Comonotone Pareto optimal allocations for law invariant robust utilities on L ¹

Abstract

We 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. The total endowment is only required to be integrable.

Appendix: Proofs of Theorems 3.8 and 3.9

In the proofs of Theorems 3.8 and 3.9, we apply the following facts about law invariant monetary utilities which apart from the sign are convex risk measures. We omit a proof since most arguments are standard and can for instance be found in [12].

Lemma A.1

Let

$$\mathcal {U}(X)=\inf_{\mathbb {Q}\in \mathcal {Q}}\left(\mathbb {E}_\mathbb {Q}[u(X)]+\alpha(\mathbb {Q})\right ), \quad X\in L^1, $$

be a law invariant robust utility as (2.1). Define

$$ U(X):= \inf_{\mathbb {Q}\in \mathcal {Q}}\left(\mathbb {E}_\mathbb {Q}\left[X\right]+\alpha(\mathbb {Q})\right), \quad X\in L^1, $$

so that $\mathcal {U}(\cdot)=U(u(\cdot))$. Then U is a proper, law invariant, $\succeq _{\rm ssd}$-monotone, upper semi-continuous, monotone, cash additive (U(X+m)=U(X)+m for all $m\in \mathbb {R}$), and concave function. Moreover, we have that

$$ U(X)\leq \mathbb {E}[X]+U(0)\quad\textit{for\ all}\ X\in L^1. $$

(A.1)

Note that (A.1) follows from $E[X]\succeq _{\rm ssd} X$ and the cash additivity of U. The next lemma is an Arzelà–Ascoli type argument which will also play a crucial role in the proofs of Theorems 3.8 and 3.9. For a proof see e.g. [9].

Lemma A.2

Let $f_{n}:\mathbb {R}\to \mathbb {R}$, $n\in \mathbb {N}$, be a sequence of increasing 1-Lipschitz-continuous functions such that f _n (0)∈[−K,K] for all $n\in \mathbb {N}$, where K≥0 is a constant. Then there are a subsequence $(f_{n_{k}})_{k\in \mathbb {N}}$ of $(f_{n})_{n\in \mathbb {N}}$ and an increasing 1-Lipschitz-continuous function $f:\mathbb {R}\to \mathbb {R}$ such that $\lim_{k\to\infty}f_{n_{k}}(x)=f(x)$ for all $x\in \mathbb {R}$.

Let $\operatorname {CFN}:= \{(f_{i})_{i=1}^{n}\in \operatorname {CF}\mid f_{1}(0)=\cdots=f_{n}(0)=0 \}$. Note that

$$ \operatorname {CF} = \bigg\{(f_i+a_i)_{i=1}^n\Bigg| (f_i)_{i=1}^n\in \operatorname {CFN}, a_i\in \mathbb {R}, \sum_{i=1}^n a_i=0\bigg\}. $$

According to Proposition 3.6, there exists a solution to (3.3) for some given weights (λ ₁,…,λ _n ) if and only if there is a solution to

$$\begin{aligned} &\mbox{maximize}\quad \sum_{i=1}^n \lambda_i\mathcal {U}_i\big(f_i(W)+a_i\big) \\ &\mbox{subject to}\quad (f_i)_{i=1}^n\in \operatorname {CFN},\ a_i\in \mathbb {R}, \\ &\qquad \quad \qquad\sum_{i=1}^n a_i=0,\ \big(f_i(W)+a_i\big)_{i=1}^n\in \mathbb {A}_c(W). \end{aligned}$$

(A.2)

