Riskiness for sets of gambles

Abstract

Aumann–Serrano (J Polit Econ 116:810–836, 2008) and Foster–Hart (J Polit Econ 117:785–814, 2009) suggest two new riskiness measures, each of which enables one to elicit a complete and objective ranking of gambles according to their riskiness. These riskiness measures were created with a risky world in mind, but not an uncertain one. We apply similar arguments to models of decision under uncertainty and develop complete and objective rankings of sets of gambles, which arise naturally in many such models. Clearly, these results extend the previous riskiness measures, and they have a natural interpretation in terms of those measures even when uncertainty does play a significant role.

Appendix

In our proofs, we will use $K_{g}$ to denote the maximal gain in the gamble $g$, and will use $L_{g}$ to denote the maximal loss in this gamble.

For each of the two riskiness measures, the Aumann–Serrano riskiness index and the Foster–Hart riskiness measure, we will first provide the proofs for $\lambda$-maximizers. Then, the proofs for the maximin and maximax decision rules will follow as special cases. We assume throughout that $G,H\in \mathbb {G}$.

1.1 Aumann–Serrano riskiness index of sets for $\lambda$ -maximizers

Fix $\lambda \in \left[ 0,1\right]$ and let $V\equiv V_{u,\lambda }$.

We will focus for a while only on utilities that are characterized by constant absolute risk aversion ($CARA$). The set containing all these utilities will be denoted by $U^\mathrm{CA}$.

Every $u\in U^\mathrm{CA}$ can be fully characterized by its risk-aversion coefficient $\alpha >0$ and will be denoted by $u_{\alpha }$. For $u=$ $u_{\alpha }$ with uncertainty-aversion coefficient $\lambda$ we use $V^{\alpha }$.

Lemma 26

$\forall g_{1},g_{2}\in G,u_{\alpha }$ either prefers $g_{1}$ to $g_{2}$ at all $w>0$, or prefers $g_{2}$ to $g_{1}$ at all $w>0$.

Proof

$u_{\alpha }$ prefers $g_{1}$ to $g_{2}$ at $w$ if and only if $E\left[ -e^{-\alpha (w+g_{1})}\right] \ge E\left[ -e^{-\alpha (w+g_{2})}\right] \Leftrightarrow E\left[ -e^{-\alpha g_{1}}\right] \ge E\left[ -e^{-\alpha g_{2}}\right]$, and this inequality is independent of $w$. $\square$

Following Lemma 26, we will perform the whole analysis of $U^\mathrm{CA}$ behavior for some arbitrary $w$, and denote $M^{\alpha }\left( G\right) \in \underset{g\in G}{\arg \max }E\left[ u_{\alpha }\left( w+g\right) \right]$ and $m^{\alpha }\left( G\right) \in \underset{g\in G}{ \arg \min }E\left[ u_{\alpha }\left( w+g\right) \right]$.

Lemma 27

$\forall \alpha$, $V^{\alpha }$ either accepts $G$ at all $w>0$, or rejects $G$ at all $w>0$.

Proof

$V^{\alpha }$ accepts $G$ at $w$ if and only if $\lambda E\left[ -e^{-\alpha (w+M^{\alpha }\left( G\right) )}\right] +(1-\lambda )E$ $\left[ -e^{-\alpha (w+m^{\alpha }\left( G\right) )}\right] >-e^{-\alpha w}$. Multiplying by $e^{\alpha w}$ yields a condition that is independent of $w$. If $\lambda E \left[ -e^{-\alpha M^{\alpha }\left( G\right) }\right] +\left( 1-\lambda \right) E\left[ -e^{-\alpha m^{\alpha }\left( G\right) }\right] \le 1$, $V^{\alpha }$ (weakly) rejects $G$ at all $w>0$. $\square$

Lemma 28

Let $u_{\alpha _{1}},u_{\alpha _{2}}\in U^\mathrm{CA}$ with $\alpha _{1}<\alpha _{2}$ such that $V^{\alpha _{1}}$ (strictly) accepts $G$ at $w$ and $V^{\alpha _{2}}$ (strictly) rejects $G$ at $w$. Then $\exists \alpha ^{*}$ s.t. $\alpha _{1}<\alpha ^{*}<\alpha _{2}$ and:

• (i) $V^{\alpha ^{*}}\left( w+G\right) =u_{\alpha ^{*}}(w)$.

• (ii) $\forall \alpha <\alpha ^{*}$, $V^{\alpha }$ accepts $G$ at $w$, and $\forall \alpha >\alpha ^{*}$, $V^{\alpha }$ rejects $G$ at $w$.

We remind here that $w$ in Lemma 28 is arbitrary and kept only for the coherence of the formulation. In fact the Lemma says that every $V^{\alpha }$ with $\alpha <\alpha ^{*}$ accepts $G$ at all $w>0$, and that every $V^{\alpha }$ with $\alpha >\alpha ^{*}$ rejects $G$ at all $w>0$.

Proof

• (i) Let $\Delta u\left( \alpha \right) \equiv V^{\alpha }\left( w+G\right) -u_{\alpha }(w)=\lambda \underset{g\in G}{\max }E\left[ u_{\alpha }\left( w+g\right) \right] +\left( 1-\lambda \right)$ $\underset{g\in G}{\min }E\left[ u_{\alpha }\left( w+g\right) \right] -u_{\alpha }(w)$. $\Delta u\left( \alpha \right)$ is continuous in $\alpha$ because the functions $E[\cdot ],\max ()$ and $\min ()$ are all continuous, and because $u_{\alpha }$ is continuous $\forall \alpha$. Moreover $\Delta u\left( \alpha _{1}\right) >0$ and $\Delta u\left( \alpha _{2}\right) <0$. The continuity of $\Delta u\left( \alpha \right)$ guarantees that $\exists \alpha ^{*}$ s.t. $\alpha _{1}<\alpha ^{*}<\alpha _{2}$ for which $\Delta u\left( \alpha ^{*}\right) =0$, i.e., $V_{\alpha ^{*}}\left( w+G\right) =u_{\alpha ^{*}}(w)$.

