Moments of utility functions and their applications

This paper presents moments and cross-moments of utility functions and measures of utility dependence. We start with an interpretation of the nth moment of a utility function, and describe methods for its assessment in practice and consistency checks that need to be satisfied for any assessed moments. We then show how moments of a utility function (i) provide a new method to determine the parameters of a given functional form of a utility function and (ii) to derive the functional form of a utility function that satisfies some given moment assessments. Next, we derive a fundamental formula that relates the expected utility of a joint distribution to the expected utility of the marginal distributions for multiattribute utility functions. We use this formulation to provide an intuitive interpretation for cross-moments of utility functions and illustrate their use in (i) constructing multiattribute utility functions that incorporate utility dependence and (ii) in providing necessary conditions for utility independence in decisions with multiple attributes. We end with a new measure of utility dependence for multiattribute utility functions and work through several examples to illustrate the approach.     

企业及市场经营效用函数形式研究

鉴于经营效用是企业制订战略决策和发展策略的重要影响因素,研究其基本效用类型,提出对应的函数形式,并在此基础上提出市场总体的综合经营效用测度方法。企业经营效用由自身经营效用和同业比较效用两部分线性合成,效用变量分别为企业单位盈利指标加权合成值及单位盈利水平与行业平均值的差,效用函数通过原点且单调,因此需将效用理论中对应于各种效用类型的对数函数、指数函数等进行坐标变换、旋转或对称。保守型和冒险型效用在定义域内分别为凹函数和凸函数,共组合为9种效用类型含81种基本函数形式,并给出各效用类型含义、经营特征和定价倾向。以市场份额作为各企业经营效用权重,构建幂平均效用合成模型作为市场总体综合经营效用测度。     

基于效用函数的模糊数的比较与排序

对相同的模糊数进行比较,不同风险偏好的决策者,会得到不同的结论.效用函数是对风险偏好的度量,因此,模糊数的比较与排序的方法,一定要结合决策者的效用函数来构造.为此,根据效用函数定义了模糊效用函数,在此基础上定义了效用序.之后,证明效用序为全序,进一步利用结构元理论对效用序进行表述.根据效用函数反映风险偏好的程度,对效用序进行分类.这样,决策者对模糊数进行比较时,依据自身对风险偏好程度来选择效用序.     

保留效用类型依存的委托代理模型

在代理人的保留效用是类型依存的情况下,代理人的租金受外部机会成本的影响.研究表明,代理人努力负效用与成本类型的单调关系,决定了最优契约安排.相比较于非类型依存的情况,代理人都可能得到租金.高类型的租金总大于0,其租金可能仅源于保留效用,也可能受保留效用与努力负效用的综合影响.为激励其努力工作,委托人必须给予该类型代理人更高的支付;另一方面,低类型代理人的努力成本低和保留效用高,总得到比高类型更多的租金.     

基于2可加模糊测度的多线性效用函数建模和求解

构建合适的多属性效用函数是多属性效用分析的关键。针对不同偏好假设,文献从可加独立、效用独立、效用依赖等分别进行了多属性效用函数构建的研究。然而,由于求解的复杂性,多属性效用理论的应用绝大部分限于可加效用函数和多乘效用函数。提出一种基于2可加模糊测度的多线性效用函数建模和求解方法。首先,证明多线性效用函数和基于模糊测度的多线性模型之间的等价性,提出利用基于模糊测度的多线性模型对多线性效用函数进行表示。其次,针对多线性模型的特点和模糊测度识别的复杂性,利用Banzhaf交互指数和2可加模糊测度对多线性模型进行表示,并利用最小方法差进行模糊测度和Banzhaf交互指数识别,进而实现多线性效用函数的求解。最后,将方法用于某可穿戴医疗设备基于顾客需求的多属性效用函数构建,确认了可行性。方法为多线性效用函数的求解提供了一种新思路。     

Objective comparisons of the optimal portfolios corresponding to different utility functions

This paper considers the effects of some frequently used utility functions in portfolio selection by comparing the optimal investment outcomes corresponding to these utility functions. Assets are assumed to form a complete market of the Black–Scholes type. Under consideration are four frequently used utility functions: the power, logarithm, exponential and quadratic utility functions. To make objective comparisons, the optimal terminal wealths are derived by integration representation. The optimal strategies which yield optimal values are obtained by the integration representation of a Brownian martingale. The explicit strategy for the quadratic utility function is new. The strategies for other utility functions such as the power and the logarithm utility functions obtained this way coincide with known results obtained from Merton’s dynamic programming approach.     

