Let X be a metric space and \(U:X^{\infty }\rightarrow \mathbb {R}\) be a continuous function satisfying the Koopmans recursion \(U(x_0,x_1,x_2,\ldots )=\varphi (x_0, U(x_1,x_2,\ldots )),\) where \(\varphi :X\times I \rightarrow I\) is a continuous function and I is an interval. Denote by \(\succeq \) a preference relation defined on the product \(X^{\infty }\) represented by a function \(U:X^{\infty }\rightarrow \mathbb {R}\), called a utility function, that means \((x_0,x_1,\ldots )\succeq (y_0,y_1,\ldots )\Leftrightarrow U(x_0,x_1,\ldots )\ge U(y_0,y_1,\ldots )\). We consider a problem when the preference relation \(\succeq \) can be represented by another utility function V satisfying the affine recursion \(V(x_0,x_1,x_2,\ldots ) = \alpha (x_0)V(x_1,x_2,\ldots )+ \beta (x_0)\). Under suitable assumptions on relation \(\succeq \) we determine the form of the functions \(\varphi \) defining the utility functions possessing the above property. The problem is reduced to solving a system of simultaneous functional equations. The subject is strictly connected to a problem of preference in economics. In this note we extend the results obtained in Zdun (Aequ Math 94, 2020).

设X为度量空间，\(U:X^{\infty }\rightarrow \mathbb {R}\)为满足库普曼递归的连续函数\(U(x_0,x_1,x_2,\ldots )=\varphi (x_0, U(x_1,x_2,\ldots ))),\)其中\(\varphi :X\times I \rightarrow I\)是连续函数，I是区间。用\(\succeq \)表示在由函数\(U:X^{\infty }\rightarrow \mathbb {R}\)表示的乘积 \(X^{\infty }\)上定义的偏好关系，称为效用函数，即\((x_0,x_1,\ldots )\succeq (y_0,y_1,\ldots )\Leftrightarrow U(x_0,x_1,\ldots )\ge U(y_0,y_1,\ldots )\)。当偏好关系\(\succeq \)可以由满足仿射递归\(V(x_0,x_1,x_2,\ldots ) = \alpha (x_0)V(x_1,x_2 ) 的另一个效用函数V表示时，我们考虑一个问题,\ldots )+ \beta (x_0)\)。在关系\(\succeq \)的适当假设下，我们确定定义具有上述属性的效用函数的函数\(\varphi \)的形式。该问题简化为求解联立函数方程组。这个主题与经济学中的偏好问题密切相关。在本文中，我们扩展了 Zdun (Aequ Math 94, 2020) 中获得的结果。