The cautious rule of combination for belief functions and some extensions

Dempster's rule plays a central role in the theory of belief functions. However, it assumes the items of evidence combined to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine non distinct items of evidence. Extensions based on triangular norms (some of which allow to define operators whose behavior is intermediate between the Dempster's rule and the cautious rule) are also introduced