## Abstract

This paper studies decisions under ambiguity when attention is paid to extreme outcomes. In a purely subjective framework, we propose an axiomatic characterization of affine capacities, which are Choquet capacities consisting in an affine transformation of a subjective probability. Our main axiom restricts the well-known Savage’s Sure-Thing Principle to a change in a common intermediate outcome. The representation result is then an affine combination of the expected utility of the valued act and its maximal and minimal utilities.

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## Notes

The difference between a GNAC and an affine capacity is not large and will be explained later in the paper.

Equivalently, one could assume that the preference \(\succsim \) satisfies a set of necessary and sufficient conditions to the existence of a CEU representation. Axiomatic characterizations of CEU preferences in a similar (purely subjective) framework may be found in, for instance, Gilboa (1987), Wakker (1989), Sarin and Wakker (1992) and Ghiradato et al. (2003).

Recall that the Dempster–Shafer updating rule defines \(\nu _E\) as \(\nu _E(A)=[\nu (A\cup E^c)-\nu (E^c)]/[1-\nu (E^c)]\).

I thank an anonymous referee for having suggested me this way of proving this point.

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## Acknowledgments

I am very grateful to two anonymous referees for their helpful comments and remarks, which have led to a substantial improvement in the quality of this work. This research has been funded by the Grenoble Applied Economics Laboratory and the University of Grenoble-Alpes.

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## Appendix 1: Proofs

### Appendix 1: Proofs

Without any essential loss of generality, we assume that the utility function *U* is normalized such that \(U(M)=1\) and \(U(0)=0\). Since the utility *U* is continuous, its range is the interval [0, 1].

Chateauneuf et al. (2007) lists some properties of essential events when there exist at least three pairwise disjoint essential events (property 4 is omitted since it is not explicitly used in our proofs).

### 1.1 Properties of essential events

###
**Property 1**

\(E\in \Sigma ^*\) implies \(E^c\in \Sigma ^*\).

###
**Property 2**

There exists a partition \(A_1,\ldots ,A_3\) of *S* such that \(A_i\in \Sigma ^*\) for any \(i=1,2,3\).

###
**Property 3**

Let \(A_1,\ldots ,A_3 \in \Sigma ^*\) be pairwise disjoints events, then \(A_i \cup A_j \in \Sigma ^*\) for any \(i,j=1,\ldots ,3\).

### 1.2 Proof of Theorem 1

###
*Proof of Lemma 1*

*Part A*
\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\). Properties A and D follow from the definition of an affine capacity.

*Part B*
\(\mathrm{(ii)}\Rightarrow \mathrm{(i)}\). Lemma 3 of Eichberger et al. (2012) shows that if property D is satisfied by a capacity \(\nu \) satisfying property A, then \(\nu \) is a GNAC. To show that \(\nu \) is an affine capacity, it remains to show that property D directly implies that the probability \(\pi \) is congruent with \(\mathcal {N}\). Let *E* be an essential event and *N* be a null event. If \(\nu \) is a GNAC satisfying property D, we have \(\nu (E\cup N)=\nu (E)\) if and only if \(a+b[\pi (E)+\pi (N)]=a+b\pi (E)\) if and only if \(\pi (N)=0\). Furthermore, \(\pi (N^c)=1\) by additivity and normalization of \(\pi \), with \(\pi (N\cup N^c)=1\). Hence, \(\pi \) is congruent with \(\mathcal {N}\) and \(\nu \) is an affine capacity.

###
*Proof of Lemma 2*

*Part A*
\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\).

By Lemma 2 of Eichberger et al. (2012), Axiom 1 implies that for any \(E\in \Sigma \) and any \(N\in \mathcal {N}\) such that \(E\cap N=\emptyset \), we have \(\nu (E\cup N)=\nu (E)\). Hence, \(\nu \) satisfies property D. Furthermore, \(\mathcal {N}\) is an ideal: CEU implies (i) \(\emptyset \in \mathcal {N}\), (ii) \(A\in \mathcal {N}\) implies \(B\in \mathcal {N}\) for any \(B\subset A\); property D with \(E\in \mathcal {N}\) implies (iii) \(A\in \mathcal {N}\) and \(B\in \mathcal {N}\) implies \(A\cup B\in \mathcal {N}\).

It remains to show that \(\nu \) is congruent with \(\mathcal {N}\).^{Footnote 4} Suppose, *per absurdum*, \(A\in \mathcal {N}\) and \(\nu (A^c) < 1\). Since \(A\cup A^c=S\) and \(\mathcal {N}\) is an ideal, we cannot have \(A^c \in \mathcal {N}\). Hence, \(A^c \in \Sigma ^*\) so property D implies \(1=\nu (S)=\nu (A \cup A^c)=\nu (A^c)\) which contradicts \(A^c \in \Sigma ^*\). Therefore, \(A \in \mathcal {N}\) implies \(\nu (A)=0\) and \(\nu (A^c)=1\).

