A primer on triangle functions II

In [32] we presented an overview of concepts, facts and results on triangle functions based on the notions of t-norm, copula, (generalized) convolution, semicopula, quasi-copula. Here, we continue our presentation. In particular, we treat the concept of duality and study a few important cases of functional equations and inequalities for triangle functions like, e.g., convolution, Cauchy's equation, dominance, and Jensen convexity

Comparison of random variables from a game-theoretic perspective

This work consists of four related parts, divided into eight chapters. A ¯rst part introduces the framework of cycle-transitivity, developed by De Baets et al. It is shown that this framework is ideally suited for describing and compar- ing forms of transitivity of probabilistic relations. Not only does it encompass most already known concepts of transitivity, it is also ideally suited to describe new types of transitivity that are encountered in this work (such as isostochas- tic transitivity and dice-transitivity). The author made many non-trivial and sometimes vital contributions to the development of this framework. A second part consists of the development and study of a new method to compare random variables. This method, which bears the name generalized dice model, was developed by De Meyer et al. and De Schuymer et al., and can be seen as a graded alternative to the well-known concept of ¯rst degree stochastic dominance. A third part involves the determination of the optimal strategies of three game variants that are closely related to the developed comparison scheme. The de¯nitions of these variants di®er from each other solely by the copula that is used to de¯ne the payo® matrix. It turns out however that the characterization of the optimal strategies, done by De Schuymer et al., is completely di®erent for each variant. A last part includes the study of some combinatorial problems that orig- inated from the investigation of the transitivity of probabilistic relations ob- tained by utilizing the developed method to compare random variables. The study, done by De Schuymer et al., includes the introduction of some new and interesting concepts in partition theory and combinatorics. A more thorough discussion, in which each section of this work is taken into account, can be found in the overview at the beginning of this manuscript. Although this work is oriented towards a mathematical audience, the intro- duced concepts are immediately applicable in practical situations. Firstly, the framework of cycle-transitivity provides an easy means to represent and compare obtained probabilistic relations. Secondly, the generalized dice model delivers a useful alternative to the concept of stochastic dominance for comparing random variables. Thirdly, the considered dice games can be viewed in an economical context in which competitors have the same resources and alternatives, and must choose how to distribute these resources over their alternatives. Finally, it must be noted that this work still leaves opportunities for future research. As immediate candidates we see, ¯rstly the investigation of the tran- sitivity of generalized dice models in which the random variables are pairwisely coupled by a di®erent copula. Secondly, the characterization of the transitivity of higher-dimensional dice models, starting with dimension 4. Thirdly, the study of the applicability of the introduced comparison schemes in areas such as mar- ket e±ciency, portfolio selection, risk estimation, capital budgeting, discounted cash °ow analysis, etc

On the dominance relation between ordinal sums of conjunctors

summary:This contribution deals with the dominance relation on the class of conjunctors, containing as particular cases the subclasses of quasi-copulas, copulas and t-norms. The main results pertain to the summand-wise nature of the dominance relation, when applied to ordinal sum conjunctors, and to the relationship between the idempotent elements of two conjunctors involved in a dominance relationship. The results are illustrated on some well-known parametric families of t-norms and copulas