When and to what extent should two belief functions be discounted?

When Dempster's rule is applied to combine two bodies of evidence, a counterintuitive result may be generated sometimes no matter the conflict is high or not. Two such kinds of emblematic examples are reviewed and analyzed further using the canonical decomposition. By generalizing these examples, a theorem is proposed to classify various counter examples into three cases. It is believed that an unreasonable result arises from the difference between their actual frames, i.e. their cores. And a discounting approach mainly focused on combining two pieces of evidence is proposed. It consists of two parts, i.e. in what cases belief functions should be discounted and to what extent they should be discounted. The former part is discussed based on the proposed theorem. As to the latter part, sequential discounting factors are defined based on evidence distance. This discounting approach is especially suitable for cases where pieces of evidence come in a sequence.