Robust normative comparisons of socially risky situations

Abstract

In this paper, we theoretically characterize robust empirically implementable normative criteria for evaluating socially risky situations. Socially risky situations are modeled as distributions, among individuals, of lotteries on a finite set of state-contingent pecuniary consequences. Individuals are assumed to have selfish Von Neumann–Morgenstern preferences for these socially risky situations. We provide empirically implementable criteria that coincide with the unanimity, over a reasonably large class of such individual preferences, of anonymous and Pareto-inclusive Von Neuman Morgenstern social rankings of risks. The implementable criteria can be interpreted as sequential expected poverty dominance. An illustration of the usefulness of the criteria for comparing the exposure to unemployment risk of different segments of the French and US workforce is also provided.

Appendix: Proofs of theorems 1–2

1.1 Proof of theorem 1

For the first implication, assume that $p\succsim _{\mathbb {U}_{1}}q$. Then, the inequality:

$$\begin{aligned} \sum _{x\in \mathbb {X}}p_{x}\sum _{i\in N}U(j_{i}^{x},y_{i}^{x})\ge \sum _{x\in \mathbb {X}}q_{x}\sum _{i\in N}U(j_{i}^{x},y_{i}^{x}) \end{aligned}$$

(16)

holds for every function $U:\Omega \times \mathbb {I}\rightarrow \mathbb {R}$ in $\mathbb {U}_{1}$. Consider, for any $k\in \{1,\ldots ,l\}$ and $t\in \mathbb {I}$, the function $V^{kt}:\Omega \times \mathbb {I}\rightarrow \mathbb {R}$ defined, for any $j\in \Omega$ and $y\in \mathbb {I}$ by:

$$\begin{aligned} V^{kt}(j,y)&= -1\text { if }y\le t\text { and }j\le k \\&= 0\text { otherwise} \end{aligned}$$

It can be checked that the function $V^{kt}$ so defined belongs to $\mathbb {U}_{1}$ for any $k\in \{1,\ldots ,l\}$ and $t\in \mathbb {I}$. Hence, inequality (16) holds for the functions $V^{kt}$ so that we have:

$$ \begin{aligned} \sum _{x\in \mathbb {X}}p_{x}\sum _{i\in N}V^{kt}(j_{i}^{x},y_{i}^{x})&\ge \sum _{x\in \mathbb {X}}q_{x}\sum _{i\in N}V^{kt}(j_{i}^{x},y_{i}^{x}) \\&\Leftrightarrow \\ \sum _{i\in N}\sum _{\{x\in \mathbb {X}:j_{i}^{x}\le k \& y_{i}^{x}\le t\}}-p_{x}&\ge \sum _{i\in N}\sum _{\{x\in \mathbb {X} :j_{i}^{x}\le k \& y_{i}^{x}\le t\}}-q_{x} \\&\Leftrightarrow \\ \sum _{i\in N}\sum _{\{x\in \mathbb {X}:j_{i}^{x}\le k \& y_{i}^{x}\le t\}}p_{x}&\le \sum _{i\in N}\sum _{\{x\in \mathbb {X} :j_{i}^{x}\le k \& y_{i}^{x}\le t\}}q_{x} \end{aligned}$$

as required by (6).

For the other implication, consider a subdivision of the interval $[0,1]$ into $r$ sub-intervals $[\rho _{h},\rho _{h+1}]$ for $h=0,\ldots ,r-1$ such that:

$$\begin{aligned} \rho _{0}&= 0 \\ \rho _{m}&= 1 \end{aligned}$$

and, for all $i\in N$, $j\in \Omega$ and $y\in \mathbb {I}$, there are $h$ and $h^{\prime }\in \{0,1,\ldots ,r\}$ such that $p(i,j,y)=\rho _{h}$ and $q(i,j,y)=\rho _{h^{\prime }}$. We this notation, we can write (16) as:

$$\begin{aligned} \sum \limits _{h=1}^{r}\sum \limits _{y=1}^{m}\sum \limits _{j=1}^{l}\Delta f_{j}(\rho _{h},y)\rho _{h}U(j,y)\ge 0 \end{aligned}$$