• (ii) W.l.o.g., let $u_{\alpha }(x)=\frac{1-e^{-\alpha \left( x-w\right) }}{ \alpha }$. If $\dot{\alpha }<\ddot{\alpha }$ then³⁴ $u_{\dot{\alpha } }(w+x)>u_{\ddot{\alpha }}(w+x)$ for all $x\ne 0$. Hence $E\left[ u_{\dot{ \alpha }}(w+g)\right] >E\left[ u_{\ddot{\alpha }}(w+g)\right]$ for all $g\in G$ and $\dot{\alpha }<\ddot{\alpha }$. Since $G$ is compact, we thus have $\underset{g\in G}{\max }E\left[ u_{\dot{\alpha }}(w+g)\right] >\underset{g\in G}{\max }E\left[ u_{\ddot{\alpha }}(w+g)\right]$ and $\underset{g\in G}{\min }E\left[ u_{\dot{\alpha }}(w+g)\right] >\underset{g\in G}{\min }E\left[ u_{ \ddot{\alpha }}(w+g)\right]$. Finally, noticing that $u_{\alpha }(w)=0$ for all $\alpha >0$, we get that $\lambda \underset{g\in G}{\max }E\left[ u_{ \dot{\alpha }}(w+g)\right] +\left( 1-\lambda \right) \underset{g\in G}{\min }E \left[ u_{\dot{\alpha }}(w+g)\right] -u_{\dot{\alpha }}(w)>\lambda \underset{ g\in G}{\max }E\left[ u_{\ddot{\alpha }}(w+g)\right] +\left( 1-\lambda \right) \underset{g\in G}{\min }E\left[ u_{\ddot{\alpha }}(w+g)\right] -u_{ \ddot{\alpha }}(w)$, i.e., $\Delta u\left( \dot{\alpha }\right) >$ $\Delta u\left( \ddot{\alpha }\right)$. So $\Delta u\left( \alpha ^{*}\right) =0$ implies that $\forall \alpha <\alpha ^{*}$ we have $\Delta u\left( \alpha \right) >0$, i.e., $V^{\alpha }$ accepts $G$ at $w$, and $\forall \alpha >\alpha ^{*}$ we have $\Delta u\left( \alpha \right) <0$, i.e., $V^{\alpha }$ rejects $G$ at $w$. $\square$

Corollary 29

There is a unique³⁵ $u\in U^\mathrm{CA}$ s.t. $u$ is indifferent to $G$ (at all $w$). This $u$ is characterized by the unique $\alpha >0$ that solves the equation $\lambda E[e^{-\alpha M^{\alpha }\left( G\right) }]+\left( 1-\lambda \right) E\left[ e^{-\alpha m^{\alpha }\left( G\right) } \right] =1$.

We now return to discuss the set of utilities $U^{*}$. We denote by $\rho _{u}(w)$ the local (absolute) risk-aversion coefficient of $u\in U^{*}$ at $w$. Let $g_{\max }^{u,w}$ be a maximizer of the expected utility of $u$ at wealth level $w$ over the set of gambles $G$, i.e., $g_{\max }^{u,w}\in \underset{g\in G}{\arg \max }E\left[ u\left( w+g\right) \right]$. Similarly let $g_{\min }^{u,w}\in \underset{g\in G}{\arg \min }E \left[ u\left( w+g\right) \right]$.

Lemma 30 compares the acceptance and rejection criteria of a general $u$ to those of $u_{\alpha ^{*}}$, and sets the ground for the main theorem. For every set $G$ we denote by $g_{\alpha ^{*}}$ the compound gamble $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ (i.e., gamble $M^{\alpha ^{*}}\left( G\right)$ with probability $\lambda$ and gamble $m^{\alpha ^{*}}\left( G\right)$ with probability $1-\lambda$), where $\alpha ^{*}$ is the unique risk-aversion coefficient of the $CARA$ utility that satisfies $V^{\alpha ^{*}}\left( w+G\right) =u(w)$ (as was shown in Lemma 28(i)).

Lemma 30

• (i) If $\rho _{u}(w)\ge \alpha ^{*}$ $\forall w>0$ then $u$ rejects $G$ $\ \forall w>0$.

• (ii) If $\rho _{u}(w)<\alpha ^{*}$ $\forall w\in \left[ \hat{w} -L_{g_{\alpha ^{*}}},\hat{w}+K_{g_{\alpha ^{*}}}\right]$ then $u$ accepts $G$ at $\hat{w}$.

Proof

• (i) Assume that $\rho _{u}(w)\ge \alpha ^{*}$ $\forall w>0$ for some $u$. Then obviously every gamble $g$ that is rejected by $u_{\alpha ^{*}}$ is rejected by $u$ either. Assume now by negation that $\exists w$ s.t. $u$ accepts $G$ at $w$. Then $u$ in fact accepts $\lambda *g_{\max }^{u,w}+\left( 1-\lambda \right) *g_{\min }^{u,w}$ at $w$, and therefore accepts $\lambda *g_{\max }^{u,w}+\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ at $w$ too.³⁶ On the other hand, we know that $u_{\alpha ^{*}}$ is indifferent to $G$ at $w$, thus (weakly) rejects $\lambda *g_{\max }^{u,w}+\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ at $w$ (remember that if $u_{\alpha ^{*}}$ is indifferent to $G$ at some $w^{\prime }$, $u_{\alpha ^{*}}$ is also indifferent to $G$ at any other $w$). So the compound gamble $\lambda *g_{\max }^{u,w}+\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ is accepted at $w$ by $u$ but rejected by $u_{\alpha ^{*}}$, in contradiction to the assumption.

• (ii) We repeat almost the same proof: now we denote $\widehat{\alpha }\equiv \max \rho _{u}(w)$ at the range $\left[ \hat{w}-L_{g_{\alpha ^{*}}},\hat{ w}+K_{g_{\alpha ^{*}}}\right]$. Let $u_{\widehat{\alpha }}$ be the $CARA$ utility with risk-aversion coefficient $\widehat{\alpha }$. Since $\widehat{\alpha }<\alpha ^{*}$ we get that $u_{\widehat{\alpha }}$ accepts $g_{\alpha ^{*}}=\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ at $\hat{ w}$, and from $\rho _{u}(w)\le \widehat{\alpha }$ $\forall w\in \left[ \hat{ w}-L_{g_{\alpha ^{*}}},\hat{w}+K_{g_{\alpha ^{*}}}\right]$ we know that also $u$ accepts $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ at $\hat{ w}$. Therefore, if $u$ were to reject $G$ at $\hat{w}$, $u$ would have to reject $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *g_{\min }^{u,\hat{w}}$ at $\hat{w}$ either. On the other hand, since $u_{\alpha ^{*}}$ is indifferent to $G$ at $\hat{w}$, and since $u_{\widehat{\alpha }}$ accepts every single gamble that $u_{\alpha ^{*}}$ accepts, we know that $u_{\widehat{\alpha }}$ accepts $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *m^{\alpha ^{*}}\left( G\right)$ at $\hat{w}$, hence accepts $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *g_{\min }^{u,\hat{w}}$ at $\hat{w}$. So the compound gamble $\lambda *M^{\alpha ^{*}}\left( G\right) +\left( 1-\lambda \right) *g_{\min }^{u,\hat{w} }$ is accepted at $\hat{w}$ by $u$ but rejected at $\hat{w}$ by $u_{ \widehat{\alpha }}$, in contradiction to the assumption.

$\square$

We are now ready for the main proposition on Aumann–Serrano riskiness of sets. Proposition 31 identifies every set of gambles $G$ with a unique compound gamble $g_{\alpha ^{*}}$ on the grounds of wealth-uniform rejection. As a result, the complete ranking of gambles induces a complete ranking of sets of gambles.

Proposition 31

For every $u\in U^{*}$, $u$ rejects $g_{\alpha ^{*}}$ $\forall w>0$ if and only if $u$ rejects $G$ $\forall w>0$. $\square$

Proof

Direction (1): $u$ rejects $g_{\alpha ^{*}}$ $\forall w>0\Rightarrow$ $u$ rejects $G$ $\forall w>0$.

