Mixed extensions of decision problems under uncertainty

Abstract

In a decision problem under uncertainty, a decision maker considers a set of alternative actions whose consequences depend on uncertain factors beyond his control. Following Luce and Raiffa (Games and decisions: introduction and critical survey. Wiley, New York, 1957), we adopt a natural representation of such a situation which takes as primitives a set of conceivable actions A, a set of states S and a consequence function $\rho {:}\,A\times S\rightarrow C$. Each action induces a map from states to consequences, a Savage act, and each mixed action induces a map from states to probability distributions over consequences, an Anscombe–Aumann act. Under an axiom of consequentialism, preferences over pure or mixed actions yield corresponding preferences over the induced acts. This observation allows us to relate the Luce–Raiffa description of a decision problem to the most common framework of modern decision theory which directly takes as primitive a preference relation over the set of all Anscombe–Aumann acts. The key advantage of the latter framework is the possibility of applying powerful convex analysis techniques as in the seminal work of Schmeidler (Econometrica 57:571–587, 1989) and the vast literature that followed. This paper shows that we can maintain the mathematical convenience of the Anscombe–Aumann framework within a description of decision problems which is closer to many applications and experiments. We argue that our framework is more expressive as it allows us to be both explicit and parsimonious about the assumed richness of the set of conceivable actions, and to directly capture preference for randomization as an expression of uncertainty aversion.

Appendices

Appendix 1: Anscombe–Aumann axioms

Here we report the axioms that are usually stated in the Anscombe–Aumann framework. In this section $\succsim _{\mathcal {F}}$ and $\succsim _{\mathcal {F}}^{\#}$ are binary relations on a convex subset $\mathcal {F}$ of $\Delta \left( C\right) ^{S}$ that contains the set $\mathcal {C}\simeq \Delta \left( C\right)$ of all constant acts.

Axiom AA.1

(Weak order) $\succsim _{\mathcal {F}}$ is complete and transitive.

Axiom AA.2

(Risk independence) For all $\gamma ,\zeta ,\xi \in \mathcal {C}$, $\gamma \sim _{\mathcal {F}}\zeta$ implies $\dfrac{1}{2}\gamma +\dfrac{1}{2} \xi \sim _{\mathcal {F}}\dfrac{1}{2}\zeta +\dfrac{1}{2}\xi$.

Axiom AA.3

(Monotonicity) For all $f,g\in \mathcal {F}$, if $f(s)\succsim _{\mathcal {F}}g(s)$ for all $s\in S$, then $f\succsim _{\mathcal {F}}g$.

Axiom AA.4

(Continuity) For all $f,g,h\in \mathcal {F}$, $\left\{ q\in \left[ 0,1\right] :qf+\left( 1-q\right) g\succsim _{\mathcal {F} }h\right\}$ and $\left\{ q\in \left[ 0,1\right] :h\succsim _{\mathcal {F} }qf+\right.$ $\left. \left( 1-q\right) g\right\}$ are closed sets.

Axiom AA.5

(Default to certainty) For all $f\in \mathcal {F}$ and $\gamma \in \mathcal {C}$, $f\not \succsim _{\mathcal {F}}^{\#}\gamma$ implies $\gamma \succ _{\mathcal {F}}f$.

Axiom AA.6

(Independence) For all $f,g,h\in \mathcal {F}$, $f\sim _{\mathcal {F}}g$ implies $\dfrac{1}{2}f+\dfrac{1}{2}h\sim _{\mathcal {F}} \dfrac{1}{2}g+\dfrac{1}{2}h$.

Axiom AA.7

(Non-triviality) There exist $f,g\in \mathcal {F}$ such that $f\succ _{\mathcal {F}}g$.

Axiom AA.8

(Uncertainty aversion) For all $f,g\in \mathcal {F}$ and $q\in \left( 0,1\right)$, $f\sim _{\mathcal {F}}g$ implies $qf+\left( 1-q\right) g\succsim _{\mathcal {F}}f$.

Axiom AA.9

(C-Independence) For all $f,g\in \mathcal {F}$, $\gamma \in \mathcal {C}$, and $q\in \left( 0,1\right]$,

$$\begin{aligned} f\succsim _{\mathcal {F}}g\iff qf+(1-q)\gamma \succsim _{\mathcal {F} }qg+(1-q)\gamma . \end{aligned}$$

Appendix 2: Proofs and related material

Throughout this appendix, if $\gamma \in \Delta \left( C\right)$ and $u\in {\mathbb {R}}^{C}$, we indifferently write

$$\begin{aligned} \text {either }{\sum \limits _{c\in C}\gamma }\left( c\right) u\left( c\right) \text { or }{\mathbb {E}}_{{\gamma }}\left[ u\right] \text { or }u\left( \gamma \right) \text {.} \end{aligned}$$

Proof of Proposition 1

• 1. For each $a\in A$, $\rho _{a}=\rho _{a}$ so that $a\approx a$ and by consequentialism $a\sim _{A}a$. Therefore $\succsim _{A}$ is reflexive and transitive, that is, a preorder. Conversely, let $\left( A,S,C,\rho \right)$ be reduced and $\succsim _{A}$ be a preorder. For every $a,b\in A$, $a\approx b$ implies (because of reduction) $a=b$, and reflexivity of $\succsim _{A}$ delivers $a\sim _{A}b$. Therefore $\succsim _{A}$ satisfies consequentialism.

• 2. Let $c,d\in C$ and $a_{c},b_{c},a_{d},b_{d}\in A$, not necessarily distinct, be such that

  $$\begin{aligned} \rho \left( a_{c},\cdot \right) =\rho \left( b_{c},\cdot \right) \equiv c\text { and }\rho \left( a_{d},\cdot \right) =\rho \left( b_{d},\cdot \right) \equiv d\text {.} \end{aligned}$$

  (23)

  These sure actions exist since all sure actions are conceivable. By consequentialism, $\succsim _{A}$ is a preorder and (23) implies $a_{c}\sim _{A}b_{c}$ and $a_{d}\sim _{A}b_{d}$. Therefore, by transitivity of $\succsim _{A}$,

  $$\begin{aligned} a_{c}\succsim _{A}a_{d}\iff b_{c}\succsim _{A}b_{d} \end{aligned}$$

  and $\succsim _{C}$ is well defined.⁴⁸ Reflexivity and transitivity of $\succsim _{C}$ follow from reflexivity and transitivity of $\succsim _{A}$.

• 3. The proof is similar to the one of the previous point, hence it is left to the reader. $\square$

Proof of Proposition 2

• 1. Fix $s\in S$. For each $c\in C$,

  $$\begin{aligned} \rho _{\delta _{a}}\left( c\mid s\right) =\delta _{a}\left( \left\{ b\in A:\rho \left( b,s\right) =c\right\} \right) =\left\{ \begin{array}{ll} 1 &{} \quad c=\rho \left( a,s\right) \\ 0 &{} \quad \text {otherwise} \end{array} \right. =\delta _{\rho \left( a,s\right) }\left( c\right) \end{aligned}$$

  that is $\rho _{\delta _{a}}\left( s\right) =\delta _{\rho \left( a,s\right) }$. Write x instead of $\delta _{x}$ if either $x\in A$ or $x\in C$, then

  $$\begin{aligned} \digamma \left( a\right) =\digamma \left( \delta _{a}\right) =\left[ \begin{array}{c} \rho _{\delta _{a}}\left( s_{1}\right) \\ \rho _{\delta _{a}}\left( s_{2}\right) \\ \vdots \\ \rho _{\delta _{a}}\left( s_{n}\right) \end{array} \right] =\left[ \begin{array}{c} \delta _{\rho \left( a,s_{1}\right) }\\ \delta _{\rho \left( a,s_{2}\right) }\\ \vdots \\ \delta _{\rho \left( a,s_{n}\right) } \end{array} \right] =\left[ \begin{array}{c} \rho \left( a,s_{1}\right) \\ \rho \left( a,s_{2}\right) \\ \vdots \\ \rho \left( a,s_{n}\right) \end{array} \right] =\rho _{a} \end{aligned}$$

  as desired.

