On the How and Why of Decision Theory. Joint confusions with Bart Lipman

Transcription

1 On the How and Why of Decision Theory Joint confusions with Bart Lipman

2 Caveats Owes much to many people with whom we have benefitted from discussing decision theory over the years. These include Larry Epstein, Tzachi Gilboa, Faruk Gul, Mark Machina, Wolfgang Pesendorfer, Ariel Rubinstein, Aldo Rustichini, Uzi Segal, Rani Spiegler and many others and the many papers I have read and absorbed and cannot even identify as responsible in part for these confusions. So no claims to originality here

3 A (positive) view of economic theory Pure / basic / high brow theory provides tools and concepts (which I ll also refer to as a language) Models of individual decision making and game theory and studies this language existence, uniqueness, robustness, etc.

4 A view of Economic Theory These concepts are of interest only if they (eventually) result in better predictions about the world. But the word predictions is typically used too narrowly; here I mean it in the broadest possible sense. We are interested in models that help us say something correct about the world that we could not say before.

5 A view of Economic Theory So what is a good theory model? One that we learn from, that increases understanding. What is understanding? Better prediction (broadly interpreted) realizing thanks to a model that an intuitive explanation A X requires an additional assumption, A&B X Or yields an additional conclusion A X&Y Or is simply internally inconsistent Results about the language (that in turn help the language feed more successfully into predictive models)

6 What does decision theory do? To understand the world, economic theory derives relationships between behavioral properties: X implies Y If people are risk neutral then in an IPV environment a set of auctions have the same revenue. If people are risk averse then they will share risks (in a particular way). If people suffer from temptation they'll seek costly commitment devices

7 What does decision theory do? Decision theory provides (some of) the language to enable making such statements precise helps us understand the models we use so as to know when they are appropriate.

8 DT makes language precise For example, it defines notions of attitudes towards risk / ambiguity / temptation and characterizes (under certain assumptions!) what it means in terms of a functional form that can be used to study behavior in other contexts.

9 DT makes language precise Identifying properties using a functional form rather than characterizing using a behavioral definition can be misleading risk = variance in quadratic model? Masatlioglu and Ok behaviorally define statusquo preferences: loss aversion functional forms violate it. Epstein behaviorally defines ambiguity aversion: convexity of capacities (a functional form definition) does not satisfy it.

10 DT characterizes behavioral properties But a completely general model that only assumes risk aversion may be too complicated to be tractable, and also too general to give useful predictions. Need structure and hence need to know what is being assumed in the structure so as to know when it is appropriate

11 Decision Theory DT develops functional forms that are tractable allow sufficient generality to capture the behavioral issue of interest, but not much more, and that therefore intentionally ignore various other behavioral issues. It is not a strive for realism per se and should not be.

12 Complexity implies tradeoffs Ideally we may want a model that is Tractable General Extendable (to dynamic and interactive environments) Thoroughly characterized Based on "realistic" assumptions And that correctly predicts choice in all environments.

13 Complexity implies tradeoffs But reality is so complex (and data so incomplete) that tradeoffs are unavoidable. Different fields (e.g., DT and BE) adopt different tradeoffs.

14 Complexity implies tradeoffs Our view: All understanding is generated by deciding which aspects of reality to ignore. Because we must make unrealistic assumptions, only by understanding them better (e.g., in what contexts they are plausible or at least orthogonal) will we know when to apply them

15 Decision theory So decision theory develops functional forms (representing decision makers) that are narrowly focused on specific behavioral properties of interest. It also provides a better understanding of what we assume when we adopt particular models (such as SEU). Does so with axioms: characterize the class of models with the same behavior (hence predictions).

