` Online Course Risk and Ambiguity Here my photo should be

Online Course Risk and Ambiguity

Peter P. Wakker
Erasmus University Rotterdam


This is an online course on decisions under risk & ambiguity (behavioral approach).


Recordings of independent interest: can be watched independently, without taking rest of course. Topics: survey on ambiguity theories, normative discussion of Allais paradox, Ellsberg paradox, Rabin paradox, reference dependence paradox, bookmaking/no-arbitrage, intuitive eplanation of Quiggin's invention of rank dependence, prescriptive application of expected utility, pessimism & insensitivity in probability weighting


Regarding the course:


Material used


Specification of material to be studied

The material to be studied is specified in detail below, organized as homework for seven meetings. For each meeting the corresponding recordings/lectures give explanations. Sections, exercises, and so on refer to the textbook unless noted otherwise.

1st meeting

Topics: decision under uncertainty (unknown probabilities); you learn to read people’s mind: measuring their subjective probabilities. Normative & descriptive perspective. de Finetti’s book making gives a rational basis to subjective probabilities, Bayesian statistics, no-arbitrage in finance, and shows that you should cancel all your small insurances, increasing your income by 0.5% for the rest of your life.

Material (homework for 2nd meeting): Preface and Introduction (read once). Study §§1.1-1.3, skip §1.4, study §§1.5-1.8. Read §1.9 once. Please pay attention to the Structural Assumption 1.2.1 (p. 17). It will be assumed throughout the course.
Exercises (with the superscript system understood): 1.2.1 (p. 16), 1.2.2, 1.2.3 (also useful to try for b and a students), 1.2.4, 1.3.1 (p. 17), 1.3.2, 1.3.3, 1.3.4, 1.3.5, 1.4.1, 1.5.1 (p. 25), 1.6.1 1.6.5, 1.6.6, and Assignment 1.6.8. A rule of the game is that for each exercise and assignment you can use every result in the book up to that point. This holds in particular for Assignment 1.6.8. For volunteers: Assignment 1.6.9—writing your own proof.

Recordings of 1st meeting


2nd meeting

Topics: decision under risk (known probabilities); you learn to read people’s heart: measuring their subjective utilities; ; the most useful classical tools: independence for rationality, indexes of risk aversion, risk premium, tractable utility functions, and applications in health economics and management science. Your risk and ambiguity attitude will be measured in an experiment (analyzed in later meetings). This course is constructive in the sense that every concept is directly linked with its empirical meaning. Thus, after this meeting, you will better “feel” what utility means.

Material (homework for 3rd meeting): study Ch. 2 §2.1 up to and including Assumption 2.1.2 but skip the rest of §2.1, §2.2, regarding §2.3 only take for granted that subjective probabilities must usually agree with objective ones if the latter are available and then can skip §2.3 (a homework exercise will btter deal with this point), 2.4, 2.5, 2.6, 2.7 (independence is an important condition), Ch. 3 §§3.1, 3.2, 3.3, 3.4, 3.5. Ch. 4 §4.1.   Some may know the material of Ch. 3 from other courses, e.g. microeconomics, and then it is easy. For others it is new and then it will be much (but useful!) work.
Exercises: Extra Exercise 2.3.3 (extra exercises are not in the book, but in the file “Extra Exercises and Assignments” in the Auxiliary Material linked above), 2.4.1((e)-(h) p. 51; write all answers (e.g., in book) and keep them readily available because we use them later in other meetings), 2.5.3 (p. 55), 2.5.5, 2.6.4 (important), 3.1.1 (p. 71), 3.3.1 (p. 74), 3.3.3 only for volunteers (but useful, being the only modeling exercise in this course).
Fill out Figures 4.1.1 - 4.1.5 (p. 96 ff.). Remember, there is no right or wrong, and only your preference counts. Again, write all answers (e.g., in book) and keep them readily available because we use them later.

Recordings of 2nd meeting


3rd meeting

Topics: simple analysis of your experimental choices. Classical theories are violated. Quiggin’s invention of first behavioral model could be defined in five minutes. Instead, we take an hour so that you “feel” the formula and the psychology captured in the maths.