(17)

where, for $h=1,\ldots ,r$, $j=1,\ldots ,l$ and $y=1,\ldots ,m$,

$$\begin{aligned} \Delta f_{j}(\rho _{h},y)&= \#\{i\in N:p(i,j,y)=\rho _{h}\}\nonumber \\&-\,\#\{i \in N:q(i,j,y)=\rho _{h}\} \end{aligned}$$

(18)

(we of course allow for the possibility that the cardinality of either of the two sets that enters in (18) be zero). We now proceed by decomposing the left hand side of (17) using Abel identity (see for instance Fishburn and Vickson 1978; eq 2.49). Doing first the decomposition with respect to the $y$-indexed summation operator yields:

$$\begin{aligned} \sum \limits _{h=1}^{r}\sum \limits _{j=1}^{l}\left[ \mathop {\displaystyle \sum }\limits _{y=1}^{m}\Delta f_{j}(\rho _{h},y)\rho _{h}U(j,m)-\mathop {\displaystyle \sum }\limits _{t=1}^{m-1}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(j,t+1)-U(j,t))\right] \ge 0 \nonumber \\ \end{aligned}$$

(19)

Decomposing (19) using Abel identity applied this time to the $j$-indexed sum operator yields:

$$\begin{aligned}&\sum \limits _{h=1}^{r}\left[ \sum \limits _{j=1}^{l}\mathop {\displaystyle \sum }\limits _{y=1}^{m}\Delta f_{j}(\rho _{h},y)\rho _{h}U(l,m)\right. \nonumber \\&\quad -\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\left( \sum \limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{m}\Delta f_{j}(\rho _{h},y)\rho _{h}\right) (U(k+1,m)-U(k,m))\nonumber \\&\quad -\mathop {\displaystyle \sum }\limits _{j=1}^{l}\mathop {\displaystyle \sum }\limits _{t=1}^{m-1}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(l,t+1)-U(l,t))\nonumber \\&\quad +\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\mathop {\displaystyle \sum }\limits _{t=1}^{m-1}\mathop {\displaystyle \sum }\limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(k+1,t+1)\nonumber \\&\quad \left. -U(k+1,t)-U(k,t+1)+U(k,t))\right] \ge 0 \end{aligned}$$

(20)

Now, using (18), one can see that that, for every $t\in \mathbb {I}$ and $k\in \Omega :$

$$\begin{aligned} \sum \limits _{h=1}^{r}\mathop {\displaystyle \sum }\limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}=\mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\le t}p(i,j,y)-\mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\le t}q(i,j,y) \end{aligned}$$

(21)

Combining this with the fact that:

$$\begin{aligned} \sum \limits _{h=1}^{r}\mathop {\displaystyle \sum }\limits _{j=1}^{l}\mathop {\displaystyle \sum }\limits _{y=1}^{m}\Delta f_{j}(\rho _{h},y)\rho _{h}\!=\!\mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}p(i,j,y)\!-\!\mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}q(i,j,y)\!=\!n-n\!=\!0 \end{aligned}$$

we can write (20) as:

$$\begin{aligned}&-\mathop {\displaystyle \sum }\limits _{g=1}^{l-1}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)-q(i,j,y))\right) (U(k+1,m)-U(k,m))\nonumber \\&\quad -\mathop {\displaystyle \sum }\limits _{k=1}^{m-1}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le l}\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)-q(i,j,y))\right) (U(l,t+1)-U(l,t))\nonumber \\&\quad +\mathop {\displaystyle \sum }\limits _{g=1}^{l-1}\mathop {\displaystyle \sum }\limits _{k=1}^{m-1}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)-q(i,j,y))\right) (U(k+1,t+1)\nonumber \\&\quad -U(k+1,t)-U(k,t+1)+U(k,t))\ge 0 \end{aligned}$$

(22)

As can be seen, having:

$$\begin{aligned} \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\le t}p(i,j,y)\le \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\le t}q(i,j,y) \end{aligned}$$

for all $j\in \Omega$ and all $k\in \mathbb {I}$ is sufficient for inequality (22) to hold for all state-dependent utility functions $U_{j}$ in $\mathbb {U}_{1}$.