Proof of Theorem 3.9

We prove the existence of a solution to (A.2). Fix a set of weights (λ ₁,…,λ _n ) satisfying the conditions (3.4). First of all, we observe that if for some j∈M the right derivative $u'_{j}$ does not attain the values $d_{H}^{j}$ and/or $d_{L}^{j}$, we can always find nonnegative numbers $\widetilde{d}_{H}^{j}$ and/or $\widetilde{d}_{L}^{j}$ in the image of $u'_{j}$ such that for the already given set of weights (λ ₁,…,λ _n ), the conditions (3.4) still hold true if we replace the $d_{H}^{j}$ and/or $d_{L}^{j}$ by $\widetilde{d}_{H}^{j}$ and/or $\widetilde{d}_{H}^{j}$. We assume that all $d_{H}^{j}$ and/or $d_{L}^{j}$ which are not attained by the corresponding $u_{j}'$ are replaced as in the described manner, and for the sake of simplicity we keep the notation $d_{H}^{j}$ and $d_{L}^{j}$. By concavity of the u _i , there is a constant k such that for all i=1,…,n, the affine functions $\mathbb {R}\ni x\mapsto d_{L}^{i} x+ k$ and $\mathbb {R}\ni x\mapsto d_{H}^{i} x+ k$ both dominate u _i .⁹ Using this, we want to show that

$$\begin{aligned} P :=\sup\bigg\{ &\sum_{i=1}^n \lambda_i\mathcal {U}_i \big (f_i(W)+a_i\big )\Bigg| (f_i)_{i=1}^n\in \operatorname {CFN},\, a_i\in \mathbb {R}, \\ & \sum_{i=1}^n a_i=0, \big(f_i(W)+a_i\big)_{i=1}^n\in \mathbb {A}_c(W) \bigg \} <\infty, \end{aligned}$$

(A.3)

and that this supremum is realized over a bounded set of comonotone allocations where the bound is given by W. More precisely, we shall prove that there exists some constant K>0 depending on W such that

$$\begin{aligned} P = \sup\bigg\{& \sum_{i=1}^n \lambda_i\mathcal {U}_i \big (f_i(W)+a_i\big )\Bigg| (f_i)_{i=1}^n\in \operatorname {CFN},\, a_i\in[-K,K], \\ & \sum_{i=1}^n a_i=0,\big(f_i(W)+a_i \big)_{i=1}^n\in \mathbb {A}_c(W){\bigg \}}. \end{aligned}$$

(A.4)

To this end, we define the functions

$$U_i(X):=\inf_{\mathbb {Q}\in \mathcal {Q}_i}\left(\mathbb {E}_\mathbb {Q}[X]+\alpha_i(\mathbb {Q})\right ), \quad X\in L^1,\ i=1,\ldots,n, $$

as in Lemma A.1. Consider $(f_{i})_{i=1}^{n}\in \operatorname {CFN}$ and $a_{i}\in \mathbb {R}$ such that $\sum _{i=1}^{n}a_{i}=0$ and $(f_{i}(W)+a_{i})_{i=1}^{n}\in \mathbb {A}_{c}(W)$. Let I:={i∈{1,…,n}∣a _i <0} and J:={1,…,n}∖I. By applying monotonicity, cash additivity and finally property (A.1) of the U _i (see Lemma A.1), we obtain

$$\begin{aligned} &\sum_{i=1}^n\lambda_i\mathcal {U}_i \big(f_i(W)+a_i\big) \\ &\quad = \sum_{i=1}^n \lambda_i U_i\Big(u_i\big(f_i(W)+a_i\big)\Big ) \\ &\quad \leq\sum_{i\in I\cap M}\lambda_i U_i\Big(d_H^i\big (f_i(W)+a_i\big )+k\Big) \\ &\qquad{}+\sum_{j\in J\cap M}\lambda_j U_j\Big(d_L^j\big (f_j(W)+a_j\big )+k\Big) + \sum_{\ell\in N}\lambda_\ell U_\ell\Big(d^\ell_H\big(f_\ell (W)+a_\ell \big)+k\Big) \\ &\quad \leq k\sum_{i=1}^n\lambda_i+ \sum_{i\in I\cap M}\lambda_i U_i\big (d_H^if_i(W)\big) + \sum_{j\in J\cap M}\lambda_j U_j\big(d_L^j f_j(W)\big) \\ &\qquad{}+ \sum_{\ell\in N}\lambda_\ell U_\ell\big(d^\ell_H f_\ell (W)\big ) \\ & \qquad{}+\left(\min_{i\in I\cap M}\lambda_id_H^i\right)\sum_{i\in I\cap M}a_i+\left(\max_{j\in J\cap M}\lambda_jd_L^j\right)\sum_{j\in J\cap M}a_j + \delta\sum_{\ell\in N}a_\ell \\ &\quad \leq k\sum_{i=1}^n\lambda_i+ \mathbb {E}\bigl[W^+\bigr] \sum_{i=1}^n \lambda_i d_H^i + \sum_{i=1}^n \lambda_i U_i(0) \\ &\qquad{}+\left(\min_{i\in I\cap M}\lambda_id_H^i\right)\sum_{i\in I\cap M}a_i+\left(\max_{j\in J\cap M}\lambda_j d_L^j\right)\sum_{j\in J\cap M}a_j +\delta\sum_{\ell\in N}a_\ell. \end{aligned}$$

