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Introduction à la théorie de la décision

Introduction à la théorie de la décision. Ferdinand M. Vieider University of Munich Home: www.ferdinandvieider.com Email: fvieider@gmail.com. Université Libre de Tunis, April 6 th , 2012. 1. 2. What is Decision Theory?.

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Introduction à la théorie de la décision

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  1. Introduction à la théorie de la décision Ferdinand M. Vieider University of Munich Home: www.ferdinandvieider.com Email: fvieider@gmail.com Université Libre de Tunis, April 6th, 2012 1

  2. 2 What is Decision Theory? • Decision Theory: studies ensemble of human decision making processes, individual and social • It mostly becomes relevant in situations with some complexity (e.g. risk, uncertainty) • It is closely related to several other fields: - operations research - linear programming - game theory - experimental economics - behavioral economics - cognitive psychology - social psychology 2

  3. 3 Main similarities and differences • Cognitive Psychology: the methodology of investigation and topics is very similar; however: rationality concepts borrowed from economics • Experimental Economics: DT methodology is very often experimental, however not exclusively so; also historically focus in individual decisions • Behavioral economics: comes closest, at least in descriptive aim; however, decision theory also encompasses rationality models, not only deviations from such models 3

  4. 4 Why experiments? • All of the disciplines just discussed make extensive use of experiments • Experiments allow to reproduce stylized situations of interest • Most importantly: one can vary one independent variable at a time • This makes it possible to isolate causal relationships (not just correlation) • Further distinctions: lab experiments versus field experiments, artificial experiments versus natural experiments 4

  5. 5 Lecture Overview • Overview of different approaches: normative, descriptive and positive • Origins of decisions theory: expected value theory to deal with risk • Introducing subjectivity: expected utility and its behavioral foundations • Expected utility's failure as a descriptive theory of choice • Descriptive theories of choice: Prospect Theory (and what it can explain) • Uncertainty, ambiguity aversion, and other puzzles (Wason, Monty Hall) 5

  6. 6 Normative, Descriptive, and Prescriptive approaches: From Expected Value to Expected Utility Theory 6

  7. 7 Different Approaches to DT • Different approaches to decision theory: normative, descriptive, and prescriptive • Normative theories describe how a perfectly rational and well-informed decision maker should behave • Descriptive analysis focuses only on actually observed behavior, and tries to find regularities • Prescriptive analysis has the aim of helping real-world decision makers in making better dec. • Are normative theories also good descriptive theories? 7

  8. 8 Descriptive Issues • At the outset, normative theories were taken for descriptive purposes as well • However: deviations from models soon emerged (falsification of theory) • Sprawling of descriptive theories that try to explain “anomalies” • Several issues that are often confounded: evidence from lab produces focus on cognitive limitations and stability of preferences • Real world: problems of awareness (“knowledge about knowledge”), then information search and processing 8

  9. 9 Prescriptive Analysis • Prescriptive analysis moves from a limited-information and processing perspective • Goal: helping to reach the best decision given the information at hand • In experiments normative and prescriptive approach often coincide (complete info) • This means that real-world situations are often very different (external validity issue) 9

  10. 10 The origins of decision theory • Historically, the concept of probabilities and how to deal with them is rather recent. • In the 1600s, Blaise Pascal and Pierre Fermat developed expected value theory • According to EVT, a prospect can be represented as its mathematical expectation: 0.5 DT 100 p*X + (1-p)*Y= 0.5*100= 50 0.5 DT 0 10

  11. 11 Example: EV normative? Choice between 2 known-probability events: 0.9 0.2 DT 10 DT 50 DT 0 DT 0 0.1 0.8 EV: 0.9*10+0.1*0=9 < 0.2*50+0.8*0=10 According to EVT, you should choose the lottery to the right. Is that your preference? Does your preference change if we increase the amounts *1000, to 10,000 & 50,000 DT? 11

  12. 12 From EV to EU • Expected value may not be a reasonable theory, even normatively, for large amounts • Also, these amounts may not be the same for everybody (wealth situation, preference) • To deal with this, we need one subjective parameter: Expected Utility Theory • In EUT, the value of a prospect is given again by its mathematical expectation, but instead of using (objective) monetary amounts we now use (subjective) utilities of those amounts 12