• 2. Fix $s\in S$. For each $c\in C$,

  $$\begin{aligned} \rho _{q\alpha +\left( 1-q\right) \beta }\left( c\mid s\right)&=\left( q\alpha +\left( 1-q\right) \beta \right) \left( \left\{ a\in A:\rho \left( a,s\right) =c\right\} \right) \\&=q\alpha \left( \left\{ a\in A:\rho \left( a,s\right) =c\right\} \right) +\left( 1-q\right) \beta \left( \left\{ a\in A:\rho \left( a,s\right) =c\right\} \right) \\&=q\rho _{\alpha }\left( c\mid s\right) +\left( 1-q\right) \rho _{\beta }\left( c\mid s\right) \end{aligned}$$

  that is, $\rho _{q\alpha +\left( 1-q\right) \beta }=q\rho _{\alpha }+\left( 1-q\right) \rho _{\beta }$.

• 3. Let D be an arbitrary nonempty subset of C. Then $\Delta \left( D\right)$ is convex in ${\mathbb {R}}^{D}$, and its extreme points are the point-masses $\left\{ \delta _{d}\right\} _{d\in D}$. Therefore $\Delta \left( D\right) ^{S}$ is convex too. Next we show that its extreme points are the vectors of point-masses. If $g\in \Delta \left( D\right) ^{S}$ is not extreme there exist $g^{\prime },g^{\prime \prime }\in \Delta \left( D\right) ^{S}$ with $g^{\prime }\ne g^{\prime \prime }$, say $0\le g^{\prime }\left( d\mid s\right) <g^{\prime \prime }\left( d\mid s\right) \le 1$, and $q\in \left( 0,1\right)$ such that $g=qg^{\prime }+\left( 1-q\right) g^{\prime \prime }$. Then $g\left( d\mid s\right) =qg^{\prime }\left( d\mid s\right) +\left( 1-q\right) g^{\prime \prime }\left( d\mid s\right) \in \left( 0,1\right)$ and $g\left( s\right)$ is not a point-mass. Conversely, if $g\in \Delta \left( D\right) ^{S}$ is not a vector of point-masses, then $g\left( s\right)$ is not a point-mass for some $s\in S$. Therefore, there exist $g^{\prime }\left( s\right) ,g^{\prime \prime }\left( s\right) \in \Delta \left( D\right)$ with $g^{\prime }\left( s\right) \ne g^{\prime \prime }\left( s\right)$ and $q\in \left( 0,1\right)$ such that $g\left( s\right) =qg^{\prime }\left( s\right) +\left( 1-q\right) g^{\prime \prime }\left( s\right)$. But this implies that $g=qg^{\prime }+\left( 1-q\right) g^{\prime \prime }$ where $g^{\prime }$ (resp. $g^{\prime \prime }$) is obtained replacing the s-th element $g\left( s\right)$ of the vector g with $g^{\prime }\left( s\right)$ (resp. $g^{\prime \prime }\left( s\right)$). Thus g is not extreme.

In particular, the generic extreme point of $\Delta \left( D\right) ^{S}$ is

$$\begin{aligned} \left[ \begin{array}{c} \delta _{e\left( s_{1}\right) }\\ \delta _{e\left( s_{2}\right) }\\ \vdots \\ \delta _{e\left( s_{n}\right) } \end{array} \right] \text { with }\left[ \begin{array}{c} e\left( s_{1}\right) \\ e\left( s_{2}\right) \\ \vdots \\ e\left( s_{n}\right) \end{array} \right] =e\in D^{S}. \end{aligned}$$

If part. Assume $\left\{ \rho _{a}\right\} _{a\in A}=C^{S}$. For each $e\in C^{S}$, there exists $a_{e}\in A$ such that $\rho _{a_{e}}=e$. For each $f\in \Delta \left( C\right) ^{S}$, set $D= {\textstyle \bigcup \nolimits _{s\in S}} \mathrm {supp}f\left( s\right)$. Then $\Delta \left( D\right) ^{S}$ is compact in ${\mathbb {R}}^{D}$ since D is finite. By the Krein–Milman Theorem, there exists $\phi \in \Delta \left( D^{S}\right)$ such that

$$\begin{aligned} f= {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \left[ \begin{array}{c} \delta _{e\left( s_{1}\right) }\\ \delta _{e\left( s_{2}\right) }\\ \vdots \\ \delta _{e\left( s_{n}\right) } \end{array} \right] = {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \left[ \begin{array}{c} \delta _{\rho \left( a_{e},s_{1}\right) }\\ \delta _{\rho \left( a_{e},s_{2}\right) }\\ \vdots \\ \delta _{\rho \left( a_{e},s_{n}\right) } \end{array} \right] = {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \left[ \begin{array}{c} \rho _{\delta _{a_{e}}}\left( s_{1}\right) \\ \rho _{\delta _{a_{e}}}\left( s_{2}\right) \\ \vdots \\ \rho _{\delta _{a_{e}}}\left( s_{n}\right) \end{array} \right] \end{aligned}$$

(24)

where the last equality follows from point 1 of this proposition. By (24), and since $\digamma$ is affine (point 2 of this proposition),

$$\begin{aligned} f= {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \rho _{\delta _{a_{e}}}= {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \digamma \left( \delta _{a_{e}}\right) =\digamma \left( {\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \delta _{a_{e}}\right) \end{aligned}$$

where ${\displaystyle \sum \limits _{e\in D^{S}}} \phi \left( e\right) \delta _{a_{e}}\in \Delta \left( A\right)$ since $D^{S}$ is finite and $\Delta \left( A\right)$ is convex. Therefore, $\digamma$ is onto.

Only if part. Assume $\digamma \left( \Delta \left( A\right) \right) =\Delta \left( C\right) ^{S}$. For each $e\in C^{S}$, there exists $\alpha \in \Delta \left( A\right)$ such that

$$\begin{aligned} \rho _{\alpha }\left( s\right) =\delta _{e\left( s\right) }\quad \forall s\in S. \end{aligned}$$

(25)

Partition the finite support of $\alpha$ into realization equivalence classes so that

$$\begin{aligned} \mathrm {supp}\alpha = {\displaystyle \bigsqcup \limits _{i=1}^{k}} A_{i} \end{aligned}$$

with $\rho _{a_{i}}=\rho _{a_{i}^{\prime }}$ for all $a_{i},a_{i}^{\prime }\in A_{i}$ and all $i=1,2,\ldots ,k$ and $\rho _{a_{i}}\ne \rho _{a_{j}}$ if $a_{i}\in A_{i}$, $a_{j}\in A_{j}$ and $i\ne j$. Moreover, for each $i=1,2,\ldots ,k$ arbitrarily select $b_{i}\in A_{i}$. By (25), for each $s\in S$,

$$\begin{aligned} \delta _{e\left( s\right) }={\sum \limits _{a\in \mathrm {supp}\alpha }} \alpha \left( a\right) \rho _{\delta _{a}}\left( s\right) ={\sum \limits _{i=1}^{k}}\left( {\sum \limits _{a_{i}\in A_{i}}}\alpha \left( a_{i}\right) \rho _{\delta _{a_{i}}}\left( s\right) \right) \end{aligned}$$

but, for each $i=1,2,\ldots ,k$, and all $a_{i}\in A_{i}$,

$$\begin{aligned} \rho _{\delta _{a_{i}}}\left( s\right) =\delta _{\rho \left( a_{i},s\right) }=\delta _{\rho \left( b_{i},s\right) }=\rho _{\delta _{b_{i}}}\left( s\right) . \end{aligned}$$

