  Beimel, Tassa and Weinreb (2008) and Farras and Padro (2010) partially
characterized access structures of ideal weighted threshold secret sharing
schemes in terms of the operation of composition. They classified
indecomposable ideal weighted threshold access structures, and proved that any
other ideal weighted threshold access structure is a composition of
indecomposable ones. It remained unclear which compositions of indecomposable
weighted threshold access structures are weighted. In this paper we fill the
gap. Using game-theoretic techniques we determine which compositions of
indecomposable ideal access structures are weighted, and obtain an if and only
if characterization of ideal weighted threshold secret sharing schemes.
