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Pessimism and optimism towards new discoveries

Abstract

In this paper, we provide an axiomatic foundation of pessimism and optimism towards ambiguity that emerges due to growing awareness. In our setup, this corresponds to a discovery of finer “descriptions” of the original contingencies (states). A decision-maker can form subjective probabilistic beliefs on the original state space and behaves as an expected utility maximizer. However, as finer contingencies are discovered, he may perceive ambiguity with respect to the newly identified states and thus be unable to extend her initial probabilistic beliefs to the expanded state space in an additive way. As a result, the decision-maker’s new beliefs are now “ambiguous” and represented by a probability distribution combined, for each refinement of an original state, with a degree of confidence in this probability estimate. We provide a parametric representation of preferences, identify the DM’s degree of ambiguity as well as his attitude towards ambiguity as captured by the degree of optimism and pessimism. We illustrate the relation of our model to some well-known capacities, such as the E-capacities introduced by Eichberger and Kelsey (Theory Decis 46:107–140, 1999), the JP-capacities of Jaffray and Philippe (Math Oper Res 22:165–185, 1997) and the NEO-additive capacities developed by Chateauneuf et al. (J Econ Theory, 137:538–567, 2007).

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Notes

  1. 1.

    In the case of Myra, if the determination of the type of the disease is based on a set of easily identifiable characteristics, such as weight and blood pressure, it might be easy to infer the relative likelihood of the different types from already available statistics. In this case, the perceived ambiguity would be close to 0. In contrast, if the type of disease is identifiable by a genetic test concerning a gene which so far has not been studied in large populations, the frequency of the two types would be difficult to identify, until more and better data are collected. In this case, the degree of ambiguity might be quite high and the probabilistic estimate will be given little, if any, weight in the decision process.

  2. 2.

    This assumption is closely related to the “reverse Bayesianism” in Karni and Viero (2013), property 2 in Definition 1 of Heifetz et al. (2013), as well as to the properties “Projections preserve posteriors” and “Conditional Independence” in Galanis (2016) and Galanis (2018). It presupposes that becoming aware does not per se convey new information about the probability of the states in S. Galanis (2016), Galanis (2018) and Piermont (2019) provide examples that challenge this property and discuss updating in this context.

  3. 3.

    Note that in our setup, the probability distributions \(\left( p\left( E_{s}\right) \right) _{s\in S}\) and \((q\left( \omega \mid E_{s}\right) )_{\omega \in \Omega }\) for \(s\in S\) are subjective. They can, of course, be informed by data and other relevant information available to the DM.

  4. 4.

    Note that Axiom 1 is the only one of our axioms that evokes the preference relation \(\succsim _{0}\). The subsequent axioms are stated only in terms of \(\succsim\). However, since these two preferences coincide on the set of measurable acts, the relation \(\succsim _{0}\) inherits all of the properties of \(\succsim\) on \(F^{S}\). In particular, Axioms 2–6 when restricted to \(F^{S}\) imply that subjective beliefs corresponding to \(\succsim _{0}\) are additive and thus, \(\succsim _{0}\) is indeed an expected utility preference as implied by our representation. An alternative approach would consist in stating these axioms separately for \(\succsim\) and for \(\succsim _{0}\). We feel that would unnecessarily overburden the paper.

  5. 5.

    If \(\beta >1\), reverse the places of \(\omega _{1}\) and \(\omega _{2}\).

  6. 6.

    As we will show, a mathematical characterization is feasible, yet it comes with an important caveat. Note that for an event with only two states, the capacity can either be convex or concave, intermediate (mixed) cases are excluded. In this particular case, more/less convex (resp., concave) capacity reflects higher/lower degree of ambiguity while the coefficient of optimism/pessimism takes values \(\alpha _{s}=1\)/\(\alpha _{s}=0\). This interpretation is implied by the well-known fact that each convex CEU preference is a Max-min expected utility preference, displaying “extreme” ambiguity aversion (\(\alpha _{s}=1\)), while each concave CEU preference is a Max-max expected utility preference, reflecting ambiguity love (\(\alpha _{s}=0\)). Hence, admitting a NEO-additive representation of a capacity on events containing two states, allowing for intermediate levels of optimism/pessimism \(\alpha _{s}\in (0,1)\) together with a degree of ambiguity \(\rho _{s}\in [0,1]\), is merely an interpretation.

  7. 7.

    Baillon et al. (2018) provide a general method for measuring ambiguity and ambiguity attitude from preferences over bets on natural events. While our method is in spirit similar to theirs, in the context of a NEO-additive capacity \(\nu _{s}\) on \(E_{s}\) characterized by two parameters, one only needs to measure 6 certainty equivalents regardless of the size of \(E_{s}\), as opposed to the \(2\left| E_{s}\right|\) as suggested by their procedure.

  8. 8.

    We adopt the convention that \(\frac{1}{0}=\infty\).

  9. 9.

    Invariance of risk attitudes is assumed in other studies on choice under growing awareness including Karni and Viero (2013), Schipper (2013), Dominiak and Tserenjigmid (2018, 2020) and Dietrich (2018). In a recent experimental study, Ma and Schipper (2017) could not find a significant evidence for changing risk attitudes of subjects making decisions in choice problems associated with different levels of awareness.

  10. 10.

    Moreover, Dominiak and Tserenjigmid (2020) show that Negative Unambiguity Independence implies Unambiguity Consistency but not Likelihood Consistency. To satisfy Likelihood Consistency, an additional axiom is required. In this regard, our Axiom 1 is stronger than their Negative Unambiguity Independence.

  11. 11.

    Recall that the set S is finite and thus, just as in the model of Eichberger and Kelsey (1999) identifies a finite partition of \(\Omega\), \(\left\{ E_{s}\right\} _{s\in S}\). Our setting provides a natural interpretation of such a partition as corresponding to the original state space.

  12. 12.

    We thank an anonymous referee for this remark.

  13. 13.

    While we directly assume the existence of such a partition, in a richer, Savage-type framework with infinitely many states, we could use the methods developed by Gul and Pesendorfer (2014) to identify the set of “ideal” events S.

  14. 14.

    In this model, we are primarily interested in the expansion of the state space. Hence, we keep the set of outcomes constant. Acts are then mappings from states to outcomes and the set of acts naturally expands with the discovery of new states. Thus, our framework, which models awareness directly through a change in the state space seems more appropriate in this context than the approach of Karni and Viero (2013), where new states have to be constructed either from new acts or new outcomes.

  15. 15.

    Piermont (2019) suggests a revision rule allowing relative likelihoods to change as awareness increases.

  16. 16.

    Karni and Viero (2015) allow for non-additivity by modelling unawareness with probabilistically sophisticated beliefs. While this captures Allais-style violations of additivity, it does not incorporate ambiguity.

  17. 17.

    For a more detailed comparison of our model to Dominiak and Tserenjigmid (2020), see Remarks 4 and 5.

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Acknowledgements

This research has been supported by IUF, Labex MME-DII and the French National Research Agency in the framework of the “Investissements d’avenir” program ANR-15-IDEX-02.

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Correspondence to Adam Dominiak.

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Submission to the Special Issue of Theory and Decision on Ambiguity and Strategic Interactions in Honor of Jürgen Eichberger.

Appendix: Proofs

Appendix: Proofs

To prove Proposition 1, we derive and prove a couple of helpful Lemmas.

Lemma 5

Axiom 4 implies:

  1. (i)

    Independence for measurable acts: For any acts f , \(g\in F\) , \(f\succsim g\) iff for any \(\lambda \in \left[ 0,1\right]\) and any measurable act \(h\in F^{S}\) , \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\) .

  2. (ii)

    Eventwise comonotonic independence: For any acts f , g and h , which are pairwise comonotonic on each \(E_{s}\) , \(f\succsim g\) iff for any \(\lambda \in \left[ 0,1\right]\) , \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\) . In particular, for any pairwise comonotonic acts f and h and g and h , \(f\succsim g\) iff for any \(\lambda \in \left[ 0,1\right]\) , \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\) .

  3. (iii)

    Extreme event independence: For all \(f,g,h\in F\) such that for every \(s\in S\) ,

    $$\begin{aligned}&{\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( h\right) \not =\varnothing , \\&{\underline{E}}_{s}\left( f\right) \cap {\underline{E}}_{s}\left( h\right) \not =\varnothing , \\&{\overline{E}}_{s}\left( g\right) \cap {\overline{E}}_{s}\left( h\right) \not =\varnothing , \\&{\underline{E}}_{s}\left( g\right) \cap {\underline{E}}_{s}\left( h\right) \not =\varnothing , \\&f\succsim g\quad \text {iff}\quad \lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h \,\text { for all } \lambda \in \left[ 0,1\right] . \end{aligned}$$
  4. (iv)

    Extreme event sensitivity: For any \(\lambda \in \left[ 0,1\right]\) and any f , g , \(h\in F\) such that \(f\sim g\) and such that for every \(s\in S\) ,

    $$\begin{aligned} {\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( h\right)&\not =&\varnothing , \\ \underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( h\right)&\not =&\varnothing ; \end{aligned}$$
  5. (v)

    if for every \(s\in S\) , \(\underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( h\right) \not =\varnothing\) , then \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\) ;

  6. (vi)

    if for every \(s\in S\) , \({\overline{E}}_{s}\left( g\right) \cap {\overline{E}}_{s}\left( h\right) \not =\varnothing\) , then \(\lambda g+\left( 1-\lambda \right) h\succsim \lambda f+\left( 1-\lambda \right) h\) .

Proof of Lemma 5:

Define \(f^{\prime }=\lambda f+\left( 1-\lambda \right) h\) and \(g^{\prime }=\lambda g+\left( 1-\lambda \right) h\).

Part \(\varvec{(i):}\) Since h is measurable, we have that for every \(s\in S\) and any two \(\omega\), \(\omega ^{\prime }\in E_{s}\), \(h\left( \omega \right) =h\left( \omega ^{\prime }\right)\). It thus follows that

$$\begin{aligned} \underline{E}_{s}\left( f\right)&= \underline{E}_{s}\left( f^{\prime }\right) , \\ \underline{E}_{s}\left( g\right)&= \underline{E}_{s}\left( g^{\prime }\right) , \\ {\overline{E}}_{s}\left( f\right)&= {\overline{E}}_{s}\left( f^{\prime }\right) , \\ {\overline{E}}_{s}\left( g\right)&= {\overline{E}}_{s}\left( g^{\prime }\right) . \end{aligned}$$

Clearly, \(\min _{\omega \in \underline{E}_{s}\left( f\right) }h\left( \omega \right) =\min _{\omega \in \underline{E}_{s}\left( g\right) }h\left( \omega \right)\) and \(\max _{\omega \in {\overline{E}}_{s}\left( f\right) }h\left( \omega \right) =\max _{\omega \in {\overline{E}}_{s}\left( g\right) }h\left( \omega \right)\) are also satisfied. By Part \(\left( i\right)\) of Axiom 4, we thus have that \(f\succsim g\) implies \(f^{\prime }\succsim g^{\prime }\) and thus,

$$\begin{aligned} \lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\text {.} \end{aligned}$$

Thus, the “if”-direction is satisfied. To show the “only if”-direction, let first \(\lambda f+\left( 1-\lambda \right) h\succ \lambda g+\left( 1-\lambda \right) h\) and suppose that \(g\succsim f\). Note, however that, by Part \(\left( ii\right)\) of Axiom 4, \(g\succsim f\) implies \(g^{\prime }\succsim f^{\prime }\) and thus,

$$\begin{aligned} \lambda g+\left( 1-\lambda \right) h\succsim \lambda f+\left( 1-\lambda \right) h\text {,} \end{aligned}$$

in contradiction to the assumption made. Suppose next that \(\lambda f+\left( 1-\lambda \right) h\sim \lambda g+\left( 1-\lambda \right) h\). By Part \(\left( i\right)\) of Axiom 4, we have \(g\not \succ f\), by part \(\left( ii\right)\), we have \(f\not \succ g\) and by Axiom 2, we obtain \(f\sim g\). It follows that \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\) implies \(f\succsim g\), demonstrating the “only if”-part of the statement.