Therefore, setting $\beta \left( b_{i}\right) ={\sum \limits _{a_{i}\in A_{i}} }\alpha \left( a_{i}\right)$ for all $i=1,2,\ldots ,k$ and $\beta \left( a\right) =0$ if $a\in A{\setminus }\left\{ b_{1},\ldots ,b_{k}\right\}$, $\beta \in \Delta \left( A\right)$ is such that

$$\begin{aligned} \delta _{e\left( s\right) }={\sum \limits _{i=1}^{k}}\left( {\sum \limits _{a_{i}\in A_{i}}}\alpha \left( a_{i}\right) \right) \rho _{\delta _{b_{i}}}\left( s\right) ={\sum \limits _{i=1}^{k}}\beta \left( b_{i}\right) \rho _{\delta _{b_{i}}}\left( s\right) ={\sum \limits _{i=1}^{k} }\beta \left( b_{i}\right) \delta _{\rho \left( b_{i},s\right) } \end{aligned}$$

and

$$\begin{aligned} \left[ \begin{array}{c} \delta _{e\left( s_{1}\right) }\\ \delta _{e\left( s_{2}\right) }\\ \vdots \\ \delta _{e\left( s_{n}\right) } \end{array} \right] ={\sum \limits _{i=1}^{k}}\beta \left( b_{i}\right) \left[ \begin{array}{c} \delta _{\rho \left( b_{i},s_{1}\right) }\\ \delta _{\rho \left( b_{i},s_{2}\right) }\\ \vdots \\ \delta _{\rho \left( b_{i},s_{n}\right) } \end{array} \right] . \end{aligned}$$

(26)

The summands $\left[ \delta _{\rho \left( b_{i},s_{1}\right) }\ \ldots \text { }\delta _{\rho \left( b_{i},s_{n}\right) }\right] ^{\intercal }$ are distinct (extreme) points in $\Delta \left( C\right) ^{S}$ and $\left[ \delta _{e\left( s_{1}\right) }\ \ldots \text { }\delta _{e\left( s_{n}\right) }\right] ^{\intercal }$ is an extreme point of $\Delta \left( C\right) ^{S}$. Then $\beta =\delta _{b}$ for some $b\in \mathrm {supp}\beta \subseteq A$, and (26) implies $e=\rho _{b}$. The arbitrary choice of e allows us to conclude that $C^{S}\subseteq \left\{ \rho _{a}\right\} _{a\in A}$ and the converse inclusion is trivial. $\square$

Proof of Corollary 1

If $\beta \ne \delta _{a}$ there exists $b\ne a$ such that $\beta \left( b\right) \ne 0$. Since the framework is reduced $\rho _{b}\ne \rho _{a}$ and there exists $s\in S$ such that $\rho \left( a,s\right) \ne \rho \left( b,s\right)$. Setting $c=\rho \left( b,s\right)$, it follows

$$\begin{aligned} \rho _{\beta }\left( c\mid s\right) \ge \beta \left( b\right) >0=\delta _{\rho \left( a,s\right) }\left( c\right) =\rho _{\delta _{a}}\left( c\mid s\right) \end{aligned}$$

and hence $\rho _{\beta }\ne \rho _{\delta _{a}}$ and $\delta _{a}\not \approx \beta$. By contrapositive $\delta _{a}\approx \beta \Longrightarrow \beta =\delta _{a}$, and the converse is trivial. $\square$

Lemma 1

Let $\left( A,S,C,\rho \right)$ be a decision framework, $\alpha \in \Delta \left( A\right)$, $u\in {\mathbb {R}}^{C}$, and $\mu \in \Delta \left( S\right)$. Then:

• 1. $\rho _{\alpha }\left( s\right) ={\sum \limits _{a\in A}}\alpha \left( a\right) \delta _{\rho \left( a,s\right) }$ for all $s\in S$;

• 2. $u\left( \rho _{\alpha }\left( s\right) \right) ={\mathbb {E}} _{\rho _{\alpha }\left( s\right) }\left[ u\right] ={\sum \limits _{a\in A} }\alpha \left( a\right) u\left( \rho \left( a,s\right) \right)$;

• 3. $\int _{S}u\left( \rho _{\alpha }\left( s\right) \right) d\mu \left( s\right) ={\mathbb {E}}_{\alpha \times \mu }\left[ u\circ \rho \right]$.

Moreover, if $\left( A,S,C,\rho \right)$ is a Luce–Raiffa framework, then

$$\begin{aligned} \begin{array}{cccc} \epsilon {:} &{} \Delta \left( C\right) &{} \rightarrow &{} \Delta _{\ell }\left( A\right) \\ &{} {\sum \limits _{c\in C}}\gamma \left( c\right) \delta _{c} &{} \mapsto &{} {\sum \limits _{c\in C}}\gamma \left( c\right) \delta _{cS} \end{array} \end{aligned}$$

is affine and bijective. In particular:

• 4. for each $\alpha ={\sum \limits _{c\in C}}\alpha \left( cS\right) \delta _{cS}\in \Delta _{\ell }\left( A\right)$ the only $\gamma \in \Delta \left( C\right)$ such that $\epsilon \left( \gamma \right) =\alpha$ is $\epsilon ^{-1}\left( \alpha \right) ={\sum \limits _{c\in C}}\alpha \left( cS\right) \delta _{c}$;

• 5. for each $\gamma \in \Delta \left( C\right)$ and each $s\in S$, $\rho _{\epsilon \left( \gamma \right) }\left( s\right) =\gamma$.

Proof

• 1. Follows from points 1 and 2 of Proposition 2. Specifically,

  $$\begin{aligned} \rho _{\alpha }=\digamma \left( {\displaystyle \sum \limits _{a\in A}} \alpha \left( a\right) \delta _{a}\right) = {\displaystyle \sum \limits _{a\in A}} \alpha \left( a\right) \digamma \left( \delta _{a}\right) ={\sum \limits _{a\in A}}\alpha \left( a\right) \rho _{\delta _{a}} \end{aligned}$$

  and therefore for each $s\in S$

  $$\begin{aligned} \rho _{\alpha }\left( s\right) ={\sum \limits _{a\in A}}\alpha \left( a\right) \rho _{\delta _{a}}\left( s\right) ={\sum \limits _{a\in A}}\alpha \left( a\right) \delta _{\rho \left( a,s\right) }. \end{aligned}$$

• 2. By definition, for all $s\in S$,

  $$\begin{aligned}&u\left( \rho _{\alpha }\left( s\right) \right) ={\mathbb {E}}_{\rho _{\alpha }\left( s\right) }\left[ u\right] ={\mathbb {E}}_{{\sum \limits _{a\in A} }\alpha \left( a\right) \delta _{\rho \left( a,s\right) }}\left[ u\right] ={\sum \limits _{a\in A}}\alpha \left( a\right) {\mathbb {E}}_{\delta _{\rho \left( a,s\right) }}\left[ u\right] \\&\quad ={\sum \limits _{a\in A}}\alpha \left( a\right) u\left( \rho \left( a,s\right) \right) . \end{aligned}$$

• 3. By definition,

  $$\begin{aligned} \int _{S}u\left( \rho _{\alpha }\left( s\right) \right) d\mu \left( s\right)= & {} {\sum \limits _{s\in S}\mu }\left( s\right) {\sum \limits _{a\in A}} \alpha \left( a\right) u\left( \rho \left( a,s\right) \right) ={\sum \limits _{a\in A}\sum \limits _{s\in S}}\alpha \left( a\right) {\mu }\left( s\right) u\left( \rho \left( a,s\right) \right) \\= & {} {\mathbb {E}} _{\alpha \times \mu }\left[ u\circ \rho \right] . \end{aligned}$$

  If $\left( A,S,C,\rho \right)$ is a Luce–Raiffa framework, since all sure actions are conceivable for each $c\in C$ there exists a sure action $a_{c}\in A$ such that $\rho \left( a_{c},\cdot \right) \equiv c$, reduction instead guarantees that such sure action is unique and we denote it by cS. In other words, the map $c\mapsto cS$ is well defined. It is also easy to check that its range is the set of all sure actions and that the map is injective. Therefore $c\hookrightarrow cS$ is an embedding of C onto the set of all sure actions. The fact that $\epsilon$ is affine and bijective immediately follows.

