Generalized Nash's choice
It is assumed that the choice function \(f\) on the nonnegative orthant \(E_+\) of the Euclidean space fulfills the well-known axioms. (I) Independence of irrelevant alternatives. If \(A\subset B\) and \(A\cap f(B)\neq \emptyset\), then \(f(A)=A\cap f(B)\). The following axioms are defined by means of a preference ordering \(\gtrsim\). But one can define them also in terms of the choice function. (H) Homogeneity. \(x\gtrsim y\Rightarrow (k_ 1x_ 1,...,k_ nx_ n)\gtrsim (k_ 1y_ 1,...,k_ ny_ n)\) for all reals \(k_ j>0\). (P) Pareto optimality. \(x\geq y\), \(x\neq y\Rightarrow x\succ y\). Let \(T\subset N\) and \(E^ T=\{x\in E_+:\) \(x_ i>0\Leftrightarrow i\in T\}\). Then: (A) Archimedean property. \(x\succ y\succ z\Rightarrow\) there exists \(\alpha\),\(\beta\in (0,1)\) such that \(\alpha x+(1-\alpha)z\gtrsim y\gtrsim \beta x+(1-\beta)z\) for all \(x,y,z\in E^ T\) and any \(T\subset N.\) Let \(M(S)\) be maximal under the inclusion \(\subset\) in the collection \(\{\) \(T\subset N:\) \(S\cap E^ T\neq \emptyset \}\). Then the main result is: Theorem. Let \(f\) on \(E_+\) satisfy the axioms I, H, P, A. Then there exists \(M\in M(S)\) and reals \(\lambda_ j>0\), \(j\in M\), such that \[ f(S)=\{x\in S:\prod_{j\in M}x_ j^{\lambda_ j}\geq \prod_{j\in M}y_ j^{\lambda_ j},\quad \forall y\in S\}\text{ for any }S. \] It is also shown (see Lemma 2 in the paper): Theorem. Let \(f\) fulfill on the positive orthant \(E_{++}\) the axioms I, H and assume there exists a pair \(x', y'\) in \(E_{++}\) such that \(x'\succ y'\) and \(z\succ y'\) in some neighborhood of \(x'\). Then there exist reals \(\lambda_ j\), \(\sum | \lambda_ j| >0\), such that \[ f(S)\subset \{x\in S:\quad \prod x_ j^{\lambda_ j}\geq \prod y_ j^{\lambda_ j},\quad \forall y\in S\}. \]
scientific article; zbMATH DE number 3943495