Material (homework for 4th meeting): of Ch. 4, study: §§4.2-4.3, 4.8.1, 4.11, 4.12. Study all Ch. 5. Although most recordings are only additional explanations of the texts in the book and in principle can be skipped, recording 3.16.Sec.4.12_Allais-pardx_ra.20mins, discussing the Allais paradox and its violation of independence, provides arguments not in the book and, from this perspective, deserves more attention.
Exercises: 4.2.1 (p. 101), 4.2.3, 4.2.7, 4.3.1-4.3.5 (p. 103 ff) (When elaborating, it is convenient to write the notation a1, a2, p1, p2, and so on, rather than to write the particular numbers that you chose. Using the recommended notation, you can readily compare your elaborations with those of others and with the elaborations of the teacher/book.), 4.8.1 (p. 115), 4.8.3, 4.8.4 (Eq. 4.8.2 you did not study, but this just concerns the result of Exercise 4.8.2), 4.12.1 p. 133, 5.1.1 (p. 148). 5.4.3 (p. 159), 5.6.1 (p. 167), 5.6.2, Assignment 5.6.3.
      This meeting your homework is different for a, b, and c students. For the empirical analysis of the data file, a-students do most work, b-students less, and c-students the least. (c-students do more theoretical work.) Hence, it is time to decide what student you want to be.
      Further homework: data Analysis: Table 4.11.2 (p. 132) gives a statistical analysis of data of 1996. Now a- and b-students are asked to redo this table for our data set. Our data set (11 students of 2009) can be found in the Auxiliary Material linked above, with filename dataset.txt. Give statistics and p-values. Where directional hypotheses were put forward (H1: Xj > aj or H1: Xj < aj or H1: PEj > j/4), you can use one-sided tests. b-students: do it only once, using a Wilcoxon rank-signed test. a-students: do it twice: (1) Using a t-test. (2) Using a t-test, but not on the raw data. Instead, renormalize the money amounts, by replacing m by (m-a0)/(a4-a0) for all money amounts m. Those money amounts m are the aj's, the bj's, the gj's, and the dj's. Thus, for instance, a0 is turned into 0 and a4 is turned into 1. The normalized b4 will exceed 1 if and only if the nonnormalized b4 exceeds the nonnormalized a4. Whenever you have a significant difference, indicate which value is bigger (i.e., the direction of your one-sided alternative hypothesis).
      Those who feel insufficiently challenged intellectually can put their teeth in the assignments on Ch. 4 in the extra exercises in the Auxiliary Material linked above.

Recordings of 3rd meeting


4th meeting

Topics: fully formalized definition of behavioral risk theory. You learn how its mathematics can be used to capture behavioral phenomena (optimism, pessimism, insenitivity). As throughout this course, every theoretical concept is immediately linked with its empirical meaning.

Material (homework for 5th meeting): study §§6.1-6.4, §7.1, §7.7 (below Eq. 7.7.6 is a remark on loss ranks, which will only be defined next meeting; skip that remark for now), §7.8 (read once), §7.9. For volunteers: §7.12 explains why “cavex” (first concave and then convex) has problems, which I suggested in a recording but did not fully explain. But you need not read this.
Exercises: 6.3.1 (p. 175), 6.4.1 (p. 177), 6.4.3.a-c, (6.4.4 is for volunteers interested in finance), 6.6.1, 7.4.1.
      In the Auxiliary Material linked above there is a link to a medical application, which in turn gives a link to a medical paper about which there will be homework in later meetings. For now, there is also a link to a file giving 1/10th of the decision tree used in the analysis in that medical application. Open that file (of jpg type), and enlarge it so much that you can see the (Dutch) terms. Thus, you can check out that real decision trees are bigger than the simplified decision tree in Figure 3.1.1.

Recordings of 4th meeting


5th meeting

Topics: actual empirical performance of behavioral risk theory. You continue to see how the maths of rank dependence can be used to capture behavioral phenomena, but now empirically realistically. This includes classical topics of the 3rd meeting, including your own performance during the experiment then. New indexes and tractable functionals.