Part (ii): If f and h are comonotonic on every \(E_{s}\), we have that for every \(s\in S\), \(\underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( h\right) \not =\varnothing\), and thus, \(\underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( f^{\prime }\right) \not =\varnothing\) and similarly, \({\overline{E}}_{s}(f)\cap {\overline{E}}_{s}(h)\not =\varnothing\), implying \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( f^{\prime }\right) \not =\varnothing\). Similarly, since g and h comonotonic on every \(E_{s}\), \(\underline{E} _{s}\left( g\right) \cap \underline{E}_{s}\left( g^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( g\right) \cap {\overline{E}} _{s}\left( g^{\prime }\right) \not =\varnothing\) obtain. It follows that for each \(s\in S\),

$$\begin{aligned} \underset{\omega \in \underline{E}_{s}\left( f\right) }{\min }h\left( \omega \right) \sim \min _{\omega \in \underline{E}_{s}\left( g\right) }h\left( \omega \right) \sim \min _{\omega \in E_{s}}h\left( \omega \right) \end{aligned}$$

and

$$\begin{aligned} \underset{\omega \in {\overline{E}}_{s}\left( f\right) }{\max }h\left( \omega \right) \sim \underset{\omega \in {\overline{E}}_{s}\left( g\right) }{\max } h\left( \omega \right) \sim \max _{\omega \in E_{s}}h\left( \omega \right) \text {.} \end{aligned}$$

Thus, the conditions of both Part (i) and Part (ii) of Axiom 4 are satisfied. Using the same arguments as in Part (i) of the proof, we thus conclude that \(f\succsim g\) iff \(f^{\prime }\succsim g^{\prime }\) iff \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\).

Note that if f, g and h are pairwise comonotonic, then they are also pairwise comonotonic on every event \(E_{s}\) and thus, the result obtains from the argument above.

Part (iii): As in the proof of Part (ii), for each \(s\in S\), \(\underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( h\right) \not =\varnothing\) implies \(\underline{E} _{s}\left( f\right) \cap \underline{E}_{s}\left( f^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}} _{s}\left( h\right) \not =\varnothing\), implies \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( f^{\prime }\right) \not =\varnothing\). Similarly, \(\underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( g^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( g\right) \cap {\overline{E}}_{s}\left( g^{\prime }\right) \not =\varnothing\) obtain. The rest of the proof is identical to that of Part (ii).

Part (iv): As in the proof of Part (ii), for each \(s\in S\), \(\underline{E}_{s}\left( f\right) \cap \underline{E} _{s}\left( h\right) \not =\varnothing\) implies \(\underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( f^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( h\right) \not =\varnothing\), implies \({\overline{E}}_{s}\left( f\right) \cap \overline{E }_{s}\left( f^{\prime }\right) \not =\varnothing\).

Part (iv-i): Similarly, for each \(s\in S\), \(\underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( h\right) \not =\varnothing\) implies \(\underline{E}_{s}\left( g\right) \cap \underline{ E}_{s}\left( g^{\prime }\right) \not =\varnothing\). We thus have:

$$\begin{aligned} \underset{\omega \in \underline{E}_{s}\left( f\right) }{\min }h\left( \omega \right) \sim \min _{\omega \in \underline{E}_{s}\left( g\right) }h\left( \omega \right) \sim \min _{\omega \in E_{s}}h\left( \omega \right) \end{aligned}$$

and

$$\begin{aligned} \max _{\omega \in E_{s}}h\left( \omega \right) \sim \underset{\omega \in {\overline{E}}_{s}\left( f\right) }{\max }h\left( \omega \right) \sim \underset{\omega \in {\overline{E}}_{s}\left( f^{\prime }\right) }{\max }h\left( \omega \right) \succsim \underset{\omega \in {\overline{E}}_{s}\left( g^{\prime }\right) }{\max }h\left( \omega \right) \end{aligned}$$

for each \(s\in S\). It follows that all conditions of Part (i) of Axiom 4 are satisfied and we thus obtain that \(f\sim g\) implies \(f^{\prime }\succsim g^{\prime }\), or \(\lambda f+\left( 1-\lambda \right) h\succsim \lambda g+\left( 1-\lambda \right) h\).

Part (iv-ii): Similarly, for each \(s\in S\), \({\overline{E}}_{s}\left( g\right) \cap {\overline{E}}_{s}\left( h\right) \not =\varnothing\) implies \({\overline{E}}_{s}\left( g\right) \cap {\overline{E}} _{s}\left( g^{\prime }\right) \not =\varnothing\). We thus have:

$$\begin{aligned} \underset{\omega \in {\overline{E}}_{s}\left( f\right) }{\max }h\left( \omega \right) =\underset{\omega \in {\overline{E}}_{s}\left( g\right) }{\max } h\left( \omega \right) =\max _{\omega \in E_{s}}h\left( \omega \right) \end{aligned}$$

and

$$\begin{aligned} \min _{\omega \in \underline{E}_{s}\left( g^{\prime }\right) }h\left( \omega \right) \succsim \underset{\omega \in \underline{E}_{s}\left( f\right) }{ \min }h\left( \omega \right) \sim \underset{\omega \in \underline{E} _{s}\left( f^{\prime }\right) }{\min }h\left( \omega \right) \sim \min _{\omega \in E_{s}}h\left( \omega \right) \end{aligned}$$

for each \(s\in S\). It follows that all conditions of Part (ii) of Axiom 4 are satisfied and we thus obtain that \(f\sim g\) implies \(g^{\prime }\succsim f^{\prime }\), or \(\lambda g+\left( 1-\lambda \right) h\succsim \lambda f+\left( 1-\lambda \right) h\). \(\square\)

Lemma 6

Axioms 2–5 imply that there exist:

  1. (i)

    an affine utility function \(U:\Delta \left( X\right) \rightarrow {\mathbb {R}}\) which is unique up to a positive-affine transformation and represents utility over constant acts, i.e., for some \(l_{f}\) and \(l_{g}\in \Delta \left( X\right)\) , \(f\left( \omega \right) =l_{f}\) for all \(\omega \in \Omega\) and \(g\left( \omega \right) =l_{g}\) for all \(\omega \in \Omega\) , \(f\succsim g\) iff \(U\left( l_{f}\right) \ge U\left( l_{g}\right)\) and

  2. (ii)

    a unique probability distribution \(\left( p\left( E_{s}\right) \right) _{s\in E_{s}}\) with \(p\left( E_{s}\right) >0\) for all \(s\in S\) such that for any two measurable acts f and \(g\in F^{S}\) , \(f\succsim g\) iff \(V\left( f\right) \ge V\left( g\right)\) , where

    $$\begin{aligned} V\left( f\right) =\sum _{s\in S}U\left( f\left( E_{s}\right) \right) p\left( E_{s}\right) \text {.} \end{aligned}$$

Proof of Lemma 6:

Follows by noting that Lemma 5, Part (i) implies independence on the set \(F^{S}\), whereas Axiom 5 implies both the monotonicity and the non-triviality axiom of Anscombe–Aumann. Thus, Axioms \(2{-}5\) on the set of measurable acts \(F^{S}\) satisfy the Anscombe–Aumann axioms. Axiom 5 implies that no \(\omega\) and thus, no \(E_{s}\) is null, so that \(p\left( E_{s}\right) >0\) for all \(s\in S\). \(\square\)

Lemma 7

Axioms 1–5 imply that there exist:

  1. (i)

    an affine utility function \(U_{0}:\Delta \left( X\right) \rightarrow {\mathbb {R}}\) which is unique up to a positive-affine transformation and represents utility over constant acts, i.e., for some \(l_{f}\) and \(l_{g}\in \Delta \left( X\right)\) , \(f\left( \omega \right) =l_{f}\) for all \(\omega \in \Omega\) and \(g\left( \omega \right) =l_{g}\) for all \(\omega \in \Omega\) , \(f\succsim _{0}g\) iff \(U_{0}\left( l_{f}\right) \ge U_{0}\left( l_{g}\right)\) and

  2. (ii)

    a unique probability distribution \(\pi :S\rightarrow \left[ 0,1\right]\) such that for any two acts f and \(g\in F_{0}\), \(f\succsim _{0}g\) iff \(V_{0}\left( f\right) \ge V_{0}\left( g\right)\), where

    $$\begin{aligned} V_{0}\left( f\right) =\sum _{s\in S}U_{0}\left( f\left( s\right) \right) \pi \left( s\right) \text {.} \end{aligned}$$

    Furthermore, \(\pi \left( s\right) =p\left( E_{s}\right) >0\) for each \(s\in S\) and \(U_{0}\) is identical up to an positive-affine transformation to U derived in Lemma 6.

Proof of Lemma 7:

Note that for any two acts \({\tilde{f}}\) and \({\tilde{g}}\in F_{0}\), there are unique measurable acts f and \(g\in F^{S}\) such that \({\tilde{f}}=\varphi \left( f\right)\) and \({\tilde{g}}=\varphi \left( g\right)\). Then, by Axiom 1, \({\tilde{f}}\succsim _{0}{\tilde{g}}\) iff \(f\succsim g\), iff, by Lemma 6,

$$\begin{aligned} \sum _{s\in S}U\left( f\left( E_{s}\right) \right) p\left( E_{s}\right) \ge \sum _{s\in S}U\left( g\left( E_{s}\right) \right) p\left( E_{s}\right) \text { .} \end{aligned}$$

By the uniqueness properties shown in Lemma 6, setting \(\pi \left( s\right) =p\left( E_{s}\right)\) and choosing \(U_{0}\) to be a positive-affine transformation of U thus gives the desired representation with the corresponding uniqueness properties. \(\square\)

Lemma 8

Axioms 1–5 imply Axioms 1–5 of Ghirardato et al. ( 2004 , p. 141).