1.2 Proof of theorem 2

Assume first that $p\succsim _{\mathbb {U}_{2}}q$ and, accordingly, that inequality (16) holds for all utility functions $U:\Omega \times \mathbb {I}\ \rightarrow \mathbb {R}$ in $\mathbb {U}_{2}$. Consider, for any $k\in \Omega$ and $t\in \mathbb {I}$, the function $\widetilde{V} ^{kt}:\Omega \times \mathbb {I}\ \rightarrow \mathbb {R}$ defined, for $j\in \Omega$ and $y\in \mathbb {I}$, by:

$$\begin{aligned} \widetilde{V}^{kt}(j,y)&= \min (y-t,0)\text { if }j\le k \\&= 0\text { otherwise} \end{aligned}$$

For a given $j\in \Omega$, the function $\widetilde{V} ^{kt}$ is the “angle” function used in the classical proof of the Hardy-Littlewood-Polya theorem made by Berge (1959). The reader can verify that the functions $\widetilde{V}^{kt}$ belong to $\mathbb {U}_{2}$. For this reason the inequality:

$$\begin{aligned} \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }p(i,j,y)\widetilde{V}^{kt}(j,y)\ge \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}q(i,j,y)\widetilde{V}^{kt}(j,y) \end{aligned}$$

(23)

holds for every $k$ and $t$. Using the definition of the functions $\widetilde{V}^{kt}$, inequality (23) writes:

$$\begin{aligned} \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }p(i,j,y)\min (y-t,0)&\ge \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}q(i,j,y)\min (y-t,0) \\&\Leftrightarrow \\ \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }p(i,j,y)\max (t-y,0)&\le \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}q(i,j,y)\max (t-y,0) \end{aligned}$$

as required by condition (8) of SEPG dominance. To obtain condition (9) of SEPG dominance, we consider, for every $k\in \Omega$, the functions $V^{k}$: $\Omega \times \mathbb {I} \rightarrow \mathbb {R}$ defined, for $j\in \Omega$ and $y\in \mathbb {I}$, by:

$$\begin{aligned} V^{k}(j,y)&= -1\text { if }j\le k \\&= 0\text { otherwise} \end{aligned}$$

These $k$-indexed functions clearly satisfy $V^{k}(j+1,y)\ge V^{k}(j,y)$ for every $y\in \mathbb {I}$ and $j=1,\ldots ,l-1$ and are all (trivially) increasing with respect to income for every $j$. It can be checked that these functions satisfy (very often trivially) the conditions imposed on the functions in $\mathbb {U}_{2}$. Hence, inequality (16) holds for any such functions $V^{k}$ so that we have, for all $k\in \Omega$:

$$\begin{aligned} \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }p(i,j,y)V^{k}(j,y)\ge \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}q(i,j,y)V^{k}(j,y) \end{aligned}$$

or

$$\begin{aligned} \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}-p(i,j,y)&\ge \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }-q(i,j,y) \\&\Leftrightarrow \\ \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}p(i,j,y)&\le \sum _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I} }q(i,j,y) \end{aligned}$$

as required by condition (9). For the other implication, it proceeds just as in the Proof of theorem 1 by writing inequality (16) in the form of (17) and by doing the Abel decomposition of (17) until one reaches condition (20). If one then goes one step further and Abel decomposes each term of (20) with respect to the inner ($y$-indexed) term, one obtains:

$$\begin{aligned}&\mathop {\displaystyle \sum }\limits _{h=1}^{r}\left[ -\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\sum \limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{m}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(k+1,m)-U(k,m))\right. \\&\quad -\mathop {\displaystyle \sum }\limits _{t=1}^{m-1}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\mathop {\displaystyle \sum }\limits _{j=1}^{l} \Delta f_{j}(\rho _{h},y)\rho _{h}(U(l,t+1)-U(l,t)) \\&\quad +\mathop {\displaystyle \sum }\limits _{v=1}^{m-2}\mathop {\displaystyle \sum }\limits _{t=1}^{v}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\mathop {\displaystyle \sum }\limits _{j=1}^{l}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(l,v+2)-2U(l,v+1)+U(l,v))\\&\quad +\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\mathop {\displaystyle \sum }\limits _{t=1}^{m-1}\mathop {\displaystyle \sum }\limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}(U(k+1,t+1)\\&\quad -U(k+1,t)-U(k,t+1)+U(k,t)) \end{aligned}$$