16 Why Axioms? Why are axioms useful? 1. Testing 2. Clarify implicit assumptions: What's driving conclusions 3. Relationship among models

17 Why axioms? 1. Testing the model Testing what are all the implications of the model. Axioms also facilitate finding examples that show the limitations of the axiom, and hence of the representation: Ellsberg urns, and more recently "Machina urns"

18 Caveats on testing Axioms are useful for testing But predictive failure of the model need not mean it should be dismissed it might help indicate contexts where it is appropriate it might help indicate contexts where the predictive failure is orthogonal to the issue of interest can have A X & Y, and even if Y is false we might still think that the model correctly illustrates the connection A X.

19 Caveats on testing: A Black Box DM The decision theoretic and axiomatic study of models aids understanding not only by enabling testing. Assume we had a black box which made perfect predictions of a person (or a distribution for a population) you feed in the choice problem, and it tells you the decision. While being a perfect predictor such a black box would certainly not be sufficient for most purposes of economics, and often would offer very little help indeed.

20 Axioms: Why characterize? Why put so much effort into fully characterizing the functional form implied by a set of behavioral properties, the axioms? Understanding the language: relations among models Understanding the world: Tells us what is driving our conclusions Understanding the world and the language: Robustness

21 2. What s driving conclusions Without characterizing "all" representations satisfying particular assumptions (risk aversion, ambiguity, status quo bias) we don t know if our conclusions stem from a particular parameterization that has that property or whether it holds for all preferences that satisfy a particular definition of that property.

22 3. Robustness axioms as a topology Due to the lack of data and the complexity of the environment one way to enhance our trust in a model is to see that results are robust. That requires knowing how to perturb the model. Axioms can help understand behavioral perturbations. In order to assess if a model holds under a perturbation we need to know the set of representations that correspond to a particular perturbation of the axioms.

23 Why care about robustness? Paucity of data / inability to test While it is obvious that robustness is an issue theorists care about, it is worth spending a moment on why it is important. If we had the ability to test every theory perfectly then robustness would be irrelevant we would simply test whether the theory is true in our world. But obviously we can't test every theory perfectly which is an added reason why robustness is important, and why theory more generally can be useful.

24 "Which" Axioms? Simple and easy to understand (i.e., "intuitive") axioms are more useful for all the above. Normatively appealing axioms in the sense that individuals should satisfy them are not important for positive economics, unless we think people do satisfy them. Naturally, there is a close connection between what we think people ought to do and what they do, but certainly we don t think these are identical.

25 Axioms are neither sufficient nor necessary for a good model Some models are well understood from their functional form. The term axiom misleadingly suggests an unarguable principle, so that an axiomatized theory is a good one. But axioms can characterize behaviors that are empirically false and uninteresting in terms of applications. Decision theory is not the only way to understand models.

26 Understanding models Much of behavioral economics shares the same ultimate goals as decision theory, but adopts a different approach. DT provides very complete analysis of the preference in a very stark environment. BE studies the implications of a functional form that has some behavioral feature of interest in a rich economic environment. Both approaches can shed light on interesting issues, they choose different tradeoffs, and can complement one another.

27 Interim Summary Due to complexity tradeoffs must be made. Decision theory (should) develop and characterize tractable functional forms, that are just general enough to capture the behavioral issue of interest. These models can be useful even when they are not perfectly predictive, and a perfectly predictive black box can fail to be useful. Axioms help understand the model: relate it to other models; know what assumptions are driving conclusions, and enable behavioral perturbations.

28 Interim Summary This perspective and conclusion suggest a few additional comments about how to do decision theory that we turn to next. Identification / uniqueness Of parameters Of the functional form The role of psychological realism

29 Identification of parameters The purpose of DT is to provide and understand models that are simplified descriptions of some aspect of reality. If the model has unidentified parameters especially if these are the parameters representing the behavior of interest then our understanding of their role is at best loose and may be misleading. For example, giving meaning to probabilities in a state-dependent EU model suffers from this problem.

30 State independence Consider a state independent A-A model: two states a, b and acts f that give lotteries f s in each state s. Assume preferences represented by some EU u and p: p a u(f a )+ p b u(f b ) We can say the agent is risk averse if (f a, f b ) (Ef a,ef b ) for all f.