Proof of Lemma 8:

Axiom 1, Weak order, of Ghirardato et al. (2004) is equivalent to our Axiom 2. Their Axiom 3, Certainty Independence, follows from our Axiom 4 as shown in Lemma 5, Part (i) by taking h to be a constant (and thus, measurable) act. Axioms 4 and 5, Monotonicity and Nondegeneracy, of Ghirardato et al. (2004) follow from our Axiom 5. \(\square\)

Lemma 9

Axioms 1–5 imply that there exist:

  1. (i)

    a utility function \(U:\Delta \left( X\right) \rightarrow {\mathbb {R}}\) which is unique up to an affine-linear transformation and represents utility over constant acts, i.e., for some \(l_{f}\) and \(l_{g}\in \Delta \left( X\right)\) , \(f\left( \omega \right) =l_{f}\) for all \(\omega \in \Omega\) and \(g\left( \omega \right) =l_{g}\) for all \(\omega \in \Omega\) , \(f\succsim g\) iff \(U\left( l_{f}\right) \ge U\left( l_{g}\right)\) ;

  2. (ii)

    a unique utility functional \(I:{\mathbb {R}}^{\Omega }\rightarrow {\mathbb {R}}\) such that \(f\succsim g\) iff \(I\left( U\left( f\left( \omega \right) \right) _{\omega \in \Omega }\right) \ge I\left( U\left( g\left( \omega \right) \right) _{\omega \in \Omega }\right)\) and such that: (a) I is constant-additive, i.e., \(I\left( {\mathbf {U}} +a\right) =I\left( {\mathbf {U}}\right) +a\); (b) I is positively homogenous, i.e., \(I\left( \lambda {\mathbf {U}}\right) =\lambda I\left( {\mathbf {U}}\right)\) and (c) I is monotonic;

  3. (iii)

    a capacity \(\nu :2^{\Omega -1}\rightarrow \left[ 0,1\right]\) such that for any two events A , \(B\subseteq \Omega\) , any outcomes x , y , z , \(w\in X\) with \(x\succsim y\) and \(w\succsim z\) and the binary acts \(f=x_{A}y\) and \(g=w_{B}z\) , \(f\succsim g\) iff

    $$\begin{aligned} \nu \left( A\right) U\left( x\right) +\left( 1-\nu \left( A\right) \right) U\left( y\right) \ge \nu \left( B\right) U\left( w\right) +\left( 1-\nu \left( B\right) \right) U\left( z\right) \text {.} \end{aligned}$$

    \(\nu \left( A\right) \in \left( 0,1\right)\) for all \(A\not =\varnothing\) .

    The capacity \(\nu\) is additive on the partition of \(\Omega\) , \(\left( E_{s}\right) _{s\in S}\) , i.e., \(\nu \left( E_{s}\right) =p\left( E_{s}\right)\) and for any s , \(s^{\prime }\in S\) , \(s\not =s^{\prime }\) , \(\nu \left( E_{s}\cup E_{s^{\prime }}\right) =\nu \left( E_{s}\right) +\nu \left( E_{s^{\prime }}\right)\) . Furthermore, for any \(A\subseteq E_{s}\) and \(B\subseteq E_{s^{\prime }}\) with \(s\not =s^{\prime }\) ,

    $$\begin{aligned} \nu \left( A\cup B\right) =\nu \left( A\right) +\nu \left( B\right) \text {.} \end{aligned}$$

Proof of Lemma 9:

Parts (i)-(ii) follow from the fact that, by Lemma 8, \(\succsim\) satisfies the Axioms 1–5 of Ghirardato et al. (2004), and thus, the conditions of Lemma 1 in Ghirardato et al. (2004, p. 141).

Part (iii): To prove this part, denote by \(x^{1}\) the best and by \(x^{0}\) the worst possible consequence in X, i.e., \(x^{1}\) is such that \(x_{\Omega }^{1}\succsim g\) for any \(g\in F\) and \(x^{0}\) is such that \(g\succsim x_{\Omega }^{0}\) for every \(g\in F\). Normalize U so that \(U\left( x^{1}\right) =1\) and \(U\left( x^{0}\right) =0\), where we slightly abuse notation and write \(U\left( x\right)\) for the utility of the lottery, which delivers x with probability 1. Part (iii) can be obtained by setting for each \(A\subseteq \Omega\), \(\nu \left( A\right) =I\left( x_{A}^{1}x^{0}\right)\) and noting that by Axiom 5, we have, \(x_{A}^{1}x^{0}\succsim x_{B}^{1}x^{0}\) whenever \(B\subseteq A\) and thus, \(\nu \left( A\right) \ge \nu \left( B\right)\), whenever \(B\subseteq A\). Since, by the definition of U, \(\nu \left( \Omega \right) =U\left( x^{1}\right) =1\) and \(\nu \left( \varnothing \right) =U\left( x^{0}\right) =0\), we thus have that \(\nu\) is a capacity on \(\Omega\). For an arbitrary outcome x, we have \(U\left( x\right) =U\left( x\right) U\left( x^{1}\right) +\left( 1-U\left( x\right) \right) U\left( x^{0}\right) \in \left[ 0,1\right]\). It follows from parts (i) and (ii) that \(x_{\Omega }\sim \left( l_{x}\right) _{\Omega }\), where \(l_{x}\) is a lottery with \(l\left( x^{1}\right) =U\left( x\right)\) and \(l\left( x^{0}\right) =1-U\left( x\right)\). Let \(x\succsim y\). By Axiom 5, we obtain: \(x_{A}y\sim \left( l_{x}\right) _{A}\left( l_{y}\right)\) and thus,

$$\begin{aligned} I\left( x_{A}y\right) &= I\left( \left( U\left( x\right) \right) _{A}\left( U\left( y\right) \right) \right) =I\left( \left( \left[ U\left( x\right) -U\left( y\right) \right] \right) _{A}0\right) +U\left( y\right) \\ &= \left[ U\left( x\right) -U\left( y\right) \right] I\left( U\left( x^{1}\right) _{A}0\right) +U\left( y\right) \\ &= \nu \left( A\right) U\left( x\right) +\left( 1-\nu \left( A\right) \right) U\left( y\right) , \end{aligned}$$

where the second equality follows by the constant-additivity of I, the third by the positive homogeneity of I and the last from the definition of \(\nu \left( A\right)\). The extension to arbitrary lotteries follows easily from the affinity of U.

The fact that \(\nu \left( A\right) \in \left( 0,1\right)\) for all non-empty A follows from Axiom 5.

To show that \(\nu\) is additive on \(\left( E_{s}\right) _{s\in S}\), note that by Lemma 6, for any f, \(g\in F^{S}\),

$$\begin{aligned} f \succsim g \text { iff } \sum _{s\in S}p\left( E_{s}\right) U\left( f\left( s\right) \right) \ge \sum _{s\in S}p\left( E_{s}\right) U\left( g\left( s\right) \right) \text {.} \end{aligned}$$
(24)

In particular, for any \(E_{s}\) consider the acts \(f=x_{E_{s}}^{1}x^{0}\) and \(g=l_{\Omega }\) for some lottery l, with \(l\left( x^{1}\right) +l\left( x^{0}\right) =1\). By (24), we get \(f\succsim g\) iff

$$\begin{aligned} p\left( E_{s}\right) \ge U\left( l\right) =l\left( x^{1}\right) \text {,} \end{aligned}$$

whereas by Part (ii), we have \(f\succsim g\) iff

$$\begin{aligned} \nu \left( E_{s}\right) \ge U\left( l\right) =l\left( x^{1}\right) \text {.} \end{aligned}$$

Since this is true for any \(l\left( x^{1}\right) \in \left[ 0,1\right]\), it must be that \(p\left( E_{s}\right) =\nu \left( E_{s}\right)\). But since p is additive, we obtain for any two s, \(s^{\prime }\in S\), \(s\not =s^{\prime }\), \(\nu \left( E_{s}\cup E_{s^{\prime }}\right) =\nu \left( E_{s}\right) +\nu \left( E_{s^{\prime }}\right)\).

Finally, consider s, \(s^{\prime }\in S\), \(s\not =s^{\prime }\), and let \(A\subseteq E_{s}\) and \(B\subseteq E_{s^{\prime }}\) and define \(f=x_{A}^{1}x^{0}\), \(h=x_{B}^{1}x^{0}\), \(f^{\prime }=\frac{1}{2}f+\frac{1}{2} h\). Note that f and h satisfy eventwise comonotonicity as in Part (ii) of Lemma 5. Consider the constant act \(g\left( \omega \right) =l\) for all \(\omega\), where \(l\left( x^{1}\right) =\nu \left( A\right)\) and \(l\left( x^{0}\right) =1-\nu \left( A\right)\). Let \(g^{\prime }=\frac{1}{2}g+\frac{1}{2}h\). Since g is constant, g and h also satisfy eventwise comonotonicity. By Part (iii) of the current Lemma, we have \(g\sim f\). Hence, by Axiom 4, we have \(g^{\prime }\sim f^{\prime }\), whereas the representation of \(\succsim\) by I implies:

$$\begin{aligned} I\left( f^{\prime }\right) =\frac{1}{2}\nu \left( A\cup B\right) =I\left( g^{\prime }\right) =\frac{1}{2}\nu \left( A\right) +\frac{1}{2}\nu \left( B\right) \text {,} \end{aligned}$$

or \(\nu \left( A\cup B\right) =\nu \left( A\right) +\nu \left( B\right)\) showing, as desired, that \(\nu\) is additively separable. \(\square\)

Lemma 10

Axioms 2–5 imply that \(f\succsim g\) iff \(V\left( f\right) \ge V\left( g\right)\), where \(V\left( f\right)\) is given by:

$$\begin{aligned} V\left( f\right) =\sum _{s\in S}p\left( E_{s}\right) \int _{E_{s}}U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}, \end{aligned}$$

where, for each \(s\in S\), \(\nu _{s}\) is a capacity on \(E_{s}\) such that for \(E\subseteq E_{s}\), \(\nu _{s}\left( E\right) =\frac{\nu \left( E\right) }{\nu \left( E_{s}\right) }\), where \(\nu\) is the capacity on \(\Omega\), U is the utility function over outcomes obtained in Lemma 9and p is the probability distribution on \(\{E_{s}\}_{s\in S}\) obtained in Lemma 6.

Proof of Lemma 10:

Normalize U so that \(U\left( x^{0}\right) =0\) and \(U\left( x^{1}\right) =1\) as in the proof of Lemma 9 and consider an arbitrary act f. For the given act f and a given \(s\in E_{s}\), order the states in \(E_{s}=\{\omega _{1}\left( s,f\right) ,\ldots ,\omega _{\left| E_{s}\right| }\left( s,f\right) \}\) in such a way that \(f\left( \omega _{n}\left( s,f\right) \right) \succ f\left( \omega _{n^{\prime }}\left( s,f\right) \right)\) implies \(n<n^{\prime }\), i.e., states within each event \(E_{s}\) are ordered in descending order according to the utility of the payoff obtained on the state. Note that for every possible realization of f , \(f\left( \omega _{n}\left( s,f\right) \right)\), Axiom 3 implies that there is a lottery \({\bar{l}}_{n}\left( s,f\right)\) such that \({\bar{l}} _{n}\left( s,f\right) \left( x^{1}\right) =\lambda _{n}\left( s,f\right)\) and \({\bar{l}}_{n}\left( s,f\right) \left( x^{0}\right) =\left( 1-\lambda _{n}\left( s,f\right) \right)\) such that \({\bar{l}}_{n}\left( s,f\right) \sim f\left( \omega _{n}\left( s,f\right) \right)\). By Axiom 5, for each \(s\in S\), \(\lambda _{n}\left( s,f\right) \ge \lambda _{n^{\prime }}\left( s,f\right)\) whenever \(n<n^{\prime }\).