Representing risk preferences in expected utility based decision models

Users of expected utility based decision models frequently find it useful or necessary to specify a functional form that represents the risk preferences of a decision maker. Having additional functional forms from which to choose would be helpful. The literature so far has provided several such functional forms for the utility function itself. The discussion presented here indicates that providing a functional form for the marginal utility function is an alternate and equally useful way to represent risk preferences. Furthermore, functional forms for marginal utility are easier to provide, and there exist functional forms for marginal utility that represent simple risk preferences for which there is no associated functional form for the utility function. Several functional forms for marginal utility are suggested, and the class of isoelastic risk preferences is identified and discussed.     

Sensitivity analysis incorporating fuzzy evaluation for scaling constants of multiattribute utility functions

A multiattribute utility function can be represented by a function of single-attribute utility functions if the decision maker’s preference satisfies additive independence or mutually utility independence. Additive independence is a preference condition stronger than mutually utility independence, and the multiattribute utility function is in the additive form if the former condition is satisfied, otherwise it is in the multiplicative form. In this paper, we propose a method for sensitivity analysis of multiattribute utility functions in multiplicative form, taking into account the imprecision of the decision maker’s judgment in the procedures for determining scaling constants (attribute weights).     

Inheritance of properties in NTU communication situations

In this paper we consider communication situations in which utility is nontransferable. We compare this model with the more familiar model of transferable utility communication situations in terms of the corresponding graph-restricted games. We extend transferable utility results on the inheritance of properties of the underlying game to the graph-restricted game to our context of nontransferable utility.Thanks are due to Peter Borm for some helpful comments.     

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.      

An efficient algorithm for mining high utility itemsets with negative item values in large databases

Utility itemsets typically consist of items with different values such as utilities, and the aim of utility mining is to identify the itemsets with highest utilities. In the past studies on utility mining, the values of utility itemsets were considered as positive. In some applications, however, an itemset may be associated with negative item values. Hence, discovery of high utility itemsets with negative item values is important for mining interesting patterns like association rules. In this paper, we propose a novel method, namely HUINIV (High Utility Itemsets with Negative Item Values)-Mine, for efficiently and effectively mining high utility itemsets from large databases with consideration of negative item values. To the best of our knowledge, this is the first work that considers the concept of negative item values in utility mining. The novel contribution of HUINIV-Mine is that it can effectively identify high utility itemsets by generating fewer high transaction-weighted utilization itemsets such that the execution time can be reduced substantially in mining the high utility itemsets. In this way, the process of discovering all high utility itemsets with consideration of negative item values can be accomplished effectively with less requirements on memory space and CPU I/O. This meets the critical requirements of temporal and spatial efficiency for mining high utility itemsets with negative item values. Through experimental evaluation, it is shown that HUINIV-Mine outperforms other methods substantially by generating much less candidate itemsets under different experimental conditions.     

通胀风险下基于HARA效用的DC型养老金计划

通货膨胀是养老基金管理过程中最直接最重要的影响因素之一. 假设通胀风险由服从几何布朗运动的物价指数来度量, 且瞬时期望通货膨胀率由Ornstein-Uhlenbeck过程来驱动. 金融市场由n+1种可连续交易的风险资产所构成, 养老基金管理者期望研究和解决通胀风险环境下DC型养老基金在累积阶段的最优投资策略问题, 以最大化终端真实财富过程的期望效用. 双曲绝对风险厌恶(HARA)效用函数具有一般的效用框架, 包含幂效用、指数效用和对数效用作为特例. 假设投资者对风险的偏好程度满足HARA效用, 运用随机最优控制理论和Legendre变换方法得到了最优投资策略的显式表达式.     

