Lars Peter Hansen Prize Lecture: Uncertainty Outside and Inside Economic Models

430 

The Nobel Prizes

deduce bounds on the expected utility consequences of alternative decisions, 

and more generally a mapping from alternative priors into alternative expected 

outcomes. Building on discussions in Walley (1991) and Berger (1994), there 

are multiple reasons to consider a family of priors. This family could represent 

the views of alternative members of an audience, but they could also capture 

the ambiguity to a single decision maker struggling with which prior should be 

used. Ambiguity aversion as conceived by Gilboa and Schmeidler (1989) and 

others confronts this latter situation by minimizing the expected utility for each 

alternative decision rule. Max-min utility gives a higher rank to a decision rule 

with the larger expected utiltiy outcome of this minimization.

38

Max-min utility has an extension whereby the minimization over a set of 

priors is replaced by a minimization over priors subject to penalization. The 

penalization limits the scope of the prior sensitivity analysis. The penalty is mea-

sured relative to a benchmark prior used as a point of reference. A discrepancy 

measure for probability distributions, for instance some of the ones I discussed 

previously, enforce the penalization. See Maccheroni et al. (2006) for a general 

analysis and Hansen and Sargent (2007) for implications using the relative en-

tropy measure that I already mentioned. Their approach leads to what is called 

variational preferences.

For either form of ambiguity aversion, with some additional regularity con-

ditions, a version of the Min-Max Theorem rationalizes a worst-case prior. The 

chosen decision rule under ambiguity aversion is also the optimal decision rule 

if this worst-case prior were instead the single prior of the decision maker. Dy-

namic counterparts to this approach do indeed imply a martingale distortion 

when compared to a benchmark prior that is among the set of priors that are en-

tertained by a decision maker. Given a benchmark prior and a dynamic formu-

lation, this worst-case outcome implies a positive martingale distortion of the 

type that I featured in Section 5. In equilibrium valuation, this positive martin-

gale represents the consequences of ambiguity aversion on the part of investors 

inside the model. This martingale distortion emerges endogenously as a way to 

confront multiple priors that is ambiguity averse or robust. In sufficiently simple 

environments, the decision maker may in effect learn the model that generates 

the data in which case the martingale may converge to unity.

There is an alternative promising approach to ambiguity aversion. A deci-

sion theoretic model that captures this aversion can be embedded in the analy-

sis of Segal (1990) and Davis and Pate-Cornell (1994), but the application to 

38 

See Epstein and Schneider (2003) for a dynamic extension that preserves a recursive 

structure to decision making.

6490_Book.indb   430

11/4/14   2:30 PM

---

Uncertainty Outside and Inside Economic Models 431

ambiguity aversion has been developed more fully in Klibanoff et al. (2005) 

and elsewhere. It is known as a smooth ambiguity model of decision making. 

Roughly speaking, distinct preference parameters dictate behavior responses 

to two different sources of uncertainty. In addition to aversion to risk given a 

model captured by one concave function, there is a distinct utility adjustment 

for ambiguity aversion that emerges when weighting alternative models using 

a Bayesian prior. While this approach does not in general imply a martingale 

distortion for valuation, as we note in Hansen and Sargent (2007), such a distor-

tion will emerge with an exponential ambiguity adjustment. This exponential 

adjustment can be motivated in two ways, either as a penalization over a fam-

ily of priors as in variational preferences or as a smooth ambiguity behavioral 

response to a single prior.

6.2  unknown Models and ambiguity aversion

I now consider an approach with an even more direct link to the analysis in 

Section 5. An important initiator of statistical decision theory, Wald (1939), ex-

plored methods that did not presume a priori weights could be assigned across 

models. Wald (1939)’s initial work generated rather substantial literatures in sta-

tistics, control theory and economics. I am interested in such an approach as a 

structured way to perform an analysis of robustness. The alternative models rep-

resented as martingales may be viewed as ways in which the benchmark prob-

ability model can be misspecified. To explore robustness, I start with a family of 

probability models represented as martingales against a benchmark model. Dis-

crepancy measures are most conveniently expressed in terms of convex func-

tions of the martingales as in Section 5. Formally the ambiguity is over models, 

or potential misspecifications of a benchmark model.

What about learning? Suppose that the family of positive martingales with 

unit expectations is a convex set. For any such martingale M in this set and some 

0 < ω < 1, construct the mixture ωM + (1 – ω) is a positive martingale with unit 

expedations. Notice that

 

ωM

t

+

τ

+(1−

ω)1

ωM

t

+(1−

ω)1

=

ωM

t

M

t

+

τ

M

t

⎛

⎝⎜

⎞

⎠⎟

+(1−

ω)1

ωM

t

+(1−

ω)1

.

 

The left-hand side is used to represent the conditional expectations operator 

between dates t + τ and t. If we interpret ω as the prior assigned to model M and 

(1 – ω) as the prior assigned to a benchmark model, then the right-hand side 

6490_Book.indb   431

11/4/14   2:30 PM

---