seems to have gone wrong.Smithnever gavethemat te ra nyconsideration andbecameabi l l ionaire , wh ereasthe thoughtful Baker remained a s truggl ing oi lman. Does this turn of eventsshow that Baker 's use of decision theory w as ir ra t ional?We can resolve some of our discomfort with this example by dis t inguish-ing between right decisions and rational decisions. Agents ' decis ions are rightif they eventuate in outcomes the agents l ike at least as well as any of the otherposs ibi l i t ies that might have occurred after they had acted. According to ours tory Smith m ade the r ight decis ion and Baker did not. I f we had complete fore-kno wled ge, indiv idual decis ion theory w ould need only one pr inciple , namely,make the right decision. Unfor tuna te ly , most of our decis ions m ust be based onwhat w e th ink might happen or on wha t w e think is likely to happen , and wecannot be certain they will result in the best outcomes possible. Yet we sti l lshould try to make choices based on the informat ion we do have and our bestassessments of the r isks involved, because that is clearly the rational approachto decis ion making. Fur thermore, once w e appreciate th e unfavorable c ircum-stancesu n d e r w h i c h m o s tof ourd ec is ions mus tbem a d e ,w e can seethatara t io-nal decis ion can be en t i r e ly p r a i s ew or thy even though i t did not turn out to bethe r ight decis ion. (How often have you heard people remark tha t a l thoughi tw astru e theyhadhopedfor abetter outcom e, they madetheon ly ra t iona l cho ice12


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INTRODUCTION

open to them at the time? O r that despi te having made a stupid decision, some-one "lucked out"?)PROBLEMS1. Set up a decision table for the fol low ing decision si tuation . Jack, w ho is nowtwenty , mustdecide whether to m a r ry his true love Ji l l immediately or no tsee her again unt i lhe is twenty-one. If hemarries her now thenhe wil l losethe mill ion dollars his uncle has left him intrust . If hewai ts to see her unti lhei stwenty-one,hewil lreceivethemoneyand canma rry Ji lla tthat timeifshe sti l l loves him. (Part of your problem is selecting an appropriate set ofacts, states, and outcomes. )

2. Pascal reasoned thati t wasbetter tolead the life of arel igious Ch rist ian thanto be a pagan, because if God exists, rel igious Christians go to Heav enandeveryone else goes to Hel l , whereas if Goddoesnotexist , the lifeo f the reli-gious Christ ian is atleast asgood asthatof thepagan . Set up adecision tablefo r this argum ent and explain why the dominance princip le su pportsPascal'sreasoning.1-3. Certainty Ignorance, and RiskSometimes we can be quite certain that our acts wi l l resul t in given outcomes.If you are in a cafeter ia and select a glass of tomato juice as your on ly d r inkthen, ingenious pranksters and uncoordinated oafsaside, that is the drink youwil l br ing to your table. Sometimes, however, you can know on ly tha t yourchoice wi l l resul t in a given outcome with a certain pro babi l i ty . If , for instance,you bet on getting a 10 in one ro l l of a pa i r of unloaded dice, you canno t becertain of winning, but you can know that your chances are 1 in 12. Final ly ,sometimes you may hav e no earthly idea about the relat ionship between an actopen to you and a possible outcome. If you have a chance to date a potent ialmate, a possible outcome is that the two of you wil l someday together pose fora photo with your great-grandchi ldren. But, offhand, i t would seem impossiblefo r you to estimate the chances of that happening if you make the date.Ifyou are m akin g a decision in which you can be certain that a l l yo ur actsare like the first example, decision theorists cal l your choice a decision undercertainty. Here all you need to do is determine which outcome you l ike best,s ince you know which act (or acts) i s certain to produce i t . That i s not alwayseasy. Even a studentw ho could becertain of getting all her courses might havea hard t ime decid ing whether to s ign up for logic and physics this term or formusic and physics this term, postponing logic unt i l next term. O r suppose youare p lan ning an auto tr ip from New York to Los Angeles wi th stops in Chicago,St . Louis, New Orleans, and Las Vegas. You can use one of many routes, butthey wi l l differ in mi leage, dr iv ing condi t ions, traffic, chances for bad wea the r ,and scenery . Even supposing that every th ing concern ing these a t t r ibutes is cer-t a in , w h i c h wi l l y o u choose?T hem athem at ica l theory ofl in e a r p r o g r a m m i n g hasbeen app l iedto m a n yproblems concern ing dec i s ions under cer ta in ty with quan t i ta t ive outcomes.

