  How can the information that a set ${X_{1},...,X_{n}}$ of random variables
contains about another random variable $S$ be decomposed? To what extent do
different subgroups provide the same, i.e. shared or redundant, information,
carry unique information or interact for the emergence of synergistic
information?
  Recently Williams and Beer proposed such a decomposition based on natural
properties for shared information. While these properties fix the structure of
the decomposition, they do not uniquely specify the values of the different
terms. Therefore, we investigate additional properties such as strong symmetry
and left monotonicity. We find that strong symmetry is incompatible with the
properties proposed by Williams and Beer. Although left monotonicity is a very
natural property for an information measure it is not fulfilled by any of the
proposed measures.
  We also study a geometric framework for information decompositions and ask
whether it is possible to represent shared information by a family of posterior
distributions.
  Finally, we draw connections to the notions of shared knowledge and common
knowledge in game theory. While many people believe that independent variables
cannot share information, we show that in game theory independent agents can
have shared knowledge, but not common knowledge. We conclude that intuition and
heuristic arguments do not suffice when arguing about information.