31 State independence Consider now the lotteries given by some f composed with p, i.e., a lottery whose probability of amount c is p a f a (c)+ p b f b (c) Observation: When the lottery given by f composed with p is a mean preserving spread of g with p then g is preferred to f.

32 State independence This observation valid even if the agent s real preferences are state dependent and we are using the wrong probabilities. E.g., if we represent preferences with ½ u(f a ) + ½ u(f b ) when they are "really" 1/3 u a (f a ) + 2/3 u(f b ) where u a = 2u.

33 State independence So these probabilities are useful even if they are "wrong". And the "real" probabilities might not be useful (in the sense of this observation).

34 State dependence If preferences are state dependent then they can be represented using any p. Assume the preferences exhibits risk aversion in the same sense ((f a,f b ) (Ef a,ef b ) for all f). Claim: There are state dependent preferences such that for all p the observation fails. That is, for any p there exist f and g such that when composed with p the former is a mean preserving spread of the latter but the agent prefers the latter.

35 State in/dependence In this sense probabilities can be given meaning in the state-independent, but not the statedependent, model. This shows the role of uniqueness. The relevant probabilities are not the ones in the person s mind. Indeed reported or "real" probabilities may not be those in the state independent representation and we need not bothered by this.

36 Identification of parameters The utility functions in EU are only unique up to positive-affine transformations. What makes this non-uniqueness different from the one in a state-dependent model?

37 Identification of parameters Separability across parameters State independent model: non-uniqueness of u is independent of State-dependent model: non-uniqueness of p depends on changing u. Similarly, in Kreps model of preferences for flexibility the state space is non-unique in a manner that depends on the utility functions, and it is therefore hard to give behavioral content to aspects of the state space.

38 The cost of identification Often identification is achieved at the expense of complexity e.g., by enriching the domain of choice. AA pin down subjective probabilities in a finite state space using lotteries as outcomes. This complexity helps highlight the importance of the issue as the complexity needed for identification increases the less likely the axiomatic characterization of the model will prove to be useful: the data is less likely to be available The model will be too complex to work with The connection to other models will be less clear

39 Identification and comparatives The goal is to develop a simple specific model that identifies a particular behavioral property. The second ingredient we think is critical in a decision theory model are comparatives, results that discuss how the parameters of the model change as the behavior of interest changes. Comparing attitudes to risk using -u"/u' is well known Other examples: more ambiguity averse, more tempted, etc. Such results are critical as they enable studying the implications of the behavioral feature.

40 Identification of the functional form A related issue of identification concerns the functional form, and not the parameters in the functional form. For example, Gul and Pesendorfer show that a specific version of Koszegi and Rabin's referencedependent choice model is equivalent to a standard choice model in which transitivity can be violated. We show that G-Ps costly temptation model is equivalent (in terms of menu choice) to a random version of Strotz' overwhelming temptation model. Sarver's model of regret has a non-empty intersection with G-P's model of temptation. Choquet EU has a non-empty intersection with MMEU.

41 Identification of the functional form One question this raises is whether these equivalent representations can be distinguished on richer domains. Just like identifying parameters, the functional form may be identifiable. Indeed in the case of temptation the models lead to distinct choice from menus, and hence are identifiable in a richer dynamic model. As far as we know it is an open question whether the K-R model is identified and distinguishable from intransitive preferences in a richer model, e.g., with uncertainty as in K-R's original model.

42 Identification of the functional form Is there a concern when the behavior of two functional forms is not distinguishable? First, regarding when two representations have nonempty (yet non trivial) intersections: It would be surprising if the same choice behavior could not arise for different reasons. If an increase in regret is like an increase in temptation for some individuals, then we cannot distinguish them. Hence for some economic problems results about the implications of increases in one are the same as for an increase in the other. This does not seem to us to be a concern.

43 Identification of the functional form What if two representations are equivalent? Distinct but equivalent representations are useful in furthering understanding about the relationships among models. They can inspire different ideas about implications of the model.

