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Theory and Decision

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01.06.2014

Additive representation of separable preferences over infinite products

verfasst von: Marcus Pivato

Erschienen in: Theory and Decision | Ausgabe 1/2014

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Abstract

Let \(\mathcal{X }\) be a set of outcomes, and let \(\mathcal{I }\) be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \((\succcurlyeq )\) on \(\mathcal{X }^\mathcal{I }\) admits an additive representation. That is: there exists a linearly ordered abelian group \(\mathcal{R }\) and a ‘utility function’ \(u:\mathcal{X }{{\longrightarrow }}\mathcal{R }\) such that, for any \(\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }\) which differ in only finitely many coordinates, we have \(\mathbf{x}\succcurlyeq \mathbf{y}\) if and only if \(\sum _{i\in \mathcal{I }} \left[u(x_i)-u(y_i)\right]\ge 0\). Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \((\succcurlyeq )\) also satisfies a weak continuity condition, then the paper shows that, for any \(\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }\), we have \(\mathbf{x}\succcurlyeq \mathbf{y}\) if and only if \({}^*\!\sum _{i\in \mathcal{I }} u(x_i)\ge {}^*\!\sum _{i\in \mathcal{I }}u(y_i)\). Here, \({}^*\!\sum _{i\in \mathcal{I }} u(x_i)\) represents a nonstandard sum, taking values in a linearly ordered abelian group \({}^*\!\mathcal{R }\), which is an ultrapower extension of \(\mathcal{R }\). The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces.

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See Sect. 5 for the formal definition of \({}^*\!\mathbb{R }\). See Anderson (1991) or Goldblatt (1998) for good introductions to nonstandard analysis.

See, e.g. Krantz et al. (1971, §1.5.2, §6.5.1 and §9.1) and (1990, §21.7), Pfanzagl (1968, §6.6 and §9.5), Narens (1974a), and Fuhrken and Richter (1991).

See e.g. Sidgwick (1884, Book IV, Chapt. I, §1, p. 411), Ramsey (1928, p. 543) and Rawls (1928, §44, p. 253). Note that ‘instrumental’ time preferences may be ethically defensible, even if ‘pure’ time preferences are not. See Sect. 6 for further discussion of this point.

It would remain infinite for any exponential discount rate less than 2 % per year.

Note that I do not assume a probability distribution on \(\mathcal{I }\).

Thus, the elements of \(\mathcal{X }\) are ‘extended alternatives’, which encode both the specific identity of a person and any ethically relevant information about her physical and mental state.

That is: for any distinct \(r,s\in \mathcal{R }\), either \(r>s\) or \(s>r\), but not both.

In Savage’s theory, this property is called the sure thing principle or Axiom P2. In axiomatic measurement theory, it is variously called (joint) independence or single cancellation. In social choice, separability is a special case of the axiom of independence of (or elimination of) indifferent individuals, which in turn is a special case of the Extended Pareto axiom.

But see Sect. 6.

Formally, \({}^*\!\mathcal{R }\) is an ultrapower of \(\mathcal{R }\) with respect to an ultrafilter \(\mathfrak{UF }\) defined over the set of all finite subsets of \(\mathcal{I }\). The precise construction of \({}^*\!\mathcal{R }\) is somewhat technical, and will be provided in Sect. 5 below.

This recalls preference relations which have been proposed by Lauwers and Vallentyne (2004) and Asheim et al. (2010); see Sect. 7.2 for discussion.

This recalls intertemporal preference relations proposed by Atsumi (1965) and Weizsäcker (1965); again see Sect. 7.2 for discussion.

But see Sect. 6.

‘Generalized’ because \(u\) might actually be a monotone increasing transformation of the ‘true’ cardinal utility function of the individuals.

In different contexts, the Archimedean property has been called continuity or substitutability.

Of course, if \((\succcurlyeq )\) itself is strictly finitary, then \(\big ( {\stackrel{\displaystyle \succcurlyeq }{{\scriptscriptstyle \mathrm fin}}}\big )\) is \((\succcurlyeq )\).

In the case \(\mathcal{I }=\mathbb{N }\), Proposition 5(a) is equivalent to Theorem 1.1 of Wakker (1986), which in turn builds on earlier, similar results by Camacho (1979a, b, 1980, 1982). For a similar result, representing separable preferences using utility averages (rather than utility sums), see Kothiyal et al. (2012).

