1. On the smooth ambiguity model and Machina’s reflection example Robert Nau Fuqua School of Business Duke University

2. Machina’s reflection example If the decision maker is not indifferent among all the alternatives, then, by symmetry, preferences among the first two and last two alternatives ought to be reflected, i.e., f 5  f 6 [resp. f 6  f 5 ] ought to imply f 8  f 7 [resp. f 7  f 8 ] Rank-dependent models do not allow this: indifference is required. But... which of the reflected patterns is more representative of ambiguity aversion? For each act there are 3 possible payoffs f 6 and f 7 are left-right reflections of f 5 and f 8, respectively.

3. Baillon et al. 2011 show that the following 4 models allow only one of the two reflected patterns: f 5  f 6 and f 8  f 7 – Maxmin expected utility (Gilboa-Schmeidler) –  maxmin expected utility (Ghirardato et. all 2004) – Variational prefererences (Maccheroni et al. 2006) – Smooth model (Klibanoff et al. 2005, Nau 2001, 2006) Should the opposite pattern be allowed, or even required? If so, it might considered as a “paradox” for those models

4. In experiments conducted by L’Haridon and Placido (2010) a plurality of subjects exhibited the opposite pattern – 46% said f 6  f 5 and f 7  f 8, contrary to the 4 models – 28% said f 5  and f 8  f 7 Baillon et al. 2011: f 6  f 5 and f 7  f 8 “can be justified in light of Ellsberg because f 6 and f 7 assign known probabilities to at least one outcome (100). Furthermore f 6 and f 7 are less exposed to ambiguity than f 5 and f 8.” Recently, Dillenberger and Segal (2014) and Dominiak and Lefort have presented examples whose parameters can be chosen to rationalize, f 6  f 5 and f 7  f 8, contrary to the other 4 models

5. Baillon et al. do also observe that the opposite pattern might be justified on the basis of aversion to mean-preserving spreads in utility. In what follows, I will argue aversion to mean-preserving spreads in the presence of multiple sources of ambiguity is rather compelling. A simple example of the smooth model will be used for demonstrations, but it captures the intuition of the other models as well.

6. Consider the following variation of Machina’s example involving the same events with different payoffs: g 1 arguably be preferred because it diversifies exposure to both risk and ambiguity. Even an EU maximizer will prefer g 1

7. Now add y to the payoff in the first two columns, which increases the objective expected payoff of both acts by ½y: These are the same choices as Machina’s example when x = y = $4000 g 3 and g 4 are equally risky from the perspective of EU theory, but the relative exposure to ambiguity is the same as in g 1 and g 2. If you preferred g 1 to g 2 at least partly on the basis of a desire to diversify your exposure to ambiguity, shouldn’t you prefer g 3 here?

8. KMM version of smooth model (2005) The probability measure  represents first-order risk The utility function u represents aversion to risk The probability measure  represents ambiguity (uncertainty about first-order risk) The utility function  represents aversion to ambiguity

9. Properties of KMM model Separates risk from risk attitude and separates ambiguity from ambiguity attitude, by analogy with SEU model In principle, requires elicitation of preferences for “2 nd -order acts”: mappings from hypothetical probability distributions to consequences. In Machina’s example, or even Ellsberg’s 2-urn problem, what second-order distribution would you use to represent your perception of ambiguity about the proportion of red balls in the unknown urn? – Binary? Uniform? Binomial...? – Was the experimenter “out to get me”?

10. My version (2001, 2006) The setting is the general state-preference framework with arbitrary smooth indifference curves in finite-dimensional state space, represented by an ordinal utility function U(x) Acts are mappings from states to amounts of money—no counterfactuals. In general, does not separate beliefs, states, and background risk Observable parameters of preferences (up to 2 nd -order effects) are the vector of local risk neutral probabilities (betting rates on states, denoted by  ) and its matrix of derivatives (denoted by D  ) Risk premium of a neutral act z is -½ z  D  z This quadratic form generalizes the Arrow-Pratt measure to settings involving background risk state-dependence, non-EU...

11. My version (2001, 2006) Ambiguity aversion is manifested as source-dependent risk aversion, as determined by the structure of D  Easy for the decision maker to choose a functional form for U(x) to represent a simple situation such as Ellsberg’s 2-run problem: – By symmetry, probabilities of all 4 states are the same. – The utility function can be given a nested form involving 2 Bernoulli utility functions that measure attitudes toward the 2 sources of risk. – Preferences among simple bets reveal which source of risk is more aversive (interpretable as more ambiguous)

12. The following functional form yields EU preferences separately for the bets on only one of the two sources of uncertainty:...but the decision need not exhibit the same risk attitude toward both.

13. If the certainty-equivalent form is used for the utility functions, the two Bernoulli utility functions u and v both have $$ as their arguments and if is easy to compare the source-dependent risk aversion that they induce: If u(v -1 (x)) is convex [concave] then the decision maker is more averse to the second [first] source of risk

14. Let the balls be renumbered {E 11, E 12, E 21, E 22 } respectively Ellsberg’s 2-urn experiment can be replicated with Machina’s single urn, by allowing only pure bets on the first digit or on the second digit:

15. Using the nested form of the utility function, the certainty equivalents for f 1 and f 2 are: Transparently, f 1 is preferred if u is less risk averse than v at every x, which is true if u(v -1 (x)) is convex.

16. Now consider Machina’s example, which involves 3 distinct payoffs:

17. Suppressing the probabilities (because they are uniform) and the outer u -1 function (because it is increasing), the comparison of f 5 and f 6 becomes equivalent to a comparison of the following two expressions: Letting w(x) = u(v -1 (x)), this becomes:

18. This yields f 5  f 6 and f 8  f 7 for all positive x and y if w(x) is convex, which is true if v(x) is more risk averse than u(x), i.e., if the decision maker is more averse toward the ambiguous source of risk than toward the unambiguous one. The opposite pattern is obtained if w(x) is concave. If w(x) is neither convex nor concave, there is not a conclusive identification of one source as being more averse, and the specific values of x and y would matter. Again, the comparison of f 5 vs. f 6 reduces to a comparison of the following expressions (bigger = better), where w(x) = u(v -1 (x)):

19. This result is intuitive if you take a closer look at the payoff table: f 5 and f 8 diversify the exposure to ambiguity so that it depends on the unknown proportions of differently-labeled balls in both urns. f 6 and f 7 “bet the farm” on just one of the two sources. This is the intuition behind all the models that favor f 5 and f 8 – Not paradoxical at all to me!

20. Why did a plurality (not a decisive one!) exhibit the opposite preferences? My conjecture: the example is tricky and the diversification angle is hard to see at first, precisely because it involves thinking of two sources of ambiguity at once.