A Smooth Model of Decision Making under Ambiguity

Source: Klibanoff, P., Marinacci, M. & Mukerji, S. (2005). Econometrica 73(6), 1849–1892.

TL;DR

Proposes and axiomatizes a decision model that cleanly separates ambiguity (a feature of the

agent's information — the set of priors deemed relevant) from ambiguity attitude (a feature of

tastes). An act f is preferred to g iff E_μ φ(E_π u∘f) ≥ E_μ φ(E_π u∘g): take expected utility

under each prior π, apply an increasing transform φ, then average over priors with a subjective

second-order measure μ. Crucially the model is smooth (differentiable indifference curves), unlike

the kinked max-min of Gilboa-Schmeidler, while still accommodating Ellsberg-type behavior.

Savage's Sure-Thing Principle (P2) fails in Ellsberg's paradox when a decision maker is uncertain

about the very odds that apply to payoff-relevant events. How can we represent preferences over acts

so that aversion to this ambiguity (Knightian/model uncertainty) is captured separately from

ordinary risk aversion, and in a tractable, differentiable form usable in economics and finance?

Two-stage evaluation: u is a von Neumann-Morgenstern utility over outcomes; the agent holds a set Π

of priors π he thinks relevant given his information, and a subjective probability μ over Π.

Representation: V(f) = E_μ φ(E_π[u∘f]) = ∫_Π φ(∫ u(f) dπ) dμ(π).

Risk attitude is governed by the shape of u (as usual); ambiguity attitude by the shape of

the second-stage transform φ. φ linear ⇒ ambiguity neutrality (collapses to subjective EU);

φ concave ⇒ ambiguity aversion (dislike of spread in expected utilities across priors).

Constant ambiguity aversion special case: φ(x) = −(1/α)e^{−αx}, with α the coefficient of

ambiguity aversion (parallel to CARA in the risk domain).

Ambiguity is defined behaviorally and characterized by properties of Π, linked to Epstein-Zhang

(2001) and Ghirardato-Marinacci (2002). Maxmin EU (Gilboa-Schmeidler 1989) emerges as the extreme

case of infinite ambiguity aversion.

Reproduces Ellsberg's paradox while keeping smooth indifference curves, so comparative statics in

ambiguity and ambiguity attitude are tractable (the well-developed machinery for risk attitudes

carries over to ambiguity attitudes).

More ambiguity-averse agents (more concave φ) tilt away from acts whose expected utility is

uncertain across priors; the paper offers two illustrative portfolio-choice applications.

A normative/positive decision-theoretic framework rather than an empirical study. It has become a

workhorse for modeling model uncertainty in finance — ambiguity premia, robust portfolio choice,

limited stock-market participation — and connects to the robust-control literature (Hansen-Sargent).

Requires specifying both the set of priors Π and a second-order measure μ — modeling choices that

drive results and are not pinned down by the data.

Empirically disentangling ambiguity aversion (φ) from risk aversion (u) is difficult.

The reduction to subjective EU under ambiguity neutrality means the extra structure only bites when

the second-order measure is non-degenerate.

Klibanoff, P., Marinacci, M. & Mukerji, S. (2005) — A Smooth Model of Decision Making under Ambiguity — Econometrica

Gilboa, I. & Schmeidler, D. (1989) — Maxmin Expected Utility with Non-Unique Prior — Journal of Mathematical Economics

Ellsberg, D. (1961) — Risk, Ambiguity, and the Savage Axioms — Quarterly Journal of Economics

Provenance: verified/generated from the paper's full text.