  13. 13 Example: EV versus EU Choice between 2 known-probability events: 0.3 0.5 DT 400 DT 200 DT 0 DT 0 0.7 0.5 EU: 0.3*u(400)+0.7*0=0.3 ?0.5*u(200)+0.5*0=? The extreme outcomes can always be normalized to 0 and 1. But how about intermediate outcomes? 13

  14. 14 Eliciting Utilities How can we elicit the missing utility? p DT 400 ~ CE DT 0 1-p We elicit either CE or p such that U(CE)=p*U(400)+(1-p)*U(0)=p Let CE=200 and elicit p (in reality easier for DM to elicit CE!) 14

  15. 15 Example reconsidered: Choice between 2 known-probability events: 0.5 0.3 €200 €400 ? €0 €0 0.5 0.7 U(0)=0, U(400)=1; assume p=0.65, then U(200)=0.65 This means that now: 0.5*U(200)+0.5*U(0)=0.325 > 0.3*U(400)+0.7*U(0)=0.3 15

  16. 16 Subjective Utility and Risk • Given the non-linearity in the utility function, preferences can change relative to EV • EUT: concavity=risk aversion. This is not universally valid! EV U(€) EU € 16

  17. 17 EV or EU? • EV is reasonable for small stakes, however most important decisions deal with large stakes • Also, many important decisions deal with non-quantitative decisions such as health states • For the latter EV cannot be defined; also: what if you have utility over money plus other things? • Expected Utility is thus generally more useful; it is however more complex, especially when combined with unknown probabilities • For the moment, we consider only utilities over monetary outcome with known probabilities 17

  18. 18 The St. Petersburg Paradox • Why concave utility? Consider the following example: • A bet is proposed to you: a fair coin is flipped until the first head come up; the amount you win at first flip is DT2, then DT4, then DT8, so that if head comes up at the kth flip you get DT2k • How much would you be willing to pay to play this game? • The Expected value of the gamble is infinite: 1/2*2+1/4*4+1/8*8... = 1+1+1... = ∞ • This goes to show that EV does not hold empirically when large amounts are at stake 18

  19. Risk Aversion and Risk seeking 19 • Risk Aversion: a prospect is considered inferior to its expected value • Risk Seeking: a prospect is preferred to its expected value • Risk Neutrality: a prospect and its expected value are equally valuable • ¡Do not confuse risk aversion with concave U! p X ? p*X+(1-p)*Y 1-p Y 19

  20. 20 Behavioral Foundations of EU • Behavioral foundations are properties of behavior (axioms) underlying a theory • They are very helpful in that a theory can be decomposed into some intuitive rule • E.g., saying that EU holds is equivalent to saying that preferences satisfy: - weak ordering - standard gamble solvability - standard gamble dominance - standard gamble consistency (or the stronger independence condition) 20

  21. 21 Independence • One of the most discussed issues is the following independence of common alternatives: p p x y ≥ x ≥ y 1-p 1-p C C How intuitive do you find this condition? 21

  22. 22 Example: Allais (common consequence) • Consider the following two choices: .10 .10 €5,000,000 €5,000,000 C A .89 €1,000,000 .01 .90 €0 €0 €1,000,000 .11 1 €1,000,000 B D .89 €0 The most common pattern is BC. This violates the independence axiom (rational: AC or BD). 22

  23. 23 Example: Compound Prospects • Consider the following two choices: 1/3 €200 1/6 1/2 €200 2/3 €0 1/6 ? €100 €100 1/3 2/3 €0 1/2 2/3 €0 Which one do you prefer? 23

  24. 24 EUT and Insurance • Under EUT, risk aversion coincides with a concave utility function, and risk seeking with a convex utility function • This does not hold generally: shortly we will see risk seeking with a concave utility function! • With a concave utility function, the expected utility of a prospect is lower than the utility of the expected value: p*U(x)+(1-p)*U(y)<U(p*x+(1-p)*y)=U(EV) • The difference between the EV of a prospect and its Certainty Equivalent is the Risk Premium 24

