Choquet积分的收敛定理

Murofushi、Sugeno等学者已对关于模糊测度的Choquet积分进行了详细的研究,但关于积分的收敛理论是不够的。本文讨论这一问题,给出Choquet积分的一些收敛定理,包括广义单调收敛定理、Fatou引理等。从中可以看出,Choquet积分与Sugeno模糊积分具有相同的收敛定理。     

On positive homogeneity and comonotonic additivity of the principle of equivalent utility under Cumulative Prospect Theory

Investigating the principle of equivalent utility under Cumulative Prospect Theory, Kałuszka and Krzeszowiec (2012) established characterizations of several important properties of the premium. It turns out that the results concerning positive homogeneity and comonotonic additivity are in general not true. The aim of this paper is to present modified and essentially generalized versions of the mentioned above results.     

A weight-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral

In the context of aggregation, the Choquet integral arises as a natural generalization of the weighted arithmetic mean. In their seminal work on bi-capacities, Grabisch and Labreuche have recently proposed an extension of the Choquet integral adapted to the situation where the values to be aggregated lie on a bipolar scale. In such a framework, they have derived generalizations of the Shapley value and of the Shapley interaction index that can be used to apprehend the main behavioral characteristics of the aggregation. Founding our approach on the intuitive notion of weight of a criterion, we propose alternative definitions of importance and interaction indices that may be more easily interpretable and we study their properties.     

Linguistic quantifiers based on Choquet integrals

The introduction of linguistic quantifiers has provided an important tool to model a large number of issues in intelligent systems. Ying [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606] recently introduced a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno’s integral. Representing linguistic quantifiers by fuzzy measures, this paper evaluates linguistic quantified propositions by the Choquet integral. Some elegant logical properties of linguistic quantifiers are derived within this approach, including a prenex normal form theorem stronger than that in Ying’s model. In addition, our Choquet integral approach to the evaluation of quantified statements is compared with others, in particular with Ying’s Sugeno integral approach.     

Robust competence assessment for job assignment

Allocating the right person to a task or job is a key issue for improving quality and performance of achievements, usually addressed using the concept of “competences”. Nevertheless, providing an accurate assessment of the competences of an individual may be in practice a difficult task. We suggest in this paper to model the uncertainty on the competences possessed by a person using a possibility distribution, and the imprecision on the competences required for a task using a fuzzy constraint, taking into account the possible interactions between competences using a Choquet integral. As a difference with comparable approaches, we then suggest to perform the allocation of persons to jobs using a robust optimisation approach, allowing to minimise the risk taken by the decision maker. We first apply this framework to the problem of selecting a candidate within n for a job, then extend the method to the problem of selecting c candidates for j jobs (c ? j) using the leximin criterion.     

基于IGAHP-熵-博弈论-Choquet积分的新型装甲装备通用质量特性评价模型研究

针对新型装甲装备通用质量特性评价准则相互关联的问题,在建立评价指标体系的基础上,结合新型装甲装备特点,明确了通用质量特性对评价的要求,运用改进群体层次分析法(Improved Group Analytic Hierarchy Process, IGAHP)确定了指标的主观权重,采用熵权法明确了指标的客观权重,并引入博弈论的思想融合了主客观权重,考虑到评价准则之间的关联性问题,引入了模糊测度的概念,运用Choquet积分实现了新型装甲装备通用质量特性评价,为新型装甲装备通用质量特性的改进研究奠定了基础。     

On (strong) α-favorability of the Wijsman hyperspace

The Banach-Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach-Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6].     

On the upper semicontinuity of Choquet capacities

The Choquet capacity T of a random closed set X on a metric space E is regarded as or related to a non-additive measure, an upper probability, a belief function, and in particular a counterpart of the distribution functions of ordinary random vectors. While the upper semicontinuity of T on the space of all closed subsets of E (hit-or-miss topology) is highly desired, T is not necessarily u.s.c. if E is not compact, e.g. E=Rⁿ. For any locally compact separable metric space E, this controversial situation can be resolved in the probabilistic context by stereographically projecting X into the Alexandroff compactification E_∞ of E with the “north pole” added to the projection. This leads to a random compact set that is defined on the same probability space, takes values in a space homeomorphic to the space of X, and possesses an equivalent probability law. Particularly, the Choquet capacity of is u.s.c. on the space of all closed subsets of E_∞. Further, consequences of the upper semicontinuity of are explored, and a proof of the equivalence between the upper semicontinuity of T and continuity from above on F(E) is provided.     

直觉模糊联盟合作博弈的非线性规划模型

本文研究联盟是直觉模糊集的合作博弈。首先,给出直觉模糊联盟的定义,并根据Choquet积分的直觉模糊形式,得到直觉模糊联盟合作博弈的区间值特征函数,进一步证明直觉模糊联盟合作博弈的区间值特征函数具有超可加性、凸性、弱超可加性. 其次根据区间数的闵可夫斯基距离、区间数的排序及损失函数的定义,建立直觉模糊联盟合作博弈的非线性规划模型,并对其求解得到最优分配. 最后给出一个具体的事例说明本文所建立的模型的合理性和有效性。     

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables, the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.