Technical note (to be read after Sect. 5): The results in this section require a specific—but quite natural—choice of ultrafilter \(\mathfrak{UF }\). Let \(\Pi :=\Pi _{\scriptscriptstyle {\mathrm{fs}}}\) be the group of all fixed-step permutations of \(\mathbb{N }\), from Example 16(c). Then define \(\mathfrak{UF }\) as in Lemma 15. Then use \(\mathfrak{UF }\) to define \({}^*\!\mathcal{R }, {}^*\!\sum \) and \(\big ({}^{^*}\!{ {\stackrel{\displaystyle \succcurlyeq }{{\scriptscriptstyle u}}}}\big )\).

Technical note (to be read after Sect. 5): Let \({\varvec{\mathcal{F }}}\) be the set of finite subsets of \(\mathcal{I }, \mathfrak{UF }\) be an ultrafilter on \({\varvec{\mathcal{F }}}\), and define \(\mathcal{R }\!:=\!\mathbb{R }^{\varvec{\mathcal{F }}}/\mathfrak{UF }\). Thus, \(\mathcal{R }\) is a hyperreal number field. Now define \({}^*\!\mathcal{R }\!:=\!\mathcal{R }^{\varvec{\mathcal{F }}}/\mathfrak{UF }\). Technically, \({}^*\!\mathcal{R }\) is another a hyperreal number field, but it is distinct from \(\mathcal{R }\). I could have referred to it as \(^{**}\mathbb{R }\), but this seemed like a notation too far.

The anonymous referee has pointed out that this property was originally introduced by Finetti (1931).

Strictly speaking, we would need to divide by the normalization factor \({}^*\!\sum _{i\in \mathcal{I }} 1\) for this to be true.

Here I adopt the usual convention that \(0\cdot \log _2(0):=0\).

To be somewhat more precise: if \((x_1,x_2,\ldots ,x_N)\) was a long sequence of independent, \(\mu \)-random variables, then, using an optimal encoding scheme, it would take approximately \(N\cdot H(\mu )\) bits to transmit complete information about the sequence \((x_1,x_2,\ldots ,x_N)\). This approximation becomes exact as \(N{\rightarrow }{\infty }\). For more information on entropy and information, see Cover and Thomas (2006).

It is possible to define, e.g. the entropy of a Borel probability measure on a compact subset of \(\mathbb{R }^N\), but only relative to some ‘baseline’ measure—typically the Lebesgue measure. The baseline measure is then the measure of maximal entropy. But this raises the question: what is the right baseline measure?

This follows from the construction of \(\mu \) in the proof of Proposition 10.

Formally, an ultrafilter \(\mathfrak{UF }\) is equivalent to a finitely additive, \(\{0,1\}\)-valued probability measure defined on all subsets of \({\varvec{\mathcal{F }}}\). The elements of \(\mathfrak{UF }\) are the ‘sets of measure 1’, and the elements of \(\mathfrak{P }\setminus \mathfrak{UF }\) are the ‘sets of measure 0’. I mean ‘almost all’ with respect to this probability measure. Just as in classical probability theory, this use of ‘almost all’ is only meaningful with respect to a particular ultrafilter. If \(\mathfrak{UF }\) and \(\mathfrak{UF }'\) are two different ultrafilters, then there will exist a subset \({\varvec{\mathcal{G }}}\subset {\varvec{\mathcal{F }}}\) such that \(\mathfrak{UF }\) judges \({\varvec{\mathcal{G }}}\) to contain ‘almost all’ elements of \({\varvec{\mathcal{F }}}\), while \(\mathfrak{UF }'\) judges \({\varvec{\mathcal{F }}}\setminus {\varvec{\mathcal{G }}}\) to contain ‘almost all’ elements of \({\varvec{\mathcal{F }}}\).

Proof: \(\big ({ {\stackrel{\displaystyle \succcurlyeq }{{\scriptscriptstyle \mathfrak{UF }}}}}\big )\) is complete by Axiom (UF). Next, \(\big ({ {\stackrel{\displaystyle \succcurlyeq }{{\scriptscriptstyle \mathfrak{UF }}}}}\big )\) is reflexive, because \({\varvec{\mathcal{F }}}\in \mathfrak{UF }\). Finally, \(\big ({ {\stackrel{\displaystyle \succcurlyeq }{{\scriptscriptstyle \mathfrak{UF }}}}}\big )\) is transitive, by Axioms (F1) and (F2).

Formally, \({}^*\!\mathcal{R }\) is called the ultrapower of \(\mathcal{R }\) modulo the ultrafilter \(\mathfrak{UF }\). It is conventional to denote ultrapower-related objects with the leading star \(*\).