  25. 25 EUT and Insurance • Under EUT, risk aversion coincides with a concave utility function. U(DT) U U(y) U(p*x+(1-p)*y) p*U(x)+(1-p)*U(y) U(x) x CE y p*x+(1-p)*y DT What is the risk premium here? 25

  26. Insurance example 26 • There is a 5% risk that your house may be flooded, potential damages are – DT100,000 • EV = – DT5,000, However, if you are risk averse, the CE is lower, e.g. CE = – DT6000 • There is a positive risk premium of DT1000; by the law of large numbers, the insurance will pay DT5000 on average, and can thus make up to DT1000 by ensuring your risk • Could you represent this problem in a graph? What changes because of the negative outcome? • When is it rational to take out insurance and when not? 26

  27. 27 Graph Insurance Example • Nothing changes: implicit reference point problem (previous wealth) U(DT) U U(y) U(p*x+(1-p)*y) p*U(x)+(1-p)*U(y) U(x) X = –€100000 CE y=0 p*x+(1-p)*y DT 27

  28. 28 Insurance and Lotteries • We can explain insurance with concave utility under EUT • In theory, we can also explain lottery play, but we need convex utility for that • However: many people take up insurance and play lottery at the same time. How can this be explained? • Under EUT, we would need convex and concave sections of the utility function • We would also need these to hold at different levels of wealth 28

  29. 29 Typical Risk Preferences • People are typically risk seeking for small probabilities (± p<0.15): lottery play • For larger probabilities, people tend to be risk averse: CE<EV • For losses, however, these findings are inverted, with risk aversion for small probabilities and risk seeking for large probabilities • EUT cannot explain such preferences, since probabilities enter the equation linearly • EUT is thus violated descriptively, so that we need a more flexible theory to explain these phenomena 29

  30. 30 Descriptive Theories of Choice: Prospect Theory 30

  31. Experimental Data: Typical CEs 31 • Using a Prospect offering either €100 or 0 with different probabilities, I asked choices between the prospect and different sure amounts • The switching point between the sure amount and the prospect indicates a person's CE • The probabilities were 0.05, 0.5, and 0.9 • Mean CEs obtained from this classroom experiment in France were: 31

  32. 32 Your (average) utility function • Remember that U(CE)=p • Thus: U(11)=0.05; U(46)=0.5; U(68)=0.9, and we can always set U(0)=0, U(100)=1 U(X) 0.9 0.5 0.05 X 11 46 68 32

  33. 33 Prospect Theory • Kahneman & Tversky (1979; Econometrica) brought psychological intuition to economics: • Risk attitudes for small amounts are driven by feelings about probability, not money • We can thus let probability be the subjective parameter, and assume utility to be linear: PV=w(p)*x+(1-w(p))*y • Linear utility seems reasonable for small monetary amounts (but not large!) • For large amount, we can combine probability weighting with utility: PU=w(p)*u(x)+(1-w(p))*u(y) 33

  34. 34 Probability Weighting: Attitudes to Risk • We have seen that CE=p*U(100); if utility is linear, then p must be transformed • Let us thus assume that CE=w(p)*100, where w represents a weighting function • From our previous results we get: - w(0.05)=11/100=0.11 - w(0.5)=46/100=0.46 - w(0.9)=68/100=0.68 • From, this, we can plot a probability weighting function assuming w(0)=0, w(1)=1 34

  35. 35 Probability Weighting Function 35

  36. Insurance and Lottery Play 36 • Notice how this function can explain contemporary insurance and lottery play through overweighting of small probabilities • Also, there are jumps at the endpoints: the possibility and certainty effects • The latter can explain the Allais paradox (common consequence effect) • It also captures common risk attitudes quite well: fourfold pattern of risk attitudes • However, with linear utility it may have problems accommodating decisions over large stakes 36

  37. 37 Utility: Attitudes towards Outcomes • We have assumed linear utility above: however, we have seen that this is not always reasonable (St. Petersburg paradox) • Even assuming concave utility, it has problems dealing with mixed gambles • Example from Rabin, Matthew (2000). Risk Aversion and Expected-Utility Theory: A Calibration Theorem. Econometrica 68 (5): If a DM turns down (.5:110; -100), then she will turn down a 50:50 of -1000 and X for all X 37