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INTRODUCTIONHowever, nei ther decision theorists nor phi losophers have paid this subjectmuch a t ten t ion , so we wi l l notcover i t fur ther in th i s book.W h e n , in a g iven dec i s ion p roblem, it is possible to ass ign p robabi l i t i esto al l the outcomes a r is ing from each act, the problem is cal leda decision underrisk.Choicesbetweenbetson faircoins,roulettewheels, ordiceare paradigmsof dec i s ions und er r i sk . But it is usua l to classify investment decisions, bets onhorse races, market ing decisions, choices ofcrops top l an t ,andm any o thers l ikethem as decisions under r isk, because even when we cannot assign an exactprobab i l i ty , say , to the stock market r is ing or to a d ro ught , it often pays to treatthose decisions as i f they were decisions under r isk, pure and simple.Final ly , when i t makes no sense to assign probabi l i t ies to the outcomesemanat ing from one or more of the acts (as in your date resul t ing in great-g r a n d c h i ld r e n ) , the decision problem is cal led a decision under ignorance.(Some decision theorists cal l i t a decision u nd er uncerta in ty .) Ignorance m ay bepart ia lor to tal ; i t may be possible to assign probabi l i t ies to some of the outcomesemanating from some of the acts , but to none emanat ing from the other acts.W e wi l l turn shor t ly to techn iques for deal ing with decisions under ignorance,but w e wi l l t reat only decisions under to tal ignorance in this book.This c lass i f i ca tiono f decisions as under cer ta in ty , r i sk , and i gn o ran ce isp la in ly an ideal izat ion. Many decisions do not fall neat ly in to one category o rano ther . Yet if the uncertain t ies in a decision are negligible, such as an uncer-ta inty as to whether thewor ld wi l l ex i s t tomorrow , theproblem isfair ly treatedas one under certain ty . And i f we can est imate upper and lower bounds on prob-abi l i t ies , we can break a decision problem into several problems under risk,solve each, andcomparethe results. Ifeachsolutionyields the samerecommen-da t i on , our inab i l i tyto assign exact probabi l i t ies wi l l no t mat t e r . On the otherh a n d , if the range of probabil i t ies is very wide, i t might be better to treat theproblem as a dec i s ion under ignorance .The classi f icat ion is phi losophical ly controversial too. Some phi losophersth ink the only certain t ies are mathematical and logical . For them there are fewt ruedecisions und er certain ty . O ther phi losophers notnecessari ly d isjo in t fromthe firstg ro up t hin kwe are never total ly ignorantof theprobabilities of theout-comes resul tantfrom an act . Thus for them there are no true decisions under igno-rance. W e w i l l learn mo re about this later wh en we stu dy subjec t ive probabi l i ty .l-3a. Some Details of Form ulation

Suppose you are decid ing w heth er to eat at GreasyPete's and are concerned thatthe food wi l l make you sick. The rele vant outcomes associated w ith yo ur act areyougetsickandyou do not getsick.Now if youtakethestatesto bePete's foodisspoiled, it is not, the act of eating at Greasy Pete'sunder the state that his foodis spoiled isnot certain to result in your getting sick. (Perhaps you eat very l i t t leof the spoi led food or have a very strong stomach.) Yet , our use of decision ta-b les p resupposes tha t we can find exact ly one o u tco me fo r each ac t - s ta te pa i r .So how can we use dec i s ion tables to represent your s imple problem?W e co u ld i n t ro d uce o u tco mes tha t themse lv es i n v o lv e e l emen t so fu n c e r -4


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INTRODUCTIONta in ty .F or e xam ple, w e could replace the outcomes of you r get ting (not get t ing)sickw i th the outcomesyou have a chance ofgettingsick,you do not. This wo uldensure nomore than one outcome per square. Or we could in troduce a morerefined d iv i s i o nof the environment in to states, using, fo r instance, th efood isspoiled but you can handle it, the food is spoiled and you cannot hand le it, andth efood is notspoiled for the last example. D ifferent decision problem s w i l l cal lfo r different combinat ions o f these approaches. In any case, because these areproblems with apply ing decision theory, w e wil l assume henceforth that eachact-state pair determines a un ique outcome.In v iew of th i s assumpt ion , we can focus all the uncer ta in ty in a decisionon the states involved. If you do not know whether eat ing at Greasy Pete's wi l lmake you sick, we wi l l take that to be because you do not know whether thefood is spoiled or whether you can h a n d l e it. A fur ther consequence is tha t , inthe case of decisions under r isk, probabi l i t ies wi l l be assigned to states ratherthanoutcomes.Again,if you eat atGreasyPete's, th eonlyw ay you can get sickis for the food to be spoiled and you to be unable to handle it. So to treat yourproblem as a decision und er r is k we mu st assign probabi l i ties to that compoundstate and the other states. Suppose the probabil i ty of the food being spoiled is70% and tha tofyou r be ing unable to handle spoi led food is 5 0 % . Then (as wewi l l learn later) the probabi l i tyof your getting rotten food at GreasyPete'sandbeing unable to handle i t is 35% , whereas the probabi l i ty that you wi l l get badfood but wi l lbe able tohand le i t i s 35% and theprobabi l i ty that you r food w i l lbe f ine i s 30%.Unless some malevolent demon hates you, your choosing to eat at GreasyPete's should not affect his food or you r abil i ty to handle i t. Th us the probabi l i -t ies assigned to the states need not reflect the acts chosen. This means that wecan use the un qua l i f ied probabi l i ty th at GreasyPete'sfood wi l l be spoi led ratherthan thep robabi l i ty tha titwi l l be spoiledgiven thatyou ea t there . On the otherhan d , if you are decid ing whether to smokeand are worr ied about dy ingo flungcancer, your acts wi l l affect your chances of entering a state y ield ing thatdreaded outcome. As y o u k n o w , the lung cancer rate is much higher amongsmokers than among the to tal populat ion of smokers and nonsmokers. Conse-quent ly , the probabi l i ty of get t ing lung cancer given that you smoke is muchhigher than the probabi l i ty that you wi l l get lung cancer n o matter w hat you do.T he latter probabil i ty iscal led theu nconditional p robabi l i ty , the former thecon-ditional probabil i ty of getting lung cancer. Plainly, in deciding whether tosmo ke , the conditional probabil i t ies are theones to use.W e say that astate isindependent of an act when the condi t ional probabi l -ity of the state given the act i s the sam e as the unco ndi t ional probabi l i ty of thatstate. Getting heads on the fl ip of a fair coin is independent of betting on heads.There being rotten food atPete's is independent of your eat ing there. But con-t ract ing lung cancer i s not independen t of smoking, earn ing good grades is notindependen t ofs t udy ing , and s u r v i v i n gamara tho nis noti n d e p e n d e n tof t ra in ingfor i t .

