On the smooth ambiguity model and
Machina’s reflection example
Robert Nau
Fuqua School of Business
Duke University
Machina’s reflection example
For each act there are 3
possible payoffs
f6 and f7 are left-right
reflections of f5 and f8,
respectively.
•
•
•
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., f5  f6 [resp. f6  f5 ] ought to
imply f8  f7 [resp. f7  f8]
Rank-dependent models do not allow this: indifference is required.
But... which of the reflected patterns is more representative of
ambiguity aversion?
• Baillon et al. 2011 show that the following 4 models allow only
one of the two reflected patterns: f5  f6 and f8  f7
– 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
• In experiments conducted by L’Haridon and Placido (2010) a
plurality of subjects exhibited the opposite pattern
– 46% said f6  f5 and f7  f8, contrary to the 4 models
– 28% said f5  and f8  f7
• Baillon et al. 2011: f6  f5 and f7  f8“can be justified in light of
Ellsberg because f6 and f7 assign known probabilities to at least
one outcome (100). Furthermore f6 and f7 are less exposed to
ambiguity than f5 and f8.”
• Recently, Dillenberger and Segal (2014) and Dominiak and Lefort
have presented examples whose parameters can be chosen to
rationalize, f6  f5 and f7  f8, contrary to the other 4 models
• 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.
• Consider the following variation of Machina’s example
involving the same events with different payoffs:
• g1 arguably be preferred because it diversifies exposure to
both risk and ambiguity.
• Even an EU maximizer will prefer g1
• 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
• g3 and g4 are equally risky from the perspective of EU theory, but
the relative exposure to ambiguity is the same as in g1 and g2.
• If you preferred g1 to g2 at least partly on the basis of a desire to
diversify your exposure to ambiguity, shouldn’t you prefer g3 here?
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
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 “2nd-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”?
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 2nd-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...
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)
• 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.
• 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
• Let the balls be renumbered {E11, E12, E21, E22} 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:
• Using the nested form of the utility function, the certainty
equivalents for f1 and f2 are:
• Transparently, f1 is preferred if u is less risk averse than v at
every x, which is true if u(v-1(x)) is convex.
• Now consider Machina’s example, which involves 3 distinct
payoffs:
• Suppressing the probabilities (because they are uniform) and
the outer u-1 function (because it is increasing), the
comparison of f5 and f6 becomes equivalent to a comparison of
the following two expressions:
• Letting w(x) = u(v-1(x)), this becomes:
• Again, the comparison of f5 vs. f6 reduces to a comparison of the
following expressions (bigger = better), where w(x) = u(v-1(x)):
• This yields f5  f6 and f8  f7 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.
• This result is intuitive if you take a closer look at the payoff table:
• f5 and f8 diversify the exposure to ambiguity so that it depends on
the unknown proportions of differently-labeled balls in both urns.
• f6 and f7 “bet the farm” on just one of the two sources.
• This is the intuition behind all the models that favor f5 and f8
– Not paradoxical at all to me!
• 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.