Choice under Uncertainty

1 Choice under UncertaintyJonathan LevinOctober 20061 IntroductionVirtually every decision is made in the face of Uncertainty . While we often rely onmodels of certain information as you ve seen in the class so far, many economicproblems require that we tackle Uncertainty head on. For instance, how should in-dividuals save for retirement when they face Uncertainty about their future income,thereturnondifferent investments, their health and their future preferences? Howshouldfirms choose what products to introduce or prices to set when demand isuncertain? What policies should governments choose when there is uncertaintyabout future productivity, growth, inflation and unemployment?Our objective in the next few classes is to develop a model of Choice behaviorunder Uncertainty . We start with the von Neumann-Morgensternexpected utilitymodel, which is the workhorse of modern economics.

2 We ll consider the foundationsof this model, and then use it to develop basic properties of preference and choicein the presence of Uncertainty : measures of risk aversion, rankings of uncertainprospects, and comparative statics of Choice under with all theoretical models, the expected utility model is not without itslimitations. One limitation is that it treats Uncertainty as objective risk that is,as a series of coinflips where the probabilities are objectively known. Of course, it shard to place an objective probability on whether Arnold Schwarzenegger wouldbe a good California governor despite the Uncertainty . In response to this, we lldiscuss Savage s (1954) approach to Choice under Uncertainty , which rather thanassuming the existence of objective probabilities attached to uncertain prospects1makes assumptions about Choice behavior and argues that if these assumptions aresatisfied, a decision-maker must act as if she is maximizing expected utility withrespect to some subjectively held probabilities.

3 Finally we ll conclude by lookingat some behavioral criticisms of the expected utility model, and where they few comments about these notes. First, I ll stick pretty close to them inlectures and problem sets, so they should be thefirstthingyoutacklewhenyou restudying. That being said, you ll probably want to consult MWG and maybeKreps at various points; I certaintly did in writing the notes. Second, I ve followedDavid Kreps style of throwing random points into the footnotes. I don t expectyou to even read these necessarily, but they might be starting points if you decideto dig a bit deeper into some of the topics. If youfind yourself in that camp, Kreps(1988) and Gollier (2001) are useful places to Expected UtilityWe start by considering the expected utility model, which dates back to DanielBernoulli in the 18th century and was formally developed by John von Neumannand Oscar Morgenstern (1944) in their bookTheory of Games and Economic Be-havior.

4 Remarkably, they viewed the development of the expected utility modelas something of a side note in the development of the theory of Prizes and LotteriesThe starting point for the model is a setXof economic problems (and for much of this class),Xwill be a set of monetarypayoffs. But it need not be. If we are considering who will win Big Game thisyear, the set of consequences might be:X={Stanford wins, Cal wins, Tie}.We represent an uncertain prospect as alotteryorprobability distributionoverthe prize space. For instance, the prospect that Stanford and Cal are equally likelyto win Big Game can be written asp=(1/2,1/2,0). Perhaps these probabilities2depend on who Stanford starts at quarterback. In that case, there might be twoprospects:porp0=(5/9,3/9,1/9), depending on who will oftenfind it useful to depict lotteries in the probability simplex, asshown in Figure 1.

5 To understand thefigure, consider the depiction of three-dimensional space on the left. The pointpis meant to represent(1/2,1/2,0)and the pointp0to represent(5/9,3/9,1/9). In general, any probability vector(p1,p2,p3)can be represented in this picture. Of course, to be a valid probabilityvector, the probabilities have to sum to 1. The triangle represents the space ofpoints(p1,p2,p3)with the property thatp1+p2+p3=1. The right hand sidepicture simply dispenses with the axes. In the rightfigure,pandp0are just asbefore. Moving up means increasing the likelihood of the second outcome, movingdown and left the likelihood of the third outcome and so p Figure 1: The Probability SimplexMore generally, given a space of consequencesX, denote the relevant set oflotteries overXasP= (X). AssumingX={x1.}