W h e n some of the states fail to be i n d e p e n d e n tof the ac t s in adec i s ion15


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INTRODUCTION

underrisk, w e should use theprobabil i t ies of the states conditional on theacts.Whenall of thestates are independent of theacts, itdoesnotmatter which prob-abil i t ies w e use; for , by the definit ion o f independence, there is no differencebetween them. Since there is no harm in a lways us ing condi t ional probab i l i ties ,fo r the sake of un i fo rmi ty , we wil l do so.Those w hoprefer uncon di t ional probabi l it ies may find i tpossible to refor-mulate their decision problems using new states that are independent of the acts.Consider the smoking decis ion again. Not everyone who smokes gets lungc a n c e r n o t even those who have been very heavy smokers s ince their teens. I tis plausible, then, that those who avoid cancer have some protective factor thatshields them from smoking's cancer- inducing effects. If there is such a factor,smoking is unlikely to be responsible for i ts presence or absence. With this inmin d, we can reform ulate the smokin g decis ion intermsof s ta tes involv ing thisnew factor . W e replace the two states you (do not) get lung cancer wi th fourstates: you have th eprotective factor and do (donot) get terminal lung cancerfrom nonsmoking causes, you do not have th eprotective factor and you do (donot) g et terminal lung cancer from nonsm oking causes. Then your smoking wil lnot affect your chances o f being in one s ta te ra ther than another . In the or iginalformula t ion , you saw smoking as actual ly determining whether you entered astate leading toy our death from lung cancer; thus you saw smoking as affectingthe probabi l i ty of being in that s ta te . On the new formulat ion, you are a l readyin a state that can lead to your death from lung cancer or you are not. If youare in the unlucky s ta te , your not smoking cannot a l ter that; but i f you smokeyou are cer ta in to die , s ince you lack the protect ive factor . You do not knowwhat state you are in, but if youknew enough about lung cancer and the factorsthatprotect those exposed to carcinogens from getting cancer, you could assignunconditional probabil i t ies to the four new states. For those states would be in-dependent of the acts of smoking or not smoking.F or a f inal bit onrefo rmula t ions , consider Joan 's problem . She is preg nan tand cann ot take care of a baby. She can abort the fetusan d thereby avoid hav ingto take care of a baby, or she can have the baby and give i t up for adoption.Ei ther course prevents the outcomeJoan takes careofababy, but Joan (and we)sense a real difference between th e means used to achieve that outcome. Thereis a simple m ethod for form ula ting Joan's cho ice so that i t becomes the truedi lemm a that she sees i t to be. W e s imp ly include act descr ipt ions in the outcomedescr ipt ions.W e no longer hav e a s ingle outcome but two:Joan has an ab ortionand doesnot take careof a baby, Joan givesher babyup for adoption and doesnot take care of it.

PROBLEMS1. Classify the fo l lowingas decis ions under cer ta inty, r isk, o r ignoran ce . Jus-tify y o u r c lass if ica t ions .a. Jones chooses his bet in a rou le t te game.b. Sm ith decides betw een seeking and not seeking as p o u s e .

