Additive representation of separable preferences over infinite products

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