Following Chateauneuf et al. (2007, p. 561), we now construct an act which is indifferent to f and for which we can identify the value of I by repeatedly using Axiom 4 and in particular, the property of Eventwise Comonotonic Independence as derived in Part (ii) of Lemma 5. Note that

$$\begin{aligned} f= & {} \left( \begin{array}{ll} f\left( \omega _{1}\left( s,f\right) \right) &{} \text {on } \omega _{1}\left( s,f\right) \\ \vdots &{} \vdots \\ f\left( \omega _{\left| E_{s}\right| }\left( s,f\right) \right) &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) _{s\in S}\sim \left( \begin{array}{ll} {\bar{l}}_{1}\left( s,f\right) &{} \text {on } \omega _{1}\left( s,f\right) \\ \vdots &{} \vdots \\ {\bar{l}}_{\left| E_{s}\right| }\left( s,f\right) &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) _{s\in S} \\&\sim \left( \left( 1-\lambda _{1}\left( s,f\right) \right) \left( \begin{array}{ll} x^{0} &{} \text {on } \omega _{1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{2}\left( s,f\right) \\ \vdots &{} \vdots \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| -1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) \right) _{s\in S} \\&+\left( \left( \lambda _{1}\left( s,f\right) -\lambda _{2}\left( s,f\right) \right) \left( \begin{array}{ll} x^{1} &{} \text {on } \omega _{1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{2}\left( s,f\right) \\ \vdots &{} \vdots \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| -1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{1}\left( s,f\right) \end{array} \right) \right) _{s\in S} \\&+\left( \left( \lambda _{2}\left( s,f\right) -\lambda _{3}\left( s,f\right) \right) \left( \begin{array}{ll} x^{1} &{} \text {on } \omega _{1}\left( s,f\right) \\ x^{1} &{} \text {on } \omega _{2}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{3}\left( s,f\right) \\ \vdots &{} \vdots \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| -1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) \right) _{s\in S}+ \ldots + \\&+\left( \left( \lambda _{\left| E_{s}\right| -1}\left( s,f\right) -\lambda _{\left| E_{s}\right| }\left( s,f\right) \right) \left( \begin{array}{ll} x^{1} &{} \text {on } \omega _{1}\left( s,f\right) \\ x^{1} &{} \text {on } \omega _{2}\left( s,f\right) \\ \vdots &{} \vdots \\ x^{1} &{} \text {on } \omega _{\left| E_{s}\right| -1}\left( s,f\right) \\ x^{0} &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) \right) _{s\in S} \\&+\left( \lambda _{\left| E_{s}\right| }\left( s,f\right) \left( \begin{array}{ll} x^{1} &{} \text {on } \omega _{1}\left( s,f\right) \\ x^{1} &{} \text {on } \omega _{2}\left( s,f\right) \\ \vdots &{} \vdots \\ x^{1} &{} \text {on } \omega _{\left| E_{s}\right| -1}\left( s,f\right) \\ x^{1} &{} \text {on } \omega _{\left| E_{s}\right| }\left( s,f\right) \end{array} \right) \right) _{s\in S}. \end{aligned}$$

In particular, note that all the acts in the sum satisfy the eventwise comonotonicity property as stated in Lemma 5, Part (ii). Furthermore, by Lemma 9, we have that for any act of the type \(x_{A}^{1}x^{0}\),

$$\begin{aligned} I\left( x_{A}^{1}x^{0}\right) =\nu \left( A\right) =\sum _{s\in S}\nu \left( A\cap E_{s}\right) \text {.} \end{aligned}$$

Thus, from the Eventwise Comonotonicty Independence, we conclude that

$$\begin{aligned} I\left( f\right) =\sum _{s\in S}\sum _{{\bar{n}}=1}^{\left| E_{s}\right| }\nu \Big ( \bigcup _{n=1}^{{\bar{n}}}\left\{ \omega _{n}\left( s,f\right) \right\} \Big ) \big [ \lambda _{{\bar{n}}}\left( s,f\right) -\lambda _{{\bar{n}} +1}\left( s,f\right) \big ], \end{aligned}$$

where as per usual convention, we set \(\lambda _{\left| E_{s}\right| +1}=0\). Since by Lemma 9, \(I\left( f\right)\) represents preferences \(\succsim\) on F, we have that

$$\begin{aligned} V\left( f\right) =\sum _{s\in S}\sum _{{\bar{n}}=1}^{\left| E_{s}\right| }\nu \Big ( \bigcup _{n=1}^{{\bar{n}}}\left\{ \omega _{n}\left( s,f\right) \right\} \Big ) \big [ \lambda _{{\bar{n}}}\left( s,f\right) -\lambda _{{\bar{n}} +1}\left( s,f\right) \big ] \end{aligned}$$

represents preferences \(\succsim\) on f. Finally, noting that by Lemma 9, \(\nu \left( E_{s}\right) =p\left( E_{s}\right)\) , and, by definition, \(\lambda _{{\bar{n}}}\left( s,f\right) =U\left( f\left( \omega _{{\bar{n}}}\left( s,f\right) \right) \right)\), we obtain:

$$\begin{aligned} V\left( f\right)= & {} \sum _{s\in S}p\left( E_{s}\right) \sum _{{\bar{n}} =1}^{\left| E_{s}\right| }\frac{\nu \big ( \bigcup _{n=1}^{{\bar{n}} }\left\{ \omega _{n}\left( s,f\right) \right\} \big ) }{\nu \left( E_{s}\right) }\big [ \lambda _{{\bar{n}}}\left( s,f\right) -\lambda _{{\bar{n}} +1}\left( s,f\right) \big ] \\= & {} \sum _{s\in S}p\left( E_{s}\right) \int _{E_s} U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}, \end{aligned}$$

where, for each \(E\subseteq E_{s}\), \(\nu _{s}\left( E\right) =\frac{\nu \left( E\right) }{\nu \left( E_{s}\right) }\).

Since \(\nu\) is unique by Lemma 9 and since \(\nu \left( E_{s}\right) >0\) for all \(s\in S\), \(\nu _{s}\) is well defined and unique for every \(s\in S\). Finally, since \(\nu \left( E\right) >0\) for all \(E\not =\varnothing\), we obtain \(\nu _{s}\left( E\right) >0\) for all \(\varnothing \not =E\subseteq E_{s}\). \(\square\)

Recall that by Axiom 5, for any \(E_{s}\), the only null event in \(E_{s}\) is the empty set \(\varnothing\) and the only universal event in \(E_{s}\) is the event \(E_{s}\) itself. We restate here the characterization of NEO-additive capacities, see Chateauneuf et al. (2007, p. 542) adapted to our setting:

Proposition 4

Let \(s \in S\) be such that \(\left| E_{s}\right| \ge 3\) and suppose that Axiom 5 holds. Then \(\nu _{s}\) on \(E_s\) is a NEO-additive capacity on \(E_{s}\) iff the following three conditions are satisfied:

  1. (i)

    for all events E , H and \(G\subseteq E_{s}\) such that \(E\cap H=E\cap G=\varnothing\) , \(E\cup H\not =E_{s}\) , \(E\cup G\not =E_{s}\) ,

    $$\begin{aligned} \nu _{s}\left( E\cup H\right) -\nu _{s}\left( H\right) =\nu _{s}\left( E\cup G\right) -\nu _{s}\left( G\right) \text {;} \end{aligned}$$
  2. (ii)

    for some E , \(H\not =\varnothing\) , such that \(E\cap H=\varnothing\) and \(E\cup H\not =E_{s}\) ,

    $$\begin{aligned} \nu \left( E\cup H\right) \le \nu \left( E\right) +\nu \left( H\right) ; \end{aligned}$$
  3. (iii)

    for some E , \(H\not =\varnothing\) , such that \(E\cap H=\varnothing\) and \(E\cup H\not =E_{s}\) ,

    $$\begin{aligned} \nu \left( E^{c}\right) +\nu \left( H^{c}\right) \le 1+\nu \left( \left( E\cup H\right) ^{c}\right) . \end{aligned}$$

Proof of Proposition 4:

See Chateauneuf et al. (2007, pp. 556–558). \(\square\)

Lemma 11

For every \(s\in S\) such that \(\left| E_{s}\right| \ge 3\), \(\nu _{s}\) as defined in Lemma  10satisfies conditions \(\left( i\right)\), \(\left( ii\right)\) and \(\left( iii\right)\) of Proposition 4and is thus, a NEO-additive capacity. That is, there is a probability distribution \(q\left( \cdot \mid E_{s}\right)\) on \(E_{s}\) with \(q\left( \omega \mid E_{s}\right) >0\) for all \(\omega \in E_{s}\) and coefficients \(\rho _{s}\in \left( 0,1\right]\), and \(\alpha _{s}\in \left[ 0,1\right]\) such that:

$$\begin{aligned} \nu _{s}\left( E\right) =\left\{ \begin{array}{ll} 1 &{} \text {if } E=E_{s}, \\ \alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}q\left( E\mid E_{s}\right) &{} \text {if } E_{s}\not =E\cap E_{s}\not =\varnothing , \\ 0 &{} \text {if } E\cap E_{s}=\varnothing , \end{array} \right. \end{aligned}$$

Moreover, \(\rho _{s}\) is unique, \(\alpha _{s}\) is unique whenever \(\rho _{s}<1\) and \(q\left( \cdot \mid E_{s}\right)\) is unique.

Proof of Lemma 11:

For the case of \(\left| E_{s}\right| \ge 4\), the statement follows from the proof of Statement (B) in Chateauneuf et al. (2007, pp. 562–565).

For the case of \(\left| E_{s}\right| =3\), the fact that the conditions of Parts (ii) and (iii) of Proposition 4 are satisfied also follows from the proof of Statement (B) in Chateauneuf et al. (2007, pp. 562–565), which does not use the assumption of the existence of four disjoint essential events.