Indeed if $u$ rejects $g_{\alpha ^{*}}$ at some $w^{\prime }$, then from Proposition 4(iii) in Hart (2011) we know that $\rho _{u}(w^{\prime }-L_{g_{\alpha ^{*}}})\ge \alpha ^{*}$. Since $u$ rejects $g_{\alpha ^{*}}$ $\forall w>0$, we get that $\rho _{u}(w)\ge \alpha ^{*}$ $\forall w>0$, and from Lemma 30(i) we conclude that $u$ rejects $G$ $\forall w>0$.

Direction (2): $u$ rejects $G$ $\forall w>0\Rightarrow$ $u$ rejects $g_{\alpha ^{*}}$ $\forall w>0$.

Assume by negation that $u$ accepts $g_{\alpha ^{*}}$ at some $w^{\prime }$. Then from Proposition 4(ii) in Hart (2011) we get that $\rho _{u}(w^{\prime }+K_{g_{\alpha ^{*}}})<\alpha ^{*}$. From the decreasing absolute risk aversion of $u$ we then get that $\forall w>w^{\prime }+K_{g_{\alpha ^{*}}}$, $\rho _{u}(w)<\alpha ^{*}$. In particular let $w^{\prime \prime }\equiv w^{\prime }+K_{g_{\alpha ^{*}}}+L_{g_{\alpha ^{*}}}$. Then $\rho _{u}(w)<\alpha ^{*}$ $\forall w\in \left[ w^{\prime \prime }-L_{g_{\alpha ^{*}}},w^{\prime \prime }+K_{g_{\alpha ^{*}}}\right]$, hence from Lemma 30(ii) we know that $u$ accepts $G$ at $w^{\prime \prime }$, in contradiction to the assumption. $\square$

As defined in the paper, $R_{\lambda }^\mathrm{AS}\left( G\right)$ is the reciprocal of the (unique) $\alpha$ such that a $CARA$ person with risk-aversion parameter $\alpha$ and uncertainty-aversion parameter $\lambda$ is indifferent between accepting and rejecting the set $G$. Since we know that this $CARA$ person has parameter $\alpha ^{*}(G)$, and since $R^\mathrm{AS}\left( g_{\alpha ^{*}(G)}\right)$ is the reciprocal of $\alpha ^{*}(G)$ (from Theorem $B$ in Aumann and Serrano (2008), while recalling that a $CARA$ person with parameter $\alpha ^{*}(G)$ is indifferent between accepting and rejecting $g_{\alpha ^{*}(G)}$), we get that $R_{\lambda }^\mathrm{AS}\left( G\right) =$ $R^\mathrm{AS}\left( g_{\alpha ^{*}(G)}\right)$.

We are now ready to prove the main theorem for the Aumann–Serrano riskiness index and the properties of this index.

Proof of Theorem 9

From Proposition 31 we know that $G\ge _{WU}^{\lambda }H$ if and only if $g_{\alpha ^{*}(G)}\ge _{WU}h_{\alpha ^{*}(H)}$. Then from Hart (2011) we get $g_{\alpha ^{*}(G)}\ge _{WU}h_{\alpha ^{*}(H)}\Leftrightarrow R_{\lambda }^\mathrm{AS}\left( g_{\alpha ^{*}}\right) \le R_{\lambda }^\mathrm{AS}\left( h_{\alpha ^{*}}\right)$ $\Leftrightarrow$ $R_{\lambda }^\mathrm{AS}\left( G\right) \le R_{\lambda }^\mathrm{AS}\left( H\right)$. $\square$

Proof of Proposition 17—for the Aumann–Serrano riskiness index

• 1. Let $\alpha ^{*}(G)\equiv \alpha ^{*}$. Since $-Ee^{-\alpha ^{*}M^{\alpha ^{*}}\left( G\right) }\ge -Ee^{-\alpha ^{*}g}$ $\ \forall g\in G$, we have $-Ee^{-\frac{\alpha ^{*}}{k}kM^{\alpha ^{*}}\left( G\right) }\ge -Ee^{-\frac{\alpha ^{*}}{k}kg}$ $\ \forall kg\in kG$, which means that a $CARA$ utility with risk-aversion coefficient $\alpha _{k}^{*}\equiv \frac{\alpha ^{*}}{k}$ prefers the gamble $kM^{\alpha ^{*}}\left( G\right) \in kG$ to any other gamble in $kG$, i.e., $M^{\alpha _{k}^{*}}\left( kG\right) =kM^{\alpha ^{*}}\left( G\right)$. Similarly, $m^{\alpha _{k}^{*}}\left( kG\right) =km^{\alpha ^{*}}\left( G\right)$. Writing down the expected utility that a $CARA$ person with risk-aversion coefficient $\alpha _{k}^{*}$ gets from accepting the set $kG$ (at $w=0$, for simplicity), and using the indifference of a $CARA$ person with risk-aversion coefficient $\alpha ^{*}$ toward $G$, we get that

  $$\begin{aligned}&\lambda E[e^{-\alpha _{k}^{*}M^{\alpha _{k}^{*}}\left( kG\right) }]+\left( 1-\lambda \right) E\left[ e^{-\alpha _{k}^{*}m^{\alpha _{k}^{*}}\left( kG\right) }\right] \\&\quad =\lambda E[e^{-\alpha ^{*}M^{\alpha ^{*}}\left( G\right) }]+\left( 1-\lambda \right) E\left[ e^{-\alpha ^{*}m^{\alpha ^{*}}\left( G\right) }\right] =1, \end{aligned}$$

  i.e., the $CARA$ utility with risk-aversion coefficient $\alpha _{k}^{*}= \frac{\alpha ^{*}}{k}$ is indifferent toward the set of gambles $kG$, thus $R_{\lambda }^\mathrm{AS}\left( kG\right)$ $=\frac{1}{\alpha _{k}^{*}}= \frac{k}{\alpha ^{*}}=$ $kR_{\lambda }^\mathrm{AS}\left( G\right)$.

• 2. Let $\alpha ^{**}$ denote the risk-aversion coefficient of a $CARA$ utility that is indifferent to $k*G$, the $k$-dilution of $G$. Expressing this indifference at some arbitrary $w$ we get that

  $$\begin{aligned} -e^{-\alpha ^{**}w}&= k\left[ \lambda \left( -Ee^{-\alpha ^{**}\left( w+M^{\alpha ^{**}}\left( G\right) \right) }\right) +\left( 1-\lambda \right) \left( -Ee^{-\alpha ^{**}\left( w+m^{\alpha ^{**}}\left( G\right) \right) }\right) \right] \\&+\left( 1-k\right) \left( -e^{-\alpha ^{**}w}\right) \\&\Rightarrow ke^{-\alpha ^{**}w}=k\left[ \lambda Ee^{-\alpha ^{**}\left( w+M^{\alpha ^{**}}\left( G\right) \right) }+\left( 1-\lambda \right) Ee^{-\alpha ^{**}\left( w+m^{\alpha ^{**}}\left( G\right) \right) }\right] \\&\Rightarrow 1=\lambda Ee^{-\alpha ^{**}M^{\alpha ^{**}}\left( G\right) }+\left( 1-\lambda \right) Ee^{-\alpha ^{**}m^{\alpha ^{**}}\left( G\right) }\text {.} \end{aligned}$$

  By the uniqueness of $\alpha ^{*}(G)$ (Corollary 29) we thus get that $\alpha ^{**}=\alpha ^{*}(G)$. That is, indifference to $k*G$ implies the same constant risk-aversion coefficient as indifference to $G$, and so $R_{\lambda }^\mathrm{AS}\left( k*G\right) = R_{\lambda }^\mathrm{AS}\left( G\right)$.