• 4. Clearly $\gamma ={\sum \limits _{c\in C}}\alpha \left( cS\right) \delta _{c}\in \Delta \left( C\right)$ and $\gamma \left( c\right) =\alpha \left( cS\right)$ for all $c\in C$, by definition of $\epsilon$ it follows $\epsilon \left( \gamma \right) ={\sum \limits _{c\in C}\gamma }\left( c\right) \delta _{cS}=\alpha$.

• 5. By definition $\epsilon \left( \gamma \right) ={\sum \limits _{c\in \mathrm {supp}\gamma }\gamma }\left( c\right) \delta _{cS}$, therefore, for each $s\in S$,

  $$\begin{aligned} \rho _{\epsilon \left( \gamma \right) }\left( s\right)= & {} \rho _{{\sum \limits _{c\in \mathrm {supp}\gamma }\gamma }\left( c\right) \delta _{cS}}\left( s\right) ={\sum \limits _{c\in \mathrm {supp}\gamma }\gamma }\left( c\right) \rho _{\delta _{cS}}\left( s\right) \nonumber \\= & {} {\sum \limits _{c\in \mathrm {supp}\gamma }\gamma }\left( c\right) \delta _{\rho \left( cS,s\right) }={\sum \limits _{c\in \mathrm {supp}\gamma }\gamma }\left( c\right) \delta _{c}=\gamma . \end{aligned}$$

  (27)

  $\square$

Proof of Theorem 1

(i) $\Longrightarrow$ (ii). Since $\succsim$ satisfies Axiom A.1, then

$$\begin{aligned} \Gamma _{\succsim }\left( \mathcal {A}\right) =\left\{ \alpha \in \mathcal {A}:\alpha \succsim \beta \ \forall \beta \in \mathcal {A}\right\} \quad \forall \mathcal {A}\in \mathcal {\wp }\left( \Delta \left( A\right) \right) \end{aligned}$$

is a rational choice correspondence and generates $\succsim$. Since $\succsim$ also satisfies Axiom A.2, then $\succsim _{\Delta \left( C\right) }$ satisfies Axioms 1, 2, and 3 of Herstein and Milnor (1953) and there exists $u\in {\mathbb {R}}^{C}$ such that, if $\gamma ,\zeta \in \Delta \left( C\right)$, then

$$\begin{aligned} \gamma \succsim _{\Delta \left( C\right) }\zeta \iff {\sum \limits _{c\in C}\gamma }\left( c\right) u\left( c\right) \ge {\sum \limits _{c\in C}\zeta }\left( c\right) u\left( c\right) \end{aligned}$$

and, by definition of $\succsim _{\Delta \left( C\right) }$ this means

$$\begin{aligned} \epsilon \left( \gamma \right) \succsim \epsilon \left( \zeta \right) \iff {\mathbb {E}}_{{\gamma }}\left[ u\right] \ge {\mathbb {E}}_{{\zeta }}\left[ u\right] . \end{aligned}$$

Therefore, if $\alpha ,\beta \in \Delta \left( A\right)$, then

Now Axiom A.3 implies

$$\begin{aligned} \alpha \geqslant _{u}\beta \implies \alpha \succsim \beta \end{aligned}$$

(28)

moreover, if $\alpha ,\beta \in \Delta _{\ell }\left( A\right)$, then $\alpha =\epsilon \left( \gamma \right)$ and $\beta =\epsilon \left( \zeta \right)$ for some $\gamma ,\zeta \in \Delta \left( C\right)$, and

$$\begin{aligned} \alpha \succsim \beta \iff \gamma \succsim _{\Delta \left( C\right) }\zeta \iff \rho _{\epsilon \left( \gamma \right) }\left( s\right) \succsim _{_{\Delta \left( C\right) }}\rho _{\epsilon \left( \zeta \right) }\left( s\right) \quad \forall s\in S\iff \alpha \geqslant _{u}\beta \end{aligned}$$

that is $\succsim$, $\succcurlyeq _{S}$, and $\geqslant _{u}$ coincide on $\Delta _{\ell }\left( A\right)$ where

This immediately implies, for each $\mathcal {A}\in \mathcal {\wp }\left( \Delta \left( A\right) \right)$ such that $\mathcal {A}\subseteq \Delta _{\ell }\left( A\right)$,

$$\begin{aligned} \Gamma _{\succsim }\left( \mathcal {A}\right)&=\left\{ \alpha \in \mathcal {A}:\alpha \succsim \beta \ \forall \beta \in \mathcal {A}\right\} \\&=\left\{ \alpha \in \mathcal {A}:\alpha \geqslant _{u}\beta \ \forall \beta \in \mathcal {A}\right\} =\Upsilon _{u}\left( \mathcal {A}\right) \end{aligned}$$

For a generic $\mathcal {A}\in \mathcal {\wp }\left( \Delta \left( A\right) \right)$, by (28),

$$\begin{aligned} \Upsilon _{u}\left( \mathcal {A}\right)&=\left\{ \alpha \in \mathcal {A} :\alpha \geqslant _{u}\beta \ \forall \beta \in \mathcal {A}\right\} \\&\subseteq \left\{ \alpha \in \mathcal {A}:\alpha \succsim \beta \ \forall \beta \in \mathcal {A}\right\} =\Gamma _{\succsim }\left( \mathcal {A}\right) . \end{aligned}$$

(ii) $\Longrightarrow$ (i). If $\succsim$ is generated by a rational choice correspondence $\Gamma :\mathcal {\wp }\left( \Delta \left( A\right) \right) \rightarrow \mathcal {\wp }\left( \Delta \left( A\right) \right)$, then $\succsim$ satisfies Axiom A.1. By (ii) there exists also $u\in {\mathbb {R}}^{C}$ such that:

$$\begin{aligned} \Upsilon _{u}\left( \mathcal {A}\right) \subseteq \Gamma \left( \mathcal {A} \right) \ \text {for all }\mathcal {A}\in \mathcal {\wp }\left( \Delta \left( A\right) \right) \text { and equality holds if }\mathcal {A}\subseteq \Delta _{\ell }\left( A\right) . \end{aligned}$$

Therefore, if $\alpha ,\beta \in \Delta \left( A\right)$,

but $\left\{ \epsilon \left( \rho _{\alpha }\left( s\right) \right) ,\epsilon \left( \rho _{\beta }\left( s\right) \right) \right\} \subseteq \Delta _{\ell }\left( A\right)$ for all $s\in S$, that is

(29)

Moreover, given $\eta ,\kappa \in \Delta \left( A\right)$

and (29) becomes

$$\begin{aligned} \left. \alpha \succcurlyeq _{S}\beta \right. \iff {\mathbb {E}}_{\rho _{\epsilon \left( \rho _{\alpha }\left( s\right) \right) }\left( w\right) }\left[ u\right] \ge {\mathbb {E}}_{\rho _{\epsilon \left( \rho _{\beta }\left( s\right) \right) }\left( w\right) }\left[ u\right] \quad \forall w,s\in S \end{aligned}$$

but, $\rho _{\epsilon \left( \gamma \right) }\equiv \gamma$ for all $\gamma \in \Delta \left( C\right)$, therefore

(30)

and $\succsim$ satisfies Axiom A.3.