It remains to consider the case of \(\left| E_{s}\right| =3\) and show that the conditions of Part (i) of Proposition 4 are satisfied in this case. Note that for such an \(E_{s}=\left\{ \omega _{1},\omega _{2},\omega _{3}\right\}\), the only three events (up to permutation) that satisfy the conditions of Part (i) of Proposition 4 are the singletons, \(E=\left\{ \omega _{2}\right\}\), \(H=\left\{ \omega _{1}\right\}\), \(G=\left\{ \omega _{3}\right\}\). Define acts f, g, h, \(f^{\prime }\) and \(g^{\prime }\) as in Example 1. Note that by Lemma 9, \(f\sim g\) implies \(\nu \left( \left\{ \omega _{1}\right\} \right) =\beta \nu \left( \left\{ \omega _{3}\right\} \right)\). Since, as shown in Example 1f, g, \(f^{\prime }\), \(g^{\prime }\) and h satisfy both conditions \(\left( i\right)\) and \(\left( ii\right)\) of Axiom 4 we obtain \(f^{\prime }\sim g^{\prime }\), or

$$\begin{aligned} \frac{1}{2}\beta \nu \left( \left\{ \omega _{1},\omega _{2}\right\} \right) + \frac{1}{2}\left( 1-\beta \right) \nu \left( \left\{ \omega _{1}\right\} \right) &= \frac{1}{2}\beta \nu \left( \left\{ \omega _{2},\omega _{3}\right\} \right) \\ \beta \nu \left( \left\{ \omega _{1},\omega _{2}\right\} \right) -\beta \nu \left( \left\{ \omega _{1}\right\} \right) &= \beta \nu \left( \left\{ \omega _{2},\omega _{3}\right\} \right) -\nu \left( \left\{ \omega _{1}\right\} \right) \\ \beta \left[ \nu \left( \left\{ \omega _{1},\omega _{2}\right\} \right) -\nu \left( \left\{ \omega _{1}\right\} \right) \right] &= \beta \nu \left( \left\{ \omega _{2},\omega _{3}\right\} \right) -\beta \nu \left( \left\{ \omega _{3}\right\} \right) \end{aligned}$$

which is equivalent to

$$\begin{aligned} \nu \left( \left\{ \omega _{1},\omega _{2}\right\} \right) -\nu \left( \left\{ \omega _{1}\right\} \right) &= \nu \left( \left\{ \omega _{2},\omega _{3}\right\} \right) -\nu \left( \left\{ \omega _{3}\right\} \right) \\ \nu \left( E\cup H\right) -\nu \left( H\right) &= \nu \left( E\cup G\right) -\nu \left( G\right) . \end{aligned}$$

Dividing both sides by \(\nu \left( E_{s}\right)\), gives the condition of Part (i) of Proposition 4 : 

$$\begin{aligned} \nu _{s}\left( E\cup H\right) -\nu _{s}\left( H\right) =\nu _{s}\left( E\cup G\right) -\nu _{s}\left( G\right) \text {.} \end{aligned}$$

Permuting the definitions of the sets E, H and G and repeating the same argument gives the desired result for all singleton sets E, H and G.

To show that \(\rho _{s}>0\) and \(q\left( \omega \mid E_{s}\right) >0\) for every \(\omega \in E_{s}\), note that Axiom 5 implies that for any \(\omega\), \(\omega ^{\prime }\in E_{s}\), \(x_{\{\omega ,\omega ^{\prime }\}}^{1}x^{0}\succ x_{\{\omega ^{\prime }\}}^{1}x^{0}\) and thus,

$$\begin{aligned} \nu _{s}\left( \left\{ \omega ,\omega ^{\prime }\right\} \right) -\nu _{s}\left( \left\{ \omega ^{\prime }\right\} \right) =\rho _{s}q\left( \omega \mid E_{s}\right) >0 \end{aligned}$$

and thus, \(\rho _{s}>0\) and \(q\left( \omega \mid E_{s}\right) >0\) holds for every \(s\in S\) and every \(\omega \in E_{s}\).

The uniqueness properties of the parameters follow from Lemma A.1 in Chateauneuf et al. (2007, p. 555). \(\square\)

Lemma 12

The representation in Part (ii) of Proposition 1satisfies Axioms 1–3.

Proof of Lemma 12:

It is obvious. \(\square\)

Lemma 13

The representation described in Part (ii) of Proposition 1satisfies Axiom 4.

Proof of Lemma 13:

Consider acts f, \(f^{\prime }\), h and \(\lambda\), which satisfy the conditions of the axiom, i.e., \(f^{\prime }=\lambda f+\left( 1-\lambda \right) h\), \(\underline{E}_{s}\left( f\right) \cap \underline{E}_{s}\left( f^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( f^{\prime }\right) \not =\varnothing\).

Let s be such that \(\left| E_{s}\right| \ge 3\). By Part (3) of Proposition 1,

$$\begin{aligned} \int _{E_s} U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s} &= \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) \\& \quad + \, \alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) + \left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) \end{aligned}$$

and

$$\begin{aligned} \int _{E_s} U\left( \lambda f\left( \omega \right) +\left( 1-\lambda \right) h\left( \omega \right) \right) \mathrm{{d}}\nu _{s}&= \left[ \begin{array}{c} \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}\left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}\left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&=\left[ \begin{array}{c} \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ + \alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in {\overline{E}} _{s}\left( f\right) }\left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in \underline{E}_{s}\left( f\right) }\left[ \lambda U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&=\left[ \begin{array}{c} \rho _{s}\left[ \lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ + \alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\overline{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \min _{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] , \end{array} \right] \end{aligned}$$

where the first equality follows from the fact that \(\nu _{s}\) is NEO-additive, the second equality follows from the fact that \(\underline{E} _{s}\left( f\right) \cap \underline{E}_{s}\left( f^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}} _{s}\left( f^{\prime }\right) \not =\varnothing\) and the third, from the fact that \(f\left( \omega \right)\) is constant on the set \({\overline{E}} _{s}\left( f\right)\), as well as on the set \(\underline{E}_{s}\left( f\right)\).

If \(\left| E_{s}\right| =2\) with \(E_{s}=\left\{ \omega _{1}^{s},\omega _{2}^{s}\right\}\), assume, w.l.o.g., that \(f\left( \omega _{1}^{s}\right) \succsim f\left( \omega _{2}^{s}\right)\). By Part (2) of Proposition 1, the evaluation of f on \(E_{s}\) is given by:

$$\begin{aligned} \int _{E_s} U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}=\nu _{s}\left( \omega _{1}^{s}\right) U\left( f\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) U\left( f\left( \omega _{2}^{s}\right) \right) \end{aligned}$$

and since by Axiom 4, \(\underline{E}_{s}\left( f\right) \cap \underline{E} _{s}\left( f^{\prime }\right) \not =\varnothing\) and \({\overline{E}}_{s}\left( f\right) \cap {\overline{E}}_{s}\left( f^{\prime }\right) \not =\varnothing\), we have

$$\begin{aligned} \lambda f\left( \omega _{1}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{1}^{s}\right) \succsim \lambda f\left( \omega _{2}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{2}^{s}\right) \text {.} \end{aligned}$$

It follows that

$$\begin{aligned}&\int _{E_s} U\left( \lambda f\left( \omega \right) +\left( 1-\lambda \right) h\left( \omega \right) \right) \mathrm{{d}}\nu _{s} \\&\qquad =\nu _{s}\left( \omega _{1}^{s}\right) U\left( \lambda f\left( \omega _{1}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) U\left( \lambda f\left( \omega _{2}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{2}^{s}\right) \right) \\&\qquad =\nu _{s}\left( \omega _{1}^{s}\right) \left[ \lambda U\left( f\left( \omega _{1}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{1}^{s}\right) \right) \right] +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \left[ \lambda U\left( f\left( \omega _{2}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{2}^{s}\right) \right) \right] \text {.} \end{aligned}$$

Finally, if \(\left| E_{s}\right| =1\) with \(E_{s}=\left\{ \omega \right\}\), then

$$\begin{aligned} U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}=U\left( f\left( \omega \right) \right) . \end{aligned}$$

(i) Consider g and \(g^{\prime }=\lambda g+\left( 1-\lambda \right) h\) which satisfy the conditions of Part (i) of Axiom 4:

$$\begin{aligned} \underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( g^{\prime }\right)&\not =\varnothing , \\ \min _{\omega \in \underline{E}_{s}\left( f\right) }h\left( \omega \right)&\sim \min _{\omega \in \underline{E}_{s}\left( g\right) }h\left( \omega \right) \text { and } \\ \underset{\omega \in {\overline{E}}_{s}\left( f\right) }{\max }h\left( \omega \right)&=\underset{\omega \in {\overline{E}}_{s}\left( f^{\prime }\right) }{ \max }h\left( \omega \right) \succsim \underset{\omega \in {\overline{E}} _{s}\left( g^{\prime }\right) }{\max }h\left( \omega \right) . \end{aligned}$$

If \(\left| E_{s}\right| \ge 3\), we have

$$\begin{aligned} \int _{E_s} U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s} &= \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) \\& \quad + \, \alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) + \left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) \end{aligned}$$

and

$$\begin{aligned} & \int_{E_s} U\left( \lambda g\left( \omega \right) +\left( 1-\lambda \right) h\left( \omega \right) \right) d\nu _{s} = \\ & \qquad =\left[ \begin{array}{c} \rho _{s}\sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\alpha _{s}\left( 1-\rho _{s}\right) \max_{\omega \in E_{s}}\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min_{\omega \in E_{s}}\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\ & \qquad=\left[ \begin{array}{c} \rho _{s}\sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ + \alpha _{s}\left( 1-\rho _{s}\right) \max_{\omega \in \overline{E} _{s}\left( g^{\prime }\right) }\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min_{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min_{\omega \in \underline{E}_{s}\left( g\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\ & \qquad \leq \left[ \begin{array}{c} \rho _{s}\left[ \lambda \sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ + \alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max_{\omega \in \bar{E }_{s}\left( g^{\prime }\right) }U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max_{\omega \in \overline{E}_{s}\left( g^{\prime }\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min_{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min_{\omega \in \underline{E}_{s}\left( g\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\ & \qquad \leq \left[ \begin{array}{c} +\rho _{s}\left[ \lambda \sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \sum_{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ + \alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max_{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max_{ \bar{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min_{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min_{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right]. \end{aligned}$$

where the first equality follows from the fact that \(\nu _{s}\) is a NEO-additive capacity; the second equality follows from the fact that \(\underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( g^{\prime }\right) \not =\varnothing\), and \(g^{\prime }\) obtains its maximum on \({\overline{E}}_{s}\left( g^{\prime }\right)\); the first inequality is a consequence of the convexity of the \(\max\), whereas the last inequality follows from \(\max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) \ge \max _{{\bar{E}}_{s}\left( g^{\prime }\right) }U\left( g\left( \omega \right) \right)\), \(\min _{\omega \in \underline{E}_{s}\left( g\right) }U\left( h\left( \omega \right) \right) =\min _{\omega \in \underline{E} _{s}\left( f\right) }U\left( h\left( \omega \right) \right)\) and \(\max _{\omega \in {\overline{E}}_{s}\left( g^{\prime }\right) }U\left( h\left( \omega \right) \right) \le \max _{\omega \in {\overline{E}}_{s}\left( f^{\prime }\right) }U\left( h\left( \omega - \right) \right) =\max _{\omega \in {\bar{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right)\).