$\square$

1.1.1 The duality axiom and the Aumann–Serrano riskiness index

Proof of Proposition 12

Take $u_{1},u_{2}\in U$ with $V_{u_{1},\lambda }\trianglerighteq V_{u_{2},\lambda }$, and take $G,H\in \mathbb {G}$ such that $G\ge _{WU}^{\lambda }H$. By Theorem 9, $R_{\lambda }^\mathrm{AS}\left( H\right) \ge R_{\lambda }^\mathrm{AS}\left( G\right)$, and so $\alpha ^{*}\left( G\right) \ge \alpha ^{*}\left( H\right)$. If $G$ is rejected by $V_{u_{2},\lambda }$ at $w$, then $\inf \rho _{u_{1}}\ge \sup \rho _{u_{2}}\ge \alpha ^{*}\left( G\right)$ (the first inequality by Lemma 18 in Hart 2010, and the second by Lemma 30(ii)), and so $\inf \rho _{u_{1}}\ge \alpha ^{*}\left( H\right)$. Thus $H$ is rejected by $V_{u_{1},\lambda }$ at $w$ by Lemma 30(i).

As for the opposite direction, assume that $G\ngeq _{WU}^{\lambda }H$, and so $\alpha ^{*}\left( G\right) <\alpha ^{*}\left( H\right)$. Take $\alpha$ and $\beta$ such that $\alpha <\alpha ^{*}\left( G\right) <\beta <\alpha ^{*}\left( H\right)$, and set $u_{1}\equiv u_{\beta }$ and $u_{2}\equiv u_{\alpha }$. Then $V_{u_{1},\lambda }\trianglerighteq V_{u_{2},\lambda }$ (by Lemma 18 in Hart 2010), $G$ is rejected by $V_{u_{2},\lambda }$ at all $w$ (by Lemma 28(ii)), while $H$ is accepted by $V_{u_{1},\lambda }$ at every $w$ (by the same lemma), and so $G\ngeq _{D}^{\lambda }H$. $\square$

1.1.2 Aumann–Serrano riskiness index of sets for maximin and maximax decision rules

We are now ready to prove Propositions 3 and 6 (1) as special cases of Theorem 9. Specifically, when DMs judge every set $G$ by their (subjective) worse gamble in the set (in terms of expected utility), $\lambda =0$, and we will prove that $R_{0}^\mathrm{AS}\left( G\right) =\underset{g\in G}{\max }R^\mathrm{AS}(g)$. Similarly, when DMs judge every set $G$ by their (subjective) best gamble in the set, $\lambda =1$, and we will prove that $R_{1}^\mathrm{AS}\left( G\right) = \underset{g\in G}{\min }R^\mathrm{AS}(g)$.

Proof of Proposition 3

Let $\bar{g}\in \underset{g\in G}{\arg \max }R^\mathrm{AS}(g)$ and $\bar{\alpha } \equiv \frac{1}{R^\mathrm{AS}(\bar{g})}$. Then by Theorem $B$ in Aumann and Serrano (2008), $u_{\bar{\alpha }}$ is indifferent to $\bar{g}$ (at all $w>0$). Therefore, every $CARA$ utility $u_{\alpha }$ with $\alpha >\bar{\alpha }$ rejects $\bar{g}$ at all $w>0$ hence rejects $G$ at all $w>0$ (judging $G$ by the worst case). Similarly, every $u_{\alpha }$ with $\alpha <\bar{\alpha }$ accepts $\bar{g}$ at all $w>0$ hence accepts $G$ at all $w>0$ (otherwise $u_{\alpha }$ rejects $m^{\alpha }\left( G\right)$ while accepting $\bar{g}$, in contradiction to $R^\mathrm{AS}(m^{\alpha }\left( G\right) )\le R^\mathrm{AS}(\bar{g})$). Then from Lemma 28, $u_{\bar{\alpha }}$ is indifferent to $G$ (at all $w>0$). Denote by $R_{0}^\mathrm{AS}\left( G\right)$ the reciprocal of the (unique) $\alpha$ such that a $CARA$ person with risk-aversion parameter $\alpha$ and uncertainty-aversion parameter $\lambda =0$ is indifferent between accepting and rejecting the set $G$. Then $R_{0}^\mathrm{AS}\left( G\right) =\frac{1}{\bar{\alpha }}=R^\mathrm{AS}(\bar{g})=\underset{ g\in G}{\max }R^\mathrm{AS}(g)$. Applying Theorem 9 we get $G\ge _{WU}H$, if and only if $\underset{h\in H}{\max }R^\mathrm{AS}(h)\ge \underset{g\in G}{\max }R^\mathrm{AS}(g)$. $\square$

Proof of Proposition 6—for the Aumann–Serrano riskiness index

Let $\underline{g}\in \underset{g\in G}{\arg \min }$ $R^\mathrm{AS}(g)$ and $\underline{\alpha }\equiv \frac{1}{R^\mathrm{AS}(\underline{g})}$. Then by Theorem $B$ in Aumann and Serrano (2008), $u_{\underline{\alpha }}$ is indifferent to $\underline{g}$ (at all $w>0$). Therefore, every $CARA$ utility $u_{\alpha }$ with $\alpha <\underline{\alpha }$ accepts $\underline{g}$ at all $w>0$ hence accepts $G$ at all $w>0$ (judging $G$ by the best case). Similarly, every $u_{\alpha }$ with $\alpha >\underline{\alpha }$ rejects $\underline{g}$ at all $w>0$ hence rejects $G$ at all $w>0$ (because rejection of $\underline{g}$ implies rejection of every $g\in G$, hence of $G$ itself). Then from Lemma 28, $u_{\underline{\alpha }}$ is indifferent to $G$ (at all $w>0$). Denote by $R_{1}^\mathrm{AS}\left( G\right)$ the reciprocal of the (unique) $\alpha$ such that a $CARA$ person with risk-aversion parameter $\alpha$ and uncertainty-aversion parameter $\lambda =1$ is indifferent between accepting and rejecting the set $G$. Then $R_{1}^\mathrm{AS}\left( G\right) =\frac{1}{\underline{\alpha }}=R^\mathrm{AS}(\underline{g })=\underset{g\in G}{\min }R^\mathrm{AS}(g)$. Applying Theorem 9 we get $G\ge _{WU}H$, if and only if $\underset{g\in G}{\min }R^\mathrm{AS}(h)\ge \underset{g\in G}{\min }R^\mathrm{AS}(g)$. $\square$