In fact, Lemma 13 is a special case of Łoś’s theorem, which roughly states that any first-order properties of any system of algebraic structures and/or \(N\)-ary relations on \(\mathcal{R }\) are ‘inherited’ by \({}^*\!\mathcal{R }\). For example, if \(\mathcal{R }\) is a linearly ordered field, then \({}^*\!\mathcal{R }\) will also be a linearly ordered field.

A positive element of \({}^*\!\mathcal{R }\) is ‘infinitesimal’ if it is smaller than every positive element of \(\mathcal{R }'\). It is ‘infinite’ if it is bigger than every element of \(\mathcal{R }'\).

For any infinite indexing set \(\mathcal{N }\) and any ultrafilter \(\varvec{\mathcal{U }\!\mathcal{F }}\) on \(\mathcal{N }\), one can obtain a hyperreal number field by taking the ultrapower \(\mathbb{R }^\mathcal{N }/\varvec{\mathcal{U }\!\mathcal{F }}\). Technically, different choices of \(\mathcal{N }\) and/or \(\varvec{\mathcal{U }\!\mathcal{F }}\) will lead to different fields, so it is somewhat inaccurate to speak of ‘the’ hyperreal number field \({}^*\!\mathbb{R }\) as if it was a single object. However, all of these fields satisfy the axioms of nonstandard analysis (e.g. they are non-Archimedean linearly ordered field extensions of the field \(\mathbb{R }\) of real numbers), hence they can be treated as ‘the same’ object for most purposes.

That is: \({\left\langle \Delta \right\rangle }:= \{\delta _1^{n_1}\cdot \delta _2^{n_2}\cdots \delta _k^{n_k}\); \(k\in \mathbb{N }\), \(\delta _1,\ldots ,\delta _k\in \Delta \), and \(n_1,\ldots ,n_k\in \mathbb{Z }\}\).

Basu and Mitra (2007a) show that a permutation group \(\Gamma \subset \Pi \) can be the symmetry group of some Paretian social welfare relation on \(\mathbb{R }^\mathbb{N }\) if and only if each single element of \(\Gamma \) has finite orbits. The condition of locally finite orbits is similar, but somewhat more restrictive.

\(\Pi _{\scriptscriptstyle {\mathrm{fs}}}\)-invariant social welfare relations on \(\mathbb{R }^\mathbb{N }\) have been considered by Lauwers (1997), Fleurbaey and Michel (2003; §4.2) and Basu and Mitra (2007a; §5).

Lauwers (1998); Basu and Mitra (2003, 2007a, b), Fleurbaey and Michel (2003; Thm. 1) and Sakai (2010; Thm.2) have analyzed this Pareto/anonymity conflict in greater detail. Seidenfeld et al. (2009) have observed an analogous conflict between permutation-invariance and the statewise dominance principle, in the setting of risky decisions.

Standardization maps all infinite hyperreals to \(\pm {\infty }\), and strips the infinitesimal part away from all finite hyperreals, leaving only the real part.

Of course, individuals can still derive (dis)utility from memory of the past, anticipation of the future, altruism/envy towards other people or the contingency of fate, as long as the relevant cognitive states are explicitly encoded in \(\mathcal{X }\).

Of particular note is Jaffray (1974b), which gives necessary and sufficient conditions for real-valued additive representations in full generality, without the use of a continuity or solvability condition.

Wakker (1986) built on Camacho’s (1979a, b, 1980, 1982) earlier theory of additive preferences over the space of arbitrary-length finite sequences in \(\mathcal{X }\). Later, Wakker and Zank (1999) developed another additive representation for infinite Cartesian powers, using measure theory. Meanwhile, Streufert (1995) has analyzed the separability structure of partially separable (but not necessarily additive) preferences on infinite Cartesian products.

See also Chipman (1971). In fact, this result had earlier been proved independently in papers by Sierpiński, Cuesta and Mendelson; see Fishburn (1974; §5) for details. Starting from Chipman’s work, Herden and Mehta (2004) have developed continuous lexicographical ordinal utility functions.

Nonstandard analysis also has other applications in economics, unrelated to the topic of this paper; see e.g. see Anderson (1991) and Stigum (1990).

See footnote 19.

Condition (i) is because \(r\) is finite. Condition (ii) ensures uniqueness by excluding binary expansions ending in an infinite sequence of \(1\)’s.

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Titel: Additive representation of separable preferences over infinite products
verfasst von: Marcus Pivato
Publikationsdatum: 01.06.2014
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 1/2014
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9391-2

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