Let next \(\left| E_{s}\right| =2\) with \(E_{s}=\left\{ \omega _{1}^{s},\omega _{2}^{s}\right\}\). If f and g are comonotonic on \(E_{s}\) , we have \(g\left( \omega _{1}^{s}\right) \succsim g\left( \omega _{2}^{s}\right)\), so that

$$\begin{aligned} \int _{E_s} U\left( g\left( \omega _{1}^{s}\right) \right) \mathrm{{d}}\nu _{s}=\nu _{s}\left( \omega _{1}^{s}\right) U\left( g\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) U\left( g\left( \omega _{2}^{s}\right) \right) \end{aligned}$$

and since \(\underline{E}_{s}\left( g\right) \cap \underline{E}_{s}\left( g^{\prime }\right) \not =\varnothing\),

$$\begin{aligned} \lambda g\left( \omega _{1}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{1}^{s}\right) \succsim \lambda g\left( \omega _{2}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{2}^{s}\right) . \end{aligned}$$

It follows that

$$\begin{aligned}&\int _{E_s} U\left( \lambda g\left( \omega \right) +\left( 1-\lambda \right) h\left( \omega \right) \right) \mathrm{{d}}\nu _{s} \nonumber \\&\quad =\nu _{s}\left( \omega _{1}^{s}\right) U\left( \lambda g\left( \omega _{1}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) U\left( \lambda g\left( \omega _{2}^{s}\right) +\left( 1-\lambda \right) h\left( \omega _{2}^{s}\right) \right) \\&\quad =\nu _{s}\left( \omega _{1}^{s}\right) \left[ \lambda U\left( g\left( \omega _{1}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{1}^{s}\right) \right) \right] +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \left[ \lambda U\left( g\left( \omega _{2}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{2}^{s}\right) \right) \right] . \nonumber \end{aligned}$$
(25)

If f and g are (non-trivially) anti-comonotonic, i.e., \(g\left( \omega _{2}^{s}\right) \succ g\left( \omega _{1}^{s}\right)\). Then, \(\min _{\omega \in \underline{E}_{s}\left( f\right) }h\left( \omega \right) \sim \min _{\omega \in \underline{E}_{s}\left( g\right) }h\left( \omega \right)\) implies that h is a constant act and thus, (25) obtains for this case, as well.

Hence, if \(f\succsim g\), we have

$$\begin{aligned} V(f)=\sum _{s\in S}p\left( E_{s}\right) \int _{E_{s}}U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}\ge \sum _{s\in S}p\left( E_{s}\right) \int _{E_{s}}U\left( g\left( \omega \right) \right) \mathrm{{d}}\nu _{s}=V(g) \end{aligned}$$

which is equivalent to

$$\begin{aligned}&V(f)= \\&\sum _{\left\{ s\in S:\left| E_{s}\right| \not =2\right\} }p\left( E_{s}\right) \Big [\rho _{s}\lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) \\&\qquad \qquad \qquad \qquad +\alpha _{s}\left( 1-\rho _{s}\right) \lambda \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \lambda \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) \Big ] \\&\qquad \qquad \qquad \qquad +\sum _{\left\{ s\in S:\left| E_{s}\right| =2\right\} }p\left( E_{s}\right) \Big [\nu _{s}\left( \omega _{1}^{s}\right) \lambda U\left( f\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \lambda U\left( f\left( \omega _{2}^{s}\right) \right) \Big ] \\&\ge \\&\sum _{\left\{ s\in S:\left| E_{s}\right| \not =2\right\} }p\left( E_{s}\right) \Big [\rho _{s}\lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) \\&\qquad \qquad \qquad \qquad +\alpha _{s}\left( 1-\rho _{s}\right) \lambda \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \lambda \min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) \Big ] \\&\qquad \qquad \qquad \qquad +\sum _{\left\{ s\in S\mid \left| E_{s}\right| =2\right\} }p\left( E_{s}\right) \Big [\nu _{s}\left( \omega _{1}^{s}\right) \lambda U\left( g\left( \omega _{1}^{s}\right) \right) +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \lambda U\left( g\left( \omega _{2}^{s}\right) \right) \Big ] \\&=V(g) \end{aligned}$$

which, in turn implies

$$\begin{aligned}&V\left( f^{\prime }\right) = \\&\sum _{\left\{ s\in S:\left| E_{s}\right| \not =2\right\} }p\left( E_{s}\right) \left[ \begin{array}{l} \rho _{s}\left[ \lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ +\alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\overline{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\lambda \right) \min _{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&\qquad \qquad \qquad +\sum _{\left\{ s\in S:\left| E_{s}\right| =2\right\} }p\left( E_{s}\right) \left[ \begin{array}{l} \nu _{s}\left( \omega _{1}^{s}\right) \left[ \lambda U\left( f\left( \omega _{1}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{1}^{s}\right) \right) \right] \\ +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \left[ \lambda U\left( f\left( \omega _{2}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{2}^{s}\right) \right) \right] \end{array} \right] \\&\ge \\&\sum _{\left\{ s\in S:\left| E_{s}\right| \not =2\right\} }p\left( E_{s}\right) \left[ \begin{array}{l} \rho _{s}\left[ \lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ +\,\alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\bar{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min _{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&\qquad \qquad \qquad +\sum _{\left\{ s\in S:\left| E_{s}\right| =2\right\} }p\left( E_{s}\right) \left[ \begin{array}{l} \nu _{s}\left( \omega _{1}^{s}\right) \big [\lambda U\left( g\left( \omega _{1}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{1}^{s}\right) \right) \big ] \\ +\left( 1-\nu _{s}\left( \omega _{1}^{s}\right) \right) \big [\lambda U\left( g\left( \omega _{2}^{s}\right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega _{2}^{s}\right) \right) \big ] \end{array} \right] \\&\ge V\left( g^{\prime }\right) \end{aligned}$$

or \(f^{\prime }\succsim g^{\prime }\) as required by Axiom 4.

Part (ii): Consider next the acts g and \(g^{\prime }=\lambda g+\left( 1-\lambda \right) h\), which satisfy the conditions of Part (ii) of Axiom 4, i.e.,

$$\begin{aligned} {\overline{E}}_{s}\left( g\right) \cap {\overline{E}}_{s}\left( g^{\prime }\right)&\not =\varnothing , \\ \min _{\omega \in \underline{E}_{s}\left( g^{\prime }\right) }h\left( \omega \right)&\succsim \min _{\omega \in \underline{E}_{s}\left( f\right) }h\left( \omega \right) =\min _{\omega \in \underline{E}_{s}\left( f^{\prime }\right) }h\left( \omega \right) , \text { and } \\ \underset{\omega \in {\overline{E}}_{s}(f)}{\max }h\left( \omega \right)\sim & {} \underset{\omega \in {\overline{E}}_{s}(g)}{\max }h\left( \omega \right) . \end{aligned}$$

Then, for a capacity \(\nu _s\) on \(E_s\), we have

$$\begin{aligned}&\int _{E_s} U\left( \lambda g\left( \omega \right) +\left( 1-\lambda \right) h\left( \omega \right) \right) d\nu _{s} \\&\qquad =\left[ \begin{array}{c} \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\, \alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&\qquad =\left[ \begin{array}{c} \rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) \left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \\ +\,\alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\overline{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in \underline{E}_{s}\left( g^{\prime }\right) }\left[ \lambda U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&\qquad \ge \left[ \begin{array}{c} \rho _{s}\left[ \lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ +\, \alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\overline{E}}_{s}(f)}U\left( h\left( \omega \right) \right) \right] \\ +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min _{\omega \in \underline{E}_{s}\left( g^{\prime }\right) }U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min _{\omega \in \underline{ E}_{s}\left( g^{\prime }\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \\&\qquad \ge \left[ \begin{array}{c} \rho _{s}\left[ \lambda \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( h\left( \omega \right) \right) \right] \\ +\, \alpha _{s}\left( 1-\rho _{s}\right) \left[ \lambda \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \max _{\omega \in {\overline{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \\ + \left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \left[ \lambda \min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\lambda \right) \min _{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) \right] \end{array} \right] \end{aligned}$$

where the first equality follows from the fact that \(\nu _{s}\) is a NEO-additive capacity; the second equality follows from the fact that \(\overline{E_{s}}\left( g\right) \cap \overline{E_{s}}\left( g^{\prime }\right) \not =\varnothing\), \(\max _{\omega \in {\overline{E}}_{s}\left( f\right) }U\left( h\left( \omega \right) \right) =\max _{\omega \in \overline{ E}_{s}\left( g\right) }U\left( h\left( \omega \right) \right)\) and that \(g^{\prime }\) obtains its minimum on \({\bar{E}}_{s}\left( g^{\prime }\right)\); the first inequality is a consequence of the concavity of the \(\min\), whereas the last inequality follows from \(\min _{\omega \in \underline{E} _{s}\left( g^{\prime }\right) }U\left( g\left( \omega \right) \right)\) \(\ge\) \(\min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right)\) and \(\min _{\omega \in \underline{E}_{s}\left( g^{\prime }\right) }U\left( h\left( \omega \right) \right)\) \(\ge\) \(\min _{\omega \in \underline{E}_{s}\left( f^{\prime }\right) }U\left( h\left( \omega \right) \right) =\min _{\omega \in \underline{E}_{s}\left( f\right) }U\left( h\left( \omega \right) \right)\).

If \(\left| E_{s}\right| =2\) with \(E_{s}=\left\{ \omega _{1}^{s},\omega _{2}^{s}\right\}\) and if f and g are comonotonic, we have \(g\left( \omega _{1}^{s}\right) \succsim g\left( \omega _{2}^{s}\right)\), so that (25) holds. If f and g are (non-trivially) anti-comonotonic, i.e., \(g\left( \omega _{2}^{2}\right) \succ g\left( \omega _{1}^{s}\right)\). Then, \(\max _{\omega \in {\overline{E}} _{s}\left( f\right) }h\left( \omega \right) \sim \max _{\omega \in \overline{E }_{s}\left( g\right) }h\left( \omega \right)\) implies that h is a constant act and again (25) obtains.

The rest of the proof is symmetric to that in Part (i) and omitted for brevity. \(\square\)

Lemma 14

The representation described in Part (ii) of Proposition 1satisfies Axiom 5.

Proof of Lemma 14:

Weak monotonicity is satisfied by Lemma 8. To show that strict monotonicity is satisfied, consider two acts f and g such that \(f\left( \omega \right) \succsim g\left( \omega \right)\) for all \(\omega \in \Omega\) and with \(f\left( \omega ^{\prime }\right) \succ g\left( \omega ^{\prime }\right)\) for some \(\omega ^{\prime }\in \Omega\). If \(\left| E_{s}\right| =2\), \(\nu _{s}\left( \omega \right) \in \left( 0,1\right)\) for \(\omega \in E_{s}\) implies

$$\begin{aligned} \int _{E_{s}}U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}>\int _{E_{s}}U\left( g\left( \omega \right) \right) \mathrm{{d}}\nu _{s}\text {.} \end{aligned}$$

If \(\left| E_{s}\right| \ge 3\) we have that since \(q_{s}\left( \omega \mid E_{s}\right) >0\) for all \(\omega \in E_{s}\),

$$\begin{aligned} \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right)\ge & {} \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) \\ \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right)\ge & {} \min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) \\ \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right)> & {} \sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) . \end{aligned}$$

Furthermore, \(\rho _{s}>0\) implies

$$\begin{aligned} \int _{E_{s}}U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu _{s}&=\rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( f\left( \omega \right) \right) \\&\quad +\alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) \\&>\rho _{s}\sum _{\omega \in E_{s}}q_{s}\left( \omega \mid E_{s}\right) U\left( g\left( \omega \right) \right) \\&\quad +\alpha _{s}\left( 1-\rho _{s}\right) \max _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) +\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) \min _{\omega \in E_{s}}U\left( g\left( \omega \right) \right) \\&=\int _{E_{s}}U\left( g\left( \omega \right) \right) \mathrm{{d}}\nu _{s}\text {.} \end{aligned}$$

Aggregating over all s and noting that \(p\left( E_{s}\right) >0\) for all s gives the desired result. \(\square\)

Proof of Proposition 1

“If-part”: Lemmata 6 and 7 show Part 1 of the Proposition, the uniqueness of U up to an affine-linear transformation, the uniqueness of \(\pi \left( s\right)\) and of \(\left( p\left( E_{s}\right) \right) _{s\in S}\) and their strict positivity, as well as the fact that \(V_{0}\left( \varphi \left( f\right) \right) =V\left( f\right)\) for all \(f\in F^{S}\). Lemma 10 shows Part 2 of the Proposition, as well as the uniqueness of the capacities \(\nu _{s}\). Finally, Lemma 11 shows Part 3 together with the uniqueness properties of \(\alpha _{s}\), \(\rho _{s}\) and \(q\left( \cdot \mid E_{s}\right)\) and the strict positivity of \(q\left( \omega \mid E_{s}\right)\).