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INTRODUCTION

c. A ve ter inar ian decides w hether to put a heal thy s tray dog to s leep or toadopt i t as his own pet.d . A student trying to satisfy degree requirements chooses among thecourses currently avai lable .e . A lawye r decides whether to accept an out-of-court se t t lemen t or to takeher client's case to tr ial .2. Set up a decision table for the last version of the Greasy Pete proble m. W hyis theoutcome d escr iptionyou do n ot getsick inmore thano nesquare? W hatis th e total probabi l i ty that you wi l l get an outcome so described?

1-4. Decision TreesIt is often more expedi t ious to ana lyze a decis ion problem as a sequence of de-c is ions taking place over t ime thanto treat it as a single one-time decision. T odo this we use a decision tree instead of a table. A decision tree is a diagramconsisting of branching l ines connected to boxes and circles. T he boxes arecalled decision nodes and represent decisions to be made at g iven po in ts inth e decision sequences. The circles are called chance nodes and represent th estates re levant to that point of the decis ion. Each l ine project ing to the r ight ofanode represents one of theactsorstates associated withi t and isusu ally labeledwith an appropriate act or state description. To demonstrate these ideas, let ususethedecision tree (show ninf igure 1-1)torepresent thed isa rmament problemdiscussed earlier. As the tree i l lustrates, outcomes are wri t ten at the tips of the

Figure 1-1t ree . This permits us to assess th e possible consequences of a choice byfo l low-ing each of the branches it genera tes to its t i p . D i sa rming , for instance, leadsto an improved soc ie ty if there is no w a r , but it leads to life as a Red if thereis o n e .

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INTRODUCTION

The prac t ica l advantage of trees over tables can be appreciated by con-s ider ing the fo l lowing example . Suppose y o u m u s t f irst choose between goingto the seashore or staying athome. If you go to the seashore, you wi l l wa i t todetermine whether i t i s ra in ing . I f i t ra ins , you wi l l dec ide whether to f i sh orstay inside. If you f ish and the f ishing is good, you wi l l be h a p p y ; if the f ishingi s no t g o o d , you wil l be d i sappoin ted . If you stay in, you wil l feel so-so. O nthe o ther hand, i f there is no r a i n , yo u wi l l sunbathe and be happy . F ina l l y i fyou s t ay home, you w i l l feel so-so. F igu re 1-2 presents thetree for this decis ion.

Figure 1-2It iseasy to go f rom a dec i s ion t ab le to adec i s i ontree. Start the t ree wi tha box wi th one l ine emanat ing f rom i t for each row of the t ab le . At the end ofeach l in e place a c i rc le with one l ine coming from i t for each colu mn of the table .Then at the t ips o f each o f these l ines wr i t e the outcome ent r ies for the square

to w h i c h t h a t t ip corresponds .It is by far more i n t e re s t i ng an d impor tan t tha t the dec i s ion tree for anyprob lem can be t ransformed in to an equiva len t dec i s ion t ab le . W e accomplishthis by co l laps ing sequences o f decis ions into one-t ime choices o fstrategies. Ast rategy (S) is a plan that determines an agent's choices u n d e r a ll re levant c i r-cums t ances . F o r example , the p l anSi: I w i l l go to the shore ; if it r a i n s , I w i l l fish; if it does no t r a i n , Iw i l l sunb a t he

is a s t ra t egy appropr ia t e for the prob lem ana lyzed by the preced ing tree. T heothe r appro priate s t rategies are:2: I w i l l go to the shore ; if it r a i n s , I w i l l stay in; if it does not , I w i l lsu n ba th e ;3: I w i l l stay at home (under a l lcircumstances).

W e c a n s i m i l a r l yf ind a seto fs t ra t eg ics that w i l l e n a b l eus to r e p r e s e n tan yo t h e rsequen t ia l d e c i s i o nas a cho iceb e tween s t r a teg i e s . Th i s r ep re s en t a t i on g ene ra l l ynecess i ta tes u s i n g more compl i ca t ed s t a t e s , as t ab le 1 -6 i l lu s t ra t e s for the sea-18


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INTRODUCTION

shore problem. In case you are wonder ing how we obtain th e entries in thesecond row of this table, notice that in this row 82 is adopted. So if it rains you1-6 Rain & Good F ishing Rain & Bad Fishing N o Ra inSiS2S3

HappySo-soSo-so

DisappointedSo-soSo-so

HappyHappySo-so

wil l stay in, and whether or not the fishing is good (for anyone), you will feelso-so. If i t does not rain, you will sunbathe and be happy.Decision theorists hav e presented m athematically r igorous fo rmu latio ns ofthe technique illustrated here and have proved that any decision tree can bereduced to a decision table in which the choices are between strategies. Sinceour intere st in this book is mo re theoretical than practical, we w ill stick with de-cision tables and not pursue the study of decision trees further.PROBLEMS

1. F ormula te the follow ing decision problem us ing a decision tree.D anny , whohas been injured by Manny in an automobile accident, has appl ied toM anny's insurance company for compensation. T he company has respondedwith an offer of $10,000, Danny is consider ing hir ing a l awye r to demand$50,000. If Danny h i res a l awye r to demand $50,000, Manny's insurancecompany wil l respond byeither offering $10,000 again or offering $25,000.If they offer $25,000, Danny p lans to take it. If they offer $10,000,D a n n ywill decide whether or not to sue. If he decides not to sue, he will get$10,000. If he decides to sue, he will win or lose. If he wins, he can expect$50,000. If he loses, he wi l lget noth ing . (T o simplify this problem, ignoreDanny's legal fees, and the emotional, tempora l , and other costs of not set-t l ing for $10,000.)2. M imickin g the method used at the end of this section, refo rmu late Dan ny'sdecis ion problem using a decis ion table .