(A.5)

Suppose that N=∅; then we further estimate

$$ \sum_{i=1}^n \lambda_i d_H^i + k\sum_{i=1}^n\lambda_i+\sum_{i=1}^n \lambda_i U_i(0) -\left(\min _{i\in I}\lambda_id_H^i-\max_{j\in J}\lambda_j d_L^j\right)a, $$

(A.6)

where a:=∑ _i∈J a _i  (≥0). If N≠∅ and ∑ _ℓ∈N a _ℓ <0, then we estimate

$$ (A.5)\leq \mathbb {E}\bigl[W^+\bigr] \sum_{i=1}^n \lambda_i d_H^i + k\sum_{i=1}^n\lambda_i+\sum_{i=1}^n \lambda_i U_i(0) -\left(\delta -\max _{j\in J\cap M}\lambda_jd_L^j\right)\tilde{a} $$

(A.7)

for $\tilde{a}:=\sum_{i\in J\cap M} a_{i}\, (\geq0)$, using (3.4). And similarly, if N≠∅ and ∑ _ℓ∈N a _ℓ ≥0, then we estimate

$$ (A.5)\leq \mathbb {E}\bigl[W^+\bigr] \sum_{i=1}^n \lambda_i d_H^i + k\sum_{i=1}^n\lambda_i+\sum_{i=1}^n \lambda_i U_i(0)- \left(\min _{i\in I\cap M}\lambda_id_H^i-\delta\right)\hat{a} $$

(A.8)

for $\hat{a}:=\sum_{i\in J\cup N} a_{i}\, (\geq0)$. Consequently we infer that

$$P\leq \mathbb {E}\bigl[W^+\bigr] \sum_{i=1}^n \lambda_i d_H^i + k\sum_{i=1}^n\lambda_i +\sum_{i=1}^n \lambda_i U_i(0) <\infty. $$

Choose any allocation $(X_{1},\ldots,X_{n})\in \mathbb {A}_{c}(W)$ (Assumption 3.1). Then we have that $P\geq\sum _{i=1}^{n}\lambda_{i}\mathcal {U}_{i}(X_{i})=:\widetilde{k}$. Letting

$$A:= \min_{i=1,\ldots,n}\lambda_id_H^i-\max_{j=1,\ldots,n}\lambda_jd_L^j $$

if N=∅, or

$$A:=\min\Big\{\delta-\max_{j\in M}\lambda_jd_L^j, \min _{i\in M}\lambda_id_H^i-\delta \Big\} $$

if N≠∅, we infer from (A.6)–(A.8) that the supremum in (A.3) is realized over allocations such that

$$|a_i|\leq\frac{|\widetilde{k}| +\mathbb {E}[W^+] \sum_{i=1}^n \lambda_i d_H^i + k\sum_{i=1}^n\lambda_i+|\sum_{i=1}^n \lambda_i U_i(0)|}{A}=:\overline{K} $$

for all i∈M, and $|\sum_{i\in N}a_{i}|\leq\overline{K}$, too. Note that A>0 due to the conditions (3.4) on the weights λ _i . In the following we argue that in the case |N|>1, we may also assume that the a _i belonging to i∈N are bounded due to the insensitivity of the cash additive $\mathcal {U}_{i}$, i∈N, to constant re-sharings of 0 among the decision makers in N. To this end, note that the choice of the λ _i and the requirement s _i =−∞ for i∈N implies

$$\begin{aligned} \sum_{i\in N}\lambda_i\mathcal {U}_i \big(f_i(W)+a_i+m_i\big) & = \sum _{i\in N}\lambda_i\mathcal {U}_i \big(f_i(W)+a_i\big) + \delta\sum_{i\in N} m_i \\ & = \sum_{i\in N}\lambda_i\mathcal {U}_i \big (f_i(W)+a_i\big ), \end{aligned}$$