Material (homework for 6th meeting): study §6.5 first 5 lines (p. 181), §6.5.1 p. 181 & 182; §§6.5.4, 6.9 until p. 200 line 3 (moral; for calculations, get the distribution functions; they automatically handle the ranking of outcomes), 7.2 (study only the neo-additive family, and read the rest once so that you can find those families back when useful), 7.4, 7.6, 7.10.
Exercises: 6.5.1 (p. 182), 6.4.2 (p. 178; best after 6.5.1), 6.5.6 (p. 188), 6.5.2 (p. 182; best done after Exercise 6.5.6). 7.2.2 (p. 210), 7.8.2 (p. 226), Extra Assignment 7.10.3 (in the Auxiliary Material linked above) and only then Exercise 7.10.1, Extra Assignment 7.2.5. For volunteers: the extra assignments for Ch. 6. (Will not be discussed.)

Recordings of 5th meeting


6th meeting

Topics: the Nobel-awarded prospect theory (Kahneman & Tversky) adds refinements to Quiggin’s theory, including loss aversion. The miraculous point is that every mathematical detail in the formulas is based on psychological insights. Then Ellsberg’s paradox (≈ homebias), ambiguity (unknown probabilities) aversion, and Gilboa-Schmeidler ambiguity theories. Ambiguity phenomena analogous to risk.

Material (homework for 7th meeting): Ch. 8 intro (pp. 234-235), §§8.1-8.6 (can skip literature discussion on pp. 244-245, although some may enjoy these kinds of fierce debates), can voluntarily read §8.7 that was discussed in a recording, §§9.1, 9.2, 9.3 up to and not including p. 257 and of the rest only Example 9.3.3, Observation 9.4.1, §9.4.2, §9.5. Read Appendix 9.8 once diagonally, mostly to know that the original 1979 prospect theory used somewhat different formulas, and only considered at most two nonzero outcomes. Part III & Ch. 10 intro (pp. 277-279), §10.1, 10.2, 10.3.1.
Exercises: 9.3.4 (p. 257), 9.4.2 (p. 262, Extra Exercise 9.5.3 (in the Auxiliary Material linked above), Exercise 10.2.1 (p. 284). A take-home exercise, which is only for a- and b-students, is in the Auxiliary Material, Medical application. In this online presentation there is no feedback on your elaborations, but it may still interest you to see into a large-scale application of decision theory.

Recordings of 6th meeting


7th meeting

Topics: new ambiguity concepts, not occurring for risk (hottest topic in decision theory in early 2020s). Applications. Empirical findings. Recent theories.

Material (last homework): §§10.4.1, 10.4.2, 10.4.3 (compare Figure 10.4.1 to Figure 7.4.1 on p. 215 to recognize the similarity, with events instead of probabilities), read Theorem 10.5.6 (p. 297) once just to know that there exists a preference foundation, Ch. 11 intro (p. 317), §11.1, 11.2, 11.8 only Procedure 11.8.1 (so, half a page). Of Ch. 12 (PT for uncertainty), all you have to know is that PT generalizes RDU for uncertainty similarly as it did for risk, with a different weighting function W- for losses than W+ for gains, and reference dependence and loss aversion coming in. In a recording I showed Figures 10 and 11 of Abdellaoui et al. (2011), which are added as file in the Auxiliary Material.

Material after 2010 (also last homework): recordings 7.8-7.12 present post-2010 material, which is not in the 2010 book. This material concerns ambiguity theories. Decision under risk has quite settled, with prospect theory most popular for advanced empirical work, and expected utility and expected value most popular for normative work and also for pragmatic simple empirical work. Decision under ambiguity did not yet reach such a state in the early 2020s, with many theories competing and final popularity still to be settled. The textbook presented an early version of my preferred descriptive theory for ambiguity: prospect theory, which is further developed in the early 2020s as source theory. But many other theories have been considered in the literature. An opiniated survey is in recordings 7.8-7.12, with material not in the book. The corresponding "extra" slides can be found in the Auxiliary Material.
Exercises: 10.5.3 (p. 295; nonnull means that all decision weights are positive), 10.5.4.

Recordings of 7th meeting