Additive representation of separable preferences over infinite products

Let $$\mathcal{X }$$ X be a set of outcomes, and let $$\mathcal{I }$$ I be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order $$(\succcurlyeq )$$ ( ≽ ) on $$\mathcal{X }^\mathcal{I }$$ X I admits an additive representation. That is: there exists a linearly ordered abelian group $$\mathcal{R }$$ R and a ‘utility function’ $$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$ u : X ⟶ R such that, for any $$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$ x , y ∈ X I which differ in only finitely many coordinates, we have $$\mathbf{x}\succcurlyeq \mathbf{y}$$ x ≽ y if and only if $$\sum _{i\in \mathcal{I }} \left[u(x_i)-u(y_i)\right]\ge 0$$ ∑ i ∈ I u ( x i ) - u ( y i ) ≥ 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 $$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$ x , y ∈ X I , we have $$\mathbf{x}\succcurlyeq \mathbf{y}$$ x ≽ y if and only if $${}^*\!\sum _{i\in \mathcal{I }} u(x_i)\ge {}^*\!\sum _{i\in \mathcal{I }}u(y_i)$$ ∗ ∑ i ∈ I u ( x i ) ≥ ∗ ∑ i ∈ I u ( y i ) . Here, $${}^*\!\sum _{i\in \mathcal{I }} u(x_i)$$ ∗ ∑ i ∈ I u ( x i ) represents a nonstandard sum, taking values in a linearly ordered abelian group $${}^*\!\mathcal{R }$$ ∗ R , which is an ultrapower extension of $$\mathcal{R }$$ 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. Copyright Springer Science+Business Media New York 2014(This abstract was borrowed from another version of this item.)