Finally, if $\alpha ,\beta \in \Delta _{\ell }\left( A\right)$, then $\alpha =\epsilon \left( \gamma \right)$ and $\beta =\epsilon \left( \zeta \right)$ for some $\gamma ,\zeta \in \Delta \left( C\right)$, and

but $\mathrm {supp}\alpha ,\mathrm {supp}\beta \subseteq \left\{ cS\right\} _{c\in C}$ and by (30)

and so $\succsim$ satisfies Axioms 2 and 3 of Herstein and Milnor (1953) on $\Delta _{\ell }\left( A\right)$, that is, Axiom A.2.$\square$

Recall that if $\gamma \in \Delta \left( C\right)$ and $u\in {\mathbb {R}}^{C}$ we indifferently write ${\mathbb {E}}_{{\gamma }}\left[ u\right]$ or $u\left( \gamma \right)$; with a similar abuse of notation, if $f\in \Delta \left( C\right) ^{S}$ we denote by $u\left( f\right)$ the element of ${\mathbb {R}}^{S}$ defined by

$$\begin{aligned} u\left( f\right) =\left[ \begin{array}{c} u\left( f\left( s_{1}\right) \right) \\ u\left( f\left( s_{2}\right) \right) \\ \vdots \\ u\left( f\left( s_{n}\right) \right) \end{array} \right] =\left[ \begin{array}{c} {\mathbb {E}}_{f\left( s_{1}\right) }\left[ u\right] \\ {\mathbb {E}}_{f\left( s_{2}\right) }\left[ u\right] \\ \vdots \\ {\mathbb {E}}_{f\left( s_{n}\right) }\left[ u\right] \end{array} \right] . \end{aligned}$$

Lemma 2

Let $\left( A,S,C,\rho \right)$ be a Luce–Raiffa framework, $\mathcal {F}=\digamma \left( \Delta \left( A\right) \right)$, $u\in {\mathbb {R}}^{C}$, and $\succsim$ be a preorder on $\Delta \left( A\right)$. Then:

• 1. $\mathcal {F}$ is a convex subset of $\Delta \left( C\right) ^{S}$ containing all constant Anscombe–Aumann acts;

• 2. if $\succsim$ satisfies Axiom A.3, then $\succsim$ satisfies mixed consequentialism;

• 3. if all bets are conceivable, $\left\{ u\left( f\right) :f\in \mathcal {F}\right\} =\left( \mathrm {co}\left( u\left( C\right) \right) \right) ^{S}=u\left( \Delta \left( C\right) \right) ^{S}$.

Moreover, if $\succsim$ satisfies mixed consequentialism, then $\succsim _{\digamma }$ is a well-defined preorder on $\mathcal {F}$ and:

• 4. for $N=1,\,3,\,4,\,6,\,7,\,9,\,\succsim _{\digamma }$ satisfies Axiom AA.N if and only if $\succsim$ satisfies Axiom A.N;

• 5. Axiom AA.2 for $\succsim _{\digamma }$ is implied by Axiom A.2 for $\succsim$ (and they are equivalent under Axiom A.4);

• 6. Axiom AA.5 for $\succsim _{\digamma }$ is equivalent to Axiom A.5 for $\succsim$ when $\succsim _{\digamma }^{\#}$ is defined by

  $$\begin{aligned} f\succsim _{\digamma }^{\#}g\iff \ f\left( s\right) \succsim _{\digamma }g\left( s\right) \quad \forall s\in S\text {;} \end{aligned}$$

• 7. Axiom AA.8 for $\succsim _{\digamma }$ implies Axiom A.8 for $\succsim ($and they are equivalent under Axioms A.1 and A.4).

Proof

• 1. Follows from the fact that $\digamma$ is affine and the fact that $\rho _{\epsilon \left( \gamma \right) }\equiv \gamma$ for all $\gamma \in \Delta \left( C\right)$.

• 2. If $\alpha ,\beta \in \Delta \left( A\right)$ and $\alpha \approx \beta$, then $\rho _{\alpha }\left( s\right) =\rho _{\beta }\left( s\right)$ for all $s\in S$; since $\succsim$ is a preorder $\epsilon \left( \rho _{\alpha }\left( s\right) \right) \succsim \epsilon \left( \rho _{\beta }\left( s\right) \right)$, that is, $\rho _{\alpha }\left( s\right) \succsim _{\Delta \left( C\right) }\rho _{\beta }\left( s\right)$ for all $s\in S$, and Axiom A.3 implies $\alpha \succsim \beta$. By a symmetric argument $\beta \succsim \alpha$.

• 3. Obviously, $\left\{ u\left( f\right) :f\in \mathcal {F}\right\} \subseteq \left( \mathrm {co}\left( u\left( C\right) \right) \right) ^{S}=u\left( \Delta \left( C\right) \right) ^{S}$. Set $U=\mathrm {co} \left( u\left( C\right) \right)$ and assume $u\left( C\right)$ is not a singleton, otherwise the result is trivial. For every vector $\varkappa =\left[ x_{1}\ \ldots \text { }x_{n}\right] ^{\intercal }\in U^{S}$ there exist $c\ne d$ in C and $q_{1},q_{2},\ldots ,q_{n}\in \left[ 0,1\right] \$such that

  $$\begin{aligned} \left[ \begin{array}{c} x_{1}\\ x_{2}\\ \vdots \\ x_{n} \end{array} \right] =\left[ \begin{array}{c} q_{1}u\left( c\right) +\left( 1-q_{1}\right) u\left( d\right) \\ q_{2}u\left( c\right) +\left( 1-q_{2}\right) u\left( d\right) \\ \vdots \\ q_{n}u\left( c\right) +\left( 1-q_{n}\right) u\left( d\right) \end{array} \right] =\left[ \begin{array}{c} u\left( q_{1}\delta _{c}+\left( 1-q_{1}\right) \delta _{d}\right) \\ u\left( q_{2}\delta _{c}+\left( 1-q_{2}\right) \delta _{d}\right) \\ \vdots \\ u\left( q_{n}\delta _{c}+\left( 1-q_{n}\right) \delta _{d}\right) \end{array} \right] . \end{aligned}$$

  Set $D=\left\{ c,d\right\}$, set $B=\left\{ a\in A:\rho _{a}\left( S\right) \subseteq \left\{ c,d\right\} \right\}$, and consider the Luce–Raiffa framework $\left( B,S,D,\rho \right)$. Since all the bets are conceivable in $\left( A,S,C,\rho \right)$, then $\left\{ \rho _{b}\right\} _{b\in B}=D^{S}$. In particular, by point 3 of Proposition 2 for each $f\in \Delta \left( D\right) ^{S}$, there exists $\beta _{f}\in \Delta \left( B\right)$ such that

  $$\begin{aligned} \beta _{f}\left( \left\{ b\in B:\rho \left( b,s\right) =c\right\} \right) =f\left( c|s\right) \quad \forall s\in S. \end{aligned}$$

  Choose $f\in \Delta \left( D\right) ^{S}$ such that $f\left( c\mid s_{i}\right) =q_{i}=1-f\left( d\mid s_{i}\right)$ for all $i=1,\ldots ,n$, and set $\alpha \left( a\right) =\beta _{f}\left( a\right)$ if $a\in B$ and $\alpha \left( a\right) =0$ if $a\in A{\setminus } B$. Then, $\alpha \in \Delta \left( A\right)$ and, for all $i=1,\ldots ,n$,

  $$\begin{aligned} \rho _{\alpha }\left( c\mid s_{i}\right)&=\beta _{f}\left( \left\{ b\in B:\rho \left( b,s_{i}\right) =c\right\} \right) =f\left( c\mid s_{i}\right) =q_{i}\\ \rho _{\alpha }\left( d\mid s_{i}\right)&=f\left( d\mid s_{i}\right) =1-q_{i} \end{aligned}$$

  so that $\rho _{\alpha }\left( s_{i}\right) =q_{i}\delta _{c}+\left( 1-q_{i}\right) \delta _{d}$ and $u\left( \rho _{\alpha }\left( s_{i}\right) \right) =q_{i}u\left( c\right) +\left( 1-q_{i}\right) u\left( d\right)$, that is, $u\left( \rho _{\alpha }\right) =\varkappa$.

