Abstract: In financial economics and decision science, the utility function is one of the most important theoretical tools to describe the risk attitudes or the satisfaction degrees of individual decision makers, and it is also a basic tool for the decision analysis under risk and uncertainty. The research on the characteristic and properties of utility functions has been an important issue in the field of decision science and risk management. Under the framework of the expected utility theory, for any two lotteries, when the preference of the decision maker does not change as the wealth in these two lotteries increases, the forms of utility functions will be given explicitly. However, in reality, a decision maker could change his/her preference when the wealth in these two lotteries increases. Hence, the concept of one-switch in the utility theory is introduced, that is, a decision maker can change his/her preference for any two lotteries once at most as the wealth in these two lotteries increases. Meanwhile, the forms of one-switch utility functions are identified explicitly.
It is a natural task to consider the notation of multiplicative one-switch corresponding to the characteristic, of which a decision maker can change his/her preference for any two lotteries once at most when the wealth in these two lotteries is multiplied by the same amount. To the best of our knowledge, there is little literature tackling this problem. Our objective is to fill the gap and shed some light on the study of this problem. Under the expected utility framework, the concept of multiplicative one-switch in the utility theory is introduced, that is, a decision maker can change his/her preference for any two lotteries once at most when the wealth in these two lotteries is multiplied by the same amount. The four forms of multiplicative one-switch utility function are obtained explicitly. They are the quadratic function of logarithm, the logarithm times power function, the power plus power function and the logarithm plus power function. Then, a wealth of relationship among these four multiplicative one-switch utility functions and the decision maker’s risk attitudes is given. More specifically, the relations among the four multiplicative one-switch utility functions and the risk aversion, the relative risk aversion coefficient as well as the risk consistency are obtained, which provide practical application backgrounds for these multiplicative one-switch utility functions. Under the framework of the multiplicative one-switch property theory, the risk consistency means that when a decision maker changes his/her preference for any two lotteries due to the wealth in these two lotteries multiplied by the same amount, the preferred lottery should have a higher logarithmic expected value. Furthermore, the concept of multiplicative strong one-switch is further introduced. It means that the preference behavior of the decision maker has the multiplicative one-switch property toward any two compound lotteries. The form of utility function, logarithm plus power, satisfying the characteristic of multiplicative strong one-switch, is also verified.
The research in this paper enriches the contents of one-switch property in the utility theory and extends the existing notation of one-switch. The families of multiplicative one-switch utility functions and the form of multiplicative strong one-switch utility functions are given. The result obtained in this paper provides a theoretical basis for applying the multiplicative one-switch utility functions to the researches on the optimal investment, and provides theoretical tools for analyzing decision-making problems with the multiplicative one-switch property under risk and uncertainty.
The existing research on the one-switch utility property focuses on the univariate utility function. It is an interesting and important problem of extending the multiplicative one-switch notion to the multi-attribute case and identifying the families of multi-attribute multiplicative one-switch utility functions. In the future work, this problem will be further considered.

Key words: utility function, multiplicative one-switch, coefficient of relative risk aversion, multiplicative strong one-switch

摘要： 在金融经济学和决策科学中,效用函数是描述决策者风险态度或满意程度最重要的理论工具之一,它也是风险和不确定性决策分析的基本工具。对效用函数特征和应用背景的研究一直是决策科学和风险管理领域的重要问题。本文提出了效用理论中的倍乘单次转换的概念,得到了具有倍乘单次转换特性的效用函数的四种具体形式,并且给出了它们与风险厌恶以及风险一致性之间的关系,从而为具有此特性的效用函数提供了实际的应用背景。以此为基础,本文进一步提出了倍乘强单次转换的概念,并证明了满足此特性的效用函数的具体形式。本文的研究丰富了效用理论中单次转换效用函数的研究内容,为分析具有倍乘单次转换特性的决策问题提供了理论工具。

关键词: 效用函数, 倍乘单次转换, 相对风险厌恶系数, 倍乘强单次转换