1-5. ReferencesIn g iving references at the end of each chapter, I will refer to works by meansof the names of their authors . Thus Luce and R a i f f a refers to the book GamesandDecisions by R. D. Luce and H. Raiffa. When Icitetw o worksby the sameauth or, I wil l either use a brief ti t le or give the author 's n am e and the pub licatio ndate of the work in question.The c lass ic general treat iseon dec is ion theory is Luce and Raiffa. It noton ly surveys severa l of the top ics not covered in this book but a lso conta insanex tens iveb i b l i o g r a p h y .T he reader w i s h i n g to do more advanced s tudy indeci-sion theory should begin wi th th i s work . VonN eumann and Morgenstern w as

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INTRODUCTION

the first major book on the subject and set the agenda for much of the field. An u m b e ro fjourna ls p ubl i sh a r t ic les on decis ion theory,but Theory and Decisionis the pr incipal interdisc ipl inary journal devoted to the subject .R a i f f a is an excel lent introduction to individual decis ion theory fo r thoseinterested in appl ica t ions of the theory to business problems. This book a lsomakes extensive use of decision trees. Eells, Jeffrey, and Levi are more ad-vanced w o r k s in the subject fo r those with a phi losophical bent. Savage and

Chernoff and Moses approach the subject from the po in t of view of statistics.Davis is a popular introduction to game theory, andS enis a classic treatiseon social choice theory . Both books conta in extensive bibl iographies .For more on problem spec if ica t ions , seeJeffrey's "Savage's Omelet," andNozick.

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Chapter2DECISIONS UNDER IGNORANCE

K A T H R Y N L A M B Ii s abr i l l ian t graduate s tuden tinphys ics .She isalso happi lymarr ied to Paul Lambi and they wa n t chi ldren. Kathrynhas a difficult decisionto make because her ideas about mothering are rather tradit ional; she believesthat sheoughtto care for her chi ldren herself dur in g their preschool y ears . Butshecannotdo thatandper form longanddel icate exp er iments. E achday resolv-ing her d i lemma becomes m ore urgent . Should she have her chi ldren now andpostpone her career? O r should shecap italizenow on herbril l iant start in orderto establish herself as a physicist and raise her family later? As Kathrynseesherdecis ion, the relevant states concern her abi l i ty seven years from now to estab-lish acareer or to be a good mother . (Shebelieves that the latter depends onher fertility and psychic energy and that both decrease withage.)W e canthusrepresent her problem as choosing between two acts: have children now andpostpone career and pursue career now and postpone children. This choice ismade against th e background of four states:

In seven years K. L. will be able to be a good mother and have a goodcareer.In seven years K. L . willnot beable to be a good mother but willbeableto have a good career.

In seven years K . L . willbeable to be agood motherbut willnot beableto have a good career.In seven years K . L. will be able neither to be a good mother nor to havea good career.

Kathryn has read several articles andbooks about delayed motherhood and feelsthat she can con fidently assign a probability (or range of probabilit ies) to her beingable to be a good mother seven years hence. But she simply has no idea of howto assign a pro bab ility to her being able successfully to start over as a ph ysi cist .Kathryn ' s is a decis ion under ignorance. Decis ion theor is ts have debatedlong and hard about how tohandle such decis ions, and today many issuescon-cerning them remain unresolved. In this chapterIwi l l present four of the alter-nat ives dec is ion theor is t s have proposed, d iscuss the i r advantages andl imi ta-t ions , andbrief ly explorethema t terofd e terminin g wh ich approach todec is ionsunder ignorance is correct .

