NOTE: CONTENT IS BEING MIGRATED OVER TO CHALK. THIS WEBSITE IS AVAILABLE FOR THOSE WHO DO NOT HAVE ACCESS TO CHALK AND MAY NOT BE FULLY UP TO DATE. THIS SESSION WILL BE REVISED.
Week 6: Decision under Risk and Uncertainty
Kahneman, Daniel and Amos Tversky (1979). “Prospect theory: An analysis of decision under risk.” Econometrica 47(2): 263-291. [This is the second most cited paper in economics in the last 30 years (with 6,164 Web of Science cites and 16,164 Google Scholar cites as of October 28, 2010).]
Tversky, Amos and Daniel Kahneman (1992). “Advances in Prospect Theory: Cumulative Representation of Uncertainty.” Journal of Risk and Uncertainty 5: 297-323. [This paper revises prospect theory in some substantial ways, as well as provides a more thorough empirical investigation of decision under risk.]
Gonzalez, Richard and George Wu (1999). “On the shape of the probability weighting function.” Cognitive Psychology 38: 129-166. [This paper provides a thorough empirical investigation and psychological interpretation of one component of prospect theory, the probability weighting function.]
Rottenstreich, Yuval and Christopher. K. Hsee (2001). “Money, kisses, and electric shocks: On the affective psychology of risk.” Psychological Science 12(3): 185-190. [This paper demonstrates an interesting violation of "probability-outcome independence", a property assumed by expected utility theory as well as porpsect theory.]
Comprehension Questions
1. The idea of expected utility has been around since Bernoulli (1738). However, it was not until 50 years ago that an axiomatic foundation for expected utility was discovered (von Neumann-Morgenstern).
a. What is an axiomatic system?
b. Why is a set of axioms useful for empiricists?
c. Consider the set of axioms for expected utility. What axiom is being violated by the common-consequence effect (Allais Paradox; Problems 1 and 2 in KT 1979) and the common-ratio effect (Problems 3 and 4 in KT 1979)?
d. Consider the two forms of prospect theory: Original Prospect Theory (OPT; KT 1979) and Cumulative Prospect Theory (CPT; TK 1992). How can these models accommodate the common-ratio and common-consequence effect outlined in 1c?
2. Prospect theory challenges some of the main tenets of economic theory.
a. What are the explicit and implicit tenets of the classical theory of decision under risk?
b. For each tenet, make sure that you understand the main demonstration that that tenet is not descriptively adequate.
3. TK 1992 propose that choice in risky situations follows a four-fold pattern (risk aversion for small probability losses and medium and large probability gains; risk seeking for small probability gains and medium and large probability losses). Assume that choices for simple prospects (p,x) can be evaluated by (5) and (6) in TK 1992 with the parameters g=.61 (gains) and .69 (losses), and a=.88 for gains and losses. What values of p give risk seeking for gains, and risk aversion for losses?
4. The rank-dependent representation used in CPT is a bit elusive. Write down how to evaluate the following gambles using CPT:
a. (.05,$100 ; .10,$50 ; .05,$25; .80,$0)
b. (.01,$99 ; .01,$98; ... ; .01,$0)
c. ("Democrats win House and Senate",$100; "Democrats win House, but not Senate",$50 ; "otherwise",$0), where the events concern the results of the 2012 National Election. d. (E1,$99 ; E2,$98, ...; E100,$0), where Ei, i=1,...,100 are possible events or states of nature
5. Gonzalez & Wu (1999) talk about attractiveness and discriminability as psychological features. How do these map into elevation and curvature of the probability weighting function, and how does the two-parameter (linear in log odds) form used by Gonzalez & Wu capture these dimensions?
Thought Questions (asterisks refer to degree of difficulty with more asterisks meaning more difficult)
*1. Articulate the psychology behind the common-ratio and common-consequence effects in non-technical terms. Does this psychology cohere well with the empirically-demonstrated functional form of OPT? with the functional form of CPT? If so, how? If not, why not?
**2. Consider the editing rules presented at the start of Section 3 in KT 1979. One view is that these editing rules were designed to eliminate violations of dominance (that OPT might generate). To what extent are editing rules still needed under CPT?
***3. What is missing from these models? In other words, what kind of risky choice behavior can these models not capture? Can they be rescued or do we need a different sort of model?
**4. The "data" used in testing choice models typically takes one of two forms: choice data (e.g., percentage of subjects choosing A over B) and cash equivalent data (e.g., the median or mean cash equivalent for a particular gamble). In what sense are these data equivalent? Would we expect to reach the same conclusions using choice data as we would using cash equivalence data? When might these data differ in terms of what conclusions are drawn?
*5. Prospect theory models are tested in various ways: direct tests of axioms (e.g., independence axiom) and parametrically through estimation. What do axiom tests show that parametric tests cannot? What do parametric tests show that axiom tests cannot? Suppose a model fails axiom tests. Does that mean we should abandon the model?
***6. Consider the individual differences found in Gonzalez & Wu. There is quite a lot of heterogeneity across subjects. Presumably, there is also variation in the weighting function for different types of gambles. Gonzalez & Wu speculate on some psychological factors that could generate intrapersonal variation. Can you think of some ways that Gonzalez & Wu don't address?
***7. OPT and CPT both seem to work on the basic "paradoxes". How would you test between these models?
**8. Why do the affect-rich outcomes lead to an increase in the possibility effect? An increase in the certainty effect? At the end of their paper, Rottenstreich and Hsee contrast a "hope and fear" account with a "dread and savoring" account. Their data leans toward the first account. Do you think their story applies widely? Are there any situations in which affect might lead to elevation differences, rather than curvature differences?