$$\begin{aligned} -\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\displaystyle \sum \limits _{v=1}^{m-2}(\mathop {\displaystyle \sum }\limits _{t=1}^{v} \mathop {\displaystyle \sum }\limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}\Delta ^{U^{2}}(k,v))\left. \right] \ge 0 \end{aligned}$$

(24)

where:

$$\begin{aligned} \Delta ^{U^{2}}(k,v)&= U(k+1,v+2)-2U(k+1,v+1)+U(k+1,v) \\&-[U(k,v+2)-2U(k,v+1)+U(k,v)] \end{aligned}$$

Noticing that:

$$\begin{aligned} \sum \limits _{h=1}^{m}\mathop {\displaystyle \sum }\limits _{j=1}^{k}\mathop {\displaystyle \sum }\limits _{t=1}^{v}\mathop {\displaystyle \sum }\limits _{y=1}^{t}\Delta f_{j}(\rho _{h},y)\rho _{h}=\mathop {\displaystyle \sum }\limits _{t=1}^{v}[\mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}[p(i,j,y)-q(i,j,y)] \end{aligned}$$

$$\begin{aligned}&= \mathop {\displaystyle \sum }\limits _{t=1}^{v}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}[p(i,j,y)-q(i,j,y)]\right) (t-y) \\&= \mathop {\displaystyle \sum }\limits _{t=1}^{v}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}[p(i,j,y)-q(i,j,y)]\right) P(t,y) \end{aligned}$$

and remembering Eq. (21) in the proof of theorem 1, one can write (24) as:

$$\begin{aligned}&-\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k }\mathop {\displaystyle \sum }\limits _{y\in \mathbb {I}}(p(i,j,y)-q(i,j,y))\right) (U(k+1,m)-U(k,m)) \\&\mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\le m-1}(p(i,j,y)-q(i,j,y))P(m-1,y)(U(l,m)-U(l,m-1)) \\&+\mathop {\displaystyle \sum }\limits _{t=1}^{m-2}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\in \Omega }\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)\!-\!q(i,j,y))P(t,y)\right) (U(l,t\!+\!2)\!-\!2U(l,t\!+\!1)\!+\!U(l,t))\\&+\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)-q(i,j,y))P(t,y)\right) (U(k+1,t+1)\\&-U(k+1,t)-U(k,t+1)+U(k,t)) \end{aligned}$$

$$\begin{aligned} -\mathop {\displaystyle \sum }\limits _{k=1}^{l-1}\mathop {\displaystyle \sum }\limits _{v=1}^{m-2}\left( \mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}(p(i,j,y)-q(i,j,y))P(t,y)\Delta _{kv}^{U^{2}}\right) \ge 0 \end{aligned}$$

(25)

For any combination of functions $U_{j}$ (for $j\in \Omega$) belonging to $\mathbb {U}_{2}$, it is sufficient for (25) to hold to have, for all $k\in \Omega$:

$$\begin{aligned} \mathop {\displaystyle \sum }\limits _{i\in N}\sum \limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\in \mathbb { I}}(p(i,j,y)-q(i,j,y))\le 0, \end{aligned}$$

and:

$$\begin{aligned} \mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}p(i,j,y)P(t,y)-\mathop {\displaystyle \sum }\limits _{i\in N}\mathop {\displaystyle \sum }\limits _{j\le k}\mathop {\displaystyle \sum }\limits _{y\le t}q(i,j,y)P(t,y)\le 0 \end{aligned}$$

for all $t\in \mathbb {I}$, which is the definition of SEPG dominance.