Aumann–Serrano riskiness index for maximin over behavioral sets

Proof of Proposition 25—for the Aumann–Serrano riskiness index

First note that if $F\!\left( a,P\right)$ is rejected by $(u,\epsilon )$ for every $\epsilon \in \left[ \underline{\epsilon },\overline{\epsilon }\right]$ and at all $w>0$, then in particular $(u,\underline{\epsilon })$ rejects $F\!\left( a,P\right)$ at all $w>0$. Likewise, for every wealth level $w$, if $(u,\underline{\epsilon })$ rejects $F\!\left( a,P\right)$ at $w$, then $F\!\left( a,P\right)$ is rejected at $w$ by $(u,\epsilon )$ for every $\epsilon \in \left[ \underline{\epsilon },\overline{\epsilon }\right]$, because $u(a,P)$ weakly decreases with $\epsilon$. Hence we see that $F\!\left( a,P\right)$ is rejected by $(u,\epsilon )$ for every $\epsilon \in \left[ \underline{\epsilon },\overline{\epsilon }\right]$ and at all $w>0$ if and only if $(u,\underline{\epsilon })$ rejects $F\!\left( a,P\right)$ at all $w>0$, i.e., if and only if $G\equiv F\!\left( a,\varphi _{\underline{ \epsilon }}(P)\right)$ is rejected by $u$ with MEU preferences at all $w>0$. So, saying that $a$ $\epsilon$-wealth-uniformly dominates $b$ under $P$ and $\left[ \underline{\epsilon },\overline{\epsilon }\right]$ is equivalent to saying that $G\equiv F\!\left( a,\varphi _{\underline{\epsilon } }(P)\right)$ wealth-uniformly dominates $H\equiv F\!\left( b,\varphi _{ \underline{\epsilon }}(P)\right)$, and from Proposition 3 we know this holds if and only if $\underset{h\in H}{\max }R^\mathrm{AS}(h)\ge \underset{g\in G}{\max }R^\mathrm{AS}(g)$, i.e., if and only if $\underset{p\in \varphi _{\underline{\epsilon }}(P)}{\max }R^\mathrm{AS}(F\!\left( b,p\right) )\ge \max \limits _{p\in \varphi _{\underline{\epsilon }}(P)}R^\mathrm{AS}(F\!\left( a,p\right) )$. $\square$

1.2 Foster–Hart riskiness measure of sets for $\lambda$-maximizers

Fix $\lambda \in \left[ 0,1\right]$ and let $V\equiv V_{u,\lambda }$.

For the log utility (i.e., $u_{\lg }(w)\equiv \log (w)$) we will denote a maximizer and a minimizer of the expected utility at wealth level $w$ by $M_{w}\left( G\right) \in \underset{g\in G}{\arg \max }E\left[ \log \left( w+g\right) \right]$ and $m_{w}\left( G\right) \in \underset{g\in G}{\arg \min }E\left[ \log \left( w+g\right) \right]$, respectively. For $u=$ $u_{\lg }$ with uncertainty-aversion coefficient $\lambda$ we use $V_{\lg }$.

Lemma 32

There is a unique $w^{\prime }$ s.t. $V_{\lg }\left( w^{\prime }+G\right) =\log (w^{\prime })$ (i.e., $u_{\lg }(w)$ is indifferent toward $G$ at $w^{\prime }$). Moreover, $u_{\lg }(w)$ accepts $G$ for every $w>w^{\prime }$ and rejects $G$ for every $w<w^{\prime }$.

Proof

Denote $\Delta u\equiv V_{\lg }\left( w+G\right) -u_{\lg }(w)=\lambda E[\log \left( w+M_{w}\left( G\right) \right) ]+\left( 1-\lambda \right)$ $E\left[ \log \left( w+m_{w}\left( G\right) \right) \right] -\log (w)$. $\Delta u$ is continuous in $w$ because the functions $E[\cdot ],\log (),\max ()$ and $\min ()$ are all continuous. By definition, $\Delta u>0$ if and only if $u_{\lg }(w)$ accepts $G$ at $w$. Let $R\left( G\right) , r\left( G\right)$ denote the gambles in $G$ with maximal and minimal values of $R^\mathrm{FH}$, respectively. Then $\forall w<r(G),$ $u_{\lg }(w)$ rejects every $g\in G$ hence rejects $G$ (i.e., $\Delta u<0$), and $\forall w>R(G),$ $u_{\lg }(w)$ accepts every $g\in G$ hence accepts³⁷ $G$ (i.e., $\Delta u>0$). From the continuity of $\Delta u$ we conclude that $\exists w^{\prime }>0$ s.t. $\Delta u=0$ at $w^{\prime }$, i.e., $V_{\lg }\left( w^{\prime }+G\right) =$ $u_{\lg }(w^{\prime })$.

Assume now by negation that $\exists w^{\prime \prime }<w^{\prime }$ s.t. $u_{\lg }(w)$ (weakly) accepts $G$ at $w^{\prime \prime }$. $u_{\lg }(w)$ is indifferent toward $\hat{g}\equiv \lambda *M_{w^{\prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime }}\left( G\right)$ at $w^{\prime }$, hence (weakly) rejects $g_{1}\equiv \lambda *M_{w^{\prime \prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime }}\left( G\right)$ at $w^{\prime }$ ($g_{1}$ is identical to $\hat{g}$ except for replacing the most preferred gamble at $w^{\prime }$ with potentially a different one). By assumption, $u_{\lg }(w)$ (weakly) accepts $G$ at $w^{\prime \prime }$, i.e., (weakly) accepts $\lambda *M_{w^{\prime \prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime \prime }}\left( G\right)$ at $w^{\prime \prime }$, hence (weakly) accepts $g_{1}$ at $w^{\prime \prime }$ (this time replacing the least preferred gamble with potentially a different one). So we get that $u_{\lg }(w)$ (weakly) accepts $g_{1}$ at $w^{\prime \prime }<w^{\prime }$ while (weakly) rejecting $g_{1}$ at $w^{\prime }$, in contradiction with the strictly decreasing absolute risk-aversion property of the log utility.

Similarly, assume now by negation that $\exists w^{\prime \prime }>w^{\prime }$ s.t. $u_{\lg }(w)$ (weakly) rejects $G$ at $w^{\prime \prime }$. $u_{\lg }(w)$ is indifferent toward $\hat{g}$ at $w^{\prime }$, hence (weakly) accepts $g_{2}\equiv \lambda *M_{w^{\prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime \prime }}\left( G\right)$ at $w^{\prime }$. By assumption, $u_{\lg }(w)$ (weakly) rejects $G$ at $w^{\prime \prime }$, i.e., (weakly) rejects $\lambda *M_{w^{\prime \prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime \prime }}\left( G\right)$ at $w^{\prime \prime }$, hence (weakly) rejects $g_{2}$ at $w^{\prime \prime }$. So we get that $u_{\lg }(w)$ (weakly) rejects $g_{2}$ at $w^{\prime \prime }>w^{\prime }$ while (weakly) accepting $g_{2}$ at $w^{\prime }$, in contradiction with the strictly decreasing absolute risk-aversion property of the log utility. $\square$

Denote $l(G)\equiv \lambda *M_{w^{\prime }}\left( G\right) +(1-\lambda )*m_{w^{\prime }}\left( G\right)$ (where $w^{\prime }$ is the unique $w$ s.t. $u_{\lg }(w)$ is indifferent toward $G$ at $w$). Then we have:

Corollary 33

$u_{\lg }(w)$ accepts $G$ at some $w$ if and only if $u_{\lg }(w)$ accepts $l\left( G\right)$ at that $w$.