  38. Prospect Theory Utility Function 38 • In PT, the utility function describes attitudes about money only, not probabilities concave U(X) X convex kink 38

  39. 39 Properties of Utility • Concave utility for gains means that even for small probabilities one can be risk averse for very large outcomes (insensitivity) • For losses one can be risk seeking for small probabilities for very large outcomes • Loss aversion: a loss is felt more than a monetarily equivalent gain • Loss aversion has been used to explain the status quo bias, endowment effect, myopic loss aversion (equity premium puzzle), etc. 39

  40. Loss Aversion 40 • Under loss aversion, “losses loom larger than gains” 0 ~ How high would the gain need to be to make you indifferent between playing and not playing the prospect? 0.5 DT ? – DT50 0.5 40

  41. Deduction of Loss Aversion 41 • Let us assume that DT 100 was elicited as gain that makes you indifference • Let us also assume that utility is linear over gains and losses, but that you are loss averse Then U(X)=Xif X≥0; and U(X)= –λ*Xif X<0 u(0)=0.5*u(100)+0.5*U(–50) 0 =0.5*100+0.5*(–λ)*50 λ*25=50 λ=2 What other assumption underlies this elicitation of the loss aversion parameter λ? 41

  42. Some functional forms 42 • A simple form for the utility that has been proposed is: • U(X)=Xα if X≥0 • U(X)= –λ*Xβ if X<0 • Can you see why the derivation of loss aversion as done before is an approximation? • Some popular functional forms for probability weighting functions are: w(p)=pφ/(pφ+(1-p)φ)1/φ w(p)= exp(-ξ (-log p)α 42

  43. Reference Point: Status Quo Bias 43 • Loss aversion is found to be the strongest phenomenon empirically • It stands and falls however on the determination of the reference point • Most of the time, the reference point is assumed to be current wealth, or the status quo • This means that people are often reluctant to switch from the status quo, no matter what that status quo is • This means that changes are perceived as gains and losses relative to status quo, with losses looming larger 43

  44. Reference Point: Endowment Effect 44 • The endowment effect was found by artificially establishing a reference point • Some people are randomly given one objects and others with a different one (e.g. mugs v. pens) • People are then given the opportunity to exchange the object in their possession • A large majority of people is found not to exchange their object • This holds true for both objects; since they have been randomly assigned, this can however not express true (average) preferences 44

  45. 45 From known to unknown probabilities: Subjective expected utility and the Ellsberg Paradox 45

  46. Unknown Probabilities 46 • We have so far only considered the case of risk, where objective probabilities are known • Good representation of situations such as lottery or well-established medical processes • However: most probabilities are unknown: stock market, entrepreneurship, education • In this case one can deduce subjective probabilities from observed decisions • Savage (1954) put forth some desirable attributes for decision making under uncertainty: Subjective Expected Utility Theory 46

  47. Ambiguity Aversion ? 20–? 20 R & B in unknown proportion 10 R 10 B 47 You are asked to choose between two urns, one 50:50, one unknown proportion of colors • First you are asked to choose which color you would like to bet on, then which urn • Which color would you rather bet on? And which urn would you prefer to bet on? • This phenomenon was discovered by Ellsberg (1961): it violates subjective expected utility theory since probabilities are the same (!) 47

  48. The Ellsberg Paradox 48 • When asked for a color preference, most people are indifferent: prr = prb; par= pab • Most people however have a strict preference for betting on the known-probability urn, no matter what which color: prr>par& prb > pab • This implies: prr + prb = 1 > par+pab; however, probabilities cannot sum to less than 1, hence the paradox • Prospect Theory has recently been adapted to deal with this: Source functions, AER 2011 48

  49. More realistic decisions under uncertainty 49 • Uncertainty has generally been studied in opposition to risk, not in its own right • Also: Ellsberg has created strong focus on 50-50 prospects • However: people react differently to different probability levels (just as for risk) • Also, people react differently to different sources of uncertainty (dislike vague probabilities, but may like uncertainties they have expertise in-->betting on football) • Applications: home bias in finance; stock market participation puzzle; 49

  50. Typical Source functions 50 50

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