“Only-If-part”: Follows from Lemmata 12, 13 and 14. \(\square\)

Proof of Lemma 1:

Note that for a NEO-additive capacity \(\nu _{s}\), we can obtain \(\left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) p\left( E_{s}\right)\) by:

$$\begin{aligned} \left( 1-\alpha _{s}\right) \left( 1-\rho _{s}\right) p\left( E_{s}\right)&=p\left( E_{s}\right) \Big [1-\alpha _{s}\left( 1-\rho _{s}\right) -\rho _{s}\left( 1-q\left( \omega _{1}\mid E_{s}\right) \right) \nonumber \\&\quad +1-\alpha _{s}\left( 1-\rho _{s}\right) -\rho _{s}\left( 1-q\left( \omega _{2}\mid E_{s}\right) \right) -1+\alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}\left( 1-q\left( \left\{ \omega _{1},\omega _{2}\right\} \mid E_{s}\right) \right) \Big ] \nonumber \\&=p\left( E_{s}\right) \Big [1-\alpha _{s}\left( 1-\rho _{s}\right) -\rho _{s}q\left( E_{s}\backslash \omega _{1}\mid E_{s}\right) -\alpha \left( 1-\rho _{s}\right) \nonumber \\&\quad -\left( 1-\rho _{s}\right) q\left( E_{s}\backslash \omega _{2}\mid E_{s}\right) +\alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}\left( q\left( E_{s}\backslash \left\{ \omega _{1},\omega _{2}\right\} \mid E_{s}\right) \right) \Big ] \nonumber \\&=\Big [CE\left( x_{E_{s}}^{1}x^{0}\right) -CE\left( x_{E_{s}\backslash \omega _{1}}^{1}x^{0}\right) \Big ]-\Big [CE\left( x_{E_{s}\backslash \omega _{2}}^{1}x^{0}\right) -CE\left( x_{E_{s}\backslash \left\{ \omega _{1},\omega _{2}\right\} }^{1}x^{0}\right) \Big ] \nonumber \\&=p\left( E_{s}\right) \Big [\nu _{s}\left( E_{s}\right) -\nu _{s}\left( E_{s}\backslash \left\{ \omega _{1}\right\} \right) \Big ]-\Big [\nu \left( E_{s}\backslash \left\{ \omega _{2}\right\} \right) -\nu \left( E_{s}\backslash \left\{ \omega _{1},\omega _{2}\right\} \right) \Big ] \nonumber \\&=p\left( E_{s}\right) \nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) . \end{aligned}$$
(26)

Similarly, \(\alpha _{s}\rho _{s}p\left( E_{s}\right)\) is given by

$$\begin{aligned} \alpha _{s}\left( 1-\rho _{s}\right) p\left( E_{s}\right)&=p\left( E_{s}\right) \Big [ \alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}q\left( \omega _{2}\mid E_{s}\right) +\alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}q\left( \omega _{1}\mid E_{s}\right) \Big ] \nonumber \\&\quad -p\left( E_{s}\right) \Big [ \alpha _{s}\left( 1-\rho _{s}\right) +\rho _{s}\big [ q\left( \omega _{1}\mid E_{s}\right) +q\left( \omega _{2}\mid E_{s}\right) \big ] \Big ] \nonumber \\&=CE\left( x_{\omega _{1}}^{1}x^{0}\right) +CE\left( x_{\omega _{2}}^{1}x^{0}\right) -CE\left( x_{\omega _{1}}^{1}x_{\omega _{2}}^{1}x^{0}\right) \nonumber \\&= \Big [ CE\left( x_{\omega _{1}}^{1}x^{0}\right) -CE\left( x^{0}\right) \Big ] -\Big [ CE\left( x_{\omega _{1}}^{1}x_{\omega _{2}}^{1}x^{0}\right) -CE\left( x_{\omega _{2}}^{1}x^{0}\right) \Big ] \nonumber \\&=p\left( E_{s}\right) \Big [ \left[ \nu _{s}\left( \left\{ \omega _{1}\right\} \right) -\nu _{s}\left( \varnothing \right) \big ] -\big [ \nu _{s}\left( \left\{ \omega _{1},\omega _{2}\right\} \right) -\nu _{s}\left( \left\{ \omega _{2}\right\} \right) \right] \Big ] \nonumber \\&=-p\left( E_{s}\right) \Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) . \end{aligned}$$
(27)

Summing up (26) and (27), we obtain \(\left( 1-\rho _{s}\right) p\left( E_{s}\right)\) as:

$$\begin{aligned} \left( 1-\rho _{s}\right) p\left( E_{s}\right) =p\left( E_{s}\right) \Big [ \nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) \Big ] \text {,} \end{aligned}$$

or

$$\begin{aligned} 1-\rho _{s}=\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) \text {.} \end{aligned}$$
(28)

Provided that \(\rho _{s}\not =1\), we can divide (27) by (28) to obtain

$$\begin{aligned} \alpha _{s}=-\frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2} \left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }\text {.} \end{aligned}$$

\(\square\)

Proof of Proposition 2:

Since \(\succsim _{0}\) and \(\succsim\) satisfy Axioms 1 through 5, \(V\left( f\right)\) derived in Proposition 1 can be written as in Equation (3) from Remark 2. Taking \(E_{s}\) as in the statement of the Proposition note that \(\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) \not =0\) implies \(\left( 1-\rho _{s}\right) \not =0\), and thus, by Equation (7),

$$\begin{aligned} \alpha _{s}=-\frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2} \left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }, \end{aligned}$$

or

$$\begin{aligned} \frac{\alpha _{s}}{1-\alpha _{s}}=\left| \frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{ \nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) }\right| . \end{aligned}$$

Set \(\alpha _s=\alpha\). Since the representation is additive in s, we can show the equivalence separately for each s. We do so first for sets \(E_{s^{\prime }}\) with \(\left| E_{s^{\prime }}\right| \ge 3\) and then for two-element sets.

(\(\mathbf {\left| E_{s^{\prime }}\right| \ge 3}\)): Suppose Axiom 6 is satisfied. Let \(s^{\prime }\in S\) with \(\left| E_{s^{\prime }}\right| \ge 3\) which satisfies \(\left( 1-\rho _{s^{\prime }}\right) \not =0\) and \(\nabla _{\omega _{1},\omega _{2}}^{2} \left[ \nu _{s}\right] \left( E_{s}\right) \not =0\). Combining (7 ) with Axiom 6 we obtain

$$\begin{aligned} \frac{\alpha }{1-\alpha }=\frac{\alpha _{s^{\prime }}}{1-\alpha _{s^{\prime }}}. \end{aligned}$$

Thus, \(\alpha =\alpha _{s^{\prime }}\). If \(\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) =0\), then \(\alpha =1\) and \(\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) \not =0\). Since for a NEO-additive capacity \(\nu _{s}\), \(\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) \le 0\), by convention, we obtain \(\frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{ \nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) }=-\infty\). Since for any capacity, \(\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) \in \left[ -1,1\right]\), Axiom 6 can only be satisfied if \(\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s^{\prime }}\right] \left( E_{s^{\prime }}\right) =0\), or \(\alpha _{s^{\prime }}=1\).

For any \(s^{\prime }\in S\) with \(\left| E_{s^{\prime }}\right| \ge 3\) and \(\left( 1-\rho _{s^{\prime }}\right) =0\), by Proposition 1, \(\alpha _{s^{\prime }}\) can be chosen arbitrary—in particular, we can set \(\alpha _{s^{\prime }}=\alpha\), and obtain the desired result.

Next, suppose that \(V\left( f\right)\) satisfies ( ). Then, by (7), for any s, \(s^{\prime }\in S\) such that \(\left| E_{s}\right| \ge 3\), \(\left| E_{s^{\prime }}\right| \ge 3\) and such that \(\left( 1-\rho _{s}\right) \not =0\), \(\left( 1-\rho _{s^{\prime }}\right) \not =0\),

$$\begin{aligned} \alpha =-\frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }=-\frac{\Delta _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( \varnothing \right) }{\nabla _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( E_{s^{\prime }}\right) -\Delta _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( \varnothing \right) }\text {.} \end{aligned}$$

If \(\alpha \not =1\), rearranging terms, we obtain

$$\begin{aligned} \frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) }=\frac{\Delta _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( \varnothing \right) }{\nabla _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( E_{s^{\prime }}\right) }\text {.} \end{aligned}$$

If \(\alpha =1\), then \(\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) =\nabla _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( E_{s^{\prime }}\right) =0\) and \(\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) =\Delta _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( \varnothing \right) =-1\) and thus,

$$\begin{aligned} \frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) }=\frac{\Delta _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( \varnothing \right) }{\nabla _{\omega _{1}^{\prime },\omega _{2}^{\prime }}^{2}\left[ \nu _{s^{\prime }}\right] \left( E_{s^{\prime }}\right) }=-\infty \end{aligned}$$

thus implying the statement of Axiom 6 for \(s^{\prime }\in S\) with \(\left| E_{s^{\prime }}\right| \ge 3\). We have thus shown the equivalence of ambiguity attitudes for sets \(E_{s^{\prime }}\) with three or more elements.

(\(\mathbf {\left| E_{s^{\prime \prime }}\right| =2}\)): Consider \(E_{s^{\prime \prime }}=\left\{ \omega _{1}^{\prime \prime },\omega _{2}^{\prime \prime }\right\}\) with capacity \(\nu _{s^{\prime \prime }}\) identified by \(\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) \in \left( 0,1\right)\) and \(\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) \in \left( 0,1\right)\) as in Proposition 1. \(\nu _{s^{\prime \prime }}\) being a NEO-additive capacity with parameter \(\alpha\) as above is equivalent to the existence of \(\rho _{s^{\prime \prime }}\in \left( 0,1\right]\), \(q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) \in \left( 0,1\right)\) and \(q\left( \omega _{2}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) =1-q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right)\) such that

$$\begin{aligned} \big [\alpha \left( 1-\rho _{s^{\prime \prime }}\right) +\rho _{s^{\prime \prime }}q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) \big ] &= \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) ,\text { and} \\ \big [\alpha \left( 1-\rho _{s^{\prime \prime }}\right) +\rho _{s^{\prime \prime }}\left[ 1-q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) \right] \big ] &= \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) . \nonumber \end{aligned}$$
(29)

Thus,

$$\begin{aligned} \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) =2\alpha +\left( 1-2\alpha \right) \rho _{s^{\prime \prime }}. \end{aligned}$$

We consider 3 cases:

Case 1: \(\alpha =\frac{1}{2}\) then (29 ) has a solution if and only if \(\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) =1\) and thus, by (15) and (16), if and only if Conditions (11) and (13) are satisfied simultaneously. In this case, any \(\rho _{s^{\prime \prime }}\in \left( 0,1\right]\) such that \(q\left( \omega _{i}^{\prime \prime }\mid E_{s}\right) =\frac{\nu _{s^{\prime \prime }}\left( \omega _{i}^{\prime \prime }\right) -\frac{1}{2}\left( 1-\rho _{s^{\prime \prime }}\right) }{\rho _{s^{\prime \prime }}}\in \left( 0,1\right)\) for \(i\in \left\{ 1,2\right\}\) gives a NEO-additive capacity with the desired properties. In particular, setting \(\rho _{s^{\prime \prime }}=1\) and \(q\left( \omega _{i}^{\prime \prime }\mid E_{s}\right) =\nu _{s^{\prime \prime }}\left( \omega _{i}^{\prime \prime }\right)\) satisfies the conditions.