44 Identification of the functional form Representations are not only a tool to represent choice but also suggest what issues to explore, e.g., what questions and comparatives to study. separation of utility and beliefs A reference-dependent representation suggests different experiments, extensions and economic applications than does an intransitive preference even if the set of behaviors they characterize is the same.

45 Psychological realism Must axioms or models be psychologically or procedurally realistic? Must they correspond to the decision making process? If decision makers say they do not have subjective beliefs, or if the beliefs they say they have are inconsistent with the ones we elicit or the ones that are consistent with their behavior, does this disprove the SEU model and make it irrelevant? Unsurprisingly our answer is NO!

46 Do not need psychological realism. We already saw that with state independent preferences the "wrong" belief might still be the "useful" one. If individuals cannot express their utility functions, does that mean their choice is intransitive?

47 Do not need psychological realism. If psychological risk aversion is distinct from decreasing marginal utility does that imply that u''/u' is not a useful measure of risk aversion? Many people would say that decreasing marginal utility of money is unrelated to risk aversion. If our goal is to study the implications of individuals disliking mean preserving spreads then we know it is useful to model this with an EU model with decreasing marginal utility. If the goal is study the difference between decreasing marginal utility and risk aversion then one needs a model with such features.

48 Psychological data Thus psychological realism of the axioms is not necessary for the model to be useful. But psychological data may be helpful in suggesting new models.

49 Psychological data can be useful Imagine that IQ was related with violations of independence. That could suggest that a cognitive model of reasoning about lotteries might be useful, resulting in models that relate IQ and income distributions.

50 Psychological data can be useful The counter-argument is that correlation doesn t mean we need a model. tall people prefer clothes that are longer Gul and Pesendorfer s discussion of complements and cue-based consumption: They argue that the reason for the complementarities is not of interest.

51 Psychological data can be useful In general we prefer simple structured models that cover as many cases as possible. Economists prefer a model of investment that derives preferences over portfolios from expected utility (not a direct preference over portfolios) We prefer models that separate the description of a lottery into probabilities and prizes over a general preferences over lotteries.

52 Psychological data can be useful It might well be that a theory of cues could unify various complementarities and help us understand them better. Just like EU gives more structure for choice under uncertainty that we find useful, cues might give structure for a class of preferences in which there are complementarities.

53 Non-choice data In general decision theorists are prone to viewing choice data as the only relevant data. As noted earlier, even "true" subjective probabilities might differ from the ones based on revealed preference, and for our purpose the latter is what is useful. But non-choice data can be useful. it can suggest more structured models

54 Non-choice data Other reasons to want non-choice data consider someone who buys glasses. Knowing that they have bad vision will enable better predicting their behavior should the price of glasses go up to the point where it is not affordable; e.g., they are more likely to buy large print books. Consider a child who prefer 6 pieces of chocolate to 3x3 pieces. We might predict differently whether that child would choose 9 pieces over 6 pieces if we knew how he did on his last multiplication test.

55 Non-choice data Of course such additional data is unhelpful if we have all possible choice data. We could then simply check if the child preferred 9 to 6 pieces. But data is scarce and given that limitation we should use non-choice data wherever it helps.

56 Summary of comments on positive DT DT should develop and characterize simple, structured models that focus on specific features of behavior. The models should be readily extendable to richer environments (dynamic, interactive). Axioms are useful for testing, relating to other models, assessing environments where the model is un/suitable. They need not be normatively appealing, but should be interpretable.

57 Summary of comments on positive DT The parameters corresponding to the behavior studied should be identified (separately from other parameters), without making the domain of choice overly complex, and comparatives should be provided to demonstrate usefulness. Distinct functional forms representing identical behavior can be interesting. Axioms and representations need not be psychologically realistic, but psychological realism and perspectives can be useful. Non-choice data can be valuable as complete choice data is not available.

58 Welfare economics THE END