*Part B*
\(\mathrm{(ii)}\Rightarrow \mathrm{(i)}\). Let *A* be an event, \(\mathcal {N}\) be a null event and *x*, *y*, *z* be any three outcomes such that \(x\succ y \succ z\). Under CEU, the act \(f\equiv y_Nx_Az\) is valued as:

If \(A\in \Sigma ^*\), property D implies \(\nu (A\cup N)=\nu (A)\) and then \(f\sim x_Az\). If \(A\in \mathcal {N}\), we also have \(\nu (A\cup N)=0\) since \(\mathcal {N}\) is an ideal (hence, \(A\cup N \in \mathcal {N}\)) and then \(f \sim x_Az\). Finally, when \(A\in \mathcal {U}\), \(\nu (A\cup N)=\nu (A)=1\) implies \(f \sim x_Az\).

###
*Proof of Lemma 3*

*Part A*
\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\). Let *n* be the number of pairwise disjoint essential events. We have two cases to treat, depending on if \(n=3\) or \(n>3\), to prove that property A holds, since \(B\cap F\ne \emptyset \) with \(B\cup G\in \Sigma ^*\) and \(F\cup G\in \Sigma ^*\) implies \(B=F\) in the former case.

*Case 1:*
\(n=3\). When there are only three pairwise disjoint essential events, property 2 implies that there exists a partition \(E_1,E_2,E_3\) of *S*, such that \(E_i\in \Sigma ^*\) for any \(i=1,2,3\), and property 3 implies \(E_i \cup E_j \in \Sigma ^*\) for any \(i,j=1,2,3\). Let \(E=E_i\cup E_j\). We have to prove that

for any \(i,j=1,2,3\) and any essential event \(G=E^c\). We know that *G* is an essential event by property 1. Let \(f_E x\equiv (x_i,E_j;x,G;x_j,E_j)\) and \(g_E x\equiv (y_i,E_i;x,G;y_j,E_j)\), with \(x_i\succ x \succ x_j\) and \(y_j\succ x \succ y_i\), such that \(f_E x\sim g_Ex\).

Let \(y\in X\) such that \(x_i\succ y \succ x_j\) and \(y_j\succ y \succ y_i\). By continuity, one can find *y* such that \(y\succ x\) or \(x\succ y\). Axiom 2 then implies \(f_E x\sim g_E x\) if and only if \(f_Ey \sim g_E y\). Under CEU, these two indifference imply

which is equivalent to Eq. (3), hence property A holds for \(B=E_i\), \(F=E_j\) and \(G=S\backslash (E_i\cup E_j)\). If \(\mathcal {N}\) contains null event(s) other than the empty set \(\{\emptyset \}\), there may be one or several partitions of *S* other than \(E_1,E_2,E_3\). Under property D, property A is automatically satisfied on these partitions hence it holds on \(\Sigma ^*\). Otherwise the same argument could be used to prove that property A holds for any partition of three essential events.

*Case 2:*
\(n>3\).

By property 2, if there exist four essential disjoint events, then there exists a partition \(A_1,\ldots ,A_n,G\) of *S* such that \(n\ge 3\) and \(A_i\in \Sigma ^*\). By property 1, \(A_n \in \Sigma ^*\) implies \(A_n ^c=\cup _{i=1}^{n-1} A_i \cup G \in \Sigma ^*\). Let

and

where

and

be such that \(f\sim g\). By EOS, \(f\sim g\) if and only if \(y_Gf \sim y_G g\), where \(\{x_{i-1},y_{i}\} \succ y \succ x_i\). By continuity of CEU preferences, such an outcome \(y\in X\) may be found. Under CEU, these two indifference imply

which is equivalent to

for any \(i=2,\ldots ,n-1\). Since this equation holds for any \(f,g\in \mathcal {A}\), hence for any partition \(A_1,\ldots ,A_n,G\) of *S* such that \(n\ge 3\) and \(A_1,\ldots ,A_n,G \in \Sigma ^*\), property A holds on \(\Sigma ^*\).

*Part B*
\(\mathrm{(ii)}\Rightarrow \mathrm{(i)}\). Since \(\succsim \) satisfies Axiom 1, Lemma 2 implies that \(\nu \) satisfies property D, \(\nu \) is congruent with \(\mathcal {N}\) and \(\mathcal {N}\) is an ideal. Therefore, by Lemma 1, \(\nu \) is an affine capacity and \(\succsim \) is represented by Eq. 2.

We prove that \(\succsim \) satisfies \(f_Ez\succsim g_Ez\) if and only if \(f_Ez'\succsim g_Ez'\) for any non-null event *E*, any two acts \(f,g\in \mathcal {A}\) and any two outcomes \(z,z' \in X\) such that for \(x,x'\in f(E)\) and \(y,y'\in g(E)\), we have \(f^{-1}(x),f^{-1}(x'),g^{-1}(y),g^{-1}(y')\in \Sigma ^*\) and \(\{x,y\} \succ \{z,z'\} \succ \{x',y'\} \).

Under representation 2, \(f_E z \succsim g_E z\) if and only if:

which is equivalent to

which holds if and only if \(f_E z'\succsim g_E z'\).

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Toquebeuf, P. Choquet expected utility with affine capacities.
*Theory Decis* **81**, 177–187 (2016). https://doi.org/10.1007/s11238-015-9529-5

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DOI: https://doi.org/10.1007/s11238-015-9529-5