Proof

The strictly decreasing absolute risk-aversion property of the log utility implies that $\exists w^{*}$ such that $u_{\lg }(w)$ is indifferent toward $l\left( G\right)$ at $w^{*}$, accepts $l\left( G\right)$ $\ \forall w>$ $w^{*}$, and rejects $l\left( G\right)$ $\ \forall w<$ $w^{*}$.³⁸ Since $u_{\lg }(w)$ is indifferent toward $G$ and therefore toward $l\left( G\right)$ at $w^{\prime }$, we get that $w^{*}=w^{\prime }$, hence $u_{\lg }(w)$ accepts both $G$ and $l\left( G\right)$ $\ \forall w>$ $w^{\prime }$, and rejects both $G$ and $l\left( G\right)$ $\ \forall w<w^{*}$. $\square$

We are now ready for the main proposition on Foster–Hart riskiness of sets, encompassing the more general set of utilities $U^{*}$. Proposition 34 identifies every set of gambles $G$ with the unique compound gamble $l(G)$ on the grounds of utility-uniform rejection. As a result, the complete ranking of gambles induces a complete ranking of sets of gambles.

Proposition 34

$\forall w>0$, every $u\in U^{*}$ rejects $l(G)$ at $w$ if and only if every $u\in U^{*}$ rejects $G$ at $w$.

Proof

Direction (1): $\forall w>0$, every $u\in U^{*}$ rejects $G$ at $w$ $\Rightarrow$ every $u\in U^{*}$ rejects $l(G)$ at $w$.

Let $w>0$ be an arbitrary wealth level. Every $u\in U^{*}$ rejects $G$ at $w$, so $u_{\lg }(w)\in U^{*}$ rejects $G$ at $w$ either, and by Corollary 33 $u_{\lg }(w)$ rejects $l(G)$ at $w$. Therefore $R^\mathrm{FH}(l(G))\ge w$ (Foster and Hart 2009, Section VI(B)), and by Lemma 10 in Hart (2011) we get that every $u\in U^{*}$ rejects $l(G)$ at $w$.

Direction (2): $\forall w>0$, every $u\in U^{*}$ rejects $l(G)$ at $w$ $\Rightarrow$ every $u\in U^{*}$ rejects $G$ at $w$.

If every $u\in U^{*}$ rejects $l(G)$ at $w$, then $u_{\lg }(w)$ rejects $l(G)$ at $w$ either, and by Corollary 33 $u_{\lg }(w)$ rejects $G$ at $w$. Assume now by negation that $\exists u^{\prime }\in U^{*}$ s.t. $u^{\prime }$ accepts $G$ at $w$. That is, $u^{\prime }$ accepts $\lambda *g_{\max }^{u^{\prime },w}+(1-\lambda )*g_{\min }^{u^{\prime },w}$ at $w$. Consequentially, $u^{\prime }$ accepts $g^{\prime }\equiv \lambda *g_{\max }^{u^{\prime },w}+(1-\lambda )*m^{w}\left( G\right)$ at $w$ too. $u_{\lg }(w)$ rejects $G$ at $w$, hence rejects $\lambda *M_{w}\left( G\right) +(1-\lambda )*m_{w}\left( G\right)$ at $w$. Consequentially, $u_{\lg }(w)$ rejects $g^{\prime }$ at $w$ too. But this means that $R^\mathrm{FH}(g^{\prime })\ge w$, and by Lemma 10 in Hart (2011) every $u\in U^{*}$ should reject $g^{\prime }$ at $w$, in contradiction to the assumption that $u^{\prime }$ accepts $g^{\prime }$ at $w$. $\square$

As defined in the paper, we denote by $R_{\lambda }^\mathrm{FH}\left( G\right)$ the (unique) wealth level $w$ at which a logarithmic utility decision maker with uncertainty-aversion parameter $\lambda$ is indifferent between accepting and rejecting the set $G$. But this wealth level is exactly $w^{\prime }$, and since the indifference of $u_{\lg }(w)$ toward $G$ at $w^{\prime }$ implies the indifference of $u_{\lg }(w)$ toward $l\left( G\right)$ at $w^{\prime }$, $w^{\prime }$ equals $R^\mathrm{FH}\left( l(G)\right)$, and we get that $R_{\lambda }^\mathrm{FH}\left( G\right) =$ $R^\mathrm{FH}\left( l(G)\right)$.

We are now ready to prove the main theorem for the Foster–Hart riskiness measure and the properties of this measure.

Proof of Theorem 16

From Proposition 34 we know that $G\ge _{UU}^{\lambda }H$ if and only if $l(G)\ge _{UU}l(H)$. Then from Hart (2011) we get $l(G)\ge _{UU}l(H)\Leftrightarrow R_{\lambda }^\mathrm{FH}\left( l(G)\right) \le R_{\lambda }^\mathrm{FH}\left( l(H)\right)$ $\Leftrightarrow$ $R_{\lambda }^\mathrm{FH}\left( G\right) \le R_{\lambda }^\mathrm{FH}\left( H\right)$. $\square$

Proof of Proposition 17—for the Foster–Hart riskiness measure

• 1. Since $M_{w}\left( G\right)$ maximizes the expected log utility at $w$ over the set of gambles $G$, we have $E\left[ \log \left( w+M_{w}\left( G\right) \right) \right] \ge E\left[ \log \left( w+g\right) \right]$ $\forall g\in G$. Adding $\log \left( k\right)$ to both sides of the inequality, we get that $E\left[ \log \left( kw+kM_{w}\left( G\right) \right) \right] \ge E\left[ \log \left( kw+kg\right) \right]$ $\forall g\in G$, i.e., $kM_{w}\left( G\right) \in kG$ maximizes the expected log utility at $kw$ over the set $kG$. Similarly, $km_{w}\left( G\right)$ minimizes the expected log utility at $kw$ over $kG$. Using the fact that $w^{\prime }$ is the (unique) solution to the equation $\log (w)=\lambda E[\log \left( w+M_{w}\left( G\right) \right) ]+\left( 1-\lambda \right) E \left[ \log \left( w+m_{w}\left( G\right) \right) \right]$, we get that the expected log utility of accepting the set $kG$ at $kw^{\prime }$ is given by

  $$\begin{aligned}&\lambda E[\log \left( kw^{\prime }+kM_{w^{\prime }}\left( G\right) \right) ]+\left( 1-\lambda \right) E\left[ \log \left( kw^{\prime }+km_{w^{\prime }}\left( G\right) \right) \right] \\&\quad =\lambda \left[ E[\log \left( w^{\prime }+M_{w^{\prime }}\left( G\right) \right) ]+\log (k)\right] +\left( 1-\lambda \right) \left[ E\left[ \log \left( w^{\prime }+m_{w^{\prime }}\left( G\right) \right) \right] \right. \\&\qquad \left. +\log (k) \right] =\log (kw^{\prime }), \end{aligned}$$

  i.e., the log utility is indifferent toward $kG$ at $kw^{\prime }$. By the definition of $R_{\lambda }^\mathrm{FH}$, we thus get that $R_{\lambda }^\mathrm{FH}\left( kG\right)$ $=kw^{\prime }=$ $kR_{\lambda }^\mathrm{FH}\left( G\right)$.