Next consider that \(\alpha \not =\frac{1}{2}\) and note that \(\rho _{s^{\prime \prime }}\) is uniquely determined by

$$\begin{aligned} 1-\rho _{s^{\prime \prime }}=\frac{\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -1}{2\alpha -1}\text {.} \end{aligned}$$

By (15) and (16), \(\rho _{s^{\prime \prime }}\le 1\) is then indeed equivalent to (11).

Furthermore, the probability of \(\omega _{1}^{\prime \prime }\) is given by

$$\begin{aligned} q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) = \frac{\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\alpha \left( 1-\rho _{s^{\prime \prime }}\right) }{\rho _{s^{\prime \prime }}}=\frac{\alpha -\left( 1-\alpha \right) \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\alpha \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) }{2\alpha -\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) }. \end{aligned}$$

Case 2: If \(\alpha <\frac{1}{2}\), and \(\rho _{s^{\prime \prime }}\le 1\), then \(\rho _{s^{\prime \prime }}\ge 0\) is equivalent to

$$\begin{aligned} 1-2\alpha &\ge \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -1\ge 0\text {, or} \nonumber \\ \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -2\alpha &\ge 0. \end{aligned}$$
(30)

Note that since \(\rho _{s^{\prime \prime }}\le 1\), we have \(\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -2\alpha>\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) +\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -1>0\). Then, \(q\left( \omega _{1}^{\prime \prime }\mid E_{s^{\prime \prime }}\right) \in \left[ 0,1\right]\) is equivalent to

$$\begin{aligned} \alpha -\left( 1-\alpha \right) \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\alpha \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) &\ge 0 \\ \alpha -\alpha \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\left( 1-\alpha \right) \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) &\ge 0\text {,} \nonumber \end{aligned}$$
(31)

which is equivalent to

$$\begin{aligned} \alpha -\alpha \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -\alpha \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) &\ge \left( 1-2\alpha \right) \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) \\ \alpha -\alpha \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) -\alpha \nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) &\ge \left( 1-2\alpha \right) \nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) \end{aligned}$$

or

$$\begin{aligned}&\underbrace{\frac{-\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( \varnothing \right) \overbrace{\nabla _{\omega _{1}^{\prime \prime },\omega _{2}^{\prime \prime }}^{2}\left[ \nu _{s^{\prime \prime }} \right] \left( E_{s^{\prime \prime }}\right) }^{=1-\nu _{s^{\prime \prime }}\left( \omega _{1}^{\prime \prime }\right) -\nu _{s^{\prime \prime }}\left( \omega _{2}^{\prime \prime }\right) }}{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }} _{=\alpha }\ge \underbrace{\frac{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) +\Delta _{\omega _{1},\omega _{2}}^{2} \left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( E_{s}\right) -\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }}_{=1-2\alpha }\max _{i\in \left\{ 1,2\right\} }\left\{ \nu _{s^{\prime \prime }}\left( \omega _{i}^{\prime \prime }\right) \right\}>0 \nonumber \\& -\frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) +\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }\ge \frac{\max _{i\in \left\{ 1,2\right\} }\left\{ \nu _{s^{\prime \prime }}\left( \omega _{i}^{\prime \prime }\right) \right\} }{\nabla _{\omega _{1}^{\prime \prime },\omega _{2}^{\prime \prime }}^{2}\left[ \nu _{s^{\prime \prime }}\right] \left( E_{s^{\prime \prime }}\right) }>0 . \end{aligned}$$
(32)

Note also that since (31) implies (30), so does (32). Hence, if \(\alpha < \frac{1}{2}\), the existence of is equivalent to condition (32). It is straightforward to see that the existence of \(\rho _{s^{\prime \prime }}\in \left( 0,1\right]\) and \(q\left( \cdot \mid E_{s^{\prime \prime }}\right) \in \left( 0,1\right)\) is equivalent to the same condition, but with a strict inequality.

Case 3: If \(\alpha >\frac{1}{2}\), an argument symmetric to that in Case 2 shows that the existence of \(\rho _{s^{\prime \prime }}\) and \(q\left( \cdot \mid E_{s^{\prime \prime }}\right)\) is equivalent to the following condition:

$$\begin{aligned} 0>-\frac{\min _{i\in \left\{ 1,2\right\} }\left\{ \nu _{s^{\prime \prime }}\left( \omega _{i}^{\prime \prime }\right) \right\} }{\nabla _{\omega _{1}^{\prime \prime },\omega _{2}^{\prime \prime }}^{2}\left[ \nu _{s^{\prime \prime }}\right] \left( E_{s^{\prime \prime }}\right) }\ge - \frac{\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) }{\nabla _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s} \right] \left( E_{s}\right) +\Delta _{\omega _{1},\omega _{2}}^{2}\left[ \nu _{s}\right] \left( \varnothing \right) } . \end{aligned}$$
(33)

Finally, note that satisfying (32) when \(\alpha < \frac{1}{2}\) and (33) when \(\alpha >\frac{1}{2}\) is equivalent to satisfying condition (11) and (12) when \(\alpha \not =\frac{1}{2}\). Combining this result with the result from Case 1 concludes the proof of the proposition. \(\square\)

Proof of Lemma 2:

Let \(\nu _{\rho _{s}}\) on \(2^{E_{s}}\) be a simple capacity on \(2^{E_{s}}\) defined as in (20). Notice that we have

$$\begin{aligned} \int _{E_{s}}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}=\rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) q\left( \omega \mid E_{s}\right) +\left( 1-\rho _{s}\right) w_{s}\left( f\right) . \end{aligned}$$

Aggregating over s with respect to p on \(\{E_{s}\}_{s\in S}\) delivers

$$\begin{aligned} \sum _{s\in S}p\left( E_{s}\right) \big [\int _{E_s}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}\big ]&=\sum _{s\in S}p\left( E_{s}\right) \Big [\rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) q\left( \omega \mid E_{s}\right) +\left( 1-\rho _{s}\right) w_{s}\left( f\right) \Big ] \\&=\sum _{s\in S}p\left( E_{s}\right) \Big [\rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) \frac{p\left( \omega \right) }{ p\left( E_{s}\right) }+\left( 1-\rho _{s}\right) w_{s}\left( f\right) \Big ] \\&=\sum _{s\in S}\Big [\rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) p\left( \omega \right) +\left( 1-\rho _{s}\right) w_{s}\left( f\right) p\left( E_{s}\right) \Big ] \\&=\int _{\Omega }U\left( f\right) \mathrm{{d}}\nu ^{EK}\text {.} \end{aligned}$$

\(\square\)

Proof of Lemma 3:

Let \(\nu _{\rho _{s}}^{JP}\) on \(2^{E_{s}}\) be a simple JP-capacity on \(2^{E_{s}}\) defined as in (22). Note that for each \(s\in S\),

$$\begin{aligned} \int _{E_{s}}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}^{JP}&= \left( 1-\alpha \right) \int _{E_{s}}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}+\alpha \int _{E_{s}}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}^{*} \\&= \left( 1-\alpha \right) \left[ \rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) q\left( \omega \mid E_{s}\right) +\left( 1-\rho _{s}\right) w_{s}\left( f\right) \right] \\&\quad +\alpha \left[ \rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) q\left( \omega \mid E_{s}\right) +\left( 1-\rho _{s}\right) b_{s}\left( f\right) \right] \\&= \rho _{s}\sum _{\omega \in E_{s}}U\left( f\left( \omega \right) \right) q\left( \omega \mid E_{s}\right) +\alpha \left( 1-\rho _{s}\right) b_{s}\left( f\right) +\left( 1-\alpha \right) \left( 1-\rho _{s}\right) w_{s}\left( f\right) . \end{aligned}$$

Thus, by aggregating over s with respect to p on \(\{E_s\}_{s\in S}\), we get

$$\begin{aligned} V\left( f\right)&=\sum _{s\in S}p\left( E_{s}\right) \left[ \rho _{s}\sum _{\omega \in E_{s}}q\left( \omega \mid E_{s}\right) U\big (f\left( \omega \right) \big )+\left( 1-\rho _{s}\right) \Big (\alpha b_{s}\left( f\right) +\left( 1-\alpha \right) w_{s}\left( f\right) \Big )\right] \\&=\sum _{s\in S}p(E_{s})\left[ \int _{E_{s}}U\left( f_{\mid E_{s}}\right) \mathrm{{d}}\nu _{\rho _{s}}^{JP}\right] \text {.} \end{aligned}$$

\(\square\)

Proof of Proposition 3:

From the definition of the generalized EK-capacity, we can write

$$\begin{aligned} \alpha \hat{\beta }_{s}\left( A\right) +\left( 1-\alpha \right) \check{\beta } _{s}\left( A\right) = {\left\{ \begin{array}{ll} 1 &{} \text {if }E_{s}\cap A=E_{s}\text {,} \\ \alpha &{} \text {if }\varnothing \not =E_{s}\cap A\not =E_{s}, \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Therefore, the Choquet integral of an f with respect to \(\nu ^{gEK}\) can be written as

$$\begin{aligned} \int _{\Omega }U\left( f\left( \omega \right) \right) \mathrm{{d}}\nu ^{gEK}=\sum _{s\in S}p\left( E_{s}\right) \left[ \rho _{s}\sum _{\omega \in E_{s}}U\big (f\left( \omega \right) \big )q\left( \omega \mid E_{s}\right) +\left( 1-\rho _{s}\right) \left[ \alpha b_{s}\left( f\right) +\left( 1-\alpha \right) w_{s}\left( f\right) \right] \right] \text {.} \end{aligned}$$

\(\square\)

Proof of Lemma 4:

Follows directly from Proposition 4. \(\square\)

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Dominiak, A., Guerdjikova, A. Pessimism and optimism towards new discoveries. Theory Decis 90, 321–370 (2021). https://doi.org/10.1007/s11238-021-09808-1

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Keywords

  • Unawareness
  • Ambiguity
  • Subjective expected utility
  • Capacities
  • Choquet expected utility
  • Parametric model
  • Confidence
  • Pessimism
  • Optimism
  • Conditional preferences