Assume $\succsim$ satisfies mixed consequentialism. Let $f,g\in \mathcal {F}$ and $\alpha _{f},\beta _{f},\alpha _{g},\beta _{g}\in \Delta \left( A\right)$, not necessarily distinct, be such that

$$\begin{aligned} \rho _{\alpha _{f}}=\rho _{\beta _{f}}=f \quad \text { and } \quad \rho _{\alpha _{g}}=\rho _{\beta _{g}}=g\text {.} \end{aligned}$$

(31)

These mixed actions exist since $\mathcal {F}=\digamma \left( \Delta \left( A\right) \right)$. By mixed consequentialism, (31) implies $\alpha _{f}\sim \beta _{f}$ and $\alpha _{g}\sim \beta _{g}$ Therefore, by transitivity of $\succsim$,

$$\begin{aligned} \alpha _{f}\succsim \alpha _{g}\iff \beta _{f}\succsim \beta _{g} \end{aligned}$$

and $\succsim _{\digamma }$ is well defined.⁴⁹ Reflexivity and transitivity of $\succsim _{\digamma }$ follow from reflexivity and transitivity of $\succsim$.

The verification of points 4, 5, and 6 is routine.

7. Assume $\succsim _{\digamma }$ satisfies Axiom AA.8. If $\alpha ,\beta \in \Delta \left( A\right)$ and $\alpha \sim \beta$, then $\rho _{\alpha } \sim _{\digamma }\rho _{\beta }$. By Axiom AA.8, $2^{-1}\rho _{\alpha } +2^{-1}\rho _{\beta }\succsim _{\digamma }\rho _{\alpha }$, therefore $\rho _{2^{-1}\alpha +2^{-1}\beta }\succsim _{\digamma }\rho _{\alpha }$ and $2^{-1} \alpha +2^{-1}\beta \succsim \alpha$, so that $\succsim$ satisfies Axiom A.8.

Conversely, assume $\succsim$ satisfies Axioms A.1 and A.4. Next we show that Axiom A.8 implies that for all $\alpha ,\beta \in \Delta \left( A\right)$ such that $\alpha \sim \beta$ and all $q\in \left( 0,1\right)$ it holds $q\alpha +\left( 1-q\right) \beta \succsim \alpha$. Per contra, assume there exist $\alpha ,\beta \in \Delta \left( A\right)$ and $p\in \left( 0,1\right)$ such that $\alpha \sim \beta$ and $p\alpha +\left( 1-p\right) \beta \prec \alpha$. Set

$$\begin{aligned} T=\left\{ t\in \left[ 0,1\right] :t\alpha +\left( 1-t\right) \beta \prec \alpha \right\} . \end{aligned}$$

Clearly $p\in T$. Moreover, by Axioms A.1 and A.4, T is open in $\left[ 0,1\right]$ and hence there exists O open in ${\mathbb {R}}$ such that $T=O\cap \left[ 0,1\right]$, but $T\subseteq \left( 0,1\right)$, therefore $T=O\cap \left( 0,1\right)$ is open in ${\mathbb {R}}$. Therefore there exists an open interval in T that contains p. The set

$$\begin{aligned} I=\bigcup \limits _{p\ni \left( q,r\right) \subseteq T}\left( q,r\right) \end{aligned}$$

is a union of pairwise overlapping open intervals, and so it is an open interval itself: $p\in I=\left( \bar{q},\bar{r}\right) \subseteq T\subseteq \left( 0,1\right)$. If $\bar{q}\in T$, there would exist $\varepsilon >0$ such that $\left( \bar{q}-\varepsilon ,\bar{q}+\varepsilon \right) \subseteq T$, and then $p\in \left( \bar{q}-\varepsilon ,\bar{r}\right) \subseteq T$, whence

$$\begin{aligned} \left( \bar{q}-\varepsilon ,\bar{r}\right) \subseteq I=\left( \bar{q},\bar{r}\right) \end{aligned}$$

a contradiction. Therefore $\bar{q}\notin T$ and (analogously) $\bar{r}\notin T$, that is,

$$\begin{aligned} \bar{q}\alpha +\left( 1-\bar{q}\right) \beta \succsim \alpha \quad \text { and } \quad \bar{r}\alpha +\left( 1-\bar{r}\right) \beta \succsim \alpha \text {.} \end{aligned}$$

Since $\left( \bar{q},\bar{r}\right) \subseteq T$ is nonempty, eventually, $\bar{q}+n^{-1}$ and $\bar{r}-n^{-1}$ belong to T, and by Axiom A.4

$$\begin{aligned} \bar{q}\alpha +\left( 1-\bar{q}\right) \beta \precsim \alpha \quad \text { and }\quad \bar{r}\alpha +\left( 1-\bar{r}\right) \beta \precsim \alpha \text {.} \end{aligned}$$

That is, $\bar{q}\alpha +\left( 1-\bar{q}\right) \beta \sim \bar{r} \alpha +\left( 1-\bar{r}\right) \beta \sim \alpha$, and Axiom A.8 implies

$$\begin{aligned}&\left( \frac{\bar{q}}{2}+\frac{\bar{r}}{2}\right) \alpha +\left( 1-\frac{\bar{q}}{2}-\frac{\bar{r}}{2}\right) \beta =\frac{1}{2}\left( \bar{q}\alpha +\left( 1-\bar{q}\right) \beta \right) \\&\quad +\frac{1}{2}\left( \bar{r}\alpha +\left( 1-\bar{r}\right) \beta \right) \succsim \bar{r} \alpha +\left( 1-\bar{r}\right) \beta \sim \alpha \end{aligned}$$

but this is a contradiction since $2^{-1}\bar{q}+2^{-1}\bar{r}\in \left( \bar{q},\bar{r}\right) \subseteq T$.

Now let $f,g\in \mathcal {F}$ be such that $f\sim _{\digamma }g$ and arbitrarily choose $q\in \left( 0,1\right)$. Let $\alpha ,\beta \in \Delta \left( A\right)$ be such that $f=\rho _{\alpha }$ and $g=\rho _{\beta }$, then $f\sim _{\digamma }g$ implies $\alpha \sim \beta$, Axioms A.1, A.4, and A.8 imply $q\alpha +\left( 1-q\right) \beta \succsim \alpha$, whence $\rho _{q\alpha +\left( 1-q\right) \beta }\succsim _{\digamma }\rho _{\alpha }$ and $qf+\left( 1-q\right) g=q\rho _{\alpha }+\left( 1-q\right) \rho _{\beta }\succsim _{\digamma }\rho _{\alpha }=f$. As wanted. $\square$

Proof of Proposition 3

It immediately follows from Lemma 2. $\square$

In order to obtain representation (16), due to Cerreia-Vioglio et al. (2011b), we need two final pieces of notation. First, given $u\in {\mathbb {R}}^{C}$, we denote by $U=\mathrm {co} \left( u\left( C\right) \right)$ the smallest interval containing $u\left( C\right)$ and by $\mathcal {G}\left( U,\Delta \left( S\right) \right)$ the set of functions $G:U\times \Delta \left( S\right) \rightarrow \left( -\infty ,\infty \right]$ such that:

• 1. G is quasiconvex,

• 2. $\inf _{\sigma \in \Delta \left( S\right) }G\left( x,\sigma \right) =x$ for all $x\in U$,

• 3. G is increasing in the first component,

• 4. the function $\varkappa \mapsto \inf _{\sigma \in \Delta \left( S\right) }G\left( \varkappa \cdot \sigma ,\sigma \right)$ is continuous on $U^{S}$.

