Introduction à la théorie de la décision Ferdinand

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

What is Decision Theory? 2 • 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 2 - social psychology

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 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

Lecture Overview 5 • 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 5 puzzles (Wason, Monty Hall)

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

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

Descriptive Issues 8 • 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 8 information search and processing

9 Prescriptive Analysis • Prescriptive analysis moves from a limitedinformation 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 The origins of decision theory • Historically, the concept of probabilities and how to deal with them is rather recent. • In the 1600 s, 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

Example: EV normative? Choice between 2 known-probability events: 0. 9 DT 10 DT 0 0. 1 EV: 0. 9*10+0. 1*0=9 < 0. 2 DT 50 DT 0 0. 8 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 11 the amounts *1000, to 10, 000 & 50, 000 DT? 11

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 Example: EV versus EU Choice between 2 known-probability events: 0. 3 DT 400 DT 0 0. 7 EU: 0. 3*u(400)+0. 7*0=0. 3 ? 0. 5 DT 200 DT 0 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 Eliciting Utilities How can we elicit the missing utility? p CE DT 400 ~ 1 -p DT 0 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 Example reconsidered: Choice between 2 known-probability events: 0. 5 € 200 0. 3 € 400 ? 0. 5 € 0 0. 7 € 0 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 Subjective Utility and Risk • Given the non-linearity in the utility function, preferences can change relative to EV U(€) EV EU € • EUT: concavity=risk aversion. This is not universally valid! 16

17 EV or EU? • EV is reasonable for small stakes, however most important decisions deal with large stakes • Also, many important decisions deal with nonquantitative 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

The St. Petersburg Paradox 18 • 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 DT 2, then DT 4, then DT 8, so that if head comes up at the kth flip you get DT 2 k • 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

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 p p*X+(1 -p)*Y X ? 1 -p Y • ¡Do not confuse risk aversion with concave 19 U!

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

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

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

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)

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 What is the risk premium here? DT 25

Insurance example 26 • There is a 5% risk that your house may be flooded, potential damages are – DT 100, 000 • EV = – DT 5, 000, However, if you are risk averse, the CE is lower, e. g. CE = – DT 6000 • There is a positive risk premium of DT 1000; by the law of large numbers, the insurance will pay DT 5000 on average, and can thus make up to DT 1000 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 26 when not?

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 CE = –€ 100000 p*x+(1 -p)*y y=0 DT 27

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

Typical Risk Preferences 29 • People are typically risk seeking for small probabilities (± p<0. 15): lottery play • For larger probabilities, people tend to be risk averse: CE

30 Descriptive Theories of Choice: Prospect Theory 30

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

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 11 46 68 X 32

Prospect Theory 33 • 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: 33 PU=w(p)*u(x)+(1 -w(p))*u(y)

Probability Weighting: Attitudes to Risk 34 • 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 Probability Weighting Function 35

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 36 stakes

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

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

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

Loss Aversion 40 • Under loss aversion, “losses loom larger than gains” 0. 5 DT ? 0 ~ – DT 50 0. 5 How high would the gain need to be to make you indifferent between playing and not playing the prospect? 40

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*100+0. 5*(–λ)*50 λ*25=50 λ=2 What other assumption underlies this elicitation of the loss aversion parameter λ? 41

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 α w(p)= exp(-ξ (-log p) 42

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 43 losses looming larger

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 44 not express true (average) preferences

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

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

Ambiguity Aversion 47 You are asked to choose between two urns, one 50: 50, one unknown proportion 20–? colors of ? 10 R 10 B 20 R & B in unknown proportion • 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

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 • Uncertainty has generally been studied in opposition to risk, not in its own right • Also: Ellsberg has created strong focus on 5050 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

Typical Source functions 50 50

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

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

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

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

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

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

Wason's Abstract Selection Task 57 • 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 57 a vowel on one side then odd number on other)

Adding context 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 <18 W (wine) >18 S (soda) 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 58 notation?

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