1、Modeling decisions under uncertaintyPervasiveness of uncertaintyCards and gambling:raise?or fold?Career decisions:which major?which career?Insurance choices:buy insurance or go uninsured?How does eating this donut affect the likelihood of heart attack?Which disease treatment to pursue:the cheaper bu
2、t higher risk one?Or the certain but more expensive option?Modeling decisions under uncertaintyConsider a“lottery”with uncertain prospects:Option x1Probability p1Option x2Probability p2Option x3Probability p3Option xNProbability pnHow should such a lottery be valued?Valuation Function:V(A)=V(x1,p1;x
3、2,p2;xN,pn)Preferred lotteries yield higher valuesModeling rationalityEconomists assume that people are rational,even under uncertaintyBut what does“rational”imply?Three Properties:1.Completeness2.Transitivity3.IndependenceModeling rationality:three properties 1.CompletenessBetween any two options,a
4、 decision can always bemade.i.e.Either V(A)V(B)V(A)V(B)and V(B)V(C)implies V(A)V(C)“If you prefer apples to bananas and bananas to coconuts,then you must prefer apples to coconuts too.”This rules out circular preferences3.IndependenceModeling rationality:three properties 1.Completeness:rules out ind
5、ecision2.Transitivity:rules out circular preferences3.IndependenceA change common to two lotteries cannot affect the preferences between those lotteriesi.e.If V(A)V(B),then p V(A)+(1-p)V(C)p V(B)+(1-p)V(C)Independence rules out impact from irrelevant alternatives Expected utility theoryThe only Valu
6、ation Function satisfying completeness,transitivity,&independence is the expected utility function V(A)=V(x1 ,p1;x2,p2;xN,pN)=p1U(x1)+p2U(x2)+pN U(xN)=E U(A)This is what we assumed when we studied healthinsurance models!Bounded rationalityBut cognitive powers may be boundedOur decisions not always c
7、onsistent with the three properties of rationalityTwo psychologists,Daniel Kahneman and Amos Tversky,posed questions to students in a lab and found inconsistenciesKahneman was awarded the 2002 Nobel Prize in Economics(Tversky passed away before 2002)Kahneman and Tverkys approachKahneman and Tversky(
8、1979)surveyed Israeli university students about decisions under uncertainty.These questions offered respondents different hypothetical lotteries to win Israeli money The inconsistencies they find with expected utility theory motivated their alternative theory of decisions under uncertainty:Prospect
9、TheoryProblem 1Win$2,500 with prob.33%$2,400 with prob.66%$0 with prob.1%Win$2,400 with prob.100%Lottery ALottery BChoose Lottery A or Lottery B82%of respondents chose Lottery B.According to expected utility theory:0.33 U(2500)+0.66 U(2400)+0.01 U(0)U(2400)0.33 U(2500)0.34 U(2400)+0.66 U(0)0.33 U(25
10、00)0.34 U(2400)Overvaluing certaintyAccording to Problem 1,majority views0.33 U(2500)0.34 U(2400)Contradiction with expected utility theory!Certainty effect:tendency to value lotteries with certain results even more than ordinary risk aversion Problem 3Win$4,000 with prob.80%$0 with prob.20%Win$3,00
11、0 with prob.100%Lottery ALottery BChoose Lottery A or Lottery B80%of respondents chose Lottery B.According to expected utility theory:0.80 U(4000)+0.20 U(0)U(3000)0.80 U(4000)U(3000)0.80 U(4000)U(3000)Overvaluing certaintyAccording to Problem 3,majority views0.80 U(4000)U(3000)Contradiction with exp
12、ected utility theory!Another example of the certainty effectProblem 5Win$5,000 with prob.0.1%$0 with prob.99.9%Win$5 with prob.100%Lottery ALottery BChoose Lottery A or Lottery B72%of respondents chose Lottery A.According to expected utility theory:0.001 U(5000)+0.999 U(0)U(5)0.001 U(5000)U(5)Overwe
13、ighting small probabilitiesBut Problem 5s result contradicts expected utility theory and diminishing returnsBecause of diminishing returns,U(5)U(0)U(5000)U(4500)So U(5)0.001 U(5000)But Problem 5 finds thatU(5)(p-0.0001)Overweighting small probabilities(0.01)0.01Evaluation:the value functionProspect
14、theorys value function v(x)is similar to expected utilitys utility function u(x),exceptv(x)allows for different risk appetites over potential losses and potential gains v(x)evaluates outcomes based on changes from a reference point,whereas u(x)evaluates final outcomes Together,these changes incorpor
15、ate loss aversionA possible value functionProspect theory&health economicsGrossman model describes health as a choice,one that agents consciously make knowing their own preferences and the probabilities out thereProspect theory reveals that this decision is far from easyThe pervasiveness of uncertai
16、nty in health means prospect theory is highly relevantA Prospect theory interventionResearchers have tried to channel prospect theory findings into“nudges”small interventions that may have outsized effectsExample:Volpp et al.(2008)propose a weight-loss regime in which dieters who achieve their goal
17、are entered into a lottery with a 1%chance of paying$100According to prospect theory,people value this chance more than a certain chance of winning$1Researchers found this program resulted in more weight loss than traditional programs ConclusionProspect theory maintains the fundamental assumption th
18、at people always act to pursue their best interest But actual rationality can be quirkier than the rationality of expected utility theory.People have used prospect theory to develop interventions that help people stay healthy Remaining question is whether or not these interventions are paternalistic should people be corrected when they make“irrational”decisions?