(A.9)

whenever $m_{i}\in \mathbb {R}$ such that ∑ _i∈N m _i =0. Hence, adding such m _i to the endowments of the decision makers in N does not affect the contribution of the allocation to P. This immediately implies that we may assume $|a_{i}|\leq\overline{K}$ for all i∈N if c _i =∞ for all i∈N, because in that case we may choose m ₁=∑ _i∈N a _i −a ₁ and m _i =−a _i for all i≠1 in (A.9), and the altered allocation satisfies the bound (meaning that $|m_{1}+a_{1}|=|\sum_{i\in N}a_{i}|\leq \overline{K}$, see above). If the set N _b ⊂N of indices i∈N such that $c_{i}\in \mathbb {R}$ is not empty, we also need to consider cash amounts that might be needed to make the endowment f _i (W)+a _i acceptable. This is the point where the assumption $\mathcal {U}_{i}(-W^{-})>-\infty$ for all i∈N _b enters (whenever |N|>1). Using the cash additivity $\mathcal {U}_{i}(Y+z)=\mathcal {U}_{i}(Y)+d_{H}^{i}z$, $z\in \mathbb {R}$, and monotonicity of $\mathcal {U}_{i}$, we obtain that f _i (W)+z is acceptable for decision maker i∈N _b whenever

$$z\geq\frac{\mathcal {U}_i(W_i)-\mathcal {U}_i(-W^-)-c_i}{d_H^i}. $$

Moreover, acceptability of f _i (W)+a _i implies (again using cash additivity and monotonicity of $\mathcal {U}_{i}$) for all i∈N _b that

$$a_i\geq\frac{\mathcal {U}_i(W_i)-c_i-\mathcal {U}_i(f_i(W_i))}{d_H^i}\geq\frac {\mathcal {U}_i(W_i)-c_i-\mathcal {U}_i(0)}{d_H^i}-\mathbb {E}[W^+] , $$

where we have used that $\mathcal {U}_{i}(f_{i}(W_{i}))\leq \mathcal {U}_{i}(W^{+})\leq \mathbb {E}[d_{H}^{i} W^{+}]+\mathcal {U}_{i}(0)$ according to (A.1). Therefore we know that there exists a constant $\widehat{K}>0$ such that the a _i , i∈N _b , are bounded from below by $-\widehat{K}$ and such that the endowments f(W _i )+a _i +m _i stay acceptable for $m_{i}=-[(a_{i}-\widehat{K})\vee0]$, i∈N _b . If N _b =N, then choosing some j∈N _b and letting $m_{i}=-[(a_{i}-\widehat{K})\vee0]$ for i∈N _b ∖{j} and m _j =−∑ _i∈N∖{j} m _i , we obtain that ∑ _i∈N m _i =0, $|a_{i}+m_{i}|\leq\widehat{K}$ for all i∈N _b ∖{j}, and

$$|a_j+m_j|\leq\bigg|\sum_{i\in N} a_i\bigg|+\bigg|\sum_{i\in N_b\setminus\{j\}} a_i+m_i\bigg| \leq\overline{K}+|N_b|\widehat{K}=:K. $$

Moreover, because a _j ≤a _j +m _j , we get that also f(W _j )+a _j +m _j is acceptable, i.e., $\mathcal {U}_{j}(f(W_{j})+a_{j}+m_{j})\geq \mathcal {U}_{j}(W_{j})-c_{j}$. If |N _u |>1 where N _u :=N∖N _b , then we choose some j∈N _u and let $m_{i}=-[(a_{i}-\widehat{K})\vee0]$ for all i∈N _b and m _i =−a _i for all i∈N _u ∖{j}, whereas m _j =−∑ _i∈N∖{j} m _i . Again, ∑ _i∈N m _i =0, and a _i +m _i =0 for all i∈N _u ∖{j}, $|a_{i}+m_{i}|\leq\widehat{K}$ for all i∈N _b , and

$$|a_j+m_j|\leq\bigg|\sum_{i\in N} a_i\bigg|+\bigg|\sum_{i\in N_b} a_i+m_i\bigg| \leq\overline{K}+|N_b|\widehat{K}=:K. $$