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DECISIONS UNDER I G NO RANCEKathryn ' s decision is one of partial ignorance, since she can assign someprobabil i t ies to the states relative to her acts . F or instance, she canappeal to thel i terature she hasread toassign ap robabi l i tyto h er being able to beboth agood

mother and a good physicist i f she pursues her career now. Despi te this I wi l lrestrict m y t r e a t m e n t to decisions under complete ignorance.2-1. Preference OrderingsThereis another compl icat ion toK athryn 's decision. Shewould ra therbe a goodmother and a good physicist , but when she contemplates the prospect of not be-ing able to bebo th , she finds herself wavering between preferr ing to be a goodmother and prefer r ing to be a good physicist . O f course, she cannot even beginto resolve her or iginal d i lemma unt i l she gets her preferences straight . Decisiontheory has very l i t t le to say about how Kathryn should do this. Rather i t concen-trates on describing wha t her preferenc es mu st be l ike i f she is to apply decisiontheory to her problem. That wi l l be our focushere.F or co n v en i en ce , I wi l l use the expression "xPy" to mean "the agentprefers x to y" and "xly" to mean "the agent is indifferent between x and y ." Iwi l l also use these abbreviat ions in d i scuss ing Kathryn ' s p references :

"m & p" for "K . L. is a good mother and a good physicist .""#z & p" for "K. L. is a good mother but not a good physicist .""m & p" for "K. L. is not a good mother but is a good phys ic i s t . "" - m & - p " fo r " K . L . i s n e i t h e r a g o o d m o t h e r n o r a g o o d p h y s i c i st ."

(We m ay presume that these are the outcomes Ka thryn cons ide rs . ) W e know that(m &p)P(m & -p), (m &p)P(-m & p) , that is, Kathryn prefers being botha good mother and a good physicist to being only one of the t w o . Do we n o talso know that (m & p)I(m & p); n a m e l y , she is ind i f ferent between be ing,on the one h a n d , a good mother whi le not a good phys ic i s t and , on the otherhand , be ing a good physicist whi lenot a good mother? N o, thatdoes no t fol lowfrom her wavering. In decision theory, an agent i s considered to be indifferentbetween tw oal ternat ives only if she has considered them and iscomple tely w i l l -ing to t rade one for the o ther . Kathrynh as arr ived at no such conclus ion . Deci-sion theor i s t s th ink that she sho u ld , and this is reflected in the r equ i remen t tha tg iven any two acts, the agent prefers one to the other or is indi f ferent betweenthem (in the sense just stated).To be more expl ici t , decision theorists have proposed a min imal set ofcondi t ions that the p references of an ideal ly rat ional agent must sat isfy. (Suchcondi t ions hav e been called vario usly condi t ions of coherence, consistency, andra t iona l i ty , but we wi l l see tha t it is debatable whether rat ional , consistent , o rcoherent agents must conform their preferences to each of these condi t ions.) Iwi l l refer to these condi t ions col lect ively as the ordering condition.Let us s tar t wi th three uncon trovers ia l componen ts of the o rd e r i n g con d i -t i o n . They requ i re the agen t not to p re fe r a thing x to one y w h i l e a l so be in gind ifferen t between them or p r e f e r r i n gy tox . These ho ld for al loutcomesx andy tha t the a g e n t has u n d e r c o n s i d e r a t io n :22


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DECISIONS UNDER IGNORANCE01. IfxPy, then no t yPx.02. IfxPy, then no txly.03. Ifxly, then notxPy and also not yPx.

In more mathematical terms, these condi t ions state that various asymmetrieshold among th e agent 's preferences.O f course, agents change their minds and may be quite rational in doingso . Condi t ionOl, for instance, is notmeant toprecludem y nowp refer r ing wr i t -ing a book to t rave l ing around the wor ld and hav in g the opposite preferencesin ten years . The ordering condi t ion deals only wi th an agent 's preferences ata moment .The n ext condi t ion requi res tha t any two o utcomes be conn ected to eachother via the agent 's preferences.

04. xPy or yPx orxly, for any relevant outcomes x and y .This is thecondition that requires Kathryn to make up her mind. (Note: I amfo l lowing thepractice, cus tomary inmathematics and logic, of interpreting "or"as meaning "a t least one and possibly both." That is why I need both the asy m-metry and connect iv i ty condi t ions. If I had interpreted "or" as "at least one butnot both" then Icould h ave used just a reinterpreted condition O 4 instead of myconditions O1-O4. Exercise: Prove m y last claim.)The next components of the ordering condition are called th e transitivityconditions.

05. IfxPy andyPz, then xPz.06. If xPy andxlz, then zPy.Ol.IfxPy andylz, then xPz.O8. If xly andylz, then xlz.

These are to holdfor al l relevant outcom esx, y , andz. Cond i t ionO5, forexam-ple, requires you to prefer apples to ice cream if you prefer apples to peachesand peaches to ice cream; condition O 7 requires you to prefer cars to boats ifyou prefer cars to horses and are indifferent between horses and boats.Exper iments can easi ly demonstrate that humans are not always able tohave transi t ivepreferences. Byadding smal l amountsofsugar to successive cupsof coffee, one can set up asequence of cupsofcoffee wi th thisproper ty: Peoplewillbe indifferent between adjacent cups, but not between the first and last. Thisshows that sometimes w e violate conditionO 8. But acarefully designed versionof this experiment can show more, namely, that w e cannot help ourselves. A llw e need do is make sure that th e increments in sugar levels between adjacentcups are below thelevelofhum an detectabi l ity . Does this mean that ourbio logyforces us to be i rrat ional? O r that condition OS's claim to be acondi t ion of ra-t ional i ty is a fraud?Speaking for myself ( for there is no consen sus among the exper tson th isissue), ne i ther a l te rnat ive is quite correc t . Sure ly w e should try for t rans i t ivepreference o rder ings when and where we can . For t rans i t ive p reference o rder-ings organize our preferences into a s imple and tractable structure. Yet the

