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Erschienen in: Theory and Decision 1/2015

01.01.2015

An axiomatization of Choquet expected utility with cominimum independence

verfasst von: Takao Asano, Hiroyuki Kojima

Erschienen in: Theory and Decision | Ausgabe 1/2015

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Abstract

This paper proposes a class of independence axioms for simple acts. By introducing the \({\mathcal {E}}\)-cominimum independence axiom that is stronger than the comonotonic independence axiom but weaker than the independence axiom, we provide a new axiomatization theorem of simple acts within the framework of Choquet expected utility. Furthermore, in order to provide the axiomatization of simple acts, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) into an infinite state space. Our axiomatization theorem relates Choquet expected utility to multi-prior expected utility through the core of a capacity that is explicitly derived within our framework. Our result in this paper also derives Gilboa (Econometrica 57:1153–1169, 1989), Eichberger and Kelsey (Theory Decis 46:107–140, 1999), and Rohde (Soc Choice Welf 34:537–547, 2010) as a corollary.

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Fußnoten
1
For the definitions of comonotonic independence and comonotonic additivity, see Sect. 3. See also Schmeidler (1986).
 
2
For the definitions of cominimum operators and comonotonic operators, see Sect. 3.
 
3
In Sect. 3, we provide the definition of a collection of sets \({\mathcal {E}}\) in detail.
 
4
In Sect. 5, we define the core of a capacity.
 
5
In Sect. 5, we explain Eichberger and Kelsey (1999) representation in detail. For the definition of E-capacities, see Footnote 15.
 
6
See Epstein (1999), Zhang (1999), or Asano (2006) that use a \(\lambda \)-system for a collection of unambiguous events.
 
7
See the proof of Proposition 3 in Kajii et al. (2007).
 
8
For example, see Schmeidler (1986).
 
9
In Kajii et al. (2007), the term “complete” is adopted from an analogy to a complete graph. For \(T \in \mathcal {F}\), let us consider an undirected graph with a vertex set \(T\) where \(\{ \omega , \omega ^{\prime } \} \subseteq T\) is an edge if there exists \(E \in {\mathcal {E}}\) such that \(\{ \omega , \omega ^{\prime } \} \subseteq E \subseteq T\). Then, this is a complete graph if and only if \(T\) is \({\mathcal {E}}\)-complete. See Kajii et al. (2007).
 
10
The asymmetric (\(\succ \)) and symmetric (\(\sim \)) parts of \(\succeq \) are defined as usual. For details, see Kreps (1988).
 
11
Equivalently, for two acts \(f, \, g \in L_0,\,f\) and \(g\) are said to be comonotonic if there are no \(\omega \) and \(\omega '\) such that \(f(\omega ) \succ f(\omega ')\) and \(g(\omega ') \succ g(\omega )\). See Schmeidler (1989).
 
12
A binary relation \(\succeq \) is a weak order if and only if \(\succ \) is asymmetric and negatively transitive, whereas a binary relation \(\succ \) is asymmetric if for all \(f,\,g\in L_{0},\,f\succ g\Rightarrow g\nsucc f\) and it is negatively transitive if for all \(f,\,g,\,h\in L_{0},\,f\nsucc g\) and \(g \nsucc h \Rightarrow f \nsucc h\).
 
13
To be precise, the notion of being simple-complete should be introduced. See Sect. 5 (Definition 5 and Theorem 3) for details.
 
14
Certainty-independence states that for all \(f,\,g \in L_0\), all \(h \in L_c\), and all \(\alpha \in [0,1],\,f \succ g\) \(\Leftrightarrow \) \(\alpha f + (1-\alpha ) h \succ \alpha g +(1-\alpha ) h\).
 
15
Let \(\{E_1, \ldots , E_n \}\) be a partition of \(\varOmega \). Let \({\mathcal {E}}=\{E_1,\ldots , E_n \}\). A capacity \(v \in {\mathbb {R}}^{\mathcal {F}}\) is called an E-capacity if there exist a finitely additive probability measure \(\pi \), a real number \(\varepsilon \in [0,1]\), and a finitely additive probability measure \(\rho \) on \({\mathcal {E}}\) such that \(v=(1-\varepsilon ) \pi + \varepsilon \sum _{i=1}^n \rho (E_i) u_{E_i}\), where \(u_{E_i}\) denotes the unanimity game on \(E_i\) for each \(i\).
 
16
Let \(\mathcal {M}(\varOmega )\) be the set of all probability measures and let \(\varepsilon \in [0,1]\). Then, the set of probability measures defined by \(\{ (1-\varepsilon ) p + \varepsilon q \, |\, q \in \mathcal {M}(\varOmega ) \}\) is called the \(\varepsilon \)-contamination of \(p\), where \(p\) is the true probability measure. Our paper derives a set of coefficients \(\varepsilon _1, \ldots , \varepsilon _n\) endogenously.
 
17
We are grateful to an anonymous referee who pointed out that Lemma 6 is very similar to a result in Chateauneuf and Jaffray (1989).
 
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Metadaten
Titel
An axiomatization of Choquet expected utility with cominimum independence
verfasst von
Takao Asano
Hiroyuki Kojima
Publikationsdatum
01.01.2015
Verlag
Springer US
Erschienen in
Theory and Decision / Ausgabe 1/2015
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI
https://doi.org/10.1007/s11238-013-9411-2

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