• 2. Let $w^{\prime \prime }$ denote the wealth level at which the log utility is indifferent to $k*G$. We then have

  $$\begin{aligned} \log (w^{\prime \prime })&= k\left[ \lambda E[\log \left( w^{\prime \prime }+M_{w^{\prime \prime }}\left( G\right) \right) ]+\left( 1-\lambda \right) E \left[ \log \left( w^{\prime \prime }+m_{w^{\prime \prime }}\left( G\right) \right) \right] \right] \\&+\left( 1-k\right) \left( \log (w^{\prime \prime })\right) \Rightarrow k\log (w^{\prime \prime })=k\left[ \lambda E[\log \left( w^{\prime \prime }+M_{w^{\prime \prime }}\left( G\right) \right) ]\right. \\&\left. +\left( 1-\lambda \right) E\left[ \log \left( w^{\prime \prime }+m_{w^{\prime \prime }}\left( G\right) \right) \right] \right] \text {.} \end{aligned}$$

  Dividing both sides of the equation by $k$ we get that the log utility is indifferent to $G$ at $w^{\prime \prime }$, and by the uniqueness of $w^{\prime }$ (Lemma 32) we get that $w^{\prime \prime }=w^{\prime }$, hence $R_{\lambda }^\mathrm{FH}(k*G)=R_{\lambda }^\mathrm{FH}(G)$.

$\square$

1.2.1 Foster–Hart riskiness measure of sets for maximin and maximax decision rules

We are now ready to prove Propositions 4 and 6 (2) as special cases of Theorem 16. Specifically, when DMs judge every set $G$ by their (subjective) worse gamble in the set (in terms of expected utility), $\lambda =0$, and we will prove that $R_{0}^\mathrm{FH}\left( G\right) =\underset{g\in G}{\max }R^\mathrm{FH}(g)$. similarly, when DMs judge every set $G$ by their (subjective) best gamble in the set, $\lambda =1$, and we will prove that $R_{1}^\mathrm{FH}\left( G\right) = \underset{g\in G}{\min }R^\mathrm{FH}(g)$.

Proof of Proposition 4

$R_{0}^\mathrm{FH}\left( G\right)$ equals the wealth level $w^{\prime }$ at which a log utility $u_{\lg }(w)$ with uncertainty-aversion parameter $\lambda =0$ is indifferent to $G$. Let $\bar{g}\in \underset{g\in G}{\arg \max } R^\mathrm{FH}(g)$. $\forall w>R^\mathrm{FH}(\bar{g})$ we have $w>R^\mathrm{FH}(g)$ $\forall g\in G$, hence by Foster and Hart (2009) $u_{\lg }(w)$ accepts every gamble in $G$ at $w$, i.e., $u_{\lg }(w)$ accepts $G$ at $w$. Conversely, $\forall w<R^\mathrm{FH}(\bar{g})$ we know that $u_{\lg }(w)$ rejects $\bar{g}$ at $w$, and therefore also rejects $G$ (judged by the worst case) at $w$. From the continuity of $u_{\lg }(w)$ we then get that $R_{0}^\mathrm{FH}\left( G\right) =R^\mathrm{FH}(\bar{g})=\underset{g\in G}{\max }R^\mathrm{FH}(g)$. Applying Theorem 16 we finally get that $G\ge _{WU}H$ if and only if $\underset{h\in H}{\max }R^\mathrm{FH}(h)\ge \underset{g\in G}{\max }R^\mathrm{FH}(g)$. $\square$

Proof of Proposition 6—for the Foster–Hart riskiness measure

$R_{1}^\mathrm{FH}\left( G\right)$ equals the wealth level $w^{\prime \prime }$ at which a log utility $u_{\lg }(w)$ with uncertainty-aversion parameter $\lambda =1$ is indifferent to $G$. Let $\underline{g}\in \underset{g\in G}{ \arg \min }R^\mathrm{FH}(g)$. $\forall w<R^\mathrm{FH}(\underline{g})$ we have $w<R^\mathrm{FH}(g)$ $\forall g\in G$, hence $u_{\lg }(w)$ rejects every gamble in $G$ at $w$, i.e., $u_{\lg }(w)$ rejects $G$ at $w$. Conversely, $\forall w>R^\mathrm{FH}(\bar{g})$ we know that $u_{\lg }(w)$ accepts $\underline{g}$ at $w$, and therefore also accepts $G$ (judged by the best case) at $w$. From the continuity of $u_{\lg }(w)$ we then get that $R_{1}^\mathrm{FH}\left( G\right) =R^\mathrm{FH}(\underline{g})=\underset{g\in G}{\min }R^\mathrm{FH}(g)$. Applying Theorem 16 we finally get that $G\ge _{WU}H$ if and only if $\underset{h\in H}{\min }R^\mathrm{FH}(h)\ge \underset{g\in G}{\min }R^\mathrm{FH}(g)$. $\square$

Foster–Hart riskiness as critical wealth

Proof of Corollary 5

Accepting a set $G$ with wealth $w\ge R^\mathrm{FH}(g)$ for every $g\in G$ guarantees no bankruptcy (Theorem 1 in Foster and Hart 2009), regardless of the way $g$ is chosen from $G$. On the other hand, if $G$ is accepted when $w<\underset{g\in G}{\max }R^\mathrm{FH}(g)$, then $\bar{g}\in \underset{g\in G}{ \arg \max }R^\mathrm{FH}(g)$ may be chosen, and then no bankruptcy cannot be guaranteed. $\square$

