Comonotone Pareto optimal allocations for law invariant robust utilities on \(L^1\)
The authors prove the existence of comonotone Pareto optimal allocations satisfying utility constraints when decision makers have probabilistic sophisticated variational preferences and thus, representing criteria in the class of law invariant robust utilities. \textit{F. Maccheroni} et al. [Econometrica 74, No. 6, 1447--1498 (2006; Zbl 1187.91066)] first introduced variational preferences and axiomatically characterized them. The connection between variational preferences and robust utilities as representing choice criteria of variational preferences on random endowments corresponding to Savage acts was found in [\textit{A. Schied} et al., in: Handbook of numerical analysis. Vol XV. Special Volume: Mathematical modeling and numerical methods in finance. Amsterdam: Elsevier/North-Holland. 29--87 (2009; Zbl 1180.91274)]. The class of law invariant robust utilities which represent probabilistic sophisticated variational preferences are of the type \[ {\mathcal U}(X)= \inf_{Q\in {\mathcal Q}}( E^Q(u(X))+\alpha (Q)),\quad X\in L^1, \] where \(u: {\mathbb R} \to {\mathbb R}\cup \{-\infty\}\) is a concave increasing utility function, \({\mathcal Q}\) is a set of probability measures which is closed under densities with the same distribution, and \(\alpha (Q)\) is a suitable law invariant penalization on \(Q\in {\mathcal Q}\). The existence of Pareto optimal allocations has so far only been established for a few subclasses of law invariant robust utilities, and for the \(L^1\) condition from Yaari preferences represented by the robust utility \[ {\mathcal U}_1(X) = \frac{1}{\alpha} \int_0^{\alpha} q_X(s) ds,\quad X\in L^1, \] for the quantile function \(q_X(s) = \inf\{x: P(X\leq x)\geq s\}\) with \(s\in (0, 1)\). The Yaari preference \({\mathcal U}_1\) is the well-known average value at risk coherent risk measure (VaR) (see [\textit{M. E. Yaari}, Econometrica 55, 95--115 (1987; Zbl 0616.90005)]). Section 2 gives the setup and definitions for the probabilistic sophisticated variational preferences for \(X, Y\in L^1\) as \(X\succeq Y \Longleftrightarrow {\mathcal U}(X) \geq {\mathcal U}(Y)\). Hence, \(X=Y\) implies \({\mathcal U}(X) = {\mathcal U}(Y)\). The probabilistic sophisticated variational preferences are proper (\({\mathcal U}(X)<\infty, \{X\in L^1: {\mathcal U}(X)> -\infty\}\neq \emptyset\)), concave (\({\mathcal U}(\lambda X+ (1-\lambda )Y)\geq \lambda {\mathcal U}(X) + (1-\lambda) {\mathcal U}(Y)\)) and upper semi-continuous (\(X_n\to X\) in \(L^1\) implies \({\mathcal U}(X)\geq \lim \sup_{n\to \infty} {\mathcal U}(X_n)\)), as characterized in Lemma 2.3 of the paper. Section 3 characterizes the Pareto optimal allocations as solutions to \[ \max \sum_{i=1}^n \lambda_i {\mathcal U}_i(X_i),\quad \text{subject to}\quad (X_1, \dots, X_n)\in A_c(W),\tag{max} \] where \(\lambda_i \geq 0\) and \(A_c(W)=\{(X_1, \dots, X_n)\in (L^1)^n: \sum_{i=1}^n X_i = W, {\mathcal U}_i(X_i)\geq {\mathcal U}_i(W_i) - c_i \}\) for \(W=W_1+\cdots + W_n\) in Proposition 3.2 under the assumption that \(A_c(W) \neq \emptyset\). Proposition 3.2 states that the Pareto optimal allocations defined by the probabilistic sophisticated variational preferences (\((X_1, \dots, X_n)\in A_c(W)\) is Pareto optimal if \((Y_1, \dots, Y_n)\in A_c(W)\) with \({\mathcal U}_i(Y_i)\geq {\mathcal U}_i(X_i)\) for all \(i\), then \({\mathcal U}_i(Y_i)= {\mathcal U}_i(X_i)\) for all \(i\)) are exactly the solutions to the maximization problem. It is known that if there exists a Pareto optimal allocation in this setting, then there is a comonotone one in Definition 3.5 and Proposition 3.6 of the paper. Section 4 states the main results of the paper in Theorem 3.8 and Theorem 3.9. Theorem 3.8 shows that there exists a comonotone solution of ({max}) for every set of strictly positive weights \(\lambda_i > 0\) (\(i=1, \dots, n\)), and Theorem 3.9 analyzes the special two cases for admitting a comonotone solution. The importance of these two theorems is concluded in Theorem 4.1 that shows the existence of acceptable comonotone Pareto optimal allocations. Section 5 gives several examples including the Yaari preferences and the bounds for Theorem 3.9. The technical proofs of Theorem 3.8 and Theorem 3.9 are given in the appendix. It would be nice to see more economic backgrounds, motivations or insights for using the larger space \(L^1\), instead of a single example from Yaari preferences.
probabilistic sophisticated variational preference
scientific article; zbMATH DE number 6369505
scientific article