Random expected utility theory with a continuum of prizes

This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries over a finite set of prizes, to the circumstances with a continuum of prizes. Let [0, M] denote this continuum of prizes; assume that each utility function is continuous, let \(C_0[0,M]\) be the set of all utility functions which vanish at the origin, and define a random utility function to be a finitely additive probability measure on \(C_0[0,M]\) (associated with an appropriate algebra). It is shown here that a random choice rule is mixture continuous, monotone, linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict our consideration to those utility functions that are continuously differentiable on [0, M] and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory.     

Optimal Control for Utility Portfolio Selection with Liability

In this paper we use stochastic optimal control theory to investigate a dynamic portfolio selection problem with liability process, in which the liability process is assumed to be a geometric Brownian motion and completely correlated with stock prices. We apply dynamic programming principle to obtain Hamilton-Jacobi-Bellman (HJB) equations for the value function and systematically study the optimal investment strategies for power utility, exponential utility and logarithm utility. Firstly, the explicit expressions of the optimal portfolios for power utility and exponential utility are obtained by applying variable change technique to solve corresponding HJB equations. Secondly, we apply Legendre transform and dual approach to derive the optimal portfolio for logarithm utility. Finally, numerical examples are given to illustrate the results obtained and analyze the effects of the market parameters on the optimal portfolios.     

An analytical framework for aggregating multiattribute utility functions

This paper proposes a procedure for aggregating individual cardinal utility functions into a social utility function that represents the preferences of all the individuals as a whole. The procedure is non-interactive and is based upon the determination of the utility consensus values. This is accomplished by minimizing a distance function model that is transformed into an Archimedean goal programming problem. The procedure is applied to a general group multilinear utility function.     

Multi-attribute non-expected utility

In management applications of risk theory, planning and decision making are typically concerned with complex multi-dimensional attributes of risk and utility trade-offs between them. This paper presents a novel approach to multi-attribute non-expected utility which is especially designed to serve application and risk management purposes. It is based on a recently developed non-expected utility model that accommodates systematic violations of expected utility of various kinds observed in risky choice experiments. In the model, the possible outcomes of risky decisions are assumed to be multi-dimensional, that is, classified, measured, compared and assessed from different economic and non-economic perspectives simultaneously. Of the risk attributes to be jointly evaluated in a decision problem, each is supposed to be utility independent of the complementary set of all the other attributes also considered. Mutual utility independence and additive independence are particularly pronounced forms of utility independence. An order-preserving preference functional exists if the agent??s risk preferences satisfy familiar rationality requirements. The functional provides a consistently scaled, multi-linear representation in terms of single-attribute probability-dependent utility functions. Finally, the formalism is applied to explain observed trade-offs between monetary benefits obtained, and fatalities incurred, in the operation of large-scale industrial systems.     

Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.     

A note on additive risk measures in rank-dependent utility

This note proves that risk measures obtained by applying the equivalent utility principle in rank-dependent utility are additive if and only if the utility function is linear or exponential and the probability weighting (distortion) function is the identity.     

效用函数理论及其应用研究

分析了效用函数理论,提出了图书效用函数,并分析其性质.得出图书最大效用化的约束条件.指导图书馆文献资源建设.     

Competitive Price Equilibrium with Consumer Reservation Utility

This paper studies a competitive price equilibrium in the market of a product category where consumers are homogeneous with a reservation utility below which they will not purchase the product. The impact of the reservation utility on the price equilibrium is of particular interest, because the reservation utility may change according to the business cycle and economic environments. Using multinomial logit model to describe market response, we study the comparative statics of the prices, profits and market shares of firms, each of which produces one brand in the product category, with respect to the reservation utility in the Nash equilibrium. It is shown that, as the reservation utility increases, the prices as well as the profits at Nash equilibrium decrease. Also, in the case of duopoly market, the firm with lower cost structure will increase its market share as the reservation utility increases.