Foster–Hart riskiness measure for maximin over behavioral sets

Proof of Proposition 25—for the Foster–Hart riskiness measure

First note that if $F\!\left( a,P\right)$ is rejected at $w$ by every $(u,\epsilon )$ with $u\in U^{*}$ and $\epsilon \in \left[ \underline{ \epsilon },\overline{\epsilon }\right]$, then in particular it is rejected by every $(u,\underline{\epsilon })$ with $u\in U^{*}$. Likewise, $\forall u\in U^{*}$, if $(u,\underline{\epsilon })$ rejects $F\!\left( a,P\right)$ at some $w$, then $F\!\left( a,P\right)$ is rejected at $w$ by $(u,\epsilon )$ for every $\epsilon \in \left[ \underline{\epsilon }, \overline{\epsilon }\right]$, because $u(a,P)$ weakly decreases with $\epsilon$. Hence we see that $F\!\left( a,P\right)$ is rejected at $w$ by every $(u,\epsilon )$ with $u\in U^{*}$ and $\epsilon \in \left[ \underline{\epsilon },\overline{\epsilon }\right]$ if and only if $F\!\left( a,P\right)$ is rejected at $w$ by every $(u,\underline{\epsilon })$ with $u\in U^{*}$, i.e., if and only if $G\equiv F\!\left( a,\varphi _{ \underline{\epsilon }}(P)\right)$ is rejected at $w$ by every $u\in U^{*}$ with MEU preferences. So, saying that $a$ $\epsilon$-utility-uniformly dominates $b$ under $P$ and $\left[ \underline{\epsilon }, \overline{\epsilon }\right]$ is equivalent to saying that $G\equiv F\!\left( a,\varphi _{\underline{\epsilon }}(P)\right)$ utility-uniformly dominates $H\equiv F\!\left( b,\varphi _{\underline{\epsilon }}(P)\right)$, and from Proposition 4 we know this holds if and only if $\underset{ h\in H}{\max }R^\mathrm{FH}(h)\ge \underset{g\in G}{\max }R^\mathrm{FH}(g)$, i.e., if and only if $\underset{p\in \varphi _{\underline{\epsilon }}(P)}{\max } R^\mathrm{FH}(F\!\left( b,p\right) )\ge \max \limits _{p\in \varphi _{\underline{ \epsilon }}(P)}R^\mathrm{FH}(F\!\left( a,p\right) )$. $\square$

1.2.2 Extrema of $R^\mathrm{FH}(g)$ in sets of gambles

We now prove that if the set of gambles $G$ is compact, and has a fixed and finite support, then $R^\mathrm{FH}(\cdot )$ gets its minimum and maximum values over every $G\in \mathbb {G}$, although $R^\mathrm{FH}(\cdot )$ is not a continuous function.

Lemma 35

If $G\subset \mathcal {G}$ is compact and has finite support, then both $\sup R^\mathrm{FH}(g)$ and $\inf R^\mathrm{FH}(g)$ are attained in $G$.

Proof

• (i) Let $\underline{\rho }$ $\equiv \inf \limits _{g\in G}R^\mathrm{FH}(g)$, and let $\left( g_{n}\right) _{n=1,2,\ldots }$ be a sequence in $G$ with $R^\mathrm{FH}\left( g_{n}\right) \rightarrow$ $\underline{\rho }$. Since $G$ is compact, and its support is finite, we can take w.l.o.g. a subsequence $g_{n}^{\prime }$ in $g_{n}$ such that $g_{n}^{\prime }\rightarrow g_{0}\in G$ and $L_{g_{n}^{\prime }}\equiv L_{0}$ for all $n$. Proposition 10 in Foster and Hart (2009) implies that $\underline{\rho }$ $=\lim R^\mathrm{FH}\left( g_{n}^{\prime }\right) =\max \left\{ R^\mathrm{FH}(g_{0}),L_{0}\right\} \ge R^\mathrm{FH}(g_{0})\ge$ $\underline{\rho }$. Therefore all inequalities are equalities, and so $\underline{\rho }$ $=R^\mathrm{FH}(g_{0})$.

• (ii) Let $\bar{\rho }\equiv \sup \limits _{g\in G}R^\mathrm{FH}(g)$, and let $\left( g_{n}\right) _{n=1,2,\ldots }$ be a sequence in $G$ with $R^\mathrm{FH}\left( g_{n}\right) \rightarrow \bar{\rho }$. W.l.o.g. let $g_{n}^{\prime }$ a subsequence in $g_{n}$ such that $R^\mathrm{FH}\left( g_{n}^{\prime }\right)$ is increasing, $g_{n}^{\prime }\rightarrow g_{0}\in G$ and $L_{g_{n}^{\prime }}\equiv L_{0}$ for all $n$. Since $L_{0}=L_{g_{n}^{\prime }}<R^\mathrm{FH}\left( g_{n}^{\prime }\right)$, and $R^\mathrm{FH}\left( g_{n}^{\prime }\right)$ increases to $\bar{\rho }$, we have $\bar{\rho }>L_{0}$. By applying Proposition 10 in Foster and Hart (2009) we get that $\bar{\rho }=\lim R^\mathrm{FH}\left( g_{n}^{\prime }\right) =\max \left\{ R^\mathrm{FH}(g_{0}),L_{0}\right\} >L_{0}$, so $\bar{\rho }=R^\mathrm{FH}(g_{0})$.

$\square$

1.3 Subadditivity and convexity

Proof of Proposition 18

From the proofs to Propositions 3 and 4 we know that $R_{0}^\mathrm{AS}\left( G\right) =\underset{g\in G}{\max }R^\mathrm{AS}(g)$ and $R_{0}^\mathrm{FH}\left( G\right) =\underset{g\in G}{\max }R^\mathrm{FH}(g)$, and from the proof to Proposition 6 we know that $R_{1}^\mathrm{AS}\left( G\right) =\underset{g\in G}{\min }R^\mathrm{AS}(g)$ and $R_{1}^\mathrm{FH}\left( G\right) =\underset{g\in G}{\min }R^\mathrm{FH}(g)$. Let $R\left( \cdot \right)$ denote either $R^\mathrm{AS}\left( \cdot \right)$ or $R^\mathrm{FH}\left( \cdot \right)$.

• 1. (i) If $\lambda =0$, then $R_{0}\left( G+H\right) =\underset{q\in \left\{ G+H\right\} }{\max }R(q)$. The subadditivity of $R\left( \cdot \right)$ as a measure of riskiness for single gambles (see Aumann and Serrano 2008, Section V.H and Foster and Hart 2009, Section V.iv) implies that for any $\bar{q}\in \underset{q\in \left\{ G+H\right\} }{\arg \max } R(q)$, $\exists \acute{g}\in G,\acute{h}\in H$ such that $R(\bar{q})\le R( \acute{g})+R(\acute{h})$. Thus $R(\bar{q})\le \underset{g\in G}{\max }R(g)$ +$\underset{h\in H}{\max }R(h)$, i.e., $R_{0}\left( G+H\right) \le R_{0}(G)+R_{0}(H)$.

• (ii) If $\lambda =1$, then $R_{1}\left( G+H\right) =\underset{q\in \left\{ G+H\right\} }{\min }R(q)$. Let $\underline{g}\in \underset{g\in G}{\arg \min }R(g)$, $\underline{h}\in \underset{h\in H}{\arg \min }R(h)$, and $\underline{q}\equiv \underline{g}+\underline{h}$. Then $R_{1}\left( G+H\right) \le R(\underline{q})$. The subadditivity of $R\left( \cdot \right)$ as a measure of riskiness for single gambles implies that $R( \underline{q})\le R(\underline{g})$ $+R(\underline{h})=R_{1}(G)+R_{1}(H)$. Thus $R_{1}\left( G+H\right) \le R_{1}(G)+R_{1}(H)$.

• 2. Convexity follows directly from subadditivity and homogeneity.

$\square$