Color distribution of three drawn balls from Ellsberg urn
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ORIGINAL RESEARCH
Color distribution of three drawn balls from Ellsberg urn Waichon Lio1 · Guangquan Cheng2 Received: 14 October 2019 / Accepted: 14 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Ellsberg urn is a complicated system with uncertainty (the unknown numbers of the colored balls) and randomness (the randomly drawn balls). By supposing that two numbers of colored balls are unknown in an Ellsberg urn, this paper applies uncertainty theory, probability theory and chance theory as rigorous mathematical tools to formulating the color distribution of the drawn balls when three balls are randomly drawn from the urn. Furthermore, an intuitive example is given to illustrate the results obtained by the mathematical method. Keywords Ellsberg urn · Uncertainty theory · Chance theory · Uncertain urn problem
1 Introduction The terminology of urns refers to a problem with one or more urns containing objects of different types. It is customary to regard the objects as balls and the object types as colors. A famous urn problem is the Pólya urn problems proposed by Eggenberger and Pólya (1923). Pólya urn assumes that each number of the colored balls is completely known to us. As a further development, Ellsberg (1961) presented Ellsberg urn problem in which the numbers of colored balls are unknown. The belief degrees of human beings are often considered to solve Ellsberg urn, and the research is often connected to psychology. As a mathematical branch to rationally handle the belief degrees of human beings, uncertainty theory was founded by Liu (2007) and perfected by Liu (2009). After that, Liu (2012) declared probability theory is a mathematical tool to model frequencies, and uncertainty theory is a mathematical tool to model belief degrees. That means, it is not suitable to apply probability theory to dealing with the belief degrees of human beings. Moreover, Liu (2019) provided an uncertain * Guangquan Cheng [email protected] Waichon Lio [email protected] 1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
College of Systems Engineering, National University of Defense Technology, Changsha 410073, China
2
urn problem to point out that probability theory fails to deal with the unknown numbers of colored balls in Ellsberg urn and it is necessary to take them as uncertain variables. Nowadays, uncertainty theory has been successfully applied in the fields such as stability of uncertain differential equation (Ma et al. 2017), heat equation (Yang and Ni 2017) and uncertain regression (Ye and Liu 2020). In some more practical situations, the system is complicated and contains not only uncertainty but also randomness. This gives Liu (2013a) motivation to found chance theory based on the chance measure and uncertain random variable. By taking some quantity as an uncertain random variable, uncertain random programming was suggested by Liu (2013b). Following that, uncertain random multiobjective programming (Zhou et al. 2014), uncertain ran
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