Hence, (A.3) and (A.4) are proved. By virtue of (A.4), we may choose a sequence $((f^{p}_{i})_{i=1}^{n} )_{p\in \mathbb {N}}\subset \operatorname {CF}$ with $f^{p}_{i}(0)\in[-K,K]$ for all i=1,…,n and $p\in \mathbb {N}$ such that $(f^{p}_{i}(W))_{i=1}^{n}\in \mathbb {A}_{c}(W)$ for all $p\in \mathbb {N}$ and

$$P=\lim_{p\to\infty}\sum_{i=1}^n\lambda_i \mathcal {U}_i\big (f^p_i(W)\big). $$

According to Lemma A.2, there exists a subsequence, which we for the sake of simplicity also denote by $((f^{p}_{i})_{i=1}^{n})$, which converges pointwise to some $(f_{i})_{i=1}^{n}\in \operatorname {CF}$. As $|f_{i}^{p}(W)|\leq|W|+K$ for all i=1,…,n and $p\in \mathbb {N}$, we may apply the dominated convergence theorem to obtain f _i (W)∈L ¹ and $\lim_{p\to\infty} \mathbb {E}[|f_{i}(W)-f_{i}^{p}(W)|]= 0$ for all i=1,…,n. By upper semi-continuity of the $\mathcal{U}_{i}$ (Lemma 2.3), we have

$$\mathcal {U}_i(W_i)-c_i\leq\limsup_{p\to\infty} \mathcal {U}_i \big (f^p_i(W)\big )\leq \mathcal {U}_i\big(f_i(W)\big) $$

and

$$P= \lim_{p\to\infty}\sum_{i=1}^n\lambda_i \mathcal {U}_i \big(f^p_i(W)\big) \leq\sum_{i=1}^n\lambda_i \limsup_{p\to\infty} \mathcal {U}_i \big(f^p_i(W)\big) \leq\sum_{i=1}^n\lambda_i \mathcal {U}_i \big(f_i(W)\big). $$

Hence, we infer that $(f_{i}(W))_{i=1}^{n}\in \mathbb {A}_{c}(W)$ (since P>−∞) and

$$P=\sum_{i=1}^n\lambda_i \mathcal {U}_i\big(f_i(W)\big). $$

For the last part of (ii), suppose that |N|=n and let (X ₁,…,X _n ) be a solution to (3.3) for the given weights. If we have $m_{i}\in \mathbb {R}$ such that $\sum_{i=1}^{n} m_{i}=0$, then the same computation as in (A.9) yields $\sum_{i=1}^{n}\lambda _{i}\mathcal {U}_{i}(X_{i}+m_{i})=\sum_{i=1}^{n}\lambda_{i}\mathcal {U}_{i}(X_{i})$. □

Proof of Theorem 3.8

Recall (A.2) and let $(f_{i})_{i=1}^{n}\in \operatorname {CFN}$, $a_{i}\in \mathbb {R}$ with $\sum_{i=1}^{n}a_{i}=0$ such that $(f_{i}(W)+a_{i})_{i=1}^{n}\in \mathbb {A}_{c}(W)$. Since in particular $\mathcal {U}_{i}(f_{i}(W)+a_{i})>-\infty$, we must have that f _i (W)+a _i ≥s _i for all i=1,…,n. Let $\widetilde{K}:=\sum_{i=1}^{n}|s_{i}|$. Then

$$-\bigl(|W|+\widetilde{K}\bigr)\leq f_i(W)+a_i= W- \bigg(\sum_{j\neq i}f_j(W)+a_j\bigg)\leq|W|+\widetilde{K}. $$

Hence, we deduce that (A.4) holds with $K:= 2\operatorname {essinf} |W|+\widetilde{K}$. The rest of the proof now follows the lines of the proof of Theorem 3.9. □