Second, for each $l\in -U$ and each $\sigma \in \Delta \left( S\right)$, we denote by $\mathcal {B}\left( l,\sigma \right)$ the set of all mixed actions that have Bayes risk level l under $\sigma$, that is,

$$\begin{aligned} \mathcal {B}\left( l,\sigma \right) =\left\{ \beta \in \Delta \left( A\right) :r\left( \beta ,\sigma \right) =l\right\} \end{aligned}$$

and by $v\left( l,\sigma \right)$ the indirect payoff $v\left( \mathcal {B}\left( l,\sigma \right) \right)$ of decision problem $\left( \mathcal {B}\left( l,\sigma \right) ,S,C,\rho \right)$, as defined in (8).

Theorem 5

Let $\left( A,S,C,\rho \right)$ be a Marschak–Radner framework and $\succsim$ a binary relation on $\Delta \left( A\right)$. The following conditions are equivalent:

• (i) $\succsim$ is a non-trivial and continuous rational preference that satisfies uncertainty aversion;

• (ii) there exist a nonconstant $u\in {\mathbb {R}}^{C}$ and $G\in \mathcal {G}\left( U,\Delta \left( S\right) \right) \rightarrow \left( -\infty ,\infty \right]$ such that, if $\alpha ,\beta \in \Delta \left( A\right)$,

  $$\begin{aligned} \alpha \succsim \beta \iff \inf _{\sigma \in \Delta \left( S\right) }G\left( {\mathbb {E}}_{\alpha \times \sigma }\left[ u\circ \rho \right] ,\sigma \right) \ge \inf _{\sigma \in \Delta \left( S\right) }G\left( {\mathbb {E}}_{\beta \times \sigma }\left[ u\circ \rho \right] ,\sigma \right) . \end{aligned}$$

  (32)

In this case, u is cardinally unique and, for each u, the minimal element of $\mathcal {G}\left( U,\Delta \left( S\right) \right)$ satisfying (32) is the indirect payoff

$$\begin{aligned} G_{u}\left( x,\sigma \right) =v\left( -x,\sigma \right) \quad \forall \left( x,\sigma \right) \in U\times \Delta \left( S\right) . \end{aligned}$$

(33)

Continuous and uncertainty averse rational preferences are thus represented by

$$\begin{aligned} V\left( \alpha \right) =\inf _{\sigma \in \Delta \left( S\right) }G\left( {\mathbb {E}}_{\alpha \times \sigma }\left[ u\circ \rho \right] ,\sigma \right) \quad \forall \alpha \in \Delta \left( A\right) \end{aligned}$$

(34)

and (16) is obtained by setting

$$\begin{aligned} R\left( \alpha ,\sigma \right) =G\left( {\mathbb {E}}_{\alpha \times \sigma }\left[ u\circ \rho \right] ,\sigma \right) \quad \forall \left( \alpha ,\sigma \right) \in \Delta \left( A\right) \times \Delta \left( S\right) \end{aligned}$$

and by choosing $\Sigma$ as the projection on $\Delta \left( S\right)$ of the domain of R. In particular, when the minimal $G_{u}$ described by (33) is considered,

$$\begin{aligned} R_{u}\left( \alpha ,\sigma \right) =v\left( r\left( \alpha ,\sigma \right) ,\sigma \right) \end{aligned}$$

is the indirect payoff of the decision problem in which the only available mixed actions are those with the same Bayes risk as $\alpha$ under $\sigma$. This is worth mentioning since the indirect payoff $v\left( l,\sigma \right)$ can be seen as a comparative index of uncertainty aversion (see Cerreia-Vioglio et al. 2011b).

Proofs of Theorems 2, 3, 4, and 5

Let $\succsim$ be a rational preference. As shown in the first part of the proof of Theorem 1, there exists $u\in {\mathbb {R}}^{C}$ such that:

• if $\gamma ,\zeta \in \Delta \left( C\right)$, then $\gamma \succsim _{\Delta \left( C\right) }\zeta \iff {\sum \limits _{c\in C}\gamma }\left( c\right) u\left( c\right) \ge {\sum \limits _{c\in C}\zeta }\left( c\right) u\left( c\right)$;

• if $\alpha ,\beta \in \Delta \left( A\right)$, then $\alpha \succcurlyeq _{S}\beta \iff \alpha \geqslant _{u}\beta \iff {\sum \limits _{a\in A} }\alpha \left( a\right) u\left( \rho \left( a,s\right) \right) \ge {\sum \limits _{a\in A}}\beta \left( a\right) u\left( \rho \left( a,s\right) \right)$ for all $s\in S$;

• if $\alpha ,\beta \in \Delta _{\ell }\left( A\right)$, then $\alpha \succsim \beta \iff \alpha \succcurlyeq _{S}\beta \iff {\sum \limits _{c\in C}} \alpha \left( cS\right) u\left( c\right) \ge {\sum \limits _{c\in C}} \beta \left( cS\right) u\left( c\right)$.

Moreover, if $\succsim$ is continuous, for every $\alpha \in \Delta \left( A\right)$ there exists $\alpha _{\ell }\in \Delta \left( C\right)$ such that $\alpha \sim \alpha _{\ell }$, more precisely, $\alpha \sim \epsilon \left( \alpha _{\ell }\right)$. In fact, choosing w and m in S so that

$$\begin{aligned} \rho _{\alpha }\left( m\right) \succsim _{\Delta \left( C\right) }\rho _{\alpha }\left( s\right) \succsim _{\Delta \left( C\right) }\rho _{\alpha }\left( w\right) \quad \forall s\in S \end{aligned}$$

(35)

it follows $\epsilon \left( \rho _{\alpha }\left( m\right) \right) \succcurlyeq _{S}\alpha \succcurlyeq _{S}\epsilon \left( \rho _{\alpha }\left( w\right) \right)$. Whence $\beta =\epsilon \left( \rho _{\alpha }\left( m\right) \right)$ and $\eta =\epsilon \left( \rho _{\alpha }\left( w\right) \right)$ belong to $\Delta _{\ell }\left( A\right)$ and, by Axiom A.3, $\beta \succsim \alpha \succsim \eta$. Therefore, the nonempty and closed (by Axiom A.4) sets

$$\begin{aligned} \left\{ q\in \left[ 0,1\right] :q\eta +\left( 1-q\right) \beta \succsim \alpha \right\} \quad \text { and } \quad \left\{ q\in \left[ 0,1\right] :\alpha \succsim q\eta +\left( 1-q\right) \beta \right\} \end{aligned}$$

cover (by Axiom A.1) the connected set $\left[ 0,1\right]$. In particular, they cannot be disjoint, and there is $q_{\ell }\in \left[ 0,1\right]$ such that $q_{\ell }\eta +\left( 1-q_{\ell }\right) \beta \sim \alpha$; convexity of $\Delta _{\ell }\left( A\right) =\epsilon \left( \Delta \left( C\right) \right)$ implies that $q_{\ell }\eta +\left( 1-q_{\ell }\right) \beta =\epsilon \left( \alpha _{\ell }\right)$ for some $\alpha _{\ell }\in \Delta \left( C\right)$.

