  Candogan et al. (2011) provide an orthogonal direct-sum decomposition of
finite games into potential, harmonic and nonstrategic components. In this
paper we study the issue of decomposing games that are strategically equivalent
from a game-theoretical point of view, for instance games obtained via
transformations such as duplications of strategies or positive affine mappings
of of payoffs. We show the need to define classes of decompositions to achieve
commutativity of game transformations and decompositions.