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DECISIONS UNDER IGNORANCEcoffee cups do not signal an irrational blind spot. The transitivity conditionscharacterize thepreferences of an ideally rational agent. We fall short of thatideal by lacking tastes sufficiently ref ined to avoid being"tricked"into a transi-t ivityfailure. But rather than dismiss a powerful decision theory based on transi-t iv i ty , we should take steps, when a decision is important, to emulate idealagents more closely. For instance, if some very important matter turned on atransitive ranking of the cups ofcoffee, we could use chemical tests to determinetheir relative "sweetness."

Wealso fail fromtime to time to have connected preferences especia llywh e n many things must be evaluated at once and our attention spans becomeovertaxed. Again Iregard theconnectivity conditionas an ideal whose impor-tance increases in proportion to the importance of the decision at hand.Those depressed by our failures atidealrationality might take comfort inthe factthatwe canoften makearational choice without satisfyingall theorder-ingconditions. For instance, if you know that you can have your first choice fordessert, you need not completely rank ice cream, cake, f ru i t , and pie againsteach other. It is enough that you decide thatcake,say, is better than all the rest.Y o u c a nleave the rest unconnectedo r wi th fai led transitivities, since, as far asgetting dessert goes, the effort of ranking them is wasted.

If an agent's preferences meet conditions O1-O8, items ordered by hispreferences divide into classes called indifference classes.They are so called be-causetheagentisindi f ferentbetween itemsin thesame classesbutprefers itemsone way or the other in one class to those in a different class. (Economists, whousua l ly restrict themselves to graphable preferences, speak of indifferencecurves rather than indifference classes. Our concept is a generalization oftheirs.) We will rank an agent's indifferenceclassesaccording to the preferencerelations that hold between their members: One indifference class ranks aboveanother just in case its members are preferred to those of the other. To illustrateth i s ,let ussuppose that anagent's preferences among ten i te m s a , b, c, d, e,f , g, h, i, a n d _ / a r e asfollows:

alb, aPc, cPd,die, elf, fig, dPh, Mi, iPj.The ordering condition permits us to derive additional information f rom thepreferences given. For example, condition O8 yieldsdlfand dig and conditionO6 yieldsbPc. This additional information tells us that the ten items divide intoranked indifference classes asfollows:

a, b [5] [4]d, e, f, g [3]h, i [2] HI

Oncew ehave divideda se t of a l t e rna t ives in toranked ind i f f e renceclasses,we canassign anumbertoeach class tha t wi l l ref lect therelative importanceo rutility o fitems in theclass to the agent. (Note: T h i s may not bepossiblew h e n24


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DECISIONSUNDER IGNORANCEthe number ofindifference classes isinfinite. See exercise 6 in the next Problem ssection.) The convention used in decision theory is to assign higher numberstohigher-ranked i tems. Other than that we are free to choose whatever numbersw ewish. W ehave used th enumbers 1-5 but we could have used 5-10, or even 1,0, 1/2, 1, 3. Any increasing sequence of five numbers wi l l do. All that isrequired isthatth enumber assigned to anitemjc calledtheutilityofx andwr i t -ten M ( X ) b e such that

a. u(x) > u(y) if and only if xPyb. u(x) =u(y) if and only if xly

fo r all ranked i tems x and y .D ecision theorists call theva r ious waysofnum beringthei temsin aprefer-ence ordering utility functions or utility scales. (More precisely, the functionsare the var ious ways of associating n umbers w ith preference order ings and the

scalesare thesequencesof numbersused.B ut no ha rm w il l resul t fromo ur usingthe terms "util i ty function"and"utilityscale"interchangeably.) U t i l ity functions(scales) that satisfy condit ions (a) and (b) are called ordinal util i ty functions(scales) because they represent only the ordering of an agent 's preferences. Anychange or transformation of an ordinal scale that preserves this representat iongives rise to anequallyacceptableordinal scale for the agent's preferences. Sup-pose ? [ M ( J C ) ] is the t ransformat ionof the ut i l i tyo fx on the w-scale into autil i tynumber on the ?(w)-scale. Then the new scale will be an acceptable o rdin al uti l i tyscale for the agent just in case

c. v v > v if and only if t(w) >t(v), for all w and v on the w-scale.F or thenthe f(w)-scale wi l lsatisfy condit ions (a) and (b)too. W e wil l cal l t rans-formations that satisfy condit ion (c) ordinal transformations.Ordinal scales represent only the relat ive ordering of the items fall ingwithin the scope of an agent's p references; they do notfurnish informat ion abouthow many i tems fall between two i tems or about the intensi ty of an agent'spreferences. Y et they do al low us to formula te some decis ion methods more con-veniently. One example is the dominance pr inciple. We can now define domi-nance in te rms of util i t ies: An act A dominates an act B if each uti l i ty numberin ro w A isgreater than or equal to its correspondent in row B and at least oneof these numbers is strict ly greater.Example. In table 2-1, with three acts and four states, actAt dominatesthe others .

