Higher-Order Ambiguity Attitudes

TL;DR

The paper introduces a model-free ambiguity preference and analyzes its one-shot and repeated applications to yield ambiguity aversion (second order) and ambiguity prudence (third order). It establishes rigorous links between these higher-order attitudes and the curvature (second and third derivatives) of capacities across Choquet Expected Utility, Variational/ Divergence preferences, Maxmin, and Smooth Ambiguity models, providing both necessary and sufficient conditions. It shows that ambiguity prudence corresponds to a positive third derivative in several frameworks and that ambiguity aversion aligns with a positive second derivative; it also connects ambiguity attitudes to an optimal insurance problem under ambiguity. The results offer a unified, testable lens for understanding how model uncertainty influences hedging, insurance demand, and disaggregation of risks in diverse decision theories, with implications for experimental validation and practical decision-making under ambiguity.

Abstract

We introduce a model-free preference under ambiguity, as a primitive trait of behavior, which we apply once as well as repeatedly. Its single and double application yield simple, easily interpretable definitions of ambiguity aversion and ambiguity prudence. We derive their implications within canonical models for decision under risk and ambiguity. We establish in particular that our new definition of ambiguity prudence is equivalent to a positive third derivative of: (i) the capacity in the Choquet expected utility model, (ii) the dual conjugate of the divergence function under variational divergence preferences and (iii) the ambiguity attitude function in the smooth ambiguity model. We show that our definition of ambiguity prudent behavior may be naturally linked to an optimal insurance problem under ambiguity.

Figures (5)

• Figure 1: Ambiguous insurance Notes: This figure illustrates the certainty equivalent $c$ (full insurance) and the acts $X$ (no insurance) and $Y$ (ambiguous insurance), where $\ell =$ € $2\mathord{,}000$.
• Figure 2: Ambiguous insurance with a large loss Notes: This figure illustrates the generalized acts $c(w_{0},K)$, $X(w_{0},K)$ and $Y(w_{0},K)$, where $\ell =$ € $2\mathord{,}000$ and $K\gg \ell$.
• Figure 3: Ambiguity averse neo-additive capacities. Notes: This figure plots two ambiguity averse neo-additive capacities. Theorems \ref{['th:CEU2nd-i']} and \ref{['th:CEU2nd-ii']} yield that a CEU DM is ambiguity averse if and only if the capacity is supermodular. To ensure the supermodularity of the neo-additive capacity, the capacity must start in the origin. This requires the equality of the parameters $a$ and $b$, as stated in Theorem \ref{['thm:ambavr-NCEU']}.
• Figure 4: Ambiguity prudent neo-additive capacities. Notes: This figure plots three neo-additive capacities with different parameters. Theorems \ref{['th:CEU3rd-i']} and \ref{['th:CEU3rd-ii']} yield that a CEU DM is prudent if and only if the third derivative of the capacity is non-negative. From Theorem \ref{['thm:ambprud-NCEU']}, we know that this is always satisfied for neo-additive capacities, also when the capacity does not start in the origin and even when the capacity is concave, as in the figure.
• Figure 5: Ordered optimal probabilities $i \mapsto p^\ast_{(i)}$ Notes: This figure plots the argmin $p^\ast$ of \ref{['eq:VPU3']}, where the $p^{\ast}_{(i)}$'s are coordinates of the vector $p^{\ast}$ in descending order. In Theorem \ref{['th:VP-prudent']}, it is proven that the VP DM is ambiguity prudent when the map $i\mapsto p^\ast_{(i)}$ is convex. This plot is an illustration of a convex map that makes the DM ambiguity prudent.

Theorems & Definitions (46)

• Lemma 2.2
• Definition 3.1
• Definition 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Remark 4.5
• Remark 4.6
• Theorem 4.7