If $\alpha ,\beta \in \Delta \left( A\right)$, then

$$\begin{aligned} \alpha \succsim \beta \iff \epsilon \left( \alpha _{\ell }\right) \succsim \epsilon \left( \beta _{\ell }\right) \iff \alpha _{\ell }\succsim _{\Delta \left( C\right) }\beta _{\ell }\iff {\sum \limits _{c\in C}}\alpha _{\ell }\left( c\right) u\left( c\right) \ge {\sum \limits _{c\in C}}\beta _{\ell }\left( c\right) u\left( c\right) \end{aligned}$$

that is, the functional

$$\begin{aligned} \begin{array}{cccc} V{:} &{} \Delta \left( A\right) &{} \rightarrow &{} {\mathbb {R}}\\ &{} \alpha &{} \mapsto &{} {\sum \limits _{c\in C}}\alpha _{\ell }\left( c\right) u\left( c\right) \end{array} \end{aligned}$$

represents $\succsim$ on $\Delta \left( A\right)$.⁵⁰

For each $\alpha \in \Delta \left( A\right)$, by Eq. (35) and Axiom A.3, $\epsilon \left( \rho _{\alpha }\left( m\right) \right) \succsim \alpha \succsim \epsilon \left( \rho _{\alpha }\left( w\right) \right)$ so that

$$\begin{aligned} \max _{s\in S}{\sum \limits _{a\in A}}\alpha \left( a\right) u\left( \rho \left( a,s\right) \right)&=\max _{s\in S}{\mathbb {E}}_{\rho _{\alpha }\left( s\right) }\left[ u\right] ={\mathbb {E}}_{\rho _{\alpha }\left( m\right) }\left[ u\right] =V\left( \epsilon \left( \rho _{\alpha }\left( m\right) \right) \right) \\&\ge V\left( \alpha \right) \\&\ge V\left( \epsilon \left( \rho _{\alpha }\left( w\right) \right) \right) ={\mathbb {E}}_{\rho _{\alpha }\left( w\right) }\left[ u\right] =\min _{s\in S}{\mathbb {E}}_{\rho _{\alpha }\left( s\right) }\left[ u\right] \\&=\min _{s\in S}{\sum \limits _{a\in A}}\alpha \left( a\right) u\left( \rho \left( a,s\right) \right) . \end{aligned}$$

Up until this point, we only assumed that $\succsim$ satisfies Axioms A.1–A.4.

Theorem 2. If, in addition, Axiom A.5 holds, then $\epsilon \left( \alpha _{\ell }\right) \precsim \alpha$ implies $\alpha \succcurlyeq _{S}\epsilon \left( \alpha _{\ell }\right)$, that is $\rho _{\alpha }\left( w\right) \succsim _{\Delta \left( C\right) } \alpha _{\ell }$, and

$$\begin{aligned} \min _{s\in S}{\sum \limits _{a\in A}}\alpha \left( a\right) u\left( \rho \left( a,s\right) \right) =\min _{s\in S}{\mathbb {E}}_{\rho _{\alpha }\left( s\right) }\left[ u\right] ={\mathbb {E}}_{\rho _{\alpha }\left( w\right) }\left[ u\right] \ge {\mathbb {E}}_{\alpha _{\ell }}\left[ u\right] =V\left( \alpha \right) . \end{aligned}$$

To sum up, $V\left( \alpha \right) =\min _{s\in S}{\sum \nolimits _{a\in A} }\alpha \left( a\right) u\left( \rho \left( a,s\right) \right)$ for all $\alpha \in \Delta \left( A\right)$, which proves (10). The converse is routine, hence omitted.$\square$

If $\succsim$ satisfies Axioms A.1–A.4, by Lemma 2, $\succsim _{\digamma }$ is well defined and satisfies Axioms AA.1–AA.4 on $\mathcal {F}=\digamma \left( \Delta \left( A\right) \right)$. For each $f=\rho _{\alpha }\in \mathcal {F}$, $\alpha \sim \epsilon \left( \alpha _{\ell }\right)$ implies $f\sim _{\digamma } \rho _{\epsilon \left( \alpha _{\ell }\right) }=\left( \alpha _{\ell }\right) _{S}$. In the Anscombe–Aumann jargon, $\alpha _{\ell }$ is a certainty equivalent $\gamma _{f}$ of $f=\rho _{\alpha }$, more precisely,

$$\begin{aligned} \left\{ \xi \in \Delta \left( C\right) :\alpha \sim \epsilon \left( \xi \right) \right\} =\left\{ \gamma \in \Delta \left( C\right) :\rho _{\alpha }\sim \gamma _{S}\right\} . \end{aligned}$$

Moreover, if $g=\rho _{\beta }\in \mathcal {F}$, then

$$\begin{aligned} f\succsim _{\digamma }g\Longleftrightarrow & {} \alpha \succsim \beta \iff \alpha _{\ell }\succsim _{\Delta \left( C\right) }\beta _{\ell }\Longleftrightarrow {\mathbb {E}}_{\alpha _{\ell }}\left[ u\right] \ge {\mathbb {E}}_{\beta _{\ell } }\left[ u\right] \\\iff & {} u\left( \gamma _{f}\right) \ge u\left( \gamma _{g}\right) . \end{aligned}$$

In particular, for every $\gamma ,\zeta \in \Delta \left( C\right)$,

$$\begin{aligned} \gamma _{S}\succsim _{\digamma }\zeta _{S}\Longleftrightarrow & {} \rho _{\epsilon \left( \gamma \right) }\succsim _{\digamma }\rho _{\epsilon \left( \zeta \right) }\Longleftrightarrow \epsilon \left( \gamma \right) \succsim \epsilon \left( \zeta \right) \iff \gamma \succsim _{\Delta \left( C\right) }\zeta \\\Longleftrightarrow & {} {\mathbb {E}}_{\gamma }\left[ u\right] \ge {\mathbb {E}}_{\zeta }\left[ u\right] \iff u\left( \gamma \right) \ge u\left( \zeta \right) . \end{aligned}$$

Set $U=\mathrm {co}\left( u\left( C\right) \right) =u\left( \Delta \left( C\right) \right)$ and $u\left( \mathcal {F}\right) =\left\{ u\left( f\right) :f\in \mathcal {F}\right\} \subseteq U^{S}\subseteq {\mathbb {R}}^{S}$, and for each $\varkappa \in u\left( \mathcal {F}\right)$ define

$$\begin{aligned} I\left( \varkappa \right) =u\left( \gamma _{f}\right) \quad \text {if }\varkappa =u\left( f\right) \text {.} \end{aligned}$$

Using the techniques of Cerreia-Vioglio et al. (2011a, b) it can be shown that $I:u\left( \mathcal {F}\right) \rightarrow {\mathbb {R}}$ is well defined, monotone and normalized (that is $I\left( x1_{S}\right) =x$ for all $x\in U$). Moreover, for each $\alpha \in \Delta \left( A\right)$, $V\left( \alpha \right) ={\mathbb {E}} _{\alpha _{\ell }}\left[ u\right] =u\left( \alpha _{\ell }\right) =I\left( u\left( \rho _{\alpha }\right) \right)$ and if $f=\rho _{\alpha } ,g=\rho _{\beta }\in \mathcal {F}$

$$\begin{aligned} f\succsim _{\digamma }g\iff & {} V\left( \alpha \right) \ge V\left( \beta \right) \iff I\left( u\left( \rho _{\alpha }\right) \right) \ge I\left( u\left( \rho _{\beta }\right) \right) \iff I\left( u\left( f\right) \right) \nonumber \\\ge & {} I\left( u\left( g\right) \right) . \end{aligned}$$

(36)

Theorems 3 and 4. If $\succsim$ satisfies Axiom A.6, then $\succsim _{\digamma }$ satisfies Axiom AA.6 by Lemma 2, which, together with (36), implies that I is affine. Theorem 3 follows by the finite dimensional versions of the Krein-Rutman Extension Theorem (see, e.g. Ok 2007, p. 496) and the Riesz Representation Theorem. The additional assumptions of Theorem 4, non-triviality of $\succsim$ and conceivability of all bets, guarantee that u is nonconstant and $\mu$ is unique.$\square$

Theorem 5. Follows from Theorem 3 of Cerreia-Vioglio et al. (2011a, b) and Lemma 2 again, which guarantees that $\left\{ u\left( f\right) :f\in \mathcal {F}\right\} =U^{S}$ when all bets are conceivable and that $\succsim _{\digamma }$ satisfies Axiom AA.8 when $\succsim$ satisfies Axioms A.1–A.4 and A.8. $\square$