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**Introduction à la théorie de la décision**

Ferdinand M. Vieider University of Munich Home: Université Libre de Tunis, April 6th, 2012 1 1

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**What is Decision Theory?**

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 2

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**Main similarities and differences**

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 3

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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 4

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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 5

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**From Expected Value to Expected Utility Theory**

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

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**Different Approaches to DT**

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 7

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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 8

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**Prescriptive Analysis**

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 9

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**The origins of decision theory**

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 10

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**Example: EV normative? Choice between 2 known-probability events:**

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 < *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 11

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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 12

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**Example: EV versus EU Choice between 2 known-probability events:**

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 ? *u(200)+0.5*0=? The extreme outcomes can always be normalized to 0 and 1. But how about intermediate outcomes? 13 13

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**Eliciting Utilities How can we elicit the missing utility? CE ~**

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 14

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**Example reconsidered:**

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)= > 0.3*U(400)+0.7*U(0)=0.3 15 15

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**Subjective Utility and Risk**

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 16

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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 17

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**The St. Petersburg Paradox**

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... = = ∞ This goes to show that EV does not hold empirically when large amounts are at stake 18 18

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**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 19

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**Behavioral Foundations of EU**

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 20

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

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**Example: Allais (common consequence)**

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 22

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**Example: Compound Prospects**

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 23

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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 24

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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 p*x+(1-p)*y y DT What is the risk premium here? 25 25

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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 26

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**Graph Insurance Example**

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 p*x+(1-p)*y y=0 DT 27 27

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**Insurance and Lotteries**

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 28

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**Typical Risk Preferences**

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 29

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**Descriptive Theories of Choice:**

30 Descriptive Theories of Choice: Prospect Theory 30 30

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**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: EV probability CE (mean) CE/EV €5 0.05 €10.96 2.14 €50 0.5 €46.48 0.93 €90 0.9 €68.37 0.76 31 31

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**Your (average) utility function**

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 32

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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 33

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**Probability Weighting: Attitudes to Risk**

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 34

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**Probability Weighting Function**

35 Probability Weighting Function 35 35

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**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 36

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**Utility: Attitudes towards Outcomes**

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 and X for all X 37 37

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**Prospect Theory Utility Function**

38 In PT, the utility function describes attitudes about money only, not probabilities U(X) concave X convex kink 38 38

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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 39

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**Loss Aversion Under loss aversion, “losses loom larger than gains” 0.5**

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 40

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**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)=X if X≥0; and U(X)= –λ*X if X<0 u(0)=0.5*u(100)+0.5*U(–50) 0 =0.5* *(–λ)*50 λ*25=50 λ=2 What other assumption underlies this elicitation of the loss aversion parameter λ? 41 41

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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 42

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**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 43

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**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 44

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**Subjective expected utility and the Ellsberg Paradox**

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

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**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 46

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Ambiguity Aversion 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 (!) 20 R & B in unknown proportion ? 20–? 10 R 10 B 47 47

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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 48

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**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 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 49

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**Typical Source functions**

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**Monty Hall's Doors and the Wason Selection Task**

51 Probability Calculus and Logical Induction: Monty Hall's Doors and the Wason Selection Task 51 51

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52 Monty Halls Doors There are three doors, one of which hides a car, and two with a goat behind You can choose a door. After you have chosen, the host opens one of the other two and reveals a goat If given the opportunity, should you switch or stay with your original choice? 52 52

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**To switch or not to switch: 1**

53 To switch or not to switch: 1 Imagine that the car is behind door 1, and the other two doors hide goats If you have chosen door 1, the host opens either door 2 or 3: In this case, switching loses the prize 53 53

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**To switch or not to switch: 2**

54 To switch or not to switch: 2 Imagine again that the car is behind door 1, and the other two doors hide goats If you have chosen door 2, the host opens door 3 for sure: Now, switching gives you the prize 54 54

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**To switch or not to switch: 3**

55 To switch or not to switch: 3 Imagine again that the car is behind door 1, and the other two doors hide goats If you have chosen door 2, the host opens door 3 for sure: Now, switching again gives you the prize 55 55

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56 Summing it up We have just seen that switching gets you the prize in 2 out of 3 cases Since the structure is symmetric if we assume the prize is behind another door, the probability of winning if switch is 2/3 This is because the door you pick at first gives a 1/3 chance; the other two doors together though give you a 2/3 chance Since the removed door is always one of the other two, you are left with a 2/3 chance by switching 56 56

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**Wason's Abstract Selection Task**

57 Wason's Abstract Selection Task There a 4 cards, all of which have a letter on one side and a number on the other Two cards show a number (4 and 7), two show a letter (O and G): 4 A 7 G Which card(s) do we need to turn over to test the logical implication: vowel-->odd (if there is a vowel on one side then odd number on other) 57 57

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**W S <18 >18 Adding context (wine) (soda)**

58 There are 4 rooms with closed doors and one person in each room You know one is older than 18, one younger, one drinks wine, and one a soda W (wine) S (soda) <18 >18 Which door(s) do we need to open to make sure nobody under drinking age drinks alcohol? How would you write the problem down in logical notation? 58 58

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Wason Revealed 59 Were your answers to the two questions above equal? Why, or why not? One potential problem lies in the formulation; different formulations of abstract task were only partially effective The most common answer is to turn around only the vowel-->confirmation bias Confirmation biases are very common, also in scientific research (how many white swans do you need to observe to conclude that all swans are white?) 59 59

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Decision theory under uncertainty

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