Additive representation of separable preferences over infinite products

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathcal{X }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">X</mi> </math> </EquationSource> </InlineEquation> be a set of outcomes, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathcal{I }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">I</mi> </math> </EquationSource> </InlineEquation> be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$(\succcurlyeq )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mo>≽</mo> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathcal{X }^\mathcal{I }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="script">X</mi> </mrow> <mi mathvariant="script">I</mi> </msup> </math> </EquationSource> </InlineEquation> admits an additive representation. That is: there exists a linearly ordered abelian group <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\mathcal{R }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">R</mi> </math> </EquationSource> </InlineEquation> and a ‘utility function’ <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>u</mi> <mo>:</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">⟶</mo> <mi mathvariant="script">R</mi> </mrow> </math> </EquationSource> </InlineEquation> such that, for any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">x</mi> <mo>,</mo> <mi mathvariant="bold">y</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">X</mi> </mrow> <mi mathvariant="script">I</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation> which differ in only finitely many coordinates, we have <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\mathbf{x}\succcurlyeq \mathbf{y}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">x</mi> <mo>≽</mo> <mi mathvariant="bold">y</mi> </mrow> </math> </EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\sum _{i\in \mathcal{I }} \left[u(x_i)-u(y_i)\right]\ge 0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> </mrow> </msub> <mfenced close="]" open="[" separators=""> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>≥</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$(\succcurlyeq )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mo>≽</mo> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> also satisfies a weak continuity condition, then the paper shows that, for any <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">x</mi> <mo>,</mo> <mi mathvariant="bold">y</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">X</mi> </mrow> <mi mathvariant="script">I</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation>, we have <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$\mathbf{x}\succcurlyeq \mathbf{y}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">x</mi> <mo>≽</mo> <mi mathvariant="bold">y</mi> </mrow> </math> </EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$${}^*\!\sum _{i\in \mathcal{I }} u(x_i)\ge {}^*\!\sum _{i\in \mathcal{I }}u(y_i)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow/> <mo>∗</mo> </msup> <mspace width="-0.166667em"/> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> </mrow> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msup> <mrow/> <mo>∗</mo> </msup> <mspace width="-0.166667em"/> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> </mrow> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$${}^*\!\sum _{i\in \mathcal{I }} u(x_i)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow/> <mo>∗</mo> </msup> <mspace width="-0.166667em"/> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> </mrow> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> represents a nonstandard sum, taking values in a linearly ordered abelian group <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$${}^*\!\mathcal{R }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow/> <mo>∗</mo> </msup> <mspace width="-0.166667em"/> <mi mathvariant="script">R</mi> </mrow> </math> </EquationSource> </InlineEquation>, which is an ultrapower extension of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$\mathcal{R }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">R</mi> </math> </EquationSource> </InlineEquation>. 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. Copyright Springer Science+Business Media New York 2014