Two-degree-of-freedom Ellsberg urn problem

Traditional model assumes there is only randomness existing in the urn problem. However, if the numbers of the colored balls are unknown, then they should be regarded as uncertain variable. Since a ball is drawn randomly, Ellsberg urn problem is essentially a complicated system with randomness and uncertainty. Instead of psychological experiment, this paper applies uncertainty theory and chance theory to provide a rigorous mathematical method for formulating the general case of a one-degree-of-freedom Ellsberg urn problem. Furthermore, a two-degree-of-freedom Ellsberg urn problem is proposed, and the formulation for the problem is given to deal with three unknown numbers of colored balls.