6 , xn}is afinite set, a lotteryoverXis a vectorp=(p1,..,pn),wherepiis the probability that outcomexioccurs. Then: (X)={(p1, .., pn):pi 0andp1+..+pn=1}3 For the rest of this section, we ll maintain the assumption thatXis afinite s easy to imagine infinite sets of outcomes for instance, any real numberbetween 0 and 1 and later we ll wave our hands and behave as if we haddeveloped the theory for larger spaces of consequences. You ll have to take iton faith that the theory goes through with only minor amendments. For those ofyou who are interested, and relatively math-oriented, I recommend taking a lookat Kreps (1988).Observe that given two lotteriespandp0, any convex combination of them: p+(1 )p0with [0,1]is also a lottery. This can be viewed simply as statingthe mathematical fact thatPis convex.

7 We can also view p+(1 )p0moreexplicitly as acompound lottery, summarizing the overall probabilities from twosuccessive events:first, a coinflip with weight ,1 that determines whetherthe lotteryporp0should be used to determine the ultimate consequences; second,either the lotteryporp0. Following up on our Big Game example, the compoundlottery is:first the quarterback decision is made, then the game is p1=1 p + (1- )p 1- pp Figure 2: A Compound LotteryFigure 2 illustrates the idea of a compound lottery as a two-stage process, and asa mathematical fact about Preference AxiomsNaturally a rational decision-maker will have preferences over consequences. Forthe purposes of the theory, however, the objects of Choice will be uncertain we will assume that the decision-maker has preferences over lotteries onthe space of consequences, that is preferences over elements start by assuming that the decision maker s preference relation onPis complete and transitive.

8 We then add two additional 1A preference relation on the space of lotteriesPiscontinuousif for anyp, p0,p00 Pwithp p0 p00, there exists some [0,1]such that p+(1 )p00 implication of the continuity axiom (sometimes called theArchimedeanaxiom) is that ifpis preferred top0, then a lottery close top(a short distanceaway in the direction ofp00for instance) will still be preferred top0. This seemseminently reasonable, though there aresituations where it might be the following example. Supposepis a gamble where you get$10for sure,p0is a gamble where you nothing for sure, andris a gamble where you get killedfor sure. Naturallypshould be strictly preferred toq, which is strictly preferred tor. But this means that there is some (0,1)such that you would be indifferentbetween getting nothing and getting$10with probability and getting killed withprobability1.

9 Given that preferences are complete, transitive and continuous, they can berepresented by a utility functionU:P R,wherep p0if and only ifU(p) U(p0). Our next axiom, which is more controversial, will allow us to say a greatdeal about the structure ofU. This axiom is the key to expected utility 2A preference relation on the space of lotteriesPsatisfiesinde-pendenceif for allp, p0,p00 Pand [0,1],wehavep p0 p+(1 )p00 p0+(1 ) independence axiom says that I preferptop0, I ll also prefer the possibilityofpto the possibility ofp0, given that the other possibility in both cases is somep00. In particular, the axiom says that if I m comparing p+(1 )p00to p0+(1 )p00, I should focus on the distinction betweenpandp0andholdthesamepreferenceind ependentlyof both andp00. This axiom is sometimes also calledthesubstitutionaxiom: the idea being that ifp00is substituted for part ofpandpart ofp0, this shouldn t change my that the independence axiom has no counterpart in standard consumertheory.

10 Suppose I prefer{2cokes,0twinkies}to{0cokes,2twink ies}.Thisdoesn t imply that I prefer{2cokes,1twinkie}to{1coke,2twinkie s}, even thoughthe latter two are averages of thefirst two bundles with{2cokes,2twinkies}.The independence axiom is a new restriction that exploits the special structure Expected UtilityWe now introduce the idea of an expected utility 3A utility functionU:P Rhas anexpected utility form(or isa von Neumann-Morgenstern utility function) if there are numbers(u1, .., un)foreach of theNoutcomes(x1, .., xn)such that for everyp P,U(p)=Pni=1pi a decision maker s preferences can be represented by an expected utilityfunction, all we need to know to pin down her preferences over uncertain outcomesare her payoffs from the certain outcomesx1,.., , an expected utility function islinearin the probabilities, meaningthat:U( p+(1 )p0)= U(p)+(1 )U(p0).