  When applied to the same game, probability theory and game theory can
disagree on calculated values of the Fisher information, the log likelihood
function, entropy gradients, the rank and Jacobian of variable transforms, and
even the dimensionality and volume of the underlying probability parameter
spaces. These differences arise as probability theory employs structure
preserving isomorphic mappings when constructing strategy spaces to analyze
games. In contrast, game theory uses weaker mappings which change some of the
properties of the underlying probability distributions within the mixed
strategy space. In this paper, we explore how using strong isomorphic mappings
to define game strategy spaces can alter rational outcomes in simple games, and
might resolve some of the paradoxes of game theory.