2-1 S S4AA 2A^

122

01

-1

333

010

25

Si


41/2

DECISIONS UNDER IGNORANCEPROBLEMS

1. Convertthe 1 to 5 scale used toranka-j intoa 20 to 25 scale, intoa - 5to 0 scale, and intoa scale whose numbers are allbetween 0 and 1.

2. What happensto anordinal scale when it istransformedbymultiplying eachn u m b e r by 1 ? B y 0? Are these acceptable transformationsof theoriginalsca le? Wh y o r wh y not?3. Show that thetransformationtof anordinalut i l i tyscale uinto ascale, t(u),produces ascale that also satisfies conditions(a) a nd(b), provided / sat isf iescondi t ion (c).4 . Show that th e f o l l ow i ng a re ordinal t r a n s fo rm a t i o n s :

5. Is thetransformation t(x)=x2 anordinaltransformation whenapplied to ascale whose numbers a re greater than o r equal to 0? What if some of then u m b e r s are negative?6 . Suppose anagent haspreferences fo rpaired items (x; y )where itemx is theamount o ftimeshe has tolivea ndyis theamounto ftimeshew i l lgothroughher death throes. Let us suppose thatxvaries continuously between0 mo-m e n t sand 20yearsandybetween0momentsand 2weeks.Theagent alwayspre fe r s to live longer and suffer less butputs anabsolute priority onlivinglonger. This means,forinstance, thatsheprefersthepair(18years;2weeks)to (17years; 11months and 28days;25 minu tes ) .Explain why it isimpossi-ble to use anordinaryf ini tenumber scale torepresent theagent'spreferences,a l t hough shedoes satisfy the ordering condition. Explain why i t is possibletorepresent this preference usingthe real numbers i f we concern ourselveswi th only preferences fo r moments of l i fe or for moments of death throes.

2-2. The Maximin RuleU s in gordinal ut i l i ty func t ions we can formulate avery simple rule fo rmakingdecisions under ignorance. The rule isknown as the maximin rule,because itte l lstheagenttocompare theminimum utilities providedbyeachact andchoosean ac t whose minimum is themaximum value for all the minimums. Inbrief,the rule says maximize theminimum.Example. In table 2-2 the minimums fo r each act are starred and the actwhose minimumis maximal is double starred.2-2AA2A}

**A4

Si S3 S45

-1*6

5

0*44

6

0*344

271*3*

26

S2


42/2

DECISIONS UNDER IGNORANCEWhen thereare two ormoreac ts whose m in imumsa remaximal , themaxi -min rule counts them as equal ly good. We can break some t ies by going to thelexical maximin rule. Thistells us to firstel iminate all the rows except the tied

ones and then cross out the minimum numbers and compare the next lowest en-t ri es . I f the ma xim um ofthese still leaves two ormore acts t ied, we repeat theprocess unt i l the tie is broken or the table isexhaus ted .Example. Ap p ly in g th e l exica l maxim in ru le stil l leaves A\ and A a , tied intable 2-3.2-3AiA 2A,A 4

Si 8 2 83 $4 S $0003

3310

5224

4131

1135

Conservat ism underl ies the maximin approach to decisions under igno-rance . Assuming that whatever one does the wors t wi l l happen , the maximinrules pickthebestof the worst . Perhaps thisis thebest stance totake whenone'spotent iallossesa reenorm ous. BeingnoJames B ond ,IsuspectIwo uldfollow thema x imin ru le were I to con templa te jum pin g from a p lane wi thout a parachute .Despi te e xamp les like these, the maximin ru les are easi ly cri t icizedon thegrounds that they prohibi tus from taking advan tage o f op por tun i t ies i nvo lv ingsl ightlosses and great gains. F or example , in table 2-4 the maximin act isA\\

2-4 Si S2AiA 2

1.501.00

1.7510 000

but choosing AI would mean giv ing up at most $.50 for a chance at$10,000.(I am assuming here that the agent ' s at t i tude toward money is fa i r ly typ ica l . )P R OBLEM S

1. F ind the m ax im in acts for tables 2-5 and 2-6. U se the lexical m axi m in ruleto break ties.2-5 2-6AiA2

Ai

124

-326

53

-10

63

5

AiA2

A,

00

3

14

0